tag:blogger.com,1999:blog-1243502645107245112019-07-22T12:09:48.119+03:00Unitary FlowUnknownnoreply@blogger.comBlogger75125tag:blogger.com,1999:blog-124350264510724511.post-44221432048701727032017-10-20T02:06:00.000+03:002017-10-21T22:11:03.667+03:00A debate inside another one<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: justify;">Tim Maudlin debated Gerard 't Hooft about his cellular automaton interpretation of quantum mechanics in a series of Facebook posts, the fourth one being here <a href="https://www.facebook.com/tim.maudlin/posts/10155699914028398">https://www.facebook.com/tim.maudlin/posts/10155699914028398</a>. Somewhere in the forest of comments I was engaged in a sort of sub-debate, with Tim, Hans, and others. Sabine was there too. The discussion was completely surrealistic, Tim and Hans completely misunderstood my point. This started by me intervening with a theoretical counterexample to a claim that all so called superdeterministic theories (in particular 't Hooft's) are not falsifiable, and of course it led to different topics. It is not known, but not a secret that <a href="https://arxiv.org/abs/1607.02076" target="_blank">the wavefunction collapse leads to violations of conservation laws</a>, and that it is possible at least in principle to remove the collapse while remaining with a single world. But removing the collapse <a href="https://arxiv.org/abs/1212.2601" target="_blank">can be seen as superdeterministic</a> (although I wouldn't call it like this, because it is based on spacetime, not on initial conditions), and I even proposed <a href="https://arxiv.org/abs/1604.05063" target="_blank">a principle to explain this, and experiments to test it</a>. I paste here most of this *debate*, because there are some parts I am interested to keep. I skipped some parts in which I was not involved.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-MgQcyxaDjGc/WekmgFYz35I/AAAAAAAABWo/-bx1UQ-k_eYCfsQ7iyuS6m4fEi8hvBchgCLcBGAs/s1600/01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="500" src="https://2.bp.blogspot.com/-MgQcyxaDjGc/WekmgFYz35I/AAAAAAAABWo/-bx1UQ-k_eYCfsQ7iyuS6m4fEi8hvBchgCLcBGAs/s1600/01.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-KzIoso7GjPU/WekmgPtD0pI/AAAAAAAABWs/lOu7SeZ_ElMEcs7brlKPz80Y5o8K1YYuQCLcBGAs/s1600/02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="816" data-original-width="504" src="https://2.bp.blogspot.com/-KzIoso7GjPU/WekmgPtD0pI/AAAAAAAABWs/lOu7SeZ_ElMEcs7brlKPz80Y5o8K1YYuQCLcBGAs/s1600/02.png" /></a></div><br /><div class="separator" style="clear: both; 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margin-right: 1em;"><img border="0" data-original-height="889" data-original-width="498" src="https://1.bp.blogspot.com/-4v2oNQOQJcI/WekmrVmrn6I/AAAAAAAABYo/KRiphwzCwLIJ7zvQL6ViGeTPx89VEWHcQCLcBGAs/s1600/36.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ntflFugcmwo/WekmrVl78fI/AAAAAAAABYw/JmU2qcwMfIcwfwIyE_KjR90nrCWtmaT5wCLcBGAs/s1600/37.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="539" data-original-width="499" src="https://4.bp.blogspot.com/-ntflFugcmwo/WekmrVl78fI/AAAAAAAABYw/JmU2qcwMfIcwfwIyE_KjR90nrCWtmaT5wCLcBGAs/s1600/37.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Q0O3mmoVIdQ/Wekmr902DpI/AAAAAAAABY4/Zxo_98M7MqgEC6Q0JLnkKcqst6M5Hc1TACLcBGAs/s1600/38.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="196" data-original-width="501" src="https://3.bp.blogspot.com/-Q0O3mmoVIdQ/Wekmr902DpI/AAAAAAAABY4/Zxo_98M7MqgEC6Q0JLnkKcqst6M5Hc1TACLcBGAs/s1600/38.png" /></a></div><br /><div class="separator" style="clear: both; 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margin-right: 1em;"><img border="0" data-original-height="650" data-original-width="500" src="https://4.bp.blogspot.com/-RvMqlSCiLRc/Wekms6HEWnI/AAAAAAAABZA/NGhjQenIfho2fkClz91SZfrn6qk4bAPlQCLcBGAs/s1600/41.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-HshYqmaprPg/Wekms5FZymI/AAAAAAAABZE/l9kffcVw6X8V4tBh9O3UBWdZZL8RGoCNQCLcBGAs/s1600/42.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="781" data-original-width="500" src="https://3.bp.blogspot.com/-HshYqmaprPg/Wekms5FZymI/AAAAAAAABZE/l9kffcVw6X8V4tBh9O3UBWdZZL8RGoCNQCLcBGAs/s1600/42.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-TKl7WlPV7M0/WekmtZE5X6I/AAAAAAAABZI/HE7wCHyd5OcVMsIdhDPI3d9FEcDTHofkgCLcBGAs/s1600/43.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="645" data-original-width="501" src="https://1.bp.blogspot.com/-TKl7WlPV7M0/WekmtZE5X6I/AAAAAAAABZI/HE7wCHyd5OcVMsIdhDPI3d9FEcDTHofkgCLcBGAs/s1600/43.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-nsJRxKM_UUI/WekmtwBsANI/AAAAAAAABZM/g86VTy2uJaUG92ilAN7H-67x7FAP6mBOgCLcBGAs/s1600/44.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="505" data-original-width="501" src="https://3.bp.blogspot.com/-nsJRxKM_UUI/WekmtwBsANI/AAAAAAAABZM/g86VTy2uJaUG92ilAN7H-67x7FAP6mBOgCLcBGAs/s1600/44.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-EL2b2lT38dU/WekmubQzEJI/AAAAAAAABZQ/Ds9_-A6A8UEQCWNMtszAf57lLqvfw-phwCLcBGAs/s1600/45.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="787" data-original-width="502" src="https://3.bp.blogspot.com/-EL2b2lT38dU/WekmubQzEJI/AAAAAAAABZQ/Ds9_-A6A8UEQCWNMtszAf57lLqvfw-phwCLcBGAs/s1600/45.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-fWu1qkjk34Q/WemHGHJyMlI/AAAAAAAABZw/uWaz9diWzlw4tW27_d0p3OgZYsrPukoegCLcBGAs/s1600/46.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="836" data-original-width="505" src="https://2.bp.blogspot.com/-fWu1qkjk34Q/WemHGHJyMlI/AAAAAAAABZw/uWaz9diWzlw4tW27_d0p3OgZYsrPukoegCLcBGAs/s1600/46.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-4Nmhesfs8dY/WekmvfbHO9I/AAAAAAAABZY/0fOimXdC9zAuE87prVPWxpwpUuo2melQwCLcBGAs/s1600/47.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="541" data-original-width="1214" height="177" src="https://2.bp.blogspot.com/-4Nmhesfs8dY/WekmvfbHO9I/AAAAAAAABZY/0fOimXdC9zAuE87prVPWxpwpUuo2melQwCLcBGAs/s400/47.png" width="400" /></a></div><br /></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-87699167474257279052017-05-11T19:55:00.001+03:002017-05-15T00:17:11.505+03:00Maudlin's "(Information) Paradox Lost" paper<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: <a href="http://arxiv.org/abs/1705.03541" target="_blank">(Information) Paradox Lost</a>. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [<a href="http://www.unitaryflow.com/2013/09/bh-paradox-1-susskind-vs-hawking.html" target="_blank">1</a>,<a href="http://www.unitaryflow.com/2013/10/black-hole-paradox-2-stretched-complementarity.html" target="_blank">2</a>,<a href="http://www.unitaryflow.com/2013/10/black-hole-paradox-3-look-for-the-information-where-you-lost-it.html" target="_blank">3</a>]. Maudlin's paper is discussed by Sabine <a href="http://backreaction.blogspot.com/2017/05/a-philosopher-tries-to-understand-black.html" target="_blank">here</a>.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I illustrate this with this animated gif:</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-NIotNcT9tjo/WRSSlCTmGTI/AAAAAAAABQo/6DH7aG-leMQUBLTx-7Efu8IoXvs1RyNOgCLcB/s1600/unitary_evaporating_black_hole.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-NIotNcT9tjo/WRSSlCTmGTI/AAAAAAAABQo/6DH7aG-leMQUBLTx-7Efu8IoXvs1RyNOgCLcB/s400/unitary_evaporating_black_hole.gif" width="325" /> </a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div style="text-align: justify;">I made this gif <a href="https://3.bp.blogspot.com/-5iP83rvuqG4/WRSUiW6bcDI/AAAAAAAABQ4/S4Gnwcb0M981ktt84xJ7wFyXluBb-hBRQCPcB/s1600/svn-log.png" target="_blank">back in 2010</a>, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to <a href="https://arxiv.org/abs/1111.4837" target="_blank">an analytic extension</a> of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from <a href="http://www.unitaryflow.com/2013/10/black-hole-paradox-3-look-for-the-information-where-you-lost-it.html" target="_blank">an older post</a> in which I say more about this:</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://i.imgur.com/BSgHYEz.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://i.imgur.com/BSgHYEz.png" height="318" width="400" /> </a></div><div class="separator" style="clear: both; text-align: center;"><b>A.</b> The standard Penrose diagram of an evaporating black hole.</div><div class="separator" style="clear: both; text-align: center;">B The diagram from the analytic solution I proposed.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">___________________________</div><div class="separator" style="clear: both; text-align: justify;">* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and <a href="https://arxiv.org/abs/1301.2231" target="_blank">my PhD thesis</a>.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-124350264510724511.post-90799148770125441822017-03-10T14:13:00.001+02:002017-03-11T17:26:35.006+02:00The Tablet of the Metalaw<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">This edition of the FQXi essay contest is called <span class="entityTitle"><a href="http://fqxi.org/community/forum/topic/2694" target="_blank">Wandering Towards a Goal</a>. My entry is called <a href="http://fqxi.org/community/forum/topic/2847" target="_blank">The Tablet of the Metalaw</a>. This is the abstract:</span></div><div style="text-align: justify;"><span class="entityTitle"><br /></span></div><blockquote class="tr_bq"><div style="text-align: justify;">Reality presents to us in multiple forms, as a multiple level pyramid. Physics is the foundation, and should be made as solid and complete as possible. Suppose we will find the unified theory of the fundamental physical laws. Then what? Will we be able to deduce the higher levels, or they have their own life, not completely depending on the foundations? At the higher levels arise goals, life, and even consciousness, which seem to be more than mere constructs of the fundamental constituents. Are all these high level structures completely reducible to the basis, or by contrary, they also affect the lower levels? Are mathematics and logic enough to solve these puzzles? Are there questions objective science can't even define rigorously? Why is there something rather than nothing? What is the world made of?</div></blockquote><div style="text-align: justify;"><br /></div><div style="text-align: justify;">At this time (2017-03-11 08.59 AM ET) my essay is in the top position, so I will immortalize this ephemeral moment in the picture below, since I expect the order will change dramatically, given that the votes will continue for nearly a month, and then the FQXi panel will add their choices:</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-qpwHaDR7O6Q/WMQWEMAxlNI/AAAAAAAABOo/CGWY0H4r7ukbo3TQsQh71NFhdZZl0ER4gCLcB/s1600/fqxi-2017-03-11%2B08.59%2BAM%2BET.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="526" src="https://1.bp.blogspot.com/-qpwHaDR7O6Q/WMQWEMAxlNI/AAAAAAAABOo/CGWY0H4r7ukbo3TQsQh71NFhdZZl0ER4gCLcB/s640/fqxi-2017-03-11%2B08.59%2BAM%2BET.png" width="550" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-36370360611077082652017-02-15T07:21:00.000+02:002017-02-15T07:21:36.890+02:00The Standard Model Algebra<div dir="ltr" style="text-align: left;" trbidi="on">arXiv link: <a href="https://arxiv.org/abs/1702.04336" target="_blank">https://arxiv.org/abs/1702.04336</a><br /><div style="text-align: justify;"><span id="goog_615638986"></span><span id="goog_615638987"></span> </div><div style="text-align: justify;">A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with a preferred Witt decomposition, and it is already implicitly present in the mathematical structure of the Standard Model. The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-1" role="math" style="display: inline-block; width: 1.545em;"><span style="display: inline-block; font-size: 120%; height: 0px; position: relative; width: 1.273em;"><span style="clip: rect(0.105em, 1001.27em, 1.323em, -1000em); left: 0em; position: absolute; top: -0.984em;"><span class="mrow" id="MathJax-Span-2"><span class="msubsup" id="MathJax-Span-3"><span style="display: inline-block; height: 0px; position: relative; width: 1.285em;"><span style="clip: rect(3.114em, 1000.46em, 4.177em, -1000em); left: 0em; position: absolute; top: -3.993em;"><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math; font-style: italic;">θ</span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span><span style="left: 0.469em; position: absolute; top: -3.843em;"><span class="mi" id="MathJax-Span-5" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">W<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.074em;"></span></span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 0.984em; width: 0px;"></span></span></span><span style="border-left: 0px solid; display: inline-block; height: 1.184em; overflow: hidden; vertical-align: -0.268em; width: 0px;"></span></span></nobr></span> given by <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-6" role="math" style="display: inline-block; width: 8.212em;"><span style="display: inline-block; font-size: 120%; height: 0px; position: relative; width: 6.829em;"><span style="clip: rect(1.159em, 1006.78em, 2.623em, -1000em); left: 0em; position: absolute; top: -2.199em;"><span class="mrow" id="MathJax-Span-7"><span class="msubsup" id="MathJax-Span-8"><span style="display: inline-block; height: 0px; position: relative; width: 1.657em;"><span style="clip: rect(3.15em, 1001.21em, 4.178em, -1000em); left: 0em; position: absolute; top: -3.993em;"><span class="mi" id="MathJax-Span-9" style="font-family: MathJax_Main;">sin</span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span><span style="left: 1.228em; position: absolute; top: -4.389em;"><span class="mn" id="MathJax-Span-10" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-11"></span><span class="mo" id="MathJax-Span-12" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-13"><span style="display: inline-block; height: 0px; position: relative; width: 1.285em;"><span style="clip: rect(3.114em, 1000.46em, 4.177em, -1000em); left: 0em; position: absolute; top: -3.993em;"><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math; font-style: italic;">θ</span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span><span style="left: 0.469em; position: absolute; top: -3.843em;"><span class="mi" id="MathJax-Span-15" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">W<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.074em;"></span></span><span style="display: inline-block; height: 3.993em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-16" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-17" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-18" style="font-family: MathJax_Main; padding-left: 0.278em;">0.25.</span></span><span style="display: inline-block; height: 2.199em; width: 0px;"></span></span></span><span style="border-left: 0px solid; display: inline-block; height: 1.479em; overflow: hidden; vertical-align: -0.369em; width: 0px;"></span></span></nobr></span></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-FZbNICCp094/WKPj3pPeFCI/AAAAAAAABNY/69qSHiBkj94fSZbPG2XDsPI2MoKy0DB8wCLcB/s1600/SMA.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="255" src="https://4.bp.blogspot.com/-FZbNICCp094/WKPj3pPeFCI/AAAAAAAABNY/69qSHiBkj94fSZbPG2XDsPI2MoKy0DB8wCLcB/s400/SMA.jpg" width="400" /></a></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-33083171441251526202016-09-07T09:30:00.000+03:002017-04-24T11:34:26.424+03:00Quantum God (short story)<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;"></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">(<a href="https://drive.google.com/open?id=0Bw6oSVcm8ehubHJxbkJFOTJzWGs" target="_blank">link to pdf version</a>) </span><br /><span class="tm6">(<a href="https://drive.google.com/file/d/0Bw6oSVcm8ehub3Vzc3lRekFzaVU/view" target="_blank">link to Italian version, translation by Erica Mannoni</a>) </span><br /><br /><span class="tm6">2033 AD. The entire population of the planet was watching, most of them through the eyes of the media, waiting for Lord Q to perform a miracle and save the world. Thousands of people gathered around his tent, meditating, praying, praising him, and hoping for the miracle. The asteroid was heading toward the Earth. All previous attempts to destroy the asteroid failed, because it was a black hole. It was detected only by the way it bent the light and the trajectories of other asteroids in the Solar System. So the asteroid continued undisturbed to threaten the Earth, and Quentin, named by his followers Lord Q, was the only hope.</span></div><div class="Normal tm7" style="text-align: justify;"><br /></div><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne joined Quentin’s group one year before, not as a believer, but as one of the last skeptics alive, by now a dying species. In the previous decades, scientific and technological progress continued to hunt God into the farthest and most obscure explanatory gaps, into oblivion. Until the emergence of Lord Q three years earlier, when everything was turned upside-down. Since then, he performed the most scientifically incredible miracles, normally attributed to a deity. Roxanne, a reputed physicist with a hobby of debunking pseudoscience, received a grant from a philanthropist who asked her to either find a scientific explanation for Quentin’s miracles, or prove that they are authentic.