Complex phase space and Weyl’s commutation relations

Journal of Physics A: Mathematical and Theoretical, 2010

We start from Wootter's construction of discrete phase spaces and Wigner functions for qubits and more generally for finite dimensional Hilbert spaces. We look at this framework from a noncommutative space perspective and we focus on the Moyal product and the differential calculus on these discrete phase spaces. In particular, the qubit phase space provides the simplest example of a four-point non-commutative phase space. We give an explicit expression of the Moyal bracket as a differential operator. We then compare the quantum dynamics encoded by the Moyal bracket to the classical dynamics : we show that the classical Poisson bracket does not satisfy the Jacobi identity thus leaving the Moyal bracket as the only consistent symplectic structure. We finally generalizes our analysis to Hilbert spaces of prime dimensions d and their associated d × d phase spaces.

2015

The date of receipt and acceptance will be inserted by the editor Abstract We de ne and study a metaplectically covariant class of pseudo-di¤erential operators acting on functions on symplectic space and general-izing a modi ed form of the usual Weyl calculus. This construction requires a precise calculation of the twisted Weyl symbol of a class of generators of the metaplectic group and the use of a ConleyZehnder type index for symplectic paths, de ned without restrictions on the endpoint. Our calcu-lus is related the usual Weyl calculus using a family of isometries of L2(Rn) on closed subspaces of L2(R2n) and to an irreducible representation of the Heisenberg algebra distinct from the usual Schrödinger representation.

Journal of Physics A: Mathematical and General, 2000

We develop a general 'classicalization' procedure that links Hilbert-space and phasespace operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in anh-Taylor series and fix parameters with the condition that coefficients ofh 0 , −ih 1 ∂/∂Q and ih 1 ∂/∂P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact. In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S(Q, P , t) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.

We develop a general 'classicalization' procedure that links Hilbert-space and phasespace operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in anh-Taylor series and fix parameters with the condition that coefficients ofh 0 , −ih 1 ∂/∂Q and ih 1 ∂/∂P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact. In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S(Q, P , t) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.

Journal of Physics A: Mathematical and General, 2001

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the-quantization, is an extension of the classical Poisson-Lie theory and can be used as an efficient tool in the quantum phase space transformation theory.

Journal of Pseudo-Differential Operators and Applications, 2012

We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.

Journal of Physics A-mathematical and General, 2005

A classical theorem of Stone and von Neumann states 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.

Journal of Physics A: Mathematical and General, 1999

The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.

Modern Physics Letters B, 1996

Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket.

Quantum Studies: Mathematics and Foundations, 2015

In this paper, we introduce the generalized Weyl operators canonically associated with the one-mode oscillator Lie algebra as unitary operators acting on the bosonic Fock space (C). Next, we establish the generalized Weyl relations and deduce a group structure on the manifold R 2 × [−π, π[×R generalizing the well-known Heisenberg one in a natural way.