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arXiv:1802.09891

Unitary Representation of Symplectic Group for Phase Point Operators on Discrete Phase Space

Daisuke Watanabe, T. Hashimoto, M. Horibe, A. Hayashi

Published Feb 27, 2018• 0 citations
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Abstract

The phase point operator $Δ(q,p)$ is the quantum mechanical counterpart of the classical phase point $(q,p)$. The discrete form of $Δ(q,p)$ was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

DOI
10.48550/arxiv.1802.09891

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Using a family of coherent state representations we obtain in a natural and coordinate independent wayan explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.

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cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper­ ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no­ tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de­ quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param­ eter subgroups of the symplectic group.8.9 One then only

•Jan 1, 2012•0 citations
The phase point operator Δ(q,p)Δ(q,p)Δ(q,p) is the quantum mechanical counterpart of the classical phase point (q,p)(q,p)(q,p). The discrete form of Δ(q,p)Δ(q,p)Δ(q,p) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.
The phase point operator Δ( q,p ) is the quantum mechanical counterpart of the classical phase point ( q,p ). The discrete form of Δ( q,p ) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective unitary representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.
The phase point operator 0(q, p) is a quantum counterpart of classical phase point (q, p) . Its discrete version was formulated for an odd number of lattice points by Cohendet et al. and even case by Leonhardt. They both have symplectic covariance which is of fundamental importance in quantum mechanics. But its explicit form of the projective representation of the symplectic group which appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation and a method to construct it using the Euclidean algorithm.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
An isomorphism is shown to exist between the group of point transformations in classical mechanics and a certain subgroup of the group of all unitary transformations in quantum mechanics. This isomorphism is used to indicate that the quantum analogs of physically significant classical expressions can be constructed uniquely in any coordinate system. There is no ambiguity in the ordering of noncommuting quantum operators, and the method of constructing the quantum analogs is covariant under general coordinate transformations. The method is actually only applicable to systems having Lagrangians which are at most quadratic in the velocities, but this includes all systems which are presently of interest in physics. The method is applied to two intrinsically nonlinear examples, one of which is the gravitational field. The correct Hamiltonian operator for a quantized version of Einstein's gravitational theory is constructed.
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space E3\mathbb{E}_{3}E3​ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group IO(3)\mathcal{I}O(3)IO(3). The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under IO(3)\mathcal{I}O(3)IO(3) is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under IO(3)\mathcal{I}O(3)IO(3). (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under IO(3)\mathcal{I}O(3)IO(3), and this is proved via the unitary transformations of quantum mechanics in this representation.
cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper­ ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no­ tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de­ quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param­ eter subgroups of the symplectic group.8.9 One then only