# Generalized Quantum Mechanics

1. Jan 23, 2008

### gel

I have a couple of questions on what is possible within quantum mechanics, and the physical justifications (if any). My question is a bit subtle and tricky to explain, but I'll try to explain as well as I can. Hopefully someone here can spread a bit of light on this.

This problem first puzzled me when thinking abount entangled systems, such as in Bell's theorem. In that case you have a pair of entangled particles, e.g. photons. Each of these is sent to an separate observer (Alice & Bob) who procede to measure their polarizations with repect to certain orientations. Let A be the result of Alice's measurement, say A=1 if she detects that the photon is polarized in the orientation she tests, A=-1 if she detects the opposite polarization (ie at 90 degrees) and A=0 if she fails to detect it. Similarly let $A^\prime$ be her result if she measures with respect to some other orientation, and $B,B^\prime$ be the results of Bob's measurement on his photon, similarly depending on which of two orientations he chooses.
Then, assuming Alice's measurement can not affect Bob's result and vice-versa, classical probability requires the following version of Bell's inequality (http://en.wikipedia.org/wiki/CHSH_inequality" [Broken]) to be satisfied

$$<AB>+<A^\prime B>+<A B^\prime> - <A^\prime B^\prime>\le 2.$$

But, as is well known, this can be violated in quantum mechanics. So far so good, this demonstrates that quantum mechanics is really saying something that isn't possible with classical mechanics (assuming locality, etc), but this doesn't allow Alice and Bob to transfer information between them instantaneously, as the distribution of Alice's measurement results does not depend on which experiment Bob performs, and vice-versa.
The issue is that the inequality above can be extended to cover quantum systems by increasing the bound from 2 to $2\sqrt{2}$ (Tsirelson's bound). This is the issue that was puzzling me. It is possible to imagine a hypothetical situation where instead of the spins of photons, some other independent measurements (i.e. no information transfer) are performed by Alice and Bob, governed by some as yet undiscovered theory, where Tsirelson's bound is broken. For example, you could imagine the situation where $A,A^\prime,B,B^\prime$ are each equal to plus or minus one with 50% probability, but $A=B, A=B^\prime, A^\prime=B, A^\prime=-B^\prime$ whenever each of these 4 different possibilities is measured. This seems physically reasonable to me, but would give 4 in the left hand side of the above inequality, breaking Tsirelson's bound.
So, my question is, once you are prepared to accept that the world is governed by quantum rules, and Bell's inequality is violated, why stop there? Why not consider situations where Tsirelson's bound is broken? More specifically,

1) Is there any physical reason why the kinds of correlations considered in my hypothetical situation above shouldn't occur in nature?
2) If there is no universally accepted physical reason, does anyone seriously study this kind of "generalized quantum mechanics" where Tsirelson's bound is broken? From google, I found such a paper (Generalized Quantum Mechanics by Bogdan Mielnik), but this didn't go very far into the physics at all, and was published way back in 1974. I also have heard some people mention non-unitary evolution (in Quantum Gravity), use of quaternions rather than complex numbers for wave functions, and non-hermitian operators for observables. Are these extensions really extending the physical systems that can be described, such as by breaking Tsirelson's bound?
3) Are there any intuitive ways of classifying which possible correlations in more complex systems are actually possible under quantum mechanics (satisfying locality), and which are not?

Last edited by a moderator: May 3, 2017
2. Jan 24, 2008

### tsirel

Try search for "Popescu-Rohrlich box".
L.J.Landau, "Experimental tests of distributivity", Lett. Math. Phys. 25:1, 47-50 (1992).
http://www.tau.ac.il/~tsirel/Research/mybound/main.html

(1) I am afraid, the only reason is the quantum theory. But, who knows?..
(2) Not much, but see the links above.
(3) Again: not much, but see the links above.

3. Jan 26, 2008

### gel

Thanks for the reply. I'll have a look through the paper.