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">When she joined the group, the believers disliked her for her skepticism, which remained unchanged even now, a year later. The only reason they tolerated her was because Quentin seemed to have a strange affection for her. She was allowed to be in his proximity all the time, and this made them dislike her even more, but they had to accept her. Quentin liked her, and was continuously amused by the suspicious look in her eyes, which was visible even when she was surprised by his miracles. For a year she followed him everywhere, witnessed him healing people, stopping natural catastrophes, wars, crime, and bringing back faith. She even saw him bringing back to life the president of the United States, killed by a rare form of cancer. But she continued to say that there must be a scientific explanation for everything.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Quentin never ceased to be intrigued by her disbelief and continued to watch her reaction as he performed his miracles. Once, he started to make flowers grow up out of nowhere and blossom in seconds, covering every piece of ground where Roxanne stepped. She was surprised, she blushed, and she told him that this is harassment. He didn’t know whether she was joking, so he stopped. It would have been the easiest thing for him to make her fall in love with him or even become a believer, but he didn’t want it to be like this. He loved her for being independent, and he would have never traded the chance to see her surprised and at the same time unimpressed by his powers for having an obedient Roxanne in place, a fervent adorer like the rest of his followers.</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6"></span><br /><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6">As Roxanne was waiting, worried to see what Quentin will do about the asteroid, Tom approached her. </span><br /><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– He will make it, don’t worry, he said. </span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– I know, she replied.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Tom was the first and only friend Roxanne made among Quentin’s followers. He used to be the leader of the third group of scientists sent by the James Randi Educational Foundation. The group tried to debunk Quentin’s miracles and find mistakes in the reports of the previous two groups, which declared that this was indeed the first authentic miracle recorded by the foundation. He saw no other choice but to accept Quentin’s miracles as true. The Foundation had to award the Psychic Prize for the first time in the history. They offered Quentin one billion dollars, which was the value of the prize at that time. Quentin donated the money to charity, of course. Tom became a believer, and in fact the most ardent one, given that he really looked for the smoke and mirrors and didn’t find any.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">During one of their first conversations, Tom told Roxanne that he believed that all these miracles can be explained if Quentin had the ability to control quantum probabilities.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">The behavior of particles is governed by quantum mechanics, which is very different from what we observe in our day-to-day life where things behave more like in classical mechanics. According to quantum mechanics, there is always an infinitesimally small probability for something apparently impossible for us to happen even in the real life. In the quantum world, if you observe that a particle – for example an atom – is in a small region of space, at a later time there is a small, but non-zero chance to find it in any other region of space. The reason is that quantum uncertainty makes the particle you know as being in a certain place to have an undetermined velocity, and therefore be able to move anywhere. Hence, at a later time, the particle will potentially be in all places simultaneously. When you observe it again, you will find it in one of these potential places. You can never know where it will be, only know the probability to find it in a given place. This probability is given by a formula called </span><i><span class="tm8">the Born rule</span></i><span class="tm6">. The probability to find the particle in a given place is small, nearly infinitesimally small, but not zero. The same works for more particles or atoms. It’s true that the probabilities become smaller and smaller as the number of involved atoms increases, but it never truly vanishes.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">So Tom told Roxanne that he thinks Quentin performs his miracles by selecting which of the potential positions of a particle becomes true. If he can make a particle be where he wants, he can move objects. He can rearrange matter at will. Roxanne remembered that the neuroscientist Adam Hobson used to entertain similar ideas some years ago, but she always considered him a crackpot. She said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– This happens only for quantum measurements, while Quentin, through his senses, makes classical observations, just like any of us. But, given that any observation we make is eventually a quantum one, maybe your hypothesis is true. However, the probability to control like this even a grain of sand is almost zero.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Tom said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Well, the chances are one in a billion of billions of billions… whatever, let’s just say a chance in a gazillion – but Quentin seems to bring those odds into existence.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– But even if he could do this, how can he influence larger objects, which behave classically due to decoherence?</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Tom said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Decoherence suppresses the probability that the object behaves in a quantum way, but that probability never becomes truly zero. So there is always a very small chance that even larger objects behave in a quantum way, and apparently Quentin has a way to make this chance happen.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne replied that quantum probabilities are just those the Born rule says they are, and she doesn’t believe anyone can really break this law even a bit. So she can’t accept Tom’s suggestion, especially since it would mean Quentin breaking them. Tom said that for him the alternative is even worse because otherwise Quentin would break other laws, which are exact.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Forced to choose between an exact physical law and a probabilistic law, Tom said, I would choose to sacrifice the probabilistic one.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– What about the many-worlds interpretation? she asked. </span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">According to the many-worlds interpretation, every possible alternative result of a quantum observation is realized in an alternative world. This way, all possibilities already existing before the observation continue to exist in independent worlds, as if the world splits in many alternative histories.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne continued: </span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Assuming the many-worlds interpretation is true, if Quentin’s miracles are explained because he controls the probabilities, this means that in the vast majority of the alternative worlds he doesn’t control them, just like any of us. And even for us, there is a very tiny chance that the possibility we wish becomes true, but that chance is so small, that it practically never happens. And even if miracles are just very improbable but still possible events, the Born rule has to remain valid in each of the alternative worlds. So the chances that Quentin remains in a world in which he always gets to make his miracles are the same – one in a gazillion.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Tom said he wants to think about this. Next day he said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– What if he suppresses the possibility of the other worlds when he controls the probabilities?</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne said she has to think about it.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Together, Roxanne and Tom analyzed every miracle made by Quentin, and indeed found that these could be achieved if Quentin had the ability to control quantum probabilities. Healing people, levitating, moving objects, all these seemed to them plausibly explainable if he would really control the quantum probabilities for the particles constituting the objects. But she was still not satisfied, she wanted to know how he does all these, and whether he really breaks the Born rule. She was sure that there must be a better explanation.</span></div><div class="Normal tm7" style="text-align: justify;"><br /></div><div class="Normal tm7" style="text-align: justify;"><span class="tm6"><span class="tm6"></span></span></div><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">In the last minutes before the asteroid was about to hit Earth, Quentin got out of his tent holding a TV set. He put the TV on a table as the TV anchor was reporting the most recent news about the asteroid. Quentin sat on the grass and started to meditate. After several minutes, his peaceful face, his entire body started to glow. He began to levitate. He raised his eyes to the sky, then his hand, and smiled warmly. Shortly after, the anchor reported in an explosion of joy that the black hole changed its place, and it was no longer a threat to Earth. After a worldwide celebration, life on Earth continued to exist as before. </span></div><span class="tm6"></span><br /><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6">A few weeks later, one night, Quentin was walking with Roxanne. After a full day of prodding him with all kinds of devices, she kept asking him all sorts of questions. He told her with a comforting voice to relax and just enjoy the night. Suddenly, she saw the stars moving in the sky, until they formed the image of her face. Really terrified, she yelled:</span><br /><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Why did you do that?</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Confused, he asked her what the problem was. She said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You just moved thousands of stars to impress me by drawing my portrait!</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– So what? he said.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You probably just killed dozens of civilizations!</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">She looked again in the sky, and the stars were back to their usual positions. He laughed:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– It was an illusion. I didn’t rearrange the stars in the sky, apparently I just bent the light rays coming from them. Or maybe I moved them back, I’m not sure…</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">They laughed, but she was still frightened.</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6"></span><br /><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6">For days, Roxanne kept studying Quentin with various high-tech devices. The money was not a problem for her sponsor. She scanned him, monitored his brain activity, recorded everything, and sent the data to specialized laboratories, for more thorough analysis. She found that he had a device implanted in his brain, following a car crash that happened three years earlier. Quentin refused to talk about the implant. But the implant seemed to do nothing relevant that would explain his abilities. It was simply a device that monitored his brain activity, collecting data from a number of places on his cortex. The implant was also stimulating some regions of his brain from time to time, but nothing relevant. </span><br /><div class="Normal tm5" style="text-align: justify;"><span class="tm6">She also found something that surprised her even more: everywhere in Quentin’s body, there were billions of tiny spheres. She didn’t see them initially, they were too small, but she eventually found them after a more detailed analysis of Quentin’s tissues. She collected several of them and sent them to a laboratory. Just like the implant in his brain, the spheres seemed to be useless, or at least they didn’t exchange energy or information with the body at all, so she was very curious about their role.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">When the result came back, she was perplexed. The tiny spheres were nanobombs, this was their only functionality. Remote controlled nanobombs, programmed to blow if they receive a certain signal, but obviously never detonated because that signal was never sent. Why on Earth would he have an implant that collects brain activity and never does anything with it, and why have billions of nanobombs also doing nothing at all? Anyway, the presence of nanobombs prevented her from trying to disable Quentin’s brain implant to see if it was the source of his powers.</span><br /><br /></div><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /><span class="tm6">Roxanne told Quentin what she found, and insisted that he should give her more explanations. He said that he will tell her more, if she promised to keep the secret.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– I have something to confess, he said. I am not the first one with these powers. Professor Adam Hobson, my uncle, who saved me after the car crash, he used to have them as well.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You mean Adam Hobson, the guy with that crazy theory about the quantum brain, who disappeared a few years ago and was never found? Roxanne said.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Quentin told her how he had the car accident, and Adam saved his life with his highly advanced surgical robots. He said he had to implant a device in Quentin’s brain. Then Adam personally conducted the recovery therapy, teaching Quentin how to gain control of his body again and how to control his thoughts. After the recovery, Adam revealed more to Quentin. He said that both of them had implants, and that the implants allowed their minds to control matter.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Quentin continued:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Uncle Adam taught me how to make miracles, by wishing them and by thinking at the changes I should be observing in the world after every miracle. He told me that he can already do this, and I will soon be able to do it too, and that both of us are godlike beings. He said there is no room for two gods, and that he will soon leave this world for a better one. He also said that I will leave this world soon too, and to tell everyone who cares about me that I will go to a better world. Then Uncle Adam activated my implant, and then he vanished in a bright explosion. It was the last time I saw him. I don’t know how this device functions. I just wish for things to happen, visualize what to expect once they happen until I have that feeling that they will do, and then they happen exactly as I visualized them. I learned that this way I can do pretty much anything.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne said:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– But this doesn’t make sense. The implant doesn’t do anything, it just collects your brain activity and stimulates it to make you feel happy. I just don’t understand…</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6"></span><br /><div class="Normal tm7" style="text-align: center;"><span class="tm6">* * *</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><span class="tm6">Quentin was doing his morning meditation, when Roxanne came with a desperate look on her face, yelling from afar:</span><br /><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You have to stop doing any miracle right now! This is gonna kill you!</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– What?! said Quentin. What do you mean? What happened?</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You know Tom’s hypothesis that the way you do your miracles is by controlling quantum probabilities? Well, you can’t control them, nobody can!</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– OK… so what? Quentin said.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– I know what’s in your head, what that implant does to you, she said.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Well, good to know you finally got it. I’m all ears. Sit down here with me…</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne caught her breath, but didn’t sit on the grass near Quentin. Instead she kept circling him, explaining:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– Whenever there’s a choice between more quantum alternatives, new worlds are created, in which each of these possibilities become reality. You can’t control which world is ours, you exist in all of them. Including in those in which your miracle doesn’t work. But when you make a miracle, you visualize the desired result, and your brain implant collects this information from your brain. Then, it compares it with what you observe afterwards. If your wish becomes real, then nothing happens.