tag:blogger.com,1999:blog-1243502645107245112024-03-13T17:07:05.294+02:00Unitary FlowCristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.comBlogger80125tag:blogger.com,1999:blog-124350264510724511.post-2576161285178474842024-01-08T22:10:00.002+02:002024-01-08T22:10:41.350+02:00Is your mind just a computation?<div>I made three videos, 46' together, about consciousness and computation.</div><div><br /></div><div style="text-align: justify;">In this series in three parts:</div><div style="text-align: justify;">Can a computer have its own mind? Is your mind just a computation? We will see what Computer Science has to say. Don't worry, it's beginner level! DIY experiment so that you can verify what I say. The proof appeals to logic and experiment, not to phenomenal experience ("what is like", the "hard problem of consciousness", qualia, the experience of feelings, emotions, pain or pleasure etc) Based on my paper <a href="http://philsci-archive.pitt.edu/22880/" target="_blank">"Does a computer think if no one is around to see it?"</a> </div><div style="text-align: justify;"><br /></div><div>In Episode 1, we will see that what we call computation is a convention, and it can be chosen in numerous ways.</div><div> </div><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="342" src="https://www.youtube.com/embed/kuziE01rh6M" width="483" youtube-src-id="kuziE01rh6M"></iframe></div><br /> </div><div><br />In Episode 2, we will make an experiment to see that what we call computation is a convention, and it can be chosen in numerous ways. We will explore some implications.</div><div> </div><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="342" src="https://www.youtube.com/embed/f1gcYMXo9qE" width="483" youtube-src-id="f1gcYMXo9qE"></iframe></div><br /> </div><div><br />In Episode 3, we will see that there is a way to know if your mind is just a computation.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="342" src="https://www.youtube.com/embed/f_GB-pC8NVQ" width="483" youtube-src-id="f_GB-pC8NVQ"></iframe></div><br /><div><br /></div><div><br /></div>Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com2tag:blogger.com,1999:blog-124350264510724511.post-8875310721753085172023-11-29T23:21:00.016+02:002023-12-04T09:51:39.664+02:00Roy Kerr vs. the singularities<div><p style="text-align: justify;"><span style="white-space: pre-wrap;"><a href="https://www.researchgate.net/profile/Roy-Kerr/publication/375744216_Do_Black_Holes_have_Singularities/links/655914f33fa26f66f411dadb/Do-Black-Holes-have-Singularities.pdf" target="_blank">This preprint by Roy Kerr</a> should be a hit (but I bet it will be ignored!) .</span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkFTWrjKuLHN-tBUyFA3P4ffAsHZHl5noGAgDoNluZtyCcYGbh6xd3f9ITm8uiXswnqimBxrtNnLSPUwXoJlW0Zx4DekxJdmpNgM0kzym_78nz4LNIycS8CADpLr1XD79feHhlMvlUmHW4B_BjQCa3pL5JRBcIjz1s04s76l8xKI0m6M8oZ-pVQn1SYAI9/s873/Kerr.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="777" data-original-width="873" height="458" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkFTWrjKuLHN-tBUyFA3P4ffAsHZHl5noGAgDoNluZtyCcYGbh6xd3f9ITm8uiXswnqimBxrtNnLSPUwXoJlW0Zx4DekxJdmpNgM0kzym_78nz4LNIycS8CADpLr1XD79feHhlMvlUmHW4B_BjQCa3pL5JRBcIjz1s04s76l8xKI0m6M8oZ-pVQn1SYAI9/w513-h458/Kerr.jpg" width="513" /></a></div><p style="text-align: justify;"><span style="white-space: pre-wrap;">Kerr (yes, who found the well-known Kerr black hole solutions) disagrees with Penrose's singularity theorem and its variations. Namely these theorems prove the existence of geodesics that can't be extended beyond a finite affine length, but Kerr finds numerous examples of inextensible light rays that don't contain singularities. These geodesics go all the way to the null infinity, and yet the affine parameter remains finite. And there are such light rays through every point of the Kerr spacetime. Only some geodesics hit the ring singularity, but this region can be replaced by a nonsingular one, perhaps matter can do this. Kerr thinks that his perfectly symmetric vacuum solution doesn't happen in reality (despite the "no-hair theorem", which is in fact a conjecture improperly called "theorem", stating that all black holes evolve into a Kerr solution), even though he thinks that black holes exist.</span><br /></p><p></p><p style="text-align: justify;"><span style="white-space: pre-wrap;">Now, how is this possible? I mean the singularity theorems, now sealed forever by a Nobel prize, prove that there are conditions that necessarily lead to singularities. That if there's a black hole, there must be a singularity beyond its horizon. Or do they?</span><br /></p><p></p><p style="text-align: justify;"><span style="white-space: pre-wrap;">This is a bit of a word play. There are more meanings of the word "singularity". Normally singularity means a place where the metric blows up. Or its inverse. Or the curvature, or any field that we think it's physical. But then we can think of excluding these points from spacetime. If these points are "in the way" of the physical fields, if the evolution equations can't go beyond such a place but they should, this would be a problem even if we exclude them from spacetime. But if these singularities are somewhere at the "edge" of spacetime, and the spacetime admits a nice foliation so that the evolution equations work fine across the entire spacetime, why would this be a problem? And yet, the other definition of singularity, the one that is actually the object of the singularity theorems, includes such cases as well. That is, as a diagnostic method, it gives numerous false positives.</span><br /></p><p></p><p style="text-align: justify;"><span style="white-space: pre-wrap;">Here's what happened. And I don't say it's a plot against General Relativity, rather an accident, perhaps welcomed by many. If your spacetime contains singularities, we can think of excluding them from spacetime. But this, as I said, doesn't solve the problem. So maybe there is a way to detect this pathology even with the singularities removed, and talk about such a spacetime as being singular anyway. And here comes into play the redefinition of singular spacetime in terms of geodesic incompleteness. And it is said in the Hawking & Ellis bible, on page 258: </span></p><p style="text-align: justify;"><span style="white-space: pre-wrap;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFZOMl4F_8bbkADUzicEi1B1rraBQ8ItlfWQgFB6UKebt2vF0szP_F7Ofydqx3mgKA5ZRkArJBu57UU-8ynQXs_8_mtTeq9LB7r_8aU2iukaIgSfcQoeotxpHBgLhWuOOwvPQSU8T9Y4neNCOMhmjXxOQCflhei-ik5KwUOCbcxfBkv8SIDWfrXlz_f04C/s865/Hawking%20&%20Ellis.