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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Contradirectional propagation and canonical transformations

Alfredo Luis, Jan Peřina

Published Feb 1, 1996• 19 citations
Publisher

Abstract

We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.

DOI
10.1088/1355-5111/8/1/005

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We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.
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.
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrodinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.
In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.