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Chinese Journal of Physics

Invariance of the Canonical Quantization Prescription under Classical Canonical Transformations

Gin-yih Tsaur, Jyhpyng Wang

Published Aug 1, 2007• 0 citations
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Abstract

Dirac's postulate of canonical quantization, [(The Symbol is abbreviated)]=-iħδ(subscript ij) for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [(The Symbol is abbreviated)]=-iħδ(subscript ij) must remain invariant under classical canonical transformations This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if ((The Symbol is abbreviated)) are related to ((The Symbol is abbreviated)) by a classical canonical transformation, then [(The Symbol is abbreviated)]=-iħδ(subscript ij) In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.

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Dirac's postulate of canonical quantization, [(The Symbol is abbreviated)]=-iħδ(subscript ij) for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [(The Symbol is abbreviated)]=-iħδ(subscript ij) must remain invariant under classical canonical transformations This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if ((The Symbol is abbreviated)) are related to ((The Symbol is abbreviated)) by a classical canonical transformation, then [(The Symbol is abbreviated)]=-iħδ(subscript ij) In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.
Following Dirac, the rules of canonical quantization include classical and quantum contact transformations of classical and quantum phase space variables. While arbitrary classical canonical coordinate transformations exist that is not the case for some analogous quantum canonical coordinate transformations. This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a ccc-number to a qqq-number. A different relationship of ccc-numbers and qqq-numbers in the procedures of Enhanced Quantization shows the compatibility of all quantum operators with all classical canonical coordinate transformations.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
As a way of studying the consequences of a possible breakdown of Lorentz invariance, the canonical commutation relations are enlarged to include time and energy as additional conjugate variables, and with this enlargement, they are shown to be invariant under the Lorentz transformation under the assumption that commutators of non-conjugate operators vanish. Although time does not commute with the energy operator, it does commute with the Hamiltonian. It is also shown that the invariance of the enlarged commutation relations holds for arbitrary non-singular linear transformations that describe a velocity boost to another uniformly moving frame, under the same assumptions about non-conjugate commutators. A covariant enlargement of the Poisson brackets, together with Dirac's interpretation, that would predict this invariance of the commutators is presented. There is a discussion of improper coordinate transformations, and a canonical time-reversal operator is introduced, and compared with the standard time-reversal operator T. It is concluded that even if the Lorentz transformation were to fail, and were to be replaced by another linear transformation, the enlarged canonical commutation relations could still hold.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real