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="205" data-original-width="865" height="125" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFZOMl4F_8bbkADUzicEi1B1rraBQ8ItlfWQgFB6UKebt2vF0szP_F7Ofydqx3mgKA5ZRkArJBu57UU-8ynQXs_8_mtTeq9LB7r_8aU2iukaIgSfcQoeotxpHBgLhWuOOwvPQSU8T9Y4neNCOMhmjXxOQCflhei-ik5KwUOCbcxfBkv8SIDWfrXlz_f04C/w531-h125/Hawking%20&%20Ellis.png" width="531" /></a></div><p></p><p style="text-align: left;"><span style="white-space: pre-wrap;">I don't want to single out this great book, it explains well the adoption of this diagnosis, and others said similar things.</span><span style="white-space: pre-wrap;"> But here I think lies the problem. Because this definition can be misunderstood (unintentionally I think) in a way that makes the singularity theorems seem about singularities even if there are no singularities in the interior of spacetime, even if the spacetime can be nicely foliated, offering a nice home to the evolution equations.</span></p><p style="text-align: justify;"><span style="white-space: pre-wrap;">The singularity theorems prove (and they indeed prove this) that there are incomplete geodesics, where incomplete means they can't be extended beyond a finite affine length. Whether all of them deserve to be called "incomplete" is also questionable. If the affine length (which is not the same as geometric length anyway) </span><span style="white-space: pre-wrap;">of a timelike or null geodesic </span><span style="white-space: pre-wrap;">is finite, but it goes to the "real edge" of spacetime, as in Kerr's paper, why should it be called incomplete? This already seeds in our minds the idea that there's something wrong with them.</span><br /></p><p style="text-align: justify;"><span style="white-space: pre-wrap;">So, one on top of another, the meaning of words shifted so that now it's widely believed that General Relativity breaks down, due to the singularities. And Kerr gives nice rich counterexamples, all in the same spacetime of a Kerr black hole. I mean, his spacetime has a singularity, but the singularity theorem doesn't even predict that singularity. It predicts some singularities, but they are false positives, they don't occur on the geodesics up to the boundary of spacetime. It doesn't predict the ring singularity, because, as </span><span style="white-space: pre-wrap;">Kerr says, </span><span style="white-space: pre-wrap;">there is no trapped surface inside the inner horizon of the Kerr black hole. So, if we cut out the spacetime around that ring, and replace that region (and the "other universe" beyond the ring) with one without singularities, we get a spacetime without singularities (and from what we know matter may do this), and yet the singularity theorems as usually cited say it has singularities (outside that region)!</span> </p><p style="text-align: justify;">I'd like to add that I was convinced as well, for a long time, that the singularity
theorems imply the kind of metric singularities that are problematic. They were the reason why I worked
to save General Relativity by reformulating it in a way that doesn't
have infinities at the singularities. And I repeated numerous times the
claim that the singularity theorems prove that the metric tensor has
singularities, assuming that they are of this kind. And I might have regarded people who didn't believe in
singularities as, let's say, not very serious. Despite being aware that
there was a step in the proof of the singularity theorems that I never understood,
namely exactly the step where from inextensibility we conclude the
existence of such singularities. Despite never being able to find a place where this step is proved for a limited person like me. And that while knowing that I didn't
understand that step, and being limited, I considered that I should trust the experts about
it, or maybe just my limited understanding of what experts say. And now, after seeing Roy Kerr's counterexamples, I think I was wrong.</p></div><div style="text-align: justify;"><p>So yes, Kerr is right, to be able to say that General Relativity breaks down because of singularities we need a proof for exactly such singularities, and the singularity theorems alone don't do the job, and there are counterexamples showing this. But of course counterexamples are a no go mainly for the more mathematically inclined (and some of them noticed this at some times, but somehow the most spread interpretation of the singularity theorems remained unaffected). Many physicists may still use the confusion between the two notions of singular spacetimes (assuming they're aware of them) to reject classical General Relativity, and at the same time they would claim that quantum gravity doesn't have this problem, again without proof, without even a theory of quantum gravity! (The only argument is that quantum fields may violate a condition in the singularity theorems, but this doesn't prove that this avoids the alleged singularity)</p><p><br />But what if somebody takes notice now of Kerr's paper, and of the disambiguation of the term "singular spacetime", and finds a singularity theorem, with different conditions evidently, that is actually about such singularities? Even so, General Relativity can be formulated in terms of finite geometric objects, which can evolve beyond the singularities, as I showed some time ago <a href="https://arxiv.org/abs/1301.2231">https://arxiv.org/abs/1301.2231</a>. This formulation is equivalent with the usual one outside the singularities, but it extends at the singularities too, at least in the usual cases. <br /></p></div><div style="text-align: justify;">So