By Maurice A. de Gosson
This e-book provides a entire mathematical learn of the operators at the back of the Born–Jordan quantization scheme. The Schrödinger and Heisenberg photos of quantum mechanics are similar provided that the Born–Jordan scheme is used. hence, Born–Jordan quantization presents the single bodily constant quantization scheme, instead of the Weyl quantization popular through physicists. during this ebook we strengthen Born–Jordan quantization from an operator-theoretical viewpoint, and study intensive the conceptual alterations among the 2 schemes. We talk about a variety of bodily prompted methods, specifically the Feynman-integral standpoint. One vital and fascinating characteristic of Born-Jordan quantization is that it's not one-to-one: there are infinitely many classical observables whose quantization is zero.
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Extra resources for Born-Jordan Quantization: Theory and Applications
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We will the symplectic structure determined by the matrix J = −In×n 0n×n occasionally also use the synonym “ symplectomorphism”, which is more common in mathematical texts. Beware: in some physics texts a “canonical transformation” has the slightly more general meaning of a transformation which preserves the form of Hamilton’s equations (see Arnol’d , Sect. 45, Footnote 76). Let H be a real-valued function; we assume for convenience that C ∞ (R2n × R), but most of what follows remains true if we assume less stringent differentiability conditions (see Abraham and Marsden  for a discussion of sufficient smoothness requirements).