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– I don’t get it, Quentin said. Nothing happens, so the device does nothing. Then how can this explain anything?</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Roxanne grabbed his shoulders and looked at him frantically:</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– But if your wish doesn’t come true, which is almost always the case, then the implant detonates the billions of bombs in your body. You explode into a bright light, and you die.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– I never died…</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– You will always find yourself, of course, in a world in which you are not killed, hence where your wish came true. Your implant is a quantum suicide device, inspired by the quantum suicide thought experiment proposed in the eighties to test the many-worlds interpretation. Gazillions of worlds are created whenever you make a miracle, and gazillions of copies of you are killed in all of these worlds! Gazillions of copies of us are left in tears… In all worlds, except in those very, very, very rare in which your wish comes true! </span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">Seeing her crying, Quentin dispersed the clouds in the sky and made out of thin air a rain of flower petals.</span></div><div class="Normal tm5" style="text-align: justify;"><span class="tm6">– See, nothing happened, dummy…</span></div><div class="Normal tm5" style="text-align: justify;"><br /></div><div class="Normal" style="text-align: justify;"><br /></div><div class="Normal" style="text-align: justify;"></div><div class="Normal tm5" style="text-align: right;"><span class="tm6"></span><i><span class="tm7">Cristi Stoica, May 17, 2016</span></i></div><div class="Normal tm5"><br /></div><div class="Normal" style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-36947983574907410192016-05-03T00:42:00.002+03:002017-12-26T15:16:05.143+02:00Are Single-World Interpretations of Quantum Theory Inconsistent?<div dir="ltr" style="text-align: left;" trbidi="on"><div><div style="text-align: justify;">A recent eprint caught my atention: <a href="https://arxiv.org/abs/1604.07422" target="_blank">Single-world interpretations of quantum theory cannot be self-consistent</a> by Daniela Frauchiger and Renato Renner. In the abstract we read</div><div style="text-align: justify;"></div><blockquote class="tr_bq"><div style="text-align: justify;"><i>We find that, in such a scenario, no single-world interpretation can be logically consistent. This conclusion extends to deterministic hidden-variable theories, such as Bohmian mechanics, for they impose a single-world interpretation. </i></div></blockquote><div style="text-align: justify;">The article contains an experiment based on <a href="https://en.wikipedia.org/wiki/Wigner's_friend" target="_blank">Wigner's friend thought experiment</a>, from which is deduced in a Theorem that there cannot exist a theory T that satisfies the following conditions:</div><blockquote class="tr_bq"><div style="text-align: justify;">(QT) <i>Compliance with quantum theory</i>: T forbids all measurement results that are forbidden by standard [non-relativistic] quantum theory (and this condition holds even if the measured system is large enough to contain itself an experimenter).<br />(SW) <i>Single-world</i>: T rules out the occurrence of more than one single outcome if an<br />experimenter measures a system once.<br />(SC) <i>Self-consistency</i>: T's statements about measurement outcomes are logically consistent (even if they are obtained by considering the perspectives of different experimenters). </div></blockquote></div><div><div style="text-align: justify;">A proof of the inconsistency of <a href="https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory" target="_blank">Bohmian mechanics</a> (discovered by de Broglie and rediscovered and further developed by David Bohm) would already be a big deal, because despite being rejected with enthusiasm by many quantum theorists, it was never actually refuted, neither by reasoning, nor by experiment. Bohmian mechanics is based on two objects: the <i>pilot-wave</i>, which is very similar to the standard wavefunction and evolves according to the Schrödinger equation, and the <i>Bohmian trajectory</i>, which is an integral curve of the current associated to the Schrödinger equation. While one would expect the Bohmian trajectory to be the trajectory of a physical particle, all observables and physical properties, including mass, charge, spin, properties like non-locality and contextuality, are attributes of the wave, and not of the Bohmian particle. This explains in part why BM is able to satisfy (QT). The pilot-wave itself evolves unitarily, not being subject to the collapse. Decoherence (first discovered by Bohm when developing this theory) plays a major role. The only role played by the Bohmian trajectory seems (to me at least) to be to point which outcome was obtained during an experiment. In other words, the pilot-wave behaves just like in the Many-Worlds Interpretation, and the Bohmian trajectory is used only to select a single-world. But the other single-worlds are equally justified, once we accepted all branches of the pilot-wave to be equally real, and the Bohmian trajectory really plays no role. I will come back later with a more detailed argumentation of what I said here about Bohmian mechanics, but I repeat, this is not a refutation of BM, rather some arguments coming from my personal taste and expectations of what a theory of QT should do. Anyway, if the result of the Frauchiger-Renner paper is correct, this will show not only that the Bohmian trajectory is not necessary, but also that it is impossible in the proposed experiment. This would be really strange, given that the Bohmian trajectory is just an integral curve of a vector field in the configuration space, and it is perfectly well defined for almost all initial configurations. This would be a counterexample given by Bohmian mechanics itself to the Frauchiger-Renner theorem. Or is the opposite true?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But when you read their paper you realize that any theory compatible with standard quantum theory (which satisfies QT and SW) has to be inconsistent, including therefore standard QT itself. Despite the fact that the paper analyzes all three options obtained by negating each of the three conditions, it is pretty transparent that the only alternative has to be Many-Worlds. In fact, even MW, where each world is interpreted as a single-world, seems to be ruled out. If correct, this may be the most important result in the foundations of QT in decades.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Recall that the <a href="https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory" target="_blank">Many-Worlds Interpretation</a> is considered by most of its supporters as being the logical consequence of the Schrödinger equation, without needing to assume the wavefunction collapse. The reason is that the unitary evolution prescribed by the Schrödinger equation contains in it all possible results of the measurement of a quantum system, in superposition. And since each possible result lies in a branch of the wavefunction that can no longer interfere with the other branches, there will be independent branches behaving as separate worlds. Although there are some important open questions in the MWI, the official point of view is that the most important ones are already solved without assuming more than the Schrödinger equation. So perhaps for them this result would add nothing. But for the rest of us, it would really be important.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">My first impulse was that there is a circularity in the proof of the Frauchiger-Renner theorem: they consider that it is possible to perform an experiment resulting in the superposition of two different classical states of a system. Here by "classical state" I understand of course still a quantum state, but one which effectively looks classical, as a measurement device is expected to be before and after the measurement. In other words, their experiment is designed so that an observer sees a superposition of a dead cat and an alive one. Their experiment is cleverly designed so that two such observations of "Schrödinger cats" lead to inconsistencies, if (SW) is assumed to be true. So my first thought was that this means they already assume MWI, by allowing an observer to observe a superposition between a classical state that "happened" and one that "didn't happen".</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But the things are not that simple, because even if a quantum state looks classical, it is still quantum. And there seem to be no absolute rule to forbid the superposition of two classical states. <a href="https://en.wikipedia.org/wiki/Einselection" target="_blank">Einselection</a> (<i>environment-induced superselection</i>) is a potential answer, but so far it is still an open problem, and at any rate, unlike the usual <a href="https://en.wikipedia.org/wiki/Superselection" target="_blank">superselection rules</a>, it is not an exact rule, but again an effective one (even if it would be proven to resolve the problem). So the standard formulation of QT doesn't actually forbid superpositions of classical states. Well, in Bohr's interpretation there are quantum and there are classical objects, and the distinction is unbreakable, so for him the extended Wigner's friend experiment proposed by Frauchiger and Renner would not make sense. But if we want to include the classical level in the quantum description, it seems that there is nothing to prevent the possibility, in principle, of this experiment.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Reading the Frauchiger-Renner paper made me think that there is an important open problem in QT, because it doesn't seem to prescribe how to deal with classical states:</div><blockquote class="tr_bq"><div style="text-align: justify;"><span style="background-color: white;"><b>Does QT allow quantum measurements of classical (macroscopic) systems, so that the resulting states are non-classical superpositions of their classical states?</b></span></div></blockquote><div style="text-align: justify;">I am not convinced that we are allowed to do this even in principle (in practice seems pretty clear it is impossible), but also I am not convinced why we are forbidden. To me, this is a big open problem. Can the answer to this question be derived logically from the principles of standard QT, or should it be added as an independent, new principle?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">My guess is that we don't have a definitive solution yet. It is therefore a matter of choice: those accepting that we are allowed to perform any quantum measurements on classical states, perhaps already accept MWI, and consider that it is a logical consequence of the Schrödinger equation. Those who think that one can't perform on classical states quantum measurements that result in Schrödinger cats, will of course object to the result of the paper of Frauchiger and Renner and consider its proof circular.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I will not rush with the verdict about the Frauchiger-Renner paper. But I think at least the open problem I mentioned deserves more attention. Nevertheless, if their result is true, it will pose a big problem not only to Bohmian mechanics, but also to standard QT. And also to my own proposed interpretation, which is based on the possibility of a single-world unitary solution of the Schrödinger equation (see my recent paper <a href="http://quanta.ws/ojs/index.php/quanta/article/view/40" target="_blank">On the Wavefunction Collapse</a> and the references therein).</div></div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-124350264510724511.post-63893581228568715882016-05-02T09:55:00.000+03:002016-05-02T10:09:12.036+03:00An attempt to refute my Big-Bang singularity solution<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">My paper in cause about Big-Bang singularities is <a href="https://arxiv.org/abs/1112.4508" target="_blank">arXiv:1112.4508</a> (<i>The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved</i>). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in <a href="https://arxiv.org/abs/1105.0201" target="_blank">arxiv:1105.0201</a>. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis <a href="https://arxiv.org/abs/1301.2231" target="_blank">arxiv:1301.2231</a>. In the paper <a href="https://arxiv.org/abs/1112.4508" target="_blank">arXiv:1112.4508</a>, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The paper attempting to refute my result is <a href="http://arxiv.org/abs/1603.02837" target="_blank">arxiv:1603.02837</a> (<i>Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities</i>, by L. Fernández-Jambrina). Both my paper and this one appeared this year in <i>International Journal of Theoretical Physics</i>. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was <u>explicitly</u> not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.</div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-47791091905043288992016-03-27T11:02:00.004+03:002016-04-07T15:21:22.256+03:00Faster than light signaling leads to paradoxes<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">You may have encountered statements like <a href="http://backreaction.blogspot.ro/2016/03/hey-bill-nye-please-stop-talking.html" target="_blank">this one made by Sabine</a>:</div><div style="text-align: justify;"><blockquote class="tr_bq"><i>Once you can send information faster than the speed of light, you can also send it back in time. If you can send information back in time, you can create inconsistent histories, that is, you can create various different pasts, a problem commonly known as “grandfather paradox:” What happens if you travel back in time and kill your grandpa? Will Marty McFly be born if he doesn’t get his mom to dance with his dad? Exactly this problem.</i></blockquote></div><div style="text-align: justify;">This is correct. Special relativity implies that, if faster than light signaling would be true, you would be able to signal to your own past, and this can lead to paradoxes. Here I will explain how exactly this can happen. This is rather elementary special relativity stuff, but I realized there is much confusion around it. First, I never saw a precise scenario in which faster than light (FTL) signaling can be used to signal back to your own past, so I will give one. Second, I have the feeling that when people make statements like this,</div><ul style="text-align: justify;"><li>they either refer to the fact that, if an observer A sends FTL signals in her own future, for another observer B it may look like sending in back in time, in B's reference frame, as in this figure:</li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Cn0Qcx0GTKs/VveApjnFLSI/AAAAAAAABII/Y_N01I-M3EMALuAAjD1nZ2muph3-XFg5Q/s1600/ftl-signaling-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="191" src="https://3.bp.blogspot.com/-Cn0Qcx0GTKs/VveApjnFLSI/AAAAAAAABII/Y_N01I-M3EMALuAAjD1nZ2muph3-XFg5Q/s320/ftl-signaling-1.png" width="320" /> </a></div><div class="separator" style="clear: both; text-align: justify;">Orange lines represent light cones, blue represent timelike curves (observers), red represents the proper space of an observer, and green represents FTL signals. While the picture represents the proper space of A as a horizontal red line, the proper space of B is oblique, due to the Lorentz transformation (relativity of simultaneity).</div><div style="text-align: justify;">The first scenario is not that paradoxical, because observer B can always reinterpret the signal from A to B as a signal going in his own future, from B to A. But even in this case, we will have the problem of who actually created the message in the first place.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><ul style="text-align: justify;"><li>or they refer to examples where the observer sends an FTL signal toward her own past, as in this figure:</li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/--pxTGlq_2VA/VveBqYQX5wI/AAAAAAAABIQ/KffZrlmLw0sp4lTs-jznwIo2WbiPL6PXw/s1600/ftl-signaling-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="249" src="https://1.bp.blogspot.com/--pxTGlq_2VA/VveBqYQX5wI/AAAAAAAABIQ/KffZrlmLw0sp4lTs-jznwIo2WbiPL6PXw/s320/ftl-signaling-2.png" width="320" /></a></div>The second scenario is the usual example of causality violation due to FTL you will find, but is refutable on the grounds that you are not allowed to send signals directly to your own past, or to receive signals directly coming from your own future.<br /><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Here is how FTL signaling would imply that one can signal back in time, <i>using only signals sent in the future and received from the past</i>, with respect to the proper reference frame:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-6g5cXgkKn4g/VveD1Q5wP5I/AAAAAAAABIc/e3pX0Rb1Xu4XFCbDRFGgjcfbu_-zbuW-g/s1600/ftl-back-signaling.