I see no reason why General Relativity is so often pronounced dead. I mean, sure, we need a quantum theory of gravity, but let's stop throwing the baby with the bathwater. There's no reason to treat like a stepchild one of the two babies, General Relativity and Quantum Theory, and favor the other one. The really naughty one ;)</div><div style="text-align: justify;"> </div><div style="text-align: justify;"><span><a name='more'></a></span> </div><div style="text-align: justify;">Update. Here is a review which should clarify that there are differences between geodesic incompleteness, the existence of curvature singularities, and global hyperbolicity:</div><div style="text-align: justify;"> <br /></div><div style="text-align: justify;"><a href="https://link.springer.com/article/10.1007/s10714-022-02973-w" target="_blank">Klaas Landsman 2022 Penrose’s 1965 singularity theorem. From geodesic incompleteness to cosmic censorship</a></div><div style="text-align: justify;"> </div><div style="text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhBAF4ZsvJUQe8KjplSo7Kf8TzRQSkZaMV2gY5BJcZ_49RoLrHKK0ZlL2Vb17fU61TQN_kFzrKes0rfflcPBBrfSf5FXH71TpvV25GxfGZgPtUOWJFqFVm5J1VHFu17lMxzlXbz9br7d3psZga4INoHNoBRccfJg6p5HNBIbdCH0YpHTMRkheaN0brdiGi/s1859/Klaas%20Landsman%202022.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1859" data-original-width="1196" height="778" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhBAF4ZsvJUQe8KjplSo7Kf8TzRQSkZaMV2gY5BJcZ_49RoLrHKK0ZlL2Vb17fU61TQN_kFzrKes0rfflcPBBrfSf5FXH71TpvV25GxfGZgPtUOWJFqFVm5J1VHFu17lMxzlXbz9br7d3psZga4INoHNoBRccfJg6p5HNBIbdCH0YpHTMRkheaN0brdiGi/w500-h778/Klaas%20Landsman%202022.png" width="500" /></a></div><br /> </div><div style="text-align: justify;"> <br /></div>Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-39272723372754904762022-04-16T10:03:00.012+03:002022-04-29T07:10:32.845+03:00An underrated gem: WAY beyond conservation laws<div><div><div><div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I think the article <a href="https://journals.aps.org/pra/export/10.1103/PhysRevA.95.012127" target="_blank">Wigner-Araki-Yanase theorem beyond conservation laws</a> by Mikko Tukiainen is an underrated gem (if we compare its content to the number of citations).</div><div style="text-align: justify;"><br /></div><div style="text-align: center;"><img alt="" height="319" 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" width="523" /></div><br /></div>Here's why I think so.<br /><br /></div><div style="text-align: justify;">First, I think the Wigner-Araki-Yanase theorem is underrated. It started with a paper by Wigner (here is <a href="https://arxiv.org/abs/1012.4372" target="_blank">an English translation</a>.), who showed that you can't have an accurate ideal spin measurement which is also repeatable. By "repeatable" it's understood that, whatever result you get, by repeating the measurement you'll get the same result. In other words, accuracy requires that the measurement disturbs the system, so the spin is no longer what you measured it to be. You can avoid this by being satisfied with a less accurate result. Wigner also showed that repeatability can be obtained and the error can be made as small as wanted, if the measuring device is large enough so that the apparatus has large uncertainty for the conserved quantities.<br /><br />Araki and Yanase generalized his result, and added some interesting observations, in particular that this limitation applies to the measurement device as well.<br /></div><br /></div><div style="text-align: justify;">Wigner was brilliant enough to know how to give a more general proof, but he wanted the idea to be understood easily. He used the conservation of angular momentum along an axis to deduce the limitation of accuracy of spin measurement along an orthogonal axis. He only uses a conservation law, but all conservation laws contribute. He had to give a simple proof, without making too many assumptions about the evolution equation. So he probably thought, spin measurement is a simple example, and also entails the existence of other spin operators that are conserved by unitary evolution and don't commute with it.<br /></div><br /></div><div style="text-align: justify;"><div data-block="true" data-editor="5bliu" data-offset-key="70fhv-0-0"><div class="_1mf _1mj" data-offset-key="70fhv-0-0"><span data-offset-key="70fhv-0-0"><span data-text="true">While all conservation laws contribute limitations, on the one hand this is an expression of the symmetries, and on the other hand, in fact, they don't do anything. The limitation is in the transformation of the total state from the state before measurement into the state after the pre-measurement, that is, just before we invoke the collapse postulate (the collapse itself breaks the conservation laws). During pre-measurement the evolution is unitary, because collapse is invoked at the end. The evolution itself constraints the possible results of the measurement. Conservation laws were originally </span></span><span><span data-offset-key="70fhv-1-0"><span data-text="true">used</span></span></span><span data-offset-key="70fhv-2-0"><span data-text="true"> as indications that we can use to find such limitations. A general proof in terms of general unitary transformations is very difficult, but you can look at a conserved quantity and deduce enough to know that the accuracy is limited if we want repeatability. So the conservation law was used to give a simple, although less general, proof. And to make it simpler, the conserved quantity had to be additive.</span></span></div><div class="_1mf _1mj" data-offset-key="70fhv-0-0"><span data-offset-key="70fhv-2-0"><span data-text="true"> </span></span></div><div class="_1mf _1mj" data-offset-key="70fhv-0-0"><span data-offset-key="70fhv-2-0"><span data-text="true">But these are just assumptions Wigner made to prove the result, and t</span></span><span data-offset-key="70fhv-2-0"><span data-text="true"><span data-offset-key="70fhv-2-0"><span data-text="true">his made me initially think that </span></span>there is nothing special or metaphysical about conservation laws in this context, despite Wigner's other very important realizations about the role of symmetry. But there is a very important lesson about symmetries (which, as we know from Emmy Noether, are the reason behind the conservation laws), as elucidated by the works of Ozawa, Loveridge, Busch, Miyadera and others.