Timeline for Relationship between canonical commutation relations and projective representations?
Current License: CC BY-SA 4.0
9 events
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| Apr 15, 2020 at 20:41 | comment | added | Francois Ziegler | If you have access to the paper, their argument for $\mathbf R^n$ is on p. 314 with reference to Kleppner (1965). | |
| Apr 15, 2020 at 20:21 | comment | added | Tim Campion | @FrancoisZiegler Thanks! For the finite-dimensional case, I'm having trouble seeing how this follows from the Stone-von Neumann theorem. Is it as obvious as they indicate? Regarding infinite dimensions, perhaps I need to be a bit more precise in my question, but I am interested in this case. The Stone-von Neumann theorem isn't available here. | |
| Apr 15, 2020 at 19:32 | comment | added | Francois Ziegler | I am very predictable :-) | |
| Apr 15, 2020 at 19:26 | comment | added | LSpice | @FrancoisZiegler, I appreciate your re-posting the edited comment so that I look prophetic. | |
| Apr 15, 2020 at 19:06 | comment | added | Francois Ziegler | Yes, to an extent. See Baggett-Kleppner (1973) whose Introduction starts: “The classification of the irreducible multiplier representations of a real vector group is an easy consequence of the Stone-von Neumann theorem. With no loss of generality one may assume that the multiplier is the exponential of an alternating bilinear form.” (And they go on to generalize to “quite arbitrary locally compact abelian groups.” That includes discrete [no continuity condition] but not $\infty$-dimensional topological vector groups...) | |
| Apr 15, 2020 at 19:00 | comment | added | LSpice | $\def\Z{\mathbb Z}\def\R{\mathbb R}$Am I right in re-formulating the question as: $\phi$ canonically gives $\overline\omega_\phi : V \otimes_{\Z} V \to \R/2\pi i\Z$ (essentially the cocycle realising the obstruction to liftability), and you're wondering under what conditions this may be lifted to $\omega : V \otimes_{\Z} V \to \R$ that factors through $V \otimes_{\Z} V \to V \otimes_{\R} V$? (EDIT: I guess the re-formulation doesn't matter, in light of @FrancoisZiegler's comment.) | |
| Apr 15, 2020 at 18:48 | history | edited | LSpice | CC BY-SA 4.0 |
DeclareMathOperator'ing
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| Apr 15, 2020 at 18:22 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 3 characters in body
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| Apr 15, 2020 at 18:14 | history | asked | Tim Campion | CC BY-SA 4.0 |