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Natural Science Forum / Physics / Research / February 2008Wanted: Defn of Equal Time Commutaion Relation
February 20, 2008 Hello, I am currently working through a copy of _The Physics of Time Reversal_ by Sachs. Sachs often refers to an "equal time commutation relation." I remember studying commutation relations in Bransden & Joachain's textbook when I took quantum mechanics, but I do not recall any mention of 'equal time' commutation relations. If any one could give me a quick definition of this form of commutation relations, I would greatly appreciate it. Thank you. LS Thomas On 23 Feb., 14:00, lek...@rocketmail.com wrote: > Sachs often refers to an "equal time commutation > relation." I remember studying commutation relations in Bransden & [quoted text clipped - 3 lines] > If any one could give me a quick definition of this form of > commutation relations, I would greatly appreciate it. You choose some system of coordinates, with a time coordinate t. You are in the Heisenberg picture, thus, your operators depend on t. Then, the commutation relations between operators with the same t are the equal time commutation relations. That's all. The point of distinguishing them is that canonical quantization _postulates_ these equal time commutation relation. Instead, commutation relations for operators at different times have to be computed, using the Heisenberg equations. On Feb 25, 2:03 pm, Ilja Schmelzer <ilja.schmel...@googlemail.com> wrote: > On 23 Feb., 14:00, lek...@rocketmail.com wrote: > [quoted text clipped - 10 lines] > the commutation relations between operators with the same t are the > equal time commutation relations. That's all. It's a little more involved than that. One cannot maintain causality unless the commutation relations for fields are imposed on a spacelike initial hypersurface. The simplest such is t=const. in a given referent. -drl leketa@rocketmail.com schrieb: > February 20, 2008 > [quoted text clipped - 8 lines] > If any one could give me a quick definition of this form of > commutation relations, I would greatly appreciate it. Standard quantum mechanics is usually done in the Schroedinger picture, in which configuration space is quantized. In terms of a relativistic point of view, this means that space-time fields are quantized at a fixed time, and hence all operators are operators at fixed, equal time. In the Heisenberg picture, operators are time-dependent, and the canonical commutation relations only apply at fixed, equal time, [q(t),p(t)]=i\hbar, whereas q(t) and p(t') at different time generally hav no simple commutation law. Arnold Neumaier. On Feb 23, 8:00 am, lek...@rocketmail.com wrote: > February 20, 2008 > [quoted text clipped - 12 lines] > > LS Thomas I don't think you would see that when considering the quantum mechanics of systems with finite degrees of freedom. yclept.ucdavis.edu/course/242/2Q_Fradkin.pdf |
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