<br /></span></span></div></div><span data-offset-key="70fhv-2-0"><span data-text="true"></span></span><br />Conservation laws are often used to deduce things without solving equations. But they don't constrain, they express the constraints of the system, since these constraints restrict the symmetries, and therefore the conservation laws. On the other hand, the symmetries of the system really capture an important aspect of the constraints, as explained in <a href="https://link.springer.com/content/pdf/10.1007/s10701-018-0138-3.pdf" target="_blank">this wonderful article by <span data-offset-key="70fhv-2-0"><span data-text="true">Loveridge, Busch, and Miyadera</span></span></a>.<br /><br />The reason why I consider <a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012127" target="_blank">Mikko Tukiainen's paper</a> important is that it seems to indicate another deeper aspect, that seems to go beyond that. He not only it gave a more general proof, but in that proof, conservation laws play no role <span data-offset-key="7jlk4-2-0"><span data-text="true">(you can read it <a href="https://arxiv.org/abs/1611.05905" target="_blank">for free here</a></span></span><span data-offset-key="7jlk4-4-0"><span data-text="true">)</span></span>. <span data-offset-key="6vmqn-4-0"><span data-text="true">He used instead the idea of quantum incompatibility, which is a way to understand the major features of quantum mechanics that distinguish it from classical mechanics (although the most useful examples are still given by conservation laws). This is neat, complements the idea based on symmetry, and it's in some </span></span><span><span data-offset-key="6vmqn-5-0"><span data-text="true">sense</span></span></span><span data-offset-key="6vmqn-6-0"><span data-text="true"> more general.</span></span><br /><br />Both the symmetries and quantum incompatibility go deep, but maybe there is a deeper reason than both of these - the full range of such limitations of measurements is still unknown. And maybe there is no general characterization of this. But anyway, I think there's more to be learned about this.<br /><br />Since both the WAY papers together have <span data-offset-key="6vmqn-4-0"><span data-text="true"></span></span><span><span data-offset-key="6vmqn-5-0"><span data-text="true">together</span></span></span><span data-offset-key="6vmqn-6-0"><span data-text="true"> </span></span>a relatively small number of citations (hundreds), I consider them underrated too. This is another mystery to me.<br /></div>Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-16666821578879817582020-03-28T21:48:00.000+02:002020-03-28T21:48:18.634+02:00The negative way to sentience (comments welcome!)<div dir="ltr" style="text-align: left;" trbidi="on">
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I wrote an essay about sentience and its relations to physics. For the moment, <a href="https://www.researchgate.net/publication/339956062_The_negative_way_to_sentience_comments_welcome" target="_blank">I keep it on ResearchGate, and I am welcoming comments</a>.</div>
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<h2 style="text-align: center;">
The negative way to sentience (comments welcome!)</h2>
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<span><b>Abstract. </b>While the materialist paradigm is credited for the incredible
success of science in describing the world, to some scientists and
philosophers there seems to be something about subjective experience
that is left out, in an apparently irreconcilable way. I show that
indeed a scientific description of reality faces a serious limitation,
which explains this position. On the other hand, to remain in the realm
of science, I explore the problem of sentient experience in an indirect
way, through its possible physical correlates. This can only be done in a
negative way, which consists in the falsification of various hypotheses
and the derivation of no-go results. The general approach I use here is
based on simple mathematical proofs about dynamical systems, which I
then particularize to several types of physical theories and
interpretations of quantum mechanics. Despite choosing this
scientifically-prudent approach, it turns out that various possibilities
to consider sentience as fundamental make empirical predictions,
ranging from some that can only be verified on a subjective basis to
some about the physical correlates of sentience, which are independently
falsifiable by objective means.</span></div>
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For the moment, <a href="https://www.researchgate.net/publication/339956062_The_negative_way_to_sentience_comments_welcome" target="_blank">you can read it and comment here</a>.</div>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-3236767638330419092019-10-27T21:02:00.000+02:002019-10-27T21:02:53.589+02:00Representation of the wave function on the three-dimensional space<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="_5pbx userContent _3576" data-ft="{"tn":"K"}" data-testid="post_message" id="js_d9" style="text-align: justify;">
My last paper<br />
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<h3>
Representation of the wave function on the three-dimensional space</h3>
<blockquote class="tr_bq">
One of the major concerns of Schrödinger, Lorentz, Einstein, and many
others about the wave function is that it is defined on the
3N-dimensional configuration space, rather than on the three-dimensional
(3D) physical space. This gives the impression that quantum mechanics
cannot have a 3D space or space-time ontology, even in the absence of
quantum measurements. In particular, this seems to affect
interpretations which take the wave function as a physical entity, in
particular, the many-worlds and the spontaneous collapse
