We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.

Physical Review AAug 19, 200915 citations

Representations of a Quantum Phase Space with General Degrees of Freedom

Asao Arai

For each integer n ≥ 2 and a parameter Λ = (θ, η) with θ and η being n × n real anti-symmetric matrices, a quantum phase space (QPS) (or a non-commutative phase space) with n degrees of freedom, denoted QPS n(Λ), is defined, where θ and η are parameters measuring non-commutativity of the QPS. Hilbert space representations of QPS n(Λ) are considered. A concept of quasi-Schrödinger representation of QPS n(Λ) is introduced. It is shown that there exists a general correspondence between representations of QPS n(Λ) and those of the canonical commutation relations with n degrees of freedom. Irreducibility of representations of QPS n(Λ) are investigated. A concept of Weyl representation of QPS n(Λ) is defined. It is proved that every Weyl representation of QPS n(Λ) on a separable Hilbert space is unitarily equivalent to a direct sum of a quasi-Schrödinger representation of the QPS n(Λ) (a uniqueness theorem). Finally representations of QPS n(Λ) which are not unitarily equivalent to any direct sum of a quasi-Schrödinger representation are described.For each integer n ≥ 2 and a parameter Λ = (θ, η) with θ and η being n × n real anti-symmetric matrices, a quantum phase space (QPS) (or a non-commutative phase space) with n degrees of freedom, denoted QPS n(Λ), is defined, where θ and η are parameters measuring non-commutativity of the QPS. Hilbert space representations of QPS n(Λ) are considered. A concept of quasi-Schrödinger representation of QPS n(Λ) is introduced. It is shown that there exists a general correspondence between representations of QPS n(Λ) and those of the canonical commutation relations with n degrees of freedom. Irreducibility of representations of QPS n(Λ) are investigated. A concept of Weyl representation of QPS n(Λ) is defined. It is proved that every Weyl representation of QPS n(Λ) on a separable Hilbert space is unitarily equivalent to a direct sum of a quasi-Schrödinger representation of the QPS n(Λ) (a uniqueness theorem). Finally representations of QPS n(Λ) which are not unitarily equivalent to any direct sum of a quasi-Schrödinger representation are described.

Jan 1, 20091 citation

Quantum/Classical Phase Space Analogies

Daniela Dragoman, Mircea Dragoman

The frontiers collectionJan 1, 20040 citations

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space

G. S. Agarwal, E. Wolf

In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$ -number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsNov 15, 1970285 citations

Quantum-mechanical operators and phase-space operators

Nandor L. Balazs, Hans–Christian Pauli

The European Physical Journal AJun 1, 19776 citations

In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto

c

-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.