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="https://1.bp.blogspot.com/-6g5cXgkKn4g/VveD1Q5wP5I/AAAAAAAABIc/e3pX0Rb1Xu4XFCbDRFGgjcfbu_-zbuW-g/s400/ftl-back-signaling.png" width="400" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The inertial observer A accelerates away from B, then sends an FTL signal at t₀. Observer B receives it at t'₀ in his proper time, then accelerates away from observer A, then sends it back, at t'₁. Observer A receives the signal at t₋₁, where t₋₁< t₀. </div><div style="text-align: justify;">So indeed FTL implies signaling back in your own past, even if FTL signals are sent only to the proper future and received only from the proper past.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Let us see how this allows paradoxes. Suppose that earlier A and B agreed on the following: if A receives the message "Yes", she sends the message "No", and if she receives "No", she sends "Yes". If B receives a signal, he just resends it without changing. Then, we have a paradox: does A send the message "Yes", or "No"? It is similar to the liar paradox, since if she sends "Yes", then she receives "Yes", so she sends "No", and so on. But it is also like grandfather's paradox, because B can send instead of a message, a killing FTL ray, to kill A or her grandfather before she was born.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So far there is no evidence of FTL signaling, except for some misunderstandings of the EPR "paradox". I don't know either of a fundamental physical law which prevents it, given that tachyonic solutions are mathematically consistent, both in special relativity, and in quantum field theory. But as we have seen, FTL would lead to time travel paradoxes.</div></div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-124350264510724511.post-24259557934724069582016-02-13T11:57:00.001+02:002016-02-29T16:42:36.348+02:00Gravitational waves, evidence of the fourth dimension of spacetime<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Most of the headlines are right: gravitational waves are a long-known prediction of General Relativity, and their detection show that the theory is correct. I waited a bit to see if an important consequence of this fact will be uncovered, but it seems it did not, so let me tell you: <i>This experiment refutes a great deal of alternatives to General Relativity proposed in the last decades</i>. You perhaps already noticed that many physicists brag on social networks or even in online articles that the detection of gravitational waves confirmed not necessarily GR, but also the alternatives to GR they endorse. But in fact this experimental result refutes those alternative theories in which the background metric of spacetime is fixed, as well as those in which space is a three-dimensional thing that is not part of a four-dimensional spacetime, as in GR is. I will discuss first the latter. Many relativists would say that such theories were already refuted, but if you talk with a supporter of such a theory, you will hear that it is not necessarily so. The idea of a 3-dimensional space still could be defended, with the price of complicating the things. But in my opinion, LIGO just put the last nail in the coffin of such theories. Because gravitational waves are waves of spacetime, and not of space. <i>They are waves of the Weyl curvature tensor, which simply vanishes in a space with less than four dimensions!</i></div><br /><div style="text-align: justify;">The number of those trying to replace GR with other theories increased very much lately. The main reasons may be that they don't know how to handle singularities, or that they don't know how to enforce to gravity the few methods we know to quantize fields, so they come up with alternative theories. While I don't think it is easy to replace GR with something that explain as much starting from as little as GR does, I agree that these alternative should be explored (by others, of course). Related to whether there is a 3-dimensional space or a 4-dimensional spacetime, you can find reasons to doubt the fourth dimension too. First, even Galilean space and time can be joined in a four-dimensional spacetime, but not as tight as in Relativity. In Relativity, indeed, Lorentz transformations mix the time and space directions, leading to length contraction and time dilation, but some think that these are sort of due to the perspective of the observer, without needing a fourth dimension. In addition, many quantities become unified in the four-dimensional spacetime, such as energy and momentum, electric and magnetic fields etc. But maybe these are all just circumstantial evidence of the fourth dimension. You can take any theory and make it satisfy some four-dimensional transformations. Especially since the evolution equations are hyperbolic, you can do this. Also, you can express any equation in Physics in curvilinear coordinates, and this doesn't mean four dimensions, neither that the invariance to diffeomorphisms means something physical. So people cooked up or even revived various alternatives to GR, in which three-dimensional space is not part of a spacetime. If such a theory does not include curvature, it will not predict gravitational waves. Also, if it admits curvature, but only of the three-dimensional space, nothing in four dimensions, it still doesn't predict gravitational waves out of this curvature. So now the proponents of alternative to GR will have to adjust their theories. Maybe some predict naturally some sort of gravitational waves, but most don't, so they will put the waves by hand. The Cotton tensor, which is somewhat analogous to the Weyl tensor in three dimensions, because its vanishing means conformal flatness, is believed sometimes to give the gravitational waves. But the Cotton tensor vanishes in vacuum, where the Ricci tensor vanishes too. So this can't give gravitational waves in three-dimensional spacetime.</div><div style="text-align: justify;"><div style="text-align: justify;"><br />What about theories with more dimensions? For instance, Kaluza theory is an extension of GR to 5 dimensions, which is able to obtain the sourceless electromagnetic field from the extra dimension. You can also obtain other gauge theories as Kaluza-type theory. Such modification predict gravitational waves too.<br /><br />What about String Theory? It is said that String Theory includes GR, so it must include gravitational waves too, isn't it? But the reason why is said to include GR is because it contains closed strings, which have spin 2, and they are identified with the still hypothetical gravitons (not even predicted by GR alone) just because they have spin 2. But if your theory has spin-2 particles, even if you call them gravitons, it doesn't mean you have included GR. String Theory usually works on fixed background, which usually is flat, or with constant curvature as in the anti-de Sitter spacetime. I am not aware of a successful way to include GR in String Theory such that gravity is an effect of spacetime curvature. If this can be done, can it predict gravitational waves in a natural way? Can it even include GR in a natural way?</div><br /></div><div style="text-align: justify;">To my surprise, the advocates of theories which don't have dynamical background, or are based on three-dimensional space, didn't take the chance to predict that there are no gravitational waves, as their theories imply. They should have done this, and they should have waited for the confirmation of their prediction by LIGO. My guess is that maybe they doubted that GR will be refuted - nobody wants to make predictions which contradict GR in regimes that can be experimentally verified. Whenever we could test the predictions of GR, they were always confirmed, so I think not even those supporting alternative theories actually believed that it will be refuted this time. So I guess that's why they didn't say that their theory predicts no gravitational waves, and that they really think that LIGO will show there are. Instead, now you can see that some claim that gravitational waves confirm their theories too. Like for examples they are waves of space alone, and not of spacetime, which is not true, unless you put them in your theory by hand (while in GR they are just there, not a mobile or replaceable part). So I expect to see a lot of papers in which it is explain that their theory was there too, along with GR, when gravitational waves were predicted.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-BYZURcHJJSA/Vr7rvXf4jbI/AAAAAAAABHg/azMs0EUUvvg/s1600/papers-are-coming.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="https://2.bp.blogspot.com/-BYZURcHJJSA/Vr7rvXf4jbI/AAAAAAAABHg/azMs0EUUvvg/s320/papers-are-coming.jpg" width="320" /></a></div></div><br />Since the model was based on calculations made using GR applied to two colliding black holes, LIGO confirmed GR (again): it confirmed <i>gravitational waves</i>, and <i>black holes</i> (again). This does not exclude though the possibility that other modifications, alternatives or extensions of GR can work out similar predictions. So further experiments may be needed. But what I can say is that the theories that remained are modifications of GR that still explain gravity as spacetime curvature, and still make use of the four-dimensional spacetime. Theories that at purpose mimic most of GR.<br /><br /><i>Space is dead, long live spacetime!</i></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-79819746169726188832016-01-12T01:53:00.001+02:002016-01-12T08:03:29.079+02:00Wavefunction collapse vs. unitary evolution, superdeterminism vs. free-will<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Today appeared <a href="http://quanta.ws/" target="_blank">Quanta</a>'s <a href="http://quanta.ws/ojs/index.php/quanta/issue/view/6" target="_blank">special issue dedicated to Feynman</a>. It is a very cool new open access journal on Quantum Mechanics.</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"> <a href="http://quanta.ws/ojs/index.php/quanta/issue/view/6" target="_blank"><img alt="http://quanta.ws/ojs/index.php/quanta/issue/view/6" border="0" src="http://quanta.ws/ojs/public/journals/1/cover_issue_6_en_US.jpg" height="320" width="320" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I am happy because it contains my article, <a href="http://quanta.ws/ojs/index.php/quanta/article/view/40" target="_blank">On the Wavefunction Collapse</a>, edited by two excellent quantum theorists, <a href="https://www.researchgate.net/profile/Eliahu_Cohen" target="_blank">Eliahu Cohen</a> and <a href="http://mattleifer.info/" target="_blank">Matt Leifer</a>. In the paper, I discuss the possibility that the unitary evolution, governed by Schrödinger's equation, allows for the apparent wavefunction collapse. <a href="http://philsci-archive.pitt.edu/4344/" target="_blank">I first wrote about this</a> idea <a href="http://fqxi.org/community/essay/winners/2008.1#Stoica" target="_blank">some years ago</a>, and its implications on free-will triggered some <a href="http://www.scottaaronson.com/papers/giqtm3.pdf" target="_blank">interesting developments</a>. There are several great difficulties with this, mostly due to the fact that quantum measurement introduce strong constraints on the solutions of Schrödinger's equation. But I hope my arguments that these constraints are not incompatible with unitary evolution are more convincing now. The article had three completely different versions. The first one was based on integral curves in the configuration space, those called by some <i>Bohmian trajectories</i>. I consider the idea of interpreting these integral curves as point-particles interesting, but in order to survive, Bohmians had to transfer more and more of the physical properties initially attributed to point-particles moving along these trajectories to the pilot wave, and I think that eventually only the pilot wave matters. So in fact the wavefunction does all the job. The second version of my paper was based on Feynman's path integrals, but I realized that my original approach, to use Schrödinger's equation, is better suited.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Note that unitary evolution is deterministic. Moreover, trying to assign reality to the wavefunction leads to non-locality, as Bell's Theorem shows, and to contextualism, as the Bell-Kochen-Specker Theorem shows. And last year I published <a href="http://arxiv.org/abs/1212.2601" target="_blank">a simple proof</a> that maintaining unitary evolution implies very fine-tuned initial conditions of the observed system and the measurement apparatus. This amounts to what is called <a href="https://en.wikipedia.org/wiki/Superdeterminism" target="_blank">superdeterminism</a>. But since nobody can see the complete initial data of the wavefunction, it is also possible to consider that the initial conditions are initially not fixed, and they are more and more constrained with each measurement. While superdeterminism forces us to admit that the property we will choose to measure one day was determined from the Big-Bang, leaving the initial conditions free, and fixing them with each measurement, allows us to choose freely what to measure. And this doesn't break causality, because you can't change the observed past, only the "yet undecided" past. The required consistency between the initial conditions can also be seen, when thinking in terms of the four-dimensional block world picture from Relativity, as a <a href="http://arxiv.org/abs/1309.2309" target="_blank">global consistency principle</a>, where "global" refers to the entire spacetime. So we have a <i>timeless picture</i>, based on the block world, but which does not contradict free-will, and a <i>temporal picture</i>, based on the delayed choice of initial conditions. These two pictures provide alternative interpretations of superdeterminism which are compatible with free-will (whatever "free-will" means).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But if there is such a thing as free-will, the free agent should at least partially be somehow above the world and outside of time, to be able to choose among the possible deterministic solutions describing the world itself. Because if it would be completely part of the solution, it could not have free-will. It is easy from here to speculate about an immortal soul and even the possibility that it is part of a supreme being, and I don't want to do this, especially since I consider myself free-will-agnostics. However, this implicit connection may be the reason why so many people are firmly either for, or against free-will.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Completely independent on this, yesterday, Sabine Hossenfelder wrote on <a href="http://backreaction.blogspot.com/" target="_blank">her blog</a> a post called <a href="http://backreaction.blogspot.com/2016/01/free-will-is-dead-lets-bury-it.html" target="_blank">Free will is dead, let’s bury it</a>, in which she made some strong affirmations against free-will and people who believe in it. That free-will is bad science. I think that we know too little about this to call it science, but this can be said also about many things which we know exist and we would want to understand better. Then she said that people who believe in free-will have existential worries and hidden agendas. I agree that when we speak about believing in something, even in a physical law, we arrive at that belief in part because of our past experiences. Otherwise, how can we explain that people can change their opinion even about physical laws? So indeed, subjectivity is involved, but this happens all the time, not only with respect to believing in free-will. Then she said "I am afraid the politically correct believe in free will hinders progress on the foundations of physics". I think that if physicists reject their peers' papers or throw away their own results for not being consistent with free will, this is rather the exception, and they do this for many other reasons, including sex, race, or simply because they have different views. At the end, she wrote "buying into the collapse of the wave-function seems a small price to pay compared to the collapse of civilization". This is a nice pun, but quantum theorists who believe in collapse do so because they can't make sense of the outcomes of measurements without collapse, not because they want to support free-will. Many of them don't even believe in free-will, while others don't believe in collapse, but still don't reject free-will. But the reason they don't accept easily alternatives to QM (in particular hidden-variable superdeterministic theories and my unitary collapse approach) is simply that standard QM works much better, and not because they want to save their illusion of free will.</div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-124350264510724511.post-40970133303961299562015-11-25T16:16:00.001+02:002015-11-25T16:19:29.573+02:00Happy 100th birthday, General Relativity!<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-nzZa3S_JBPk/VlXB1-9cpKI/AAAAAAAABFk/PPhddlRwJJc/s1600/Einstein%2BEquation.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-gTZculQTd7U/VlXB9al_e6I/AAAAAAAABFw/dSqEhDVhm-E/s1600/La%2BMadre%2BTerra%2Bby%2BPietro%2BCascella.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="390" src="http://2.bp.blogspot.com/-gTZculQTd7U/VlXB9al_e6I/AAAAAAAABFw/dSqEhDVhm-E/s400/La%2BMadre%2BTerra%2Bby%2BPietro%2BCascella.