Pinball with disks and rays

Joseph O'Rourke proposed a beautiful problem on Math Overflow:

Let every point of $\mathbb Z^2$ be surrounded by a mirrored disk of radius $r\lt 1/2$, except leave the origin (0,0) unoccupied by a disk.
    Q. Is it the case that every disk can be hit by a lightray emanating from the origin and reflecting off the mirrored disks?

Here is an example

I liked the solution I gave, so I would like to share it here. It assumes the radius to be about $1/3$ or less. The idea is that, instead of light ray and reflection, to think in terms of rings and ropes connecting them. We assume the rings and the rope satisfying suitable idealizations. The following picture shows that any ring in the lattice $\mathbb Z^2$ is reachable.

Moreover, we can use this method to connect with the origin all rings in the plane, with a single rope.

Of course, if the radius gets close to $1$, the rope becomes overlapped with the rings, and this solution will no longer work. But we can still use it to find a correct solution. We start with a radius of $1/3$, then gradually increase it. At some point, the rope will become tangent to one or more rings. In this case, just wrap it more, using the moves in the following picture

Violation of Heisenberg's uncertainty principle, or evidence for Quantum Mechanics?

A recent paper entitled  Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements  appeared (see the arXiv link to the paper).

This paper was presented to the public in some articles which gave the wrong idea that quantum uncertainty is disproved  (see for example Heisenberg uncertainty principle stressed in new test, with the subtitle "Pioneering experiments have cast doubt on a founding idea of the branch of physics called quantum mechanics.").

I will argue that the experiment presented in the paper actually supports Quantum Mechanics. This may be not quite explicit in the paper, but also there is nothing against the standard view on quantum mechanics in it.

Heisenberg originally stated his principle in terms of measurement-disturbance relationship (MDR). This is how he understood it at that time. The uncertainty principle which was proven theoretically, either in the context of wave mechanics, or from the non-commutativity of the operators, is correct, and it's correctness is acknowledged by the paper. This is called Heisenberg's uncertainty principle (HUP), and is very different from MDR.

The paper refers to previous theoretical works which disprove MDR, and present experimental evidence purported to confirm the violation of the MDR.

Why do I claim that the violation of MDR supports Quantum Mechanics? Because, if MDR would be correct, it would be enough to explain quantum uncertainty. Recall that even Heisenberg originally thought that the uncertainty is due to disturbance caused by measurement. If the states would behave as they are due to the measurement disturbance, then we could consider them classical, and extract Born's probability rule as we calculate probabilities in statistical mechanics. But we know this is not true. Quantum states exhibit properties which can't be explained by classical mechanisms. Among these, HUP plays an important role, together with entanglement. The service made by this paper is that it shows that the wrong version of the uncertainty principle can be violated. The authors seem to me to support the HUP:

"These two readings of the uncertainty principle are typically taught side-by-side, although only the modern one [HUP] is given rigorous proof."

and

"Our work conclusively shows that, although correct for uncertainties in states [HUP], the form of Heisenberg's precision limit is incorrect if naively applied to measurement [MDR]."

[1] Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements

[2] arXiv link to the paper

[3] Heisenberg uncertainty principle stressed in new test

Analytic Reissner–Nordström singularity

This paper can be downloaded for the following 30 days from the Physica Scripta website.

Link to the arXiv version.

Abstract:

An analytic extension of the Reissner–Nordström solution at and beyond the singularity is presented. The extension is obtained by using new coordinates in which the metric becomes degenerate at r = 0. The metric is still singular in the new coordinates, but its components become finite and smooth. Using this extension it is shown that the charged and non-rotating black hole singularities are compatible with the global hyperbolicity and with the conservation of the initial value data. Geometric models for electrically charged particles are obtained.

Is semi-classical gravity wrong?

Semi-classical gravity is not considered fundamental, yet it escaped to experimental falsification. Of course, maybe we don't have yet the technology, or at least ideas of experiments we can do, to falsify it. It would be nice to be able to differentiate it experimentally from various quantum gravity approaches. But theoretically, it stands pretty well: being the most straightforward union between general relativity and quantum mechanics, it inherits their successes.

Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?

One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).

Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).

About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here, here, and at arxiv:1309.2309.

I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.

Local Hidden Variables Correlations

Number of experiments: Number of tests:

To generate random angular momentum variables, press "Generate Random Data". In this case the correlations will be the classical ones (along the blue line), as it is predicted. To test it on random generated orientations and plot the result, press "Plot Data". Alternatively, if anybody believes that a local hidden variable theory can provide a set of data which gives the correlation of Quantum Mechanics, he or she can paste the data instead of randomly generating it, and then plot it. For example, Joy Christian claims in arxiv:0806.3078v2, page 4, that an experiment he describes can provide a list of angular momenta which gives correlations = -cos of the angle between the two orientations chosen by Alice and Bob (the green curve). He claims by this that his local hidden variables can reproduce the outcomes of the EPR-Bohm experiment.

More exactly, Joy saids that after an experiment involving balls which explode in halves which have total angular momentum 0, a list of angular momenta can be collected. The second part of his experiment is to randomly generate on a computer pairs of directions a and b in space, and calculate the result using equation (16) from his paper, page 4. He then claims that the result will be -cos of the angle between the two orientations (the green curve), rather than the linear function represented in blue.

My application does exactly the second part of Joy's experiment. If Joy Christian or anybody else can produce this kind of data, they he can test it in this application. I already provided a mathematical proof that the only possible correlation depending only on the angle between a and b is the linear one, but there are people who don't trust the mathematical proof. Therefore, I challenge them to produce the data which will contradict my proof by counterexample.

Given that the output of the first part of Joy's experiment is just a list of angular momenta, you can produce it by performing the first part of Joy's experiment. But I will not require anybody to get the data only by actually making the experiment. Joy can produce the list by any means he wants, I will not constrain him to make the experiment. Just to provide a list of angular momenta which give his prediction.

I used JavaScript, so that anyone can easily verify the source code.

P.S. For the moment, there is a problem viewing the results in Internet Explorer, so please use Mozilla, Chrome or Opera instead.

P.P.S. Thanks to Florin Moldoveanu for the uniformization of the random generator.

P.P.P.S. This test may look similar to the Quantum Randi Challenge of Sascha Vongehr, but its purpose was different, as it can be seen from the text. I fully agree with Vongehr, just that my script has a different purpose, although it can be used for the Quantum Randi Challenge too. Vongehr's challenge appeared few months earlier than mine, in connection to J.C.'s claimed disproof of Bell's theorem. Mine refers only to J.C.'s proposed experiment in which he claimed that classical mechanics reproduces Bell's correlations, and appeared from a discussion I had with J.C. on FQXi's blog, where I also gave a proof that the kind of correlations he proposed can't be obtained in that way. The discussion started on Florin Moldoveanu's post, before he turned completely against J.C., and continued here, here, here, and here. For the "disproof", I shown there that J.C.'s proposal works only for singlet state, and it is in fact the quaternion formulation from Peres's book, hence no locality involved; I gave then many other arguments, but I will not repeat the discussion here. The only reason I made the script in this post was to show that the classical proposed experiment of J.C. (also inspired from the same book by Peres) can't give the same correlations as Bell's. He rejected the math proof (both Peres's and mine), so I made this script and asked him to introduce at that time, or after the experiment will be performed, the data he claimed will show the same correlations as Bell's. My point was that no data can give the Bell correlations by using the data as generated in his proposed experiment. I maintain this page only because I said I will wait for him to come with the data, but I am no longer interested in that topic.