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Progress of Theoretical Physics

On the Time Operator in Quantum Mechanics. II: -- General Prescription in One Dimensional Case --

Toru Goto, S. Naka, K. Yamaguchi

Published Dec 1, 1981• 9 citations
Publisher
DOI
10.1143/ptp.66.1915

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Representations of discrete symmetry operators (DSO's) connected with space (𝒫), time (T), and generalized charge (𝒞) are considered. It is shown that if one writes a DSO as exp (iπΩ) × a phase transformation, then (under certain conditions on Ωs) to each DSO there corresponds a set of Ωs which is closed with respect a Lie algebra, which is isomorphic to the Lie algebra of generators of rotation in an n-dimensional Euclidean space; where n is the number of commuting observables that changes sign under this DSO in a given representation (e.g. linear momentum representation). In the particular case of the (T𝒞𝒫) operation, there are six Ωs, of which two are diagonal, viz. the generalized charge Q, and spin projection along the z axis Sz; corresponding Euclidean group is four-dimensional. For the sake of completeness, the representations are also given for the following cases: (i) nonrelativistic quantum mechanics, (ii) quantum theory of free fields, in terms of field operators.

Journal of Mathematical Physics•Jan 1, 1966•6 citations
In a previous paper we used the method of finite elements to formulate consistent, unitary, discrete-time quantum-mechanical systems having one degree of freedom. In this paper we extend the treatment to systems having two degrees of freedom. The proof of consistency is more delicate: It is nontrivial to show that there are no operator-ordering problems and that independent degrees of freedom remain independent at subsequent lattice sites. The construction of purely bosonic lattice systems is a straightforward application of the finite-element prescription. However, it is surprising that quantum-mechanical systems having interacting fermions and bosons are only unitary if the interaction is modified on the lattice by a term which vanishes as the lattice spacing approaches zero. The modified interaction is determined by an interesting nonlinear condition. In all cases we give an explicit formula for the lattice transfer operator.
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.
Representations of discrete symmetry operators (DSO's) connected with space (𝒫), time (T), and generalized charge (𝒞) are considered. It is shown that if one writes a DSO as exp (iπΩ) × a phase transformation, then (under certain conditions on Ωs) to each DSO there corresponds a set of Ωs which is closed with respect a Lie algebra, which is isomorphic to the Lie algebra of generators of rotation in an n-dimensional Euclidean space; where n is the number of commuting observables that changes sign under this DSO in a given representation (e.g. linear momentum representation). In the particular case of the (T𝒞𝒫) operation, there are six Ωs, of which two are diagonal, viz. the generalized charge Q, and spin projection along the z axis Sz; corresponding Euclidean group is four-dimensional. For the sake of completeness, the representations are also given for the following cases: (i) nonrelativistic quantum mechanics, (ii) quantum theory of free fields, in terms of field operators.