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><span class="fbPhotosPhotoCaption" data-ft="{"tn":"K"}" id="fbPhotoSnowliftCaption" tabindex="0"><span class="hasCaption">"<a href="http://www.icranet.org/MG12/CASCELLA_LA_MADRE_TERRA.pdf" target="_blank">La Madre Terra</a>" by <a href="https://en.wikipedia.org/wiki/Pietro_Cascella" target="_blank">Pietro Cascella</a>, made for ICRANet. </span></span><span class="fbPhotosPhotoCaption" data-ft="{"tn":"K"}" id="fbPhotoSnowliftCaption" tabindex="0"><span class="hasCaption"><span class="fbPhotosPhotoCaption" data-ft="{"tn":"K"}" id="fbPhotoSnowliftCaption" tabindex="0"><span class="hasCaption">You can see Einstein's equation, which he translated into the metaphor "marble = wood". </span></span>My guess is that it symbolizes Einstein's idea that everything (matter, life, not just the earth) emerges from the perfect geometry of spacetime. I took this photo at the Marco Besso Foundation exhibition in Rome, during the <a href="http://www.icra.it/MG/mg14/" target="_blank">XIV-th Marcel Grossman conference</a>.<br /> <a href="http://www.icranet.org/MG12/CASCELLA_LA_MADRE_TERRA.pdf" rel="nofollow nofollow" target="_blank"></a></span></span> </div><div class="separator" style="clear: both; text-align: justify;"><a href="http://3.bp.blogspot.com/-gTZculQTd7U/VlXB9al_e6I/AAAAAAAABFs/FNH8jSuDxTE/s1600/La%2BMadre%2BTerra%2Bby%2BPietro%2BCascella.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-76851210031682847372015-10-20T23:28:00.002+03:002015-10-20T23:34:34.849+03:00Quantum Measurement and Initial Conditions<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">My paper <a href="http://link.springer.com/article/10.1007/s10773-015-2829-2" target="_blank">Quantum Measurement and Initial Conditions</a>, recently published in <a href="http://link.springer.com/journal/10773" target="_blank">International Journal of Theoretical Physics</a>:</div><div style="text-align: justify;"><br /></div><blockquote class="tr_bq"><div style="text-align: justify;">Quantum measurement finds the observed system in a collapsed state, rather than in the state predicted by the Schrödinger equation. Yet there is a relatively spread opinion that the wavefunction collapse can be explained by unitary evolution (for instance in the decoherence approach, if we take into account the environment). In this article it is proven a mathematical result which severely restricts the initial conditions for which measurements have definite outcomes, if pure unitary evolution is assumed. This no-go theorem remains true even if we take the environment into account. The result does not forbid a unitary description of the measurement process, it only shows that such a description is possible only for very restricted initial conditions. The existence of such restrictions of the initial conditions can be understood in the four-dimensional block universe perspective, as a requirement of global self-consistency of the solutions of the Schrödinger equation.</div></blockquote><a href="http://arxiv.org/abs/1212.2601" target="_blank">The arXiv link</a><a href="http://arxiv.org/abs/1212.2601" target="_blank"></a>. </div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-124350264510724511.post-54067911383938978132015-06-11T20:00:00.000+03:002015-06-11T20:45:40.305+03:00FQXi essay contest 2015 results<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">The results of this year's FQXi essay contest are out.</div><div style="text-align: justify;">The theme was <span class="essayPageTitle">2015 "<a href="http://fqxi.org/community/forum/topic/2282" target="_blank">Trick or Truth: the Mysterious Connection Between Physics and Mathematics</a>".</span></div><div style="text-align: justify;"><span class="essayPageTitle"><a href="http://fqxi.org/community/forum/category/31424?sort=community" target="_blank">Here is the list of all the essays from this contest</a>.</span></div><div style="text-align: justify;"><span class="essayPageTitle"><a href="http://fqxi.org/community/essay/winners/2015.1" target="_blank">And here is the list of the winning essays</a>.</span></div><div style="text-align: justify;"><span class="essayPageTitle">My essay is named </span><a href="http://fqxi.org/community/forum/topic/2383" target="_blank"><span class="entityTitle">"And the math will set you free</span></a><span class="entityTitle">", and <a href="http://fqxi.org/community/essay/winners/2015.1#Stoica" target="_blank">won the third prize</a>.</span></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"></div><div style="text-align: justify;">This is the list of the winning essays:<br /><a href="http://fqxi.org/community/essay/winners/2015.1#Wenmackers">Sylvia Wenmackers</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#S%C3%A9guin">Marc Séguin</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Leifer">Matthew Saul Leifer</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Stoica">Cristinel Stoica</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Maudlin">Tim Maudlin</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Smolin">Lee Smolin</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Wharton">Ken Wharton</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Wise">Derek K Wise</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Bolognesi">Tommaso Bolognesi</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Burov">Alexey Burov, Lev Burov</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Magnusdottir">Sophia Magnusdottir</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Yanofsky">Noson S. Yanofsky</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Fillion">Nicolas Fillion</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Garfinkle">David Garfinkle</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Dantas">Christine Cordula Dantas</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Gibbs">Philip Gibbs</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Durham">Ian Durham</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Mujumdar">Anshu Gupta Mujumdar, Tejinder Singh</a> • <a href="http://fqxi.org/community/essay/winners/2015.1#Walker">Sara Imari Walker</a></div></div>Unknownnoreply@blogger.com5tag:blogger.com,1999:blog-124350264510724511.post-88758799074460785672015-05-08T13:58:00.002+03:002015-05-08T14:37:53.505+03:00The top 5 finalist essays, FQXi essay contest 2015<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Here are the top 5 essays from the 40 finalists of this year's FQXi essay contest, based on the community ratings. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://fqxi.org/community/forum/category/31424?sort=community" target="_blank"><img alt="http://fqxi.org/community/forum/category/31424?sort=community" border="0" height="310" src="http://4.bp.blogspot.com/-hCVJPUtnhrU/VUyU7RFgfSI/AAAAAAAAAvQ/oSUTlOq6DfE/s400/fqxi-finalists-2015.jpg" width="400" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Unofficially, since FQXi didn't announce yet which of the more than 200 essays are the 40 finalists, although the announcement was expected since April 22. <a href="http://fqxi.org/community/forum/topic/2383" target="_blank">My essay</a> is on the fourth place<a href="http://fqxi.org/community/forum/topic/2397" target="_blank"></a>.<br /><br />The finalists will be judged by a jury, who will decide the awards until June 6, 2015.</div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-124350264510724511.post-50249546598236419282015-04-21T20:42:00.002+03:002015-04-22T23:44:15.957+03:00Singular General Relativity (my PhD Thesis) at Minkowski Institute Press<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: justify;">My Ph.D. Thesis Singular General Relativity was published at <a href="http://www.minkowskiinstitute.org/mip/books/stoica.html" target="_blank">Minkowski Institute Press</a> and <a href="http://www.amazon.com/gp/product/1927763347/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1927763347&linkCode=as2&tag=unitflow-20&linkId=42GUEPW4L2G5A2CN" target="_blank">can be ordered at Amazon</a>.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div style="text-align: center;"><a href="http://www.amazon.com/gp/product/1927763347/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1927763347&linkCode=as2&tag=unitflow-20&linkId=42GUEPW4L2G5A2CN" target="_blank"><img alt="http://www.amazon.com/gp/product/1927763347/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1927763347&linkCode=as2&tag=unitflow-20&linkId=42GUEPW4L2G5A2CN" border="0" src="http://www.minkowskiinstitute.org/mip/books/StoicaPhD.jpg" height="400" width="266" /></a></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-124350264510724511.post-23600608261786807322015-03-16T20:17:00.002+02:002017-05-16T15:32:22.036+03:00The Monty Hall problem, retold<div dir="ltr" style="text-align: left;" trbidi="on"><h2 style="text-align: justify;">The Monty Hall problem</h2><div style="text-align: justify;">The <a href="http://en.wikipedia.org/wiki/Monty_Hall_problem" target="_blank">Monty Hall problem</a> is inspired by an American television game show. There are three doors, and behind one of them, the host of the show, Monty, hides a car. Each of the other two doors hides a goat.</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://upload.wikimedia.org/wikipedia/commons/3/3f/Monty_open_door.svg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/3/3f/Monty_open_door.svg" height="177" width="320" /></a></div><div style="text-align: justify;">The contestant is asked to pick a door, so that if she finds the car, she wins the game (and the car). Since there are three doors, chances are $1/3$ that she picked the door behind which is the car. But Monty doesn't open yet the door, but he opens one of the remaining doors, revealing a goat. He then asks the contestant either to keep her original choice, or to switch to the other unopened door. The problem is, what should the contestant do?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The first instinct of anybody may be to think that since there are only two remaining doors, it doesn't matter if you switch the door or not, because the chances are $1/2$ in both ways. However, Marilyn vos Savant explained that if the contestant switches the doors, the chances are $2/3$. while if she doesn't switch them, the chances are $1/3$. This is counterintuitive, and the legend says that <a href="http://en.wikipedia.org/wiki/Monty_Hall_problem" target="_blank">not even Paul Erdős understood it</a>. You can find on <a href="http://en.wikipedia.org/wiki/Monty_Hall_problem" target="_blank">Wikipedia</a> some solutions of this puzzle.</div><h2 style="text-align: justify;">An equivalent puzzle</h2><div style="text-align: justify;">I will present another, simpler puzzle, and show that it is equivalent to the Monty Hall problem.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Consider again three doors, one hiding a car. The contestant is asked to pick either one of the three doors, or two of them. What is the best choice?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Obviously, the contestant should better choose two doors, rather than one. Since if she thinks that the car is behind door number three, choosing also door number one will only double the chances to win.</div><div style="text-align: justify;"></div><br /><div style="text-align: justify;">But how is this related to the Monty Hall problem? Well, it is, because if you play the Monty Hall problem, you can pick two doors, but don't tell Monty, you just tell you picked the remaining one. When Monty asks if you want to switch, then you switch to the other two doors, and since one is already open, you choose the remaining one. This means that choosing a door and switching is equivalent to choosing the other two doors.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So the Monty Hall problem is actually equivalent to having to choose one or two doors. Not switching is equivalent to choosing one door, and switching is equivalent to choosing two doors. So switching gives indeed probability $2/3$.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-124350264510724511.post-79062472931329901922015-03-14T21:33:00.002+02:002015-03-16T07:50:57.895+02:00Round squares exist<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Bertrand Russell said that there are no round squares. But there are. Here are two solutions.</div><h2 style="text-align: justify;">A circle-square</h2><div style="text-align: justify;">This is a square that is circle:</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" height="218" width="400" /></a></div><br />To make it, first make a paper circle and a paper square, with equal perimeters:<br /><br /><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-hzj5qBOeCbQ/VQSK4n5ncTI/AAAAAAAAAs4/5RGB_zalkxE/s1600/round%2Bsquare%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-hzj5qBOeCbQ/VQSK4n5ncTI/AAAAAAAAAs4/5RGB_zalkxE/s1600/round%2Bsquare%2B1.png" height="215" width="400" /></a></div><br />Fold them a bit: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-piHiWTXeuCg/VQSK45fnFRI/AAAAAAAAAs8/3bMrWmCbAS8/s1600/round%2Bsquare%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-piHiWTXeuCg/VQSK45fnFRI/AAAAAAAAAs8/3bMrWmCbAS8/s1600/round%2Bsquare%2B2.png" height="223" width="400" /></a></div><br />Then paste their edges together:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" height="175" width="320" /></a></div><br /><br /><br /><br />The common boundary forms a square that is circle. It is a square, because in the blue surface it has right angles and equal straight edges. It is a circle, because in the red surface its points are at equal distance from a point. In fact, its points are at equal distance from the center even in space, because the red surface is ruled, and all the lines pass through the same point. So the common boundary is also a line on the surface of a sphere.<br /><br /><h2 style="text-align: justify;">Round squares in non-Euclidean geometry</h2><div style="text-align: justify;">Consider for example the geometry on a sphere. On a sphere, polygons are made of the straightest lines on the sphere, which are arcs of the big circles. So, there are <a href="http://en.wikipedia.org/wiki/Square#Non-Euclidean_geometry" target="_blank">squares on a sphere</a></div><div style="text-align: justify;"><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/600px-Square_on_sphere.svg.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/600px-Square_on_sphere.svg.png" height="319" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Image from <a href="http://en.wikipedia.org/wiki/Square#Non-Euclidean_geometry" target="_blank">Wikipedia</a></td></tr></tbody></table>This is a square, since its edges are the shortest and straightest lines on the sphere, they have equal lengths, and its angles are all equal. If one gradually increases the size of the square, the angles increase too. At some point, the angles become $180^\circ$, and the edges become aligned, forming one single big circle:<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Tetragonal_dihedron.png/600px-Tetragonal_dihedron.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Tetragonal_dihedron.png/600px-Tetragonal_dihedron.png" height="320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Image from <a href="http://en.wikipedia.org/wiki/Square#Non-Euclidean_geometry" target="_blank">Wikipedia</a></td></tr></tbody></table><br /><div style="text-align: justify;">So, is it a circle? Is a square? Is a circle and a a square!</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://ytimg.googleusercontent.com/vi/0IBZocFkXGY/0.jpg" src="http://www.youtube.com/embed/0IBZocFkXGY?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div style="text-align: justify;"><br /></div><br /><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-11924561868077600112015-03-14T08:45:00.002+02:002017-05-17T21:16:00.835+03:00A problem with towers of coins<div dir="ltr" style="text-align: left;" trbidi="on"><h2 style="text-align: justify;">The problem</h2><div style="text-align: justify;"><i>In how many ways you can arrange $p$ coins in a sequence of $q$ towers?</i> (it doesn't matter whether the coins can be flipped or rotated).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">For example, here is one way to arrange $12$ coins into a sequence of $5$ towers. The problem asks to count all these ways.</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-nyFyVmcvZ54/VQPSNSyFe-I/AAAAAAAAAsM/Vv6GjpncBHY/s1600/coin-towers-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="81" src="https://4.bp.blogspot.com/-nyFyVmcvZ54/VQPSNSyFe-I/AAAAAAAAAsM/Vv6GjpncBHY/s1600/coin-towers-1.png" width="400" /></a></div><h2 style="text-align: justify;">Motivation</h2><div style="text-align: justify;">I arrived at this problem by being inspired by my yesterday's post, <a href="http://www.unitaryflow.com/2015/03/a-combinatorial-problem-with-balls-and-boxes.html" target="_blank">A combinatorial problem with balls and boxes</a>. The problem was to count the number of ways you can place $k$ balls in $n$ boxes. The answer is <i>$n-1+k$ choose $k$</i>, which is $\displaystyle{\frac{(n-1+k)!}{(n-1)!k!}}$.