Symplectic Covariance of Phase Point Operator on Discrete Space

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.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.

Mar 1, 20170 citations

Symplectic Covariance Properties

Maurice A. de Gosson

Fundamental theories of physicsJan 1, 20160 citations

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

Daisuke Watanabe, T. Hashimoto, M. Horibe +1 more

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.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.

arXiv:1802.09891Feb 27, 20180 citations

Symplectic Invariance of Weyl Transforms

Springer eBooksMar 31, 20060 citations

Covariant projective representation of symplectic group on discrete phase space

Daisuke Watanabe, T. Hashimoto, M. Horibe +1 more

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 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.

Journal of Physics Conference SeriesApr 1, 20192 citations

The symplectic group and invariants of quantum systems

Vladimir I. Man’ko

Journal of Mathematical SciencesMar 1, 19872 citations

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.