interpretations, and some versions of the pilot wave theory. Here, a
representation of the many-particle states is given, as multilayered
fields defined on the three-dimensional physical space. This
representation is equivalent to the usual representation on the
configuration space, but it makes it explicit that it is possible to
interpret the wave functions as defined on the physical space. As long
as only unitary evolution is involved, the interactions are local. I
intended this representation to capture and formalize the nonexplicit
and informal intuition of many working quantum physicists, who, by
considering the wave function sometimes to be defined on the
configuration space and sometimes on the physical space, may seem to
researchers in the foundations of quantum theory as adopting an
inconsistent view about its ontology. This representation does not aim
to solve the measurement problem, and it allows for Schrödinger cats
just like the usual one. But, it may help various interpretations to
solve these problems, through inclusion of the wave function as (part
of) their primitive ontology. In appendices, it is shown how the multilayered field representation can be extended to quantum field
theory.</blockquote>
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<a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.042115">https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.042115</a></div>
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Free arXiv version <a href="https://arxiv.org/abs/1906.12229">https://arxiv.org/abs/1906.12229</a></div>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0tag: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">
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
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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>
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I illustrate this with this animated gif:</div>
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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>
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<b>A.</b> The standard Penrose diagram of an evaporating black hole.</div>
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B The diagram from the analytic solution I proposed.</div>
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* 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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
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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>
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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>
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com1tag: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">
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<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 />
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<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>
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<span class="tm6">* * *</span></div>
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<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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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<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>
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<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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
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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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com1tag: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">
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The results of this year's FQXi essay contest are out.</div>
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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>
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<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>
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<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>
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<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>
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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Here are the top 5 essays from the 40 finalists of this year's FQXi essay contest, based on the community ratings. </div>
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<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>
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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 />
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The finalists will be judged by a jury, who will decide the awards until June 6, 2015.</div>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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The Monty Hall problem</h2>
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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>
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<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>
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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>
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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>
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An equivalent puzzle</h2>
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I will present another, simpler puzzle, and show that it is equivalent to the Monty Hall problem.</div>
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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>
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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>
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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>