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So I asked myself, since the result is of the form <i>"$p$ choose $q$"</i>, couldn't I modify the problem so that the result will be the sum over $q$, which is known to be $2^p$? But to do this, boxes and balls should be replaced with objects of the same nature, and playing the role of a box or a ball to be determined by the configuration.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I will tell you a solution by reducing to the problem with boxes and balls, and then a simpler, direct solution.</div><h2 style="text-align: left;">Solution based on the balls and boxes problem</h2>Let's identify two distinct roles in a sequence of towers of coins. We color each coin that starts a tower with blue, and the others with red, as below.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-mT6N2PSvaas/VQPS4MfAn6I/AAAAAAAAAsU/p4LXtqeaPPc/s1600/coin-towers-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="81" src="https://2.bp.blogspot.com/-mT6N2PSvaas/VQPS4MfAn6I/AAAAAAAAAsU/p4LXtqeaPPc/s1600/coin-towers-2.png" width="400" /></a></div><br />We can now consider that the blue coins are boxes, and the red coins are balls, and reduce to the previous problem. The number of possible ways to put $k$ balls in $n$ boxes is $n-1+k$ choose $k$, which is also $n-1+k$ choose $n-1$<i>. </i>In our case, the number of boxes equals the number of towers, so it is $n$, and the number of balls is $p-n$. So, the number of possible ways to arrange $p$ coins in $n$ towers is $n-1+(p-n)=p-1$ choose $n-1$. Since we can have any number of towers, from $n=1$ to $n=p$, we have to sum accordingly, and the total number is $\sum_{n=1}^p \left(<br />\begin{array}{c}<br />p-1\\n-1<br />\end{array} \right)=\sum_{q=0}^{p-1} \left(<br />\begin{array}{c}<br />p-1\\q<br />\end{array} \right)=2^{p-1}.$<br /><i><br /></i>This solution is based on the problem of balls in boxes, which inspired the very problem. But since we've got $2^{p-1}$, shouldn't be a simpler and direct way to count all possible configurations?<br /><h2 style="text-align: left;">Simpler solution</h2>Rather than coloring the coins as previously, let's color the even towers with red, and the odd towers with blue.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Iu1yNncyDXA/VQPWkDbu9bI/AAAAAAAAAsg/Qs7pDRC0dqk/s1600/coin-towers-3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="81" src="https://4.bp.blogspot.com/-Iu1yNncyDXA/VQPWkDbu9bI/AAAAAAAAAsg/Qs7pDRC0dqk/s1600/coin-towers-3.png" width="400" /></a></div><br />We see now that any sequence of colors of the $p$ coins starting with blue corresponds to a way to arrange them in towers, and conversely. For example, the above arrangement corresponds to the sequence <span style="color: blue;">BB<span style="color: red;">RRRR</span>BBB<span style="color: red;">R</span>BB</span>. The first coin has to be blue, but each of the other $p-1$ can be chosen in two ways. Hence, the number of all such sequences is $2^{p-1}$.<br /><br /><br /></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-71448640481860302192015-03-13T21:49:00.003+02:002017-05-20T22:26:04.575+03:00A problem with balls and boxes<div dir="ltr" style="text-align: left;" trbidi="on"><h2 style="text-align: justify;">The problem</h2><div style="text-align: justify;"></div><div style="text-align: justify;">Combinatorial problems can be simply to state, and difficult to solve. But this one has a surprisingly simple solution, if you reframe it a bit. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The problem is: <i>in how many ways you can place $k$ identical balls in $n$ distinct boxes?</i> We assume that each box is large enough so that you can place all balls in it, so we have to count also the cases with empty boxes. You have to place all balls in boxes.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Yesterday, a friend and fellow physicist told me the problem, he needed to solve it in order to count some quantum states, but this is not relevant here. He solved it before, but forgot how. He found an ingenious way to see what happens if we add a new box or a ball. This would lead to some recurrence formula, which involved summing both over the number of balls, and the number of boxes. So he asked me to help him with these calculations. This is a problem of induction, which anyone should be able to resolve in high school, but I considered that all these calculations were too tedious for me, especially since I wanted to have lunch. So I replied that I would rather prefer to find a direct way to the solution, by framing it differently.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Before reading the solution, I would like to ask you to solve it yourself.</div><div style="text-align: justify;"></div><h2 style="text-align: justify;">The solution</h2>We can reframe the problem like this. We can arrange the boxes one next to another, like the carts of a train. Then we get something like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-_zC02f_pYYQ/VQM7EFhr7KI/AAAAAAAAArc/TqruYhhrxkc/s1600/boxes-balls-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="71" src="https://3.bp.blogspot.com/-_zC02f_pYYQ/VQM7EFhr7KI/AAAAAAAAArc/TqruYhhrxkc/s1600/boxes-balls-1.png" width="400" /></a></div><br />Now we can invent a notation for each configuration: we denote every space between boxes with a square, and every ball with a circle. Here's what we get:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-7b-5VBmrnVY/VQM-SQp5zvI/AAAAAAAAAr4/Hs5yVzfAYmY/s1600/boxes-balls-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="50" src="https://1.bp.blogspot.com/-7b-5VBmrnVY/VQM-SQp5zvI/AAAAAAAAAr4/Hs5yVzfAYmY/s1600/boxes-balls-2.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br />The sequence starts with a separator, because the first box is empty. Then there are four balls in the second box. There are two successive separators because the third box is empty. Then there's a box with two balls, and the last contains only one ball.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div>It is easy to see now that we have $n-1+k$ objects, $k$ of them being the balls, and $n-1$ of them being the separators. Since the conditions of the problem allow to place them in any order, the problem becomes equivalent with choosing $k$ out of $n-1+k$. So the result is<i> $n-1+k$ choose $k$</i>, which is $\displaystyle{\frac{(n-1+k)!}{(n-1)!k!}}$.<br /><br />You may try to solve it by double induction, and at the end the result may look more complicated, unless you are able to apply some formulas to bring it in this simple form. </div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-83458015168311738672015-02-14T11:04:00.003+02:002015-02-14T11:04:30.407+02:00Men are classical, women are quantum<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Man can be understood in the framework of classical physics, but for woman you'll need quantum physics.</div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-35094687465930059142014-11-05T01:18:00.000+02:002014-11-05T09:30:46.111+02:0050 Years of Misunderstanding Bell's Theorem<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Precisely 50 years ago, Bell's paper "<a href="http://www.informationphilosopher.com/solutions/scientists/bell/Bell_On_EPR.pdf" target="_blank">On the Einstein Podolsky Rosen Paradox</a>", containing his <a href="http://en.wikipedia.org/wiki/Bell%27s_theorem" target="_blank">famous theorem</a> was received by the journal Physics. Today is <a href="http://www.ria.ie/john-bell-day.aspx" target="_blank">John Bell Day</a>.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Bell's theorem is one of the most influential result in physics, despite the fact that it is a negative result. Contrary to what many people believe, Bell was actually searching for a hidden variable theory, and he found instead some severe limitations of such theories. The limitation expressed by Bell's theorem celebrated today is that hidden variable theories have to be nonlocal. The outcome of measurements are correlated in a way which seems to ignore the separation in space. Some misunderstand this result as rejecting determinism, or as rejecting any kind of hidden variables, or at least as proving that any theory which describes the quantum world using hidden variables has to rely on instantaneous communication.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Maybe others searching for a hidden variables description of quantum phenomena hit the same wall Bell hit, but rather than having the same revelation as Bell, they ignored it and continued to search for a replacement or completion of quantum mechanics. For example, Einstein had all the data to find Bell's theorem almost 30 years before Bell. The paper coauthored by Einstein, Podolsky and Rosen, "<a href="http://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777" target="_blank">Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?</a>" shown that entanglement allows nonlocal correlations. But Einstein disliked nonlocality because it seemed to violate special relativity. So he concluded that quantum mechanics was incomplete, by interpreting those correlations as revealing that Heisenberg's uncertainty principle can be trespassed. So Einstein and coauthors hit the same wall as Bell, only that they considered that the problem could be solved by completing quantum mechanics. Bell's theorem clarifies their findings in showing that no matter how you put it, the world is nonlocal (if Bell's inequality is violated, as it <a href="http://en.wikipedia.org/wiki/Bell_test_experiments" target="_blank">was confirmed by experiments</a>).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Almost 30 years later, Bell understood nonlocality as the major consequence of the EPR "paradox", and expressed it in the form of his theorem. Today, at 50 years after Bell clarified the problem, there are so many who consider that Einstein was a crackpot in what concerns quantum mechanics, and Bell defeated him. Today it is easy for any student who took a class of quantum mechanics or philosophy of physics, to consider that he has a better understanding of quantum mechanics that Einstein, and to feel superior to him (true story, just search the physics blogs and forums and you will see many examples). Most often they believe (as they are taught) that quantum mechanics is so radically different because it is not deterministic, and that what Einstein searched was a deterministic theory. And that EPR suggested this, and Bell rejected it. This is so unfair for EPR, but also for Bell.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The truth is that despite the 10 hot years of discoveries in quantum mechanics, when nearly every aspect was understood, and the foundations were laid down, nobody before Einstein, Podolsky and Rosen found that "paradox", which is true and relevant. It is unfair to consider the EPR an attack against quantum mechanics, as it is seen by many since the beginning. Rather, it is a most important discovery, which could only be made because three rebels were not satisfied with Bohr's prescriptions. Moreover, in almost 30 years since the EPR paper, nobody solved their "paradox". Not even Bohr, who rushed to respond too quickly with an article bearing <a href="http://journals.aps.org/pr/pdf/10.1103/PhysRev.48.696" target="_blank">the same name as the EPR one</a>. And the solution was found by Bell, who was a supporter of hidden variables, and maybe he wouldn't find it either, without the reformulation of the EPR argument due to the main exponent of hidden variable theories at that time, David Bohm.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Now, the reader may think that I am defending the hidden variables, by praising hidden variables theorists like Einstein, Bohm, and Bell. I actually don't defend hidden variables, and I don't say this just because of the witch hunt against "Bohmians". I just want to emphasize that without these "crackpots", we would not have today the understanding of entanglement and nonlocality which allows scientists to put at use the "magic" of quantum mechanics at work in quantum computing, quantum information, quantum cryptography, and other recent hot areas.</div><div style="text-align: justify;"></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Actually, to be honest, among Einstein, Bohm, and Bell, only the first two are considered a lacking understanding of quantum mechanics, and Bell is considered as the one who defeat them, so he is celebrated, while the other two are not. But this is only because Bell is perceived as being, because of his theorem, against hidden variables, while in fact he was also searching for a hidden variable theory.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Moreover, for some reason, many consider that Bell's theorem is only about hidden variable theories, while in fact it is about any quantum theory or interpretation which describes quantum correlations as are observed in nature, and therefore violates Bell's inequality. Including therefore standard quantum mechanics. So, quantum mechanics is nonlocal too, and no Copenhagen Interpretation, no Many Worlds Interpretation, no Decoherence Interpretation can make it otherwise. Similarly, quantum mechanics is contextual too, despite the fact that the Bell-Kochen-Specker theorem is considered to apply to hidden variable theories only.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But why some tend to consider only hidden variable theories guilty of the sins of nonlocality and contextuality? Maybe because they just want to reject such theories? Or could it be because they believe that it makes no sense to think about what happens between measurements (as Bohr teaches us)? Or because nearly everyone, when first learning about quantum mechanics, has the instinct of finding a local realist explanation, and fails, of course, and then denies having this sin by throwing stones at those who seem to have it? I think this is fine, since this is what we should do, we should question everything, and that <a href="http://rationalwiki.org/wiki/Extraordinary_claims_require_extraordinary_evidence" target="_blank">the persistence with which we should question a claim has to increase with the degree by which that claim contradicts what we learned before</a>, as is the case of quantum mechanics.</div><div style="text-align: justify;"><br />For lack of time, for the rush of getting published, for the fear of getting rejected for having unorthodox views, we tend to eat much more than we can digest, and actually we cease digesting. This is why misunderstanding are propagated even at the top of the scientific community. Misunderstandings concerning quantum mechanics and Bell's theorem prevent us from seeing both the truth, and the amazing beauty of quantum mechanics, which is transformed into a mere tool to calculate probabilities, and any attempt at understanding it is regarded with disdain.<br /><br /></div><div style="text-align: justify;">I find very fortunate the fact that Tim Maudlin wrote for the 50th anniversary of Bell's theorem a paper named "<a href="http://iopscience.iop.org/1751-8121/47/42/424010/pdf/1751-8121_47_42_424010.pdf" target="_blank">What Bell did</a>", in which he explains that Bell's result is that indeed our world, hence quantum mechanics, is nonlocal. He makes a thorough and in my opinion probably the most down to earth analysis of the meaning of the EPR paper and of Bell's theorem, and how they are misunderstood. He identifies a cluster of misunderstandings that are propagated among physicists and philosophers of physics. This is one of the cases when a philosopher really can help physicists understand physics. I'll leave you the pleasure to read it.</div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-39539041677171851162014-11-04T11:58:00.001+02:002014-11-05T09:43:07.819+02:00Happy Birthday, Nature!<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Exactly 145 years ago, on 4 November 1869, the first number of <a href="http://www.nature.com/" target="_blank">Nature</a> appeared.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">According to the current Romanian prime minister Victor Ponta, Nature is controlled (unofficially) by the current president of Romania, Traian Băsescu, with the main purpose <a href="http://www.nature.com/news/romanian-prime-minister-accused-of-plagiarism-1.10845" target="_blank">to accuse Ponta of plagiarism</a>. A possible explanation is that some collaborators of Băsescu traveled in time to create the journal. And then they also founded 145 years ago a secret society of scientists, who kept making great scientific discoveries. This also explains why most scientific discoveries were made in the last 1.5 centuries. The reason to make these scientific breakthroughs is not to advance the world, but to publish them in Nature, or in the other journals citing Nature, to make it <a href="http://www.nature.com/nature/about/" target="_blank">world's most cited journal</a>, so that, when the time comes, Nature's accusations against Ponta will have greater impact. And to invent rules that it is dishonest to <a href="http://integru.org/reviews/ponta-2003-phd-thesis" target="_blank">copy text from other books and articles without attributing it explicitly when you write your PhD thesis</a>. The second reason would be that the research made by this secret society of genii secretly led by the mastermind Băsescu will eventually lead to the discovery of time travel [<a href="http://www.nature.com/news/2011/110922/full/news.2011.554.html" target="_blank">1</a>,<a href="http://www.nature.com/news/neutrino-experiment-replicates-faster-than-light-finding-1.9393" target="_blank">2</a>,<a href="http://www.nature.com/news/neutrinos-not-faster-than-light-1.10249" target="_blank">3</a>,<a href="http://www.nature.com/ncomms/2014/140619/ncomms5145/full/ncomms5145.html" target="_blank">4</a>], which would allow him to send his people 145 years back in time...