Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization
Abstract
:Autant de quantifications, autant de religions (Jean Leray (Private communication (1988)), 1906–1998)
1. Introduction
2. The Moyal Star Product
- is the space of all functions such that for every there exist and such that ;
- is the space of all functions such that there exists such that for every there exist such that ;
- is the Shubin class [11]: () if ; it is sometimes called the “GLS symbol class” in the older literature.
3. Born–Jordan Quantization
4. The Dirac Correspondence
- (Q1)
- We have for ;
- (Q2)
- (the identity operator on );
- (Q3)
- When A is real, is a symmetric operator defined on ;
- (Q4)
- For , we have and .
5. Ehrenfest’s Theorem: Schrödinger Picture
6. Ehrenfest’s Theorem: Heisenberg Picture
7. Comments and Discussion
Author Contributions
Funding
Conflicts of Interest
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de Gosson, M.; Luef, F. Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization. Quantum Rep. 2019, 1, 71-81. https://doi.org/10.3390/quantum1010008
de Gosson M, Luef F. Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization. Quantum Reports. 2019; 1(1):71-81. https://doi.org/10.3390/quantum1010008
Chicago/Turabian Stylede Gosson, Maurice, and Franz Luef. 2019. "Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization" Quantum Reports 1, no. 1: 71-81. https://doi.org/10.3390/quantum1010008