Using general construction of star-product the q-deformed Wigner-Weyl-Moyal quantization procedure is elaborated. The q-deformed Groenewold kernel determining the product of quantum observables is given in explicit form for small nonlinearities corresponding to nonlinear vibrations of classical and quantum q-oscillators. The deformations of Groenewold kernel related to general kinds of nonlinear vibrations described by f-oscillators are considered.
States of nonlinear quantum oscillators (f-oscillators) are considered in the Weyl-Wigner-Moyal representation and the tomographic probability representation, where the states are described by standard probability distributions instead of wave functions or density matrices. The evolving integrals of motion for classical and quantum f-oscillators are found and the solution for the Liouville equation associated with the probability distribution on the phase space for this oscillator is obtained along with the solution of Moyal equation for quantum f-oscillator, which provide the solutions for pa…
The formalism of generalized Wigner transformations developped in a previous paper, is applied to kinetic equations of the Lindblad type for quantum harmonic oscillator models. It is first applied to an oscillator coupled to an equilibrium chain of other oscillators having nearest-neighbour interactions. The kinetic equation is derived without using the so called rotating-wave approximation. Then it is shown that the classical limit of the corresponding phase-space equation is independent of the ordering of operators corresponding to the inverse of the generalized Wigner transformation, provid…
Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more symmetrical definition for the Moyal products, because they lead to simpler differential equations. In addition, we demonstrate how to use this approach to determine the Hermitian counterpart for a Pseudo-Hermitian Hamiltonian. We illustrate our suggestions with the explicitly solvable example of the -x^4-potential and the ubiquitous harmonic oscillator in a complex cubic potential.
The Hilbert-Schmidt operator formulation of non-commutative quantum mechanics in 2D Moyal plane is shown to allow one to construct Schwinger's SU(2) generators. Using this the SU(2) symmetry aspect of both commutative and non-commutative harmonic oscillator are studied and compared. Particularly, in the non-commutative case we demonstrate the existence of a critical point in the parameter space of mass and angular frequency where there is a manifest SU(2) symmetry for a unphysical harmonic oscillator Hamiltonian built out of commuting (unphysical yet covariantly transforming under SU(2)) posit…
Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for a complete set of observables formed from the photon creation and annihilation operators. These Moyal solutions incorporate the asymptotics of the classical limit in a simple explicit form. An unusual feature of these solutions …
We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase-space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the non-commutativity leads to a deformation of the classical phase-space: instead of being a vector space it becomes a manifold, the topology of which is given by the commutator relations. It is shown in fact that the classical phase-space, for a semi-simple Lie algebra, becomes a homogenous symplectic manifold. The symplectic product is also deformed. We finally make some comments on how to …
We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic alge…
The quantum deformation of the Poisson bracket is the Moyal bracket. We construct quantum deformation of the Dirac bracket for systems which admit global symplectic basis for constraint functions. Equivalently, it can be considered as an extension of the Moyal bracket to second-class constraints systems and to gauge-invariant systems which become second class when gauge-fixing conditions are imposed.
The Moyal formalism for a particle can be derived from the Moyal formalism for a spin. This is done by contracting the group of rotations to the oscillator group. A new derivation is given for the contraction of the spin Wigner-kernel to the Wigner kernel of a particle.
Groenewold-Moyal twists, integrable spin-chains and AdS/CFT
Instituto Galego de Física de Altas Enerxías (IGFAE), and Departamento de Física de Partículas, Universidade de Santiago de Compostela（加利西亚高等物理研究所（IGFAE），圣地亚哥-德孔波斯特拉大学粒子物理系）
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 func…
The von Neumann evolution equation for density matrix and the Moyal equation for the Wigner function are mapped onto evolution equation for optical tomogram of quantum state. The connection with known evolution equation for symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for the optical tomograms. Classical Liouville equation for optical tomogram is obtained. Example of parametric oscillator is considered in detail.
Quantization of constraint systems within the Weyl-Wigner-Groenewold-Moyal framework is discussed. Constraint dynamics of classical and quantum systems is reformulated using the skew-gradient projection formalism. The quantum deformation of the Dirac bracket is generalized to match smoothly the classical Dirac bracket in and outside of the constraint submanifold in the limit $\hbar \to 0$.