</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Now, presidential elections are taking place in Romania. Two days ago was the first round, and now <a href="http://en.wikipedia.org/wiki/Romanian_presidential_election,_2014" target="_blank">Ponta is the favorite to become the new president</a> after the second round, in 12 days. Ponta has now the chance to forbid time travel, to change the history back to its track, which is a world without the Nature journal and all that scientific research in it, a world without physicists who can discover time travel. Or quite the opposite, he may actually make use of time travel, to set our history back to 25 years ago, when people overthrew Ceaușescu, and the communists were forced to disguise themselves as social-democrats to continue to keep the power.</div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-124350264510724511.post-16648689396811331322014-10-06T23:05:00.000+03:002014-10-08T20:22:35.270+03:00Dots plus dots equal spheres<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">I took this photo in a bus in Pisa. We can see a pattern of spheres.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-rQlhWpj3o9s/VDLt4ADB2zI/AAAAAAAAAjI/jlucrKy8lPs/s1600/20140914_120154.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-rQlhWpj3o9s/VDLt4ADB2zI/AAAAAAAAAjI/jlucrKy8lPs/s1600/20140914_120154.jpg" height="300" width="400" /></a></div><br />Here is how to obtain it. We overlap these dots<br /><br /><div class="separator" style="clear: both;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-RO7nPu4UT6E/VDLmDHRaQ0I/AAAAAAAAAio/lMUf6yXlXUY/s1600/circle-illusion-tile-front-small.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-RO7nPu4UT6E/VDLmDHRaQ0I/AAAAAAAAAio/lMUf6yXlXUY/s1600/circle-illusion-tile-front-small.png" /></a></div><br />over a shrunk version of theirs<br /><br /><div style="text-align: center;"><a href="https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2F2.bp.blogspot.com%2F-r6MdDbUsw1Q%2FVDLmDGmlWrI%2FAAAAAAAAAiw%2F8SqeqVl4_Ug%2Fs1600%2Fcircle-illusion-tile-back-small.png&container=blogger&gadget=a&rewriteMime=image%2F*" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-r6MdDbUsw1Q/VDLmDGmlWrI/AAAAAAAAAiw/8SqeqVl4_Ug/s1600/circle-illusion-tile-back-small.png" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">and we get the following pattern:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-_lrRrf3yKLg/VDTWAiO5GnI/AAAAAAAAAjw/rLKzF896aww/s1600/circle-illusion3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-_lrRrf3yKLg/VDTWAiO5GnI/AAAAAAAAAjw/rLKzF896aww/s1600/circle-illusion3.png" height="312.5" width="500" /></a></div><br /></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-49202878711505497112014-10-01T12:53:00.000+03:002014-12-17T10:21:23.873+02:00Living in a vector<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Vectors are present in all domains of fundamental physics, so if you want to understand physics, you will need them. You may think you know them, but the truth is that they appear in so many guises, that nobody really knows everything about them. But vectors are a gate that allows you to enter the Cathedral of physics, and once you are inside, they can guide you in all places. That is, <i>special</i> and <i>general relativity</i>, <i>quantum mechanics</i>, <i>particle physics</i>, <i>gauge theory</i>... all these places need vectors, and once you master the vectors, they become much simpler (if you don't know them and are interested, read this post).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The Cathedral has many gates, and vectors are just one of them. You can enter through groups, sets and relations, functions, categories, through all sorts of objects or structures from algebra, geometry, even logic. I decided to show you now the way of vectors, because I think is fast and deep in the same time, but remember, this is a matter of choice. And vectors will lead us, inevitably, to the other gates too.<br /><br />I will explain some elementary and not so elementary things about vectors, but you have to read and practice, because here I just give some guidelines, a big picture. The reason I am doing this is that when you study, you may get lost in details and miss the essential.<br /><br /><h2 style="text-align: justify;">Very basic things</h2><br />A vector can be understood in many ways. One way is to see it as <i>a way to specify how to move from one point to another</i>. A vector is like an arrow, and if you place the arrow in that point, you find the destination point. To find the new position for any point, just place the vector in that point, and the tip of the vector will show you the new position. You can compose more such arrows, and what you'll get is another vector, their sum. You can also subtract them, just place their origins in the same point, and the difference is the vector obtained by joining their tips with another arrow.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-9j_3jIRvs4A/VCf6bFsMEvI/AAAAAAAAAgU/lsKY5rQSBpo/s1600/vector-add-sub.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-9j_3jIRvs4A/VCf6bFsMEvI/AAAAAAAAAgU/lsKY5rQSBpo/s1600/vector-add-sub.gif" height="192" width="400" /></a></div></div><div style="text-align: justify;"></div><div style="text-align: justify;"><br />Once you fix a reference position, an <b>origin</b>, you can specify any position, by the vector that tells you how to move from origin to that position. You can see that vector as being the difference between the destination, and the starting position.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-gm7hhKPA53M/VCmX8TaChJI/AAAAAAAAAgw/sdoG1GXIoC4/s1600/position-vector.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-gm7hhKPA53M/VCmX8TaChJI/AAAAAAAAAgw/sdoG1GXIoC4/s1600/position-vector.gif" height="255" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-DToOUKutJZM/VCf8Q4JcrfI/AAAAAAAAAgg/ZKwCt9sFgDM/s1600/position-vector.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div>You can add and subtract vectors. You can multiply them with numbers. Those numbers are from a field $\mathbb{K}$, and we can take for example $\mathbb{K}=\mathbb{R}$, or $\mathbb{K}=\mathbb{C}$, and are called <b>scalars</b>. A <b>vector space</b> is a set of vectors, so that no matter how you add them and scale them, the result is from the same set. The vector space is real (complex), if the scalars are real (complex) numbers. A sum of rescaled vectors is named <b>linear combination</b>. You can always pick a <b>basis</b>, or a <b>frame</b>, a set of vectors so that any vector can be written as a linear combination of the basis vectors, in a unique way.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-L0Cu8iSQCIk/VCqmflpehkI/AAAAAAAAAhg/TqKZntzKpgc/s1600/vector-basis.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-L0Cu8iSQCIk/VCqmflpehkI/AAAAAAAAAhg/TqKZntzKpgc/s1600/vector-basis.gif" height="245" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-QCF-9ygA5fs/VCm_moXIHkI/AAAAAAAAAhQ/MPLOkGAPOh0/s1600/vector-basis.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><h2>Vectors and functions</h2><br />Consider a vector $v$ in an $n$-dimensional space $V$, and suppose its components in a given basis are $(v^1,\ldots,v^n)$. You can represent any vector $v$ as a function $f:\{1,\ldots,n\}\to\mathbb{K}$ given by $f(i)=v^i$. Conversely, any such function defines a unique vector. In general, if $S$ is a set, then the set of the functions $f:S\to\mathbb{K}$ form a vector space, which we will denote by $\mathbb{K}^S$. The cardinal of $S$ gives the dimension of the vector space, so $\mathbb{K}^{\{1,\ldots,n\}}\cong\mathbb{K}^n$. So, if $S$ is an infinite set, we will have an infinite dimensional vector space. For example, the scalar fields on a three dimensional space, that is, the functions $f:\mathbb{R}^3\to \mathbb{R}$, form an infinite dimensional vector space. Not only the vector spaces are not limited to $2$ or $3$ dimensions, but infinite dimensional spaces are very natural too.<br /><br /><h2>Dual vectors</h2><br />If $V$ is a $\mathbb{K}$-vector space, a linear functions $f:V\to\mathbb{K}$ is a function satisfying $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u,v\in V,\alpha\in\mathbb{K}$. The linear functions $f:V\to\mathbb{K}$ form a vector space $V^*$ named the <b>dual space</b> of $V$.<br /><br /><h2>Tensors</h2><br />Consider now two sets, $S$ and $S'$, and a field $\mathbb{K}$. The <b>Cartesian product</b> $S\times S'$ is defined as the set of pairs $(s,s')$, where $s\in S$ and $s'\in S'$. The functions defined on the Cartesian product, $f:S\times S'\to\mathbb{K}$, form a vector space $\mathbb{K}^{S\times S'}$, named the <b>tensor product</b> of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, $\mathbb{K}^{S\times S'}=\mathbb{K}^{S}\otimes\mathbb{K}^{S'}$. If $(e_i)$ and $(e'_j)$ are bases of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, then $(e_ie'j)$, where $e_ie'_j(s,s')=e_i(s)e'_j(s')$, is a basis of $\mathbb{K}^{S\times S'}$. Any vector $v\in\mathbb{K}^{S_1\times S_2}$ can be uniquely written as $v=\sum_i\sum_j \alpha_{ij} e_ie'j$.<br /><br />Also, the set of functions $f:S\to\mathbb{K}^{S'}$ is a vector space, which can be identified with the tensor product $\mathbb{K}^{S}\otimes(\mathbb{K}^{S'})^*$. <br /><br />The vectors that belong to tensor products of vector spaces are named <b>tensors</b>. So, tensors are vectors with some extra structure.<br /><br />The tensor product can be defined easily for any kind of vector spaces, because any vector space can be thought of as a space of functions. The tensor product is associative, so we can define it between multiple vector spaces. We denote the tensor product of $n>1$ copies of $V$ by $V^{\otimes n}$. We can check that for $m,n>1$, $V^{\otimes (m+n)}=V^{\otimes {m}}\otimes V^{\otimes {n}}$. This can work also for $m,n\geq 0$, if we define $V^1=V$, $V^0=\mathbb{K}$. So, vectors and scalars are just tensors.<br /><br />Let $U$, $V$ be $\mathbb{K}$-vector spaces. A <b>linear operator</b> is a function $f:U\to V$ which satisfies $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u\in U,v\in V,\alpha\in\mathbb{K}$. The operator $f:U\to V$ is in fact a tensor from $U^*\otimes V$.<br /><br /><h2>Inner products </h2><br />Given a basis, any vector can be expressed as a set of numbers, the <b>components </b>of the vector. But the vector is independent of this numerical representation. The basis can be chosen in many ways, and in fact, any non-zero vector can have any components (provided not all are zero) in a well chosen basis. This shows that <b>any two non-zero vectors play identical roles</b>, which may be a surprise. This is a key point, since a common misconception when talking about vectors is that they have definite intrinsic sizes and orientations, or that they can make an angle. But in fact the sizes and orientations are relative to the frame, or to the other vectors. Moreover, you can say that from two vectors, one is larger than the other, only if they are collinear. Otherwise, no matter how small is one of them, we can easily find a basis in which it becomes larger than the other. <b>It makes no sense to speak about the size, or magnitude, or length of a vector, as an intrinsic property.</b><br /><br />But wait, one may say, there is a way to define the size of a vector! Consider a basis in a two-dimensional vector space, and a vector $v=(v^1,v^2)$. Then, the size of the vector is given by Pythagoras's theorem, by $\sqrt{(v^1)^2+(v^2)^2}$. The problem with this definition is that, if you change the basis, you will obtain different components, and different size of the vector. To make sure that you obtain the same size, you should allow only certain bases. To speak about the size of a vector, and about the angle between two vectors, you need an additional object, which is called <b>inner product</b>, or <b>scalar product</b>. Sometimes, for example in geometry and in relativity, it is called <b>metric</b>.<br /><br />Choosing a basis gives a default inner product. But the best way is to define the inner product, and not to pick a special basis. Once you have the inner product, you can define angles between vectors too. But size and angles are not intrinsic properties of vectors, they depend on the scalar product too.<br /><br />The inner product between two vectors $u$ and $v$, defined by a basis, is $u\cdot v = u^1 v^1 + u^2 v^2 + \ldots + u^n v^n$. But in a different basis, it will have a general form $u\cdot v=\sum_i\sum_j g_{ij} u^i v^j$, where $g_{ij}=g_{ji}$ can be seen as the components of a symmetric matrix. These components change when we change the basis, they form the components of a tensor from $V^*\otimes V^*$. Einstein had the brilliant idea to omit the sum signs, so the inner product looks like $u\cdot v=g_{ij} u^i v^j$, where you know that since $i$ and $j$ appear both in upper and in lower positions, we make them run from $1$ to $n$ and sum. This is a thing that many geometers hate, but physicists find it very useful and compact in calculations, because the same summation convention appears in many different situations, which to geometers appear to be different, but in fact are very similar.<br /><br />Given a basis, we can define the inner product by choosing the coefficients $g_{ij}$. And we can always find another basis, in which $g_{ij}$ is diagonal, that is, it vanishes unless $i=j$. And we can rescale the basis so that $g_{ii}$ are equal to $-1$, $1$, or $0$. Only if $g_{ii}$ are all $1$ in some basis, the size of the vector is given by the usual Pythagoras's theorem, otherwise, there will be some minus signs there, and even some terms will be omitted (corresponding to $g_{ii}=0$).<br /><h2>Quantum mechanics </h2><br />Quantum particles are described by Schrödinger's equation. Its solutions are, for a single elementary particle, complex functions $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}$, or more general, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$, named <b>wavefunctions</b>. They describe completely the states of the quantum particle. They form a vector space $H$ which also has a <b>hermitian product</b> (a complex scalar product so that $h_{ij}=\overline{h_{ji}}$), and is named the <b>Hilbert space </b>(because in the infinite dimensional case also satisfies an additional property which we don't need here), or the <b>state space</b>. Linear transformations of $H$ which preserve the complex scalar product are named <b>unitary transformations</b>, and they are the complex analogous of rotations.<br /><br />The wavefunctions are represented in a basis as functions of positions, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$. The element of the position basis represent <b>point particles</b>. But we can make a unitary transformation and obtain another basis, made of functions of the form $e^{i (k_x x + k_y y + k_z z)}$, which represent pure <b>waves</b>. Some observations use one of the bases, some the other, and here is why there is a duality between waves and point particles.<br /><br />For more elementary particles, the state space is the tensor product of the state spaces of the individual particles. A tensor product of the form $|\psi\rangle\otimes|\psi'\rangle$ represents <b>separable states</b>, which can be observed independently. If the system can't be written like this, but only as a sum, the particles are <b>entangled</b>. When we measure them, the outcomes are correlated.<br /><br />The evolution of a quantum system is described by Schrödinger's equation. Basically, the state rotates, by a unitary transformation. Only such transformations conserve the probabilities associated to the wavefunction.<br /><br />When you <b>measure</b> the quantum systems, you need an observable. One can see an <b>observable</b> as defining a decomposition of the state space, in perpendicular subspaces. After the observation, the state is found to be in one of the subspaces. We can only know the subspace, but not the actual state vector. This is strange, because the system can, in principle, be in any possible state, but the measurement finds it to be only in one of these subspaces (we say it <b>collapsed</b>). This is the <b>measurement problem</b>. The things become even stranger, if we realize that if we measure another property, the corresponding decomposition of the state space is different. In other words, if you look for a point particle, you find a point particle, and if you look for a wave, you find a wave. This seems as if the unitary evolution given by the Schrödinger's equation is broken during observations. Perhaps the wavefunction remains intact, but to us, only one of the components continues to exist, corresponding to the subspace we obtained after the measurement. In the <b>many worlds interpretation</b> the universes splits, and all outcomes continue to exist, in new created universes. So, not only the state vector contains the universe, but it actually contains many universes.