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Canonical Transformations and Spectra of Quantum Operators

Boris Leaf

The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one-sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one-dimensional systems. The properties of one-sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one-dimensional case for which the one-sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one-dimensional observable. Generalization to higher dimensions is also discussed.The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one-sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one-dimensional systems. The properties of one-sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one-dimensional case for which the one-sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one-dimensional observable. Generalization to higher dimensions is also discussed.

Journal of Mathematical PhysicsNov 1, 196935 citations

On the relation between classical and quantum observables

Abhay Ashtekar

Communications in Mathematical PhysicsFeb 1, 198033 citations

Canonical Transformations for Quantum Systems

WORLD SCIENTIFIC eBooksMar 1, 20100 citations

Wigner distribution functions and the representation of a non-bijective canonical transformation in quantum mechanics

Rainer Dirl, Peter Kasperkovitz, M. Moshińsky

In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1 , where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1 , where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.

Journal of Physics A Mathematical and GeneralApr 21, 198815 citations

Canonical Transformations for Quantum Systems

WORLD SCIENTIFIC eBooksAug 17, 20150 citations

The ambiguity group for canonical transformations in classical mechanics and its role in their representation in quantum mechanics

M. Moshińsky, T. H. Seligman

Springer eBooksApr 6, 20081 citation

Ambiguities in canonical transformations of classical systems and the spectra of quantum observables

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