Is quantum field theory a generalization of quantum mechanics?
A. V. Stoyanovsky
备注：
7 pages; minor corrections
发表信息：
Short version: in: Proceedings of the International Workshop "Idempotent and tropical mathematics and problems of mathematical physics", G. L. Litvinov, V. P. Maslov, S. N. Sergeyev eds., Moscow, 2007, p. 58--62.
We construct a mathematical model analogous to quantum field theory, but without the notion of vacuum and without measurable physical quantities. This model is a direct mathematical generalization of scattering theory in quantum mechanics to path integrals with multidimensional trajectories (whose mathematical interpretation has been given in a previous paper). In this model the normal ordering of operators in the Fock space is replaced by the Weyl-Moyal algebra. This model shows to be useful in proof of various results in quantum field theory: one first proves these results in the mathematical model and then "translates" them into the usual language of quantum field theory by more or less "ugly" procedures.
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Hartree--Fock theory in quantum mechanics is reviewed, from the proposal of the Hartree--Fock approximation right after quantum mechanics was formulated to its applications in modern physics. This includes the description of traditional Hartree--Fock theory in quantum chemistry, its generalizations of various kinds, and its importance for predicting the presence of symmetry breaking, or the absence thereof.
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A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schrödinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in δ-potential well, linear potential, fractional Bohr atom and fractional oscilla…
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The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g. momentum, energy, etc), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E {\bf 66}, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits t…