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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>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com3tag:blogger.com,1999:blog-124350264510724511.post-79062472931329901922015-03-14T21:33:00.002+02:002020-02-26T23:07:41.051+02:00Round squares exist<div dir="ltr" style="text-align: left;" trbidi="on">
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Bertrand Russell said that there are no round squares. But there are. Here are two solutions.</div>
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A circle-square</h2>
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This is a square that is circle:</div>
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<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" height="218" src="https://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" width="400" /></a></div>
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To make it, first make a paper circle and a paper square, with equal perimeters:<br />
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<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" height="215" src="https://2.bp.blogspot.com/-hzj5qBOeCbQ/VQSK4n5ncTI/AAAAAAAAAs4/5RGB_zalkxE/s1600/round%2Bsquare%2B1.png" width="400" /></a></div>
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Fold them a bit: <br />
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<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" height="223" src="https://3.bp.blogspot.com/-piHiWTXeuCg/VQSK45fnFRI/AAAAAAAAAs8/3bMrWmCbAS8/s1600/round%2Bsquare%2B2.png" width="400" /></a></div>
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Then glue their edges together:<br />
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<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" height="175" src="https://2.bp.blogspot.com/-dvR0q9kiKnA/VQSK4aEGCeI/AAAAAAAAAs0/oJC4qfKyD7k/s1600/round%2Bsquare%2B3.png" width="320" /></a></div>
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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 />
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<h2 style="text-align: justify;">
Round squares in non-Euclidean geometry</h2>
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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>
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<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>
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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 />
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<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>
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So, is it a circle? Is it a square? It's a circle and a a square!</div>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0tag:blogger.com,1999:blog-124350264510724511.post-11924561868077600112015-03-14T08:45:00.002+02:002020-01-25T12:40:52.112+02:00A problem with towers of coins<div dir="ltr" style="text-align: left;" trbidi="on">
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The problem</h2>
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<i>In how many ways you can arrange $p$ coins in a sequence of towers?</i> (it doesn't matter whether the coins can be flipped or rotated, and we assume that there are no empty towers).</div>
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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>
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<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>
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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>
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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>
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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 />
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<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>
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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 />
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<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>
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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 />
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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>
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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>
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<br /></div>
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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>
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<br /></div>
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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>
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Before reading the solution, I would like to ask you to solve it yourself.</div>
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</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 />
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<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>
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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 />
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<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>
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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 />
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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 />
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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>
Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@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">
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Man can be understood in the framework of classical physics, but for woman you'll need quantum physics.</div>
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Cristi Stoicahttp://www.blogger.com/profile/00577217435388643300noreply@blogger.com0