<br /><br /><br />I have a proposed explanation for some strange quantum features, in [<a href="http://fqxi.org/community/essay/winners/2008.1#Stoica" target="_blank">1</a>, <a href="http://arxiv.org/abs/1309.2309" target="_blank">2</a>, <a href="http://fqxi.org/community/essay/winners/2013.1#Stoica" target="_blank">3</a>], and in these videos:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/8H-nuIbkzfw?feature=player_embedded' frameborder='0' /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/IBYDBJRtZRg?feature=player_embedded' frameborder='0' /></div><br /><h2>Special relativity</h2><br />An example when there is a minus signs in the Pythagoras's theorem is given by the <b>theory of relativity</b>, where the squared size of a vector is $v\cdot v=-(v^t)^2+(v^x)^2+(v^y)^2+(v^z)^2$.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-qE7yhFPn3t8/VCqv3QAOSJI/AAAAAAAAAh4/qWtp34N4PNI/s1600/lightcone.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-qE7yhFPn3t8/VCqv3QAOSJI/AAAAAAAAAh4/qWtp34N4PNI/s1600/lightcone.gif" height="291" width="400" /></a></div><br />This inner product is named the <b>Lorentz metric</b>. Special relativity takes place in the <b>Minkowski spacetime</b>, which has four dimensions. A vector $v$ is named <b>timelike </b>if $v\cdot v < 0$, <b>spacelike</b> if $v\cdot v > 0$, and <b>null</b> or <b>lightlike</b> if $v\cdot v = 0$. A particle moving with the speed of light is described by a lightlike vector, and one moving with an inferior speed, by a timelike vector. Spacelike vectors would describe faster than light particles, if they exist. Points in spacetime are named <b>events</b>. Events can be simultaneous, but this depends on the frame. Anyway, to be simultaneous in a frame, two events have to be separated by a spacelike interval. If they are separated by a lightlike or timelike interval, they can be connected causally, or joined by a particle with a speed equal to, respectively smaller than the speed of light.<br /><br />In Newtonian mechanics, the laws remain unchanged to translations and rotations in space, translations in time, and inertial movements of the frame - together they form the <b>Galilei transformations</b>. However, electromagnetism disobeyed. In fact, this was the motivation of the research of Einstein, Poincaré, Lorentz, and FitzGerald. Their work led to the discovery of special relativity, according to which the correct transformations are not those of Galilei, but those of Poincaré, which preserve the distances given by the Lorentz metric.<br /><br /><h2>Curvilinear coordinates</h2><br />A basis or a frame of vectors in the Minkowski spacetime allows us to construct Cartesian coordinates. However, if the observer's motion is accelerated (hence the observer is <b>non-inertial</b>), her frame will rotate in time, so Cartesian coordinates will have to be replaced with curved coordinates. In curved coordinates, the coefficients $g_{ij}$ depend on the position. But in special relativity they have to satisfy a flatness condition, otherwise spacetime will be curved, and this didn't make much sense back in 1905, when special relativity was discovered.<br /><br /><h2>General relativity</h2><br />Einstein remarked that to a non-inertial observer, inertia looks similar to gravity. So he imagined that a proper choice of the metric $g_{ij}$ may generate gravity. This turned out indeed to be true, but the choice of $g_{ij}$ corresponds to a curved spacetime, and not a flat one.<br /><br />One of the problems of general relativity is that it has <b>singularities</b>. Singularities are places where some of the components of $g_{ij}$ become infinite, or where $g_{ij}$ has, when diagonalized, some zero entries on the diagonal. For this reason, many physicist believe that this problem indicates that general relativity should be replaced with some other theory, to be discovered. Maybe it will be solved when we will replace it with a theory of quantum gravity, like string theory or loop quantum gravity. But until we will know what is the right theory of quantum gravity, general relativity can actually deal with its own singularities (while the ones mentioned above did not solve this problem). I will not describe this here, but you can read <a href="http://www.unitaryflow.com/2013/12/phd-thesis-defended.html" target="_blank">my articles about this</a>, and also <a href="http://fqxi.org/community/forum/topic/1357" target="_blank">this essay</a>, and these posts about the <b>black hole information paradox</b> [<a href="http://www.unitaryflow.com/2013/09/bh-paradox-1-susskind-vs-hawking.html" target="_blank">1</a>, <a href="http://www.unitaryflow.com/2013/10/black-hole-paradox-2-stretched-complementarity.html" target="_blank">2</a>, <a href="http://www.unitaryflow.com/2013/10/black-hole-paradox-3-look-for-the-information-where-you-lost-it.html" target="_blank">3</a>]. And watch this video<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/M8SErmAxLDU?feature=player_embedded' frameborder='0' /></div><br /><br /><h2>Vector bundles and forces</h2><br />We call <b>fields</b> the functions defined on the space or the spacetime. We have seen that fields valued in vector spaces are actually vector spaces. On a flat space $M$ which looks like a vector space, the fields valued in vector spaces can be thought of as being valued in the same vector space, for example $f:M\to V$. But if the space is curved, or if it has nontrivial topology, we are forced to consider that at each point there is another copy of $V$. So, such a field will be more like $f(x)\in V_x$, where $V_x$ is the copy of the vector space $V$ at the point $x$. Such fields still form a vector space. The union of all $V_x$ is called a <b>vector bundle</b>. The fields are also called <b>sections</b>, and $V_x$ is called the <b>fiber</b> at $x$.<br /><br />Now, since $V_x$ are copies of $V$ at each point, there is no invariant way to identify each $V_x$ with $V$. In other words, $V_x$ and $V$ can be identified, for each $x$, up to a linear transformation of $V$. We need a way to move from $V_x$ to a neighboring $V_{x+d x}$. This can be done with a <b>connection</b>. Also, moving a vector from $V_x$ along a closed curve reveals that, when returning to $V_x$, the vector is rotated. This is explained by the presence of a <b>curvature</b>, which can be obtained easily from the connection.<br /><br />Connections behave like <b>potentials</b> of force fields. And a <b>force field</b> corresponds to the curvature of the connection. This makes very natural to use vector bundles to describe forces, and this is what <b>gauge theory</b> does.<br /><br />Forces in the <b>standard model </b>of particles are described as follows. We assume that there is a typical complex vector space $V$ of dimension $n$, endowed with a hermitian scalar product. The connection is required to preserve this hermitian product when moving among the copies $V_x$. The set of linear transformations that preserve the scalar product is named <b>unitary group</b>, and is denoted by $U(n)$. The subset of transformations having the determinant equal to $1$ is named the <b>special unitary group</b>, $SU(n)$. The <b>electromagnetic force</b> corresponds to $U(1)$, the <b>weak force</b> to $SU(2)$, and the <b>strong force</b> to $SU(3)$. Moreover, all particles turn out to correspond to vectors that appear in the representations of the gauge groups on vector spaces. <br /><br /><h2>What's next?</h2><br />Vectors are present everywhere in physics. We see that they help us understand quantum mechanics, special and general relativity, and the particles and forces. They seem to offer a unitary view of fundamental physics.<br /><br />However, up to this point, <b>we don't know how to unify</b><br /><ul><li>unitary evolution and the collapse of the wavefunction</li><li>the quantum level with the mundane classical level</li><li>quantum mechanics and general relativity</li><li>the electroweak and strong forces (we know though how to combine the electromagnetic and weak forces, in the unitary group $U(2)$)</li><li>the standard model forces and gravity </li></ul></div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-124350264510724511.post-45235587873993595782014-09-27T10:37:00.002+03:002014-10-14T14:46:20.234+03:00The unreasonable beauty of mathematics in the natural sciences*<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Imagine a man and a woman, seeing and liking each other at a party or club or so. They start talking, the mutual attraction is obvious, but they want to be casual for two minutes. So they exchange informal formalities about doesn't matter what. Then he asks her: "so, what do you do?", and she replies "I'm a poet". What if the guy would say something like "I hate poetry!", or even declare proudly "I never knew how to use letters to write words and stuff, and I don't care!". Or imagine she's a musician, and he says "I hate music!". There are two things we can say about that kind of guy. First, he is very rude, he never ever deserves a second chance with that girl or any other human being for that matter. He should be isolated, kept outside society. Second, or maybe this should be first, how on earth can he be proud for being illiterate!</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">You probably guessed that this story is true. OK, In my case it was about math instead of poetry, and the genders are reversed. This happened to me or to anyone in the same situation quite often. There is no political correctness when it comes about math, maybe because one tends to believe that if you like math, you have no feelings, and such a remark wouldn't hurt you. And I actually was never offended when a girl said such outrageous things like that she hates math. Because whenever a girl told me she hates math, I knew she calls math something that really is boring and ugly, and not what I actually call math. Because math as I know it is poetry, is music, and is a wonderful goddess.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The story continues, years later. You talk about physics, with people interested in physics, or even with physicists. And you say something about this being just a mathematical consequence of that, or that certain phenomenon can be better understood if we consider it as certain mathematical object. It happens sometimes that your interlocutor becomes impatient and says that this is only math, and you were discussing physics, that math has no power there, and so on. Or that math is at best just a tool, and it actually obscures the real picture, or even that it limits our power of understanding.</div><div style="text-align: justify;"></div><div style="text-align: justify;">People got the wrong picture that math is about numbers, or letters that stand for unknown numbers, or being extremely precise and calculating a huge number of decimals, or being very rigid and limited. In fact, math is just the study of relations. You will be surprised, but this is actually the mathematical definition of math. Numbers come into math only incidentally, as they come into music, when you indicate the duration or the tempo. Math is just a qualitative description of relations, and by relations we can understand a wide rainbow of things. I will detail this another time.</div><div style="text-align: justify;"></div><div style="text-align: justify;">Imagine you wake up and you don't remember where you are, or who you are, like you were just born. You are surrounded by noise, which hurts your ears and your brain, meaningless random violent noise. You run desperately, trying to avoid it, but it is everywhere. And you finally find a spot where everything becomes suddenly wonderful: the noise becomes music, a celestial, beautiful music, and everything starts making sense. You are in a wonderful Cathedral, and you are tempted to call what you are listening "music of the spheres". The same music played earlier, but you were in the wrong place, where the acoustics was bad, or the sounds reached your ear in the wrong order, because of the relative positions of the instruments. Or maybe your ears were not yet tuned to the music. The point is that what seemed to be ugly noise, suddenly became so wonderful.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So, when someone says "I hate math!", all I hear is "I am in the Cathedral you call wonderful, but in the wrong place, where the celestial music becomes ugly violent noise!".</div><div style="text-align: justify;"></div><div style="text-align: justify;">If you are interested in physics, you entered the Cathedral. But if you hate math, you will not last here, and maybe it is better to get out immediately! And if you are still interested in physics, come inside slowly, carefully choosing your steps, to avoid being assaulted by the music of the spheres, to allow it gently to enter in your mind, and to open your eyes. Choose carefully what you read, what lectures you watch, and ask questions. Don't be shy, any question you will ask is the right question for your current position, and for your next step. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">There are some places in the Cathedral where the music is really beautiful. If you meet people there, to share the music, to dance, you will feel wonderful. If not, you will feel lonely. So you will want to share that place, you will want to invite your friends to join you.</div><div style="text-align: justify;"></div><div style="text-align: justify;">The reason I love physics, is that I want to find these places. The reason I read blogs and papers, is that I want them to help me find such places. The reason I write papers, and I blog about this, is that I would like to share my places with others. I attend conferences (four so far this year) because they are like concerts, where you get the chance to listen some wonderful music, and to play your own.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But these are just words. I would like to write more posts in which I show the unreasonable beauty of math in physics, with concrete examples. Judging by the statistics, I have a few readers; judging by the number of comments, I don't really touch many of them. I know sometimes I am too serious, or too brief when I should explain more, especially when mathematical subtleties are involved. I am not very good at explaining abstract things to non-specialists, but I want to learn. I would like to write better, to be more useful, so, I would like to encourage comments and suggestions. Ask me to clarify, to explain, to detail, to simplify. Tell me what you would like to understand.</div><div style="text-align: justify;"><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-T4PNJx2CtXA/VCZqi0atcsI/AAAAAAAAAfE/bGaeXCTou4c/s1600/kochen-specker.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-T4PNJx2CtXA/VCZqi0atcsI/AAAAAAAAAfE/bGaeXCTou4c/s1600/kochen-specker.png" height="303" width="320" /></a></div><br /></div><div style="text-align: justify;">To start, I would like to write about vectors. They are so fundamentals in all areas of physics and mathematics, so I think it's a good idea to start with them. You may think they are too simple, and that you know all about them from high school, but you don't know the whole story. Later, when I will say something about quantum mechanics and relativity, they will be necessary (after all, according to quantum mechanics, the state of the universe is a vector). On the other hand, if you will understand them well, you will be around half of the way to understand some modern physics.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">______________________</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">* You surely guessed that the title is a reference to Wigner's brilliant and insightful lecture, <a href="http://math.northwestern.edu/~theojf/FreshmanSeminar2014/Wigner1960.pdf" target="_blank">The unreasonable effectiveness of mathematics in the natural sciences</a>.<br /><br /><br /><h3>Update, October 14, 2014</h3>I just watched <a href="http://thecolbertreport.cc.com/videos/xj9d66/edward-frenkel" target="_blank">an episode of the Colbert Report</a>, where the mathematician <a href="http://en.wikipedia.org/wiki/Edward_Frenkel" target="_blank">Edward Frenkel</a> was invited in April this year. It was about Frenkel's <a href="http://www.loveandmathbook.com/" target="_blank">new book</a> and about <a href="http://www.imdb.com/title/tt1530994/" target="_blank">his movie</a>. He discusses at some point precisely the fact that it is so acceptable to hate math, as opposed to hating music or painting. Here is what he says for <a href="http://online.wsj.com/news/articles/SB10001424127887324165204579026930747839174" target="_blank">The Wall Street Journal</a>:<br /><blockquote class="tr_bq">It's like teaching an art class where they only tell you how to paint a fence but they never show you Picasso. People say 'I'm bad at math,' but what they're really saying is 'I was bad at painting the fence.'</blockquote>Also see this video:<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/PcAZWgvFc1Q?feature=player_embedded' frameborder='0' /></div></div></div>Unknownnoreply@blogger.com0