By Maurice A. de Gosson
ISBN-10: 3319279009
ISBN-13: 9783319279008
This e-book provides a finished mathematical examine of the operators in the back of the Born–Jordan quantization scheme. The Schrödinger and Heisenberg images of quantum mechanics are similar provided that the Born–Jordan scheme is used. hence, Born–Jordan quantization presents the one bodily constant quantization scheme, in place of the Weyl quantization frequent via physicists. during this e-book we increase Born–Jordan quantization from an operator-theoretical perspective, and research extensive the conceptual alterations among the 2 schemes. We speak about quite a few bodily stimulated methods, specifically the Feynman-integral viewpoint. 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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M. B. Fairlie, The algebra of Weyl symmetrized polynomials and its quantum extension. Comm. Math. Phys. 136(3), 487–499 (1991) 7. H. G. Sutcliffe, Unique Hamiltonian operators via Feynman path integrals. J. Math. Phys. 11(2), 391–393 (1970) 8. H. McCoy, On the function in quantum mechanics which corresponds to a given function in classical mechanics. Proc. Natl. Acad. Sci. A. 18(11), 674–676 (1932) 9. L. Mehta, Phase-space formulation of the dynamics of canonical variables. J. Math. Phys. 5(1), 677–686 (1963) 10.
2 The Action Functional The notion of action functional is usually defined using the Lagrangian formalism [2, 4, 8]; this approach has certain conceptual advantages and sometimes makes the notion more tractable. We adopt here a slightly different point of view, using directly the Hamiltonian formalism. Let x and x be two arbitrary points in Rn . 19) has a unique solution s −→ (x(s), p(s)); we denote by phase space Rnx × Rnp × Rt : we have the lift to the extended (s) = (x(s), p(s), s) , t ≤ s ≤ t.
13). The definition of quantization given above is very simple; in many texts one adds supplementary restrictive assumptions, see for instance Niederle and Tolar [11] or Przanowski and Tosiek [12]. We will not discuss such conditions here, because they are actually unnecessary complications of the theory. We also note that it is often required in texts on quantization that one should in some way recover the classical observable by taking the “limit” → 0. 26) but this notation is of course, mathematically speaking, rather formal, to say the least.
Born-Jordan Quantization: Theory and Applications by Maurice A. de Gosson
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