Timeline for Toy Models of Quantum Mechanics
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2013 at 1:34 | answer | added | J Tyson | timeline score: 2 | |
| Mar 25, 2011 at 15:39 | answer | added | Rob Spekkens | timeline score: 13 | |
| Mar 21, 2011 at 12:07 | answer | added | John Sidles | timeline score: 4 | |
| Mar 21, 2011 at 8:03 | answer | added | Stefan Waldmann | timeline score: 7 | |
| Mar 21, 2011 at 5:59 | answer | added | Anatoly Kochubei | timeline score: 5 | |
| Mar 21, 2011 at 2:36 | history | edited | NebulousReveal | CC BY-SA 2.5 |
added 79 characters in body; edited body
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| Mar 21, 2011 at 0:46 | comment | added | Amritanshu Prasad | Historical note: in his book The Theory of Groups and Quantum Mechanics Hermann Weyl actually did consider these toy models of quantum mechanics. The case of $\mathbf Z_2$ models electron spin. He speculated that other finite groups may arise in the context of nuclear physics. It was Mackey who developed the theory of Heisenberg groups in the context of locally compact abelian groups (A Theorem of Stone and von Neumann, Duke Math. J., 1949). | |
| Mar 20, 2011 at 22:58 | answer | added | Abdelmalek Abdesselam | timeline score: 5 | |
| Mar 20, 2011 at 21:20 | answer | added | Greg Kuperberg | timeline score: 15 | |
| Mar 20, 2011 at 20:25 | comment | added | Qiaochu Yuan | I don't think I understand the distinction you're making between "clarify our understanding" and "providing any new insight." | |
| Mar 20, 2011 at 17:06 | comment | added | Steve Flammia | I think what you are after is better called a toy theory, i.e. a theory which is distinct from regular quantum mechanics, but still exhibits structural similarities that helps us gain insight into the standard theory. This would preclude other examples (harmonic oscillator, transverse field Ising model, quantum walks on graphs etc.) which would qualify as "toys" in the sense that they are simple to analyze models of regular quantum mechanics. | |
| Mar 20, 2011 at 16:34 | answer | added | Steve Flammia | timeline score: 12 | |
| Mar 20, 2011 at 14:06 | comment | added | Theo Johnson-Freyd | (continuation) yourself whether you want the things corresponding to "polynomials" to allow divided powers or not. You may want to identify one-parameter families of operators with their derivatives at the identity --- you may want Lie theory --- but not every Lie algebra integrates to an algebraic group, and in positive characteristic you don't have as good access to notions like "compact" or "negative definite" that can assure that they do. Which is all to say: "physics in characteristic p" is less a "toy" and more a hard area of research. (Even p-adic --- char=0 --- physics is worth doing!) | |
| Mar 20, 2011 at 14:02 | comment | added | Theo Johnson-Freyd | Toy models are certainly valuable, but I disagree that "quantum mechanics over a finite field" is a "toy" with which we can "better understand 'regular' quantum mechanics". Indeed, many of the standard axioms in QM have a lot to do with structure on the complex numbers: positivity, for example, or unitarity, or self-adjointness. It's not at all clear what the corresponding constructions should be in positive characteristic. Integrals are also less-well behaved, or anyway more nuanced. If you want to write down a theory of differential equations, among other things you have to ask (continued) | |
| Mar 20, 2011 at 13:55 | comment | added | Qfwfq | @Snark: topologists and physicists usually use the notation " $\mathbb{Z}_n$ " for the ring $\mathbb{Z}/n$. | |
| Mar 20, 2011 at 8:42 | comment | added | Julien Puydt | Don't you really mean $\mathbb F_2$ (the prime characteristic 2 field), instead of $\mathbb Z_2$ (the completion of $\mathbb Z$ along 2 -- which is of characteristic 0)? | |
| Mar 20, 2011 at 6:25 | answer | added | Daniel Litt | timeline score: 18 | |
| Mar 20, 2011 at 5:43 | answer | added | Bjørn Kjos-Hanssen | timeline score: 6 | |
| Mar 20, 2011 at 5:02 | history | asked | NebulousReveal | CC BY-SA 2.5 |