The orbit method of Kirillov is used to derive the p-mechanical brackets [quant-ph/0212101]. They generate the quantum (Moyal) and classic (Poisson) brackets on respective orbits corresponding to representations of the Heisenberg group. The extension of p-mechanics to field theory is made through the De Donder-Weyl Hamiltonian formulation. The principal step is the substitution of the Heisenberg group with Galilean. Keywords: Classic and quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, representation theory, De Donder-We…
Field theories on the Groenewold-Moyal(GM) plane are studied using the Lehmann-Symanzik-Zimmermann(LSZ) formalism. The example of real scalar fields is treated in detail. The S-matrix elements in this non-perturbative approach are shown to be equal to the interaction representation S-matrix elements. This is a new non-trivial result: in both cases, the S-operator is independent of the noncommutative deformation parameter $θ_{μν}$ and the change in scattering amplitudes due to noncommutativity is just a time delay. This result is verified in two different ways. But the off-shell Green's functio…
A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory, for the case when the interaction is deformed. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady state solution of the equation for the density matrix in the number representation is obtained and shown that it satisfies the detailed balance condition. The equilibrium energy of the deformed harmonic oscillator, …
We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploit the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the -genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner …
Motivated by the recent developments of gauge-covariant methods in the phase-space, a systematic method is presented aiming at the generalisation of the Moyal star-product to a non-Abelian gauge covariant one at any order. Such an expansion contains some dressing of the bare particle model by the gauge-fields explicitly, and might serve as a drastically simplifying tool for the elaborations of gauge-covariant quantum transport models. In addition, it might be of fundamental importance for the mathematical elaborations of gauge theory using the strict or deformation quantisation principles. A f…
Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex c…
The Groenewold-Moyal plane is the algebra A_θ(R^(d+1)) of functions on R^(d+1) with the star-product as the multiplication law, and the commutator [x_μ,x_ν] =i θ_{μν} between the coordinate functions. Chaichian et al. and Aschieri et al. have proved that the Poincare group acts as automorphisms on A_θ(R^(d+1))$ if the coproduct is deformed. (See also the prior work of Majid, Oeckl and Grosse et al). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Aschieri et al. In this paper we show that for this new action, the Bose and Fermi commutation r…
We use a deformed differential structure to obtain a curved metric by a deformation quantization of the flat space-time. In particular, by setting the deformation parameters to be equal to physical constants we obtain the Friedmann-Robertson-Walker (FRW) model for inflation and a deformed version of the FRW space-time. By calculating classical Einstein-equations for the extended space-time we obtain non-trivial solutions. Moreover, in this framework we obtain the Moyal-Weyl, i.e. a constant non-commutative space-time, by a consistency condition.
We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time 't' is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period $\mathcal{T}$ is studied in Heisenberg picture to compute the expression of geometric phase-shift.
We propose a new approach to the quantization of the damped harmonic oscillator in the framework of deformation quantization. The quantization is performed in the Schrödinger picture by a star-product induced by a modified "Poisson bracket". We determine the eigenstates in the damped regime and compute the transition probability between states of the undamped harmonic oscillator after the system was submitted to dissipation.
Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or P…
The orbit method of Kirillov is used to derive the p-mechanical brackets [math-ph/0007030, quant-ph/0212101]. They generate the quantum (Moyal) and classic (Poisson) brackets on respective orbits corresponding to representations of the Heisenberg group. The extension of p-mechanics to field theory is made through the De Donder--Weyl Hamiltonian formulation. The principal step is the substitution of the Heisenberg group with Galilean. Keywords: Classic and quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, representation th…
Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows establishing a $c$-equivalence between the Moyal and the n…
In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jay…
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equati…
The unified $ (p,q; α,γ, l)$-deformation of a number of well-known deformed oscillator algebras is introduced.The deformation is constructed by imputing new free parameters into the structure functions and by generalizing the defining relations of these algebras. The generalized Jordan-Schwinger and Holstein-Primakoff realizations of the $U_{pq}^{αγl}(su(2))$ algebra by the generalized $ (p,q; α,γ, l)$-deformed operators are found. The generalized $ (p,q; α,γ, l)$-deformation of the two-dimensional conformal field theory is established. By introducing the $ (p,q; α,γ, l)$-operator product expa…
$C_λ$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_λ$ is the cyclic group of order $λ$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed algebras admitting Casimir operators analogous to those of their undeformed counterparts are looked for, yielding three new algebraic structures. One of them includes the Brzeziński {\em et al.} …
For the two-parameter $p,q$-deformed Heisenberg algebra introduced recently and in which, instead of usual commutator of $X$ and $P$ in the l.h.s. of basic relation $[X,P] = {\rm i}\hbar$, one uses the $p,q$-commutator, we established interesting properties. Most important is the realizability of the $p,q$-deformed Heisenberg algebra by means of the appropriate deformed oscillator algebra. Another uncovered property is special extension of the usual mutual Hermitian conjugation of the creation and annihilation operators, namely the so-called $η(N)$-pseudo-Hermitian conjugation rule, along with…
