Weyl Wigner Moyal Research Articles

The non-disjoint ontic states of the Grassmann ontological model, transformation contextuality, and the single qubit stabilizer subtheory

We show that it is possible to construct a preparation non-contextual ontological model that does not exhibit ‘transformation contextuality’ for single qubits in the stabilizer subtheory. In particular, we consider the ‘blowtorch’ map and show that it does not exhibit transformation contextuality under the Grassmann Wigner–Weyl–Moyal (WWM) qubit formalism. Furthermore, the transformation in this formalism can be fully expressed at order , where it satisfies all of Kolmogorov’s axioms of classical probability theory, and so does not qualify as a candidate quantum phenomenon. In particular, we find that the Grassmann WWM formalism at order corresponds to an ontological model governed by an additional set of constraints arising from the relations defining the Grassmann algebra. Due to this additional set of constraints, the allowed probability distributions in this model do not form a single convex set when expressed in terms of disjoint ontic states and so cannot be mapped to models whose states form a single convex set over disjoint ontic states. However, expressing the Grassmann WWM ontological model in terms of non-disjoint ontic states corresponding to the monomials of the Grassmann algebra results in a single convex set. We further show that a recent result by Lillystone et al that proves a broad class of preparation and measurement non-contextual ontological models must exhibit transformation contextuality lacks the generality to include the ontological model considered here; Lillystone et al’s result is appropriately limited to ontological models whose states produce a single convex set when expressed in terms of disjoint ontic states. Therefore, we prove that for the qubit stabilizer subtheory to be captured by a preparation, transformation and measurement non-contextual ontological theory, it must be expressed in terms of non-disjoint ontic states, unlike the case for the odd-dimensional single-qudit stabilizer subtheory.

Measurement contextuality and Planck’s constant

Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck’s constant, ℏ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ℏ—all orders higher than ℏ0 can be interpreted as quantum corrections to the order ℏ0 term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner–Weyl–Moyal (WWM) formalism that require higher than order ℏ0 terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is closely related to orders of ℏ as a resource within the WWM formalism. This offers an explanation for why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres–Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, qubit states exhibit contextuality when measured by qubit Pauli observables regardless of the state being measured and so the Weyl symbol of these observables lack an order ℏ0 contribution altogether. On the other hand, odd-dimensional qudit states exhibit contextuality when measured by qudit observables depending on the state measured and so odd-dimensional qudit observables generally possess non-zero order ℏ0 terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ℏ1 contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ℏ1 contributions do not exhibit measurement contextuality.

The Phase Space Formulation of Time-Symmetric Quantum Mechanics

Time-symmetric quantum mechanics can be described in the Weyl–Wigner–Moyal phase space formalism by using the properties of the cross-terms appearing in the Wigner distribution of a sum of states. These properties show the appearance of a strongly oscillating interference between the pre-selected and post-selected states. It is interesting to note that the knowledge of this interference term is sufficient to reconstruct both states. Quanta 2015; 4: 27–34.

Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

We propose a self-consistent theoretical framework for a wide class of physical systems characterized by a finite space of states which allows us, within several mathematical virtues, to construct a discrete version of the Weyl–Wigner–Moyal (WWM) formalism for finite-dimensional discrete phase spaces with toroidal topology. As a first and important application from this ab initio approach, we initially investigate the Robertson–Schrödinger (RS) uncertainty principle related to the discrete coordinate and momentum operators, as well as its implications for physical systems with periodic boundary conditions. The second interesting application is associated with a particular uncertainty principle inherent to the unitary operators, which is based on the Wiener–Khinchin theorem for signal processing. Furthermore, we also establish a modified discrete version for the well-known Heisenberg–Kennard–Robertson (HKR) uncertainty principle, which exhibits additional terms (or corrections) that resemble the generalized uncertainty principle (GUP) into the context of quantum gravity. The results obtained from this new algebraic approach touch on some fundamental questions inherent to quantum mechanics and certainly represent an object of future investigations in physics.

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.

Moyal and tomographic probability representations for f-oscillator quantum states

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 the Moyal equation for the quantum f-oscillator, which provide solutions for the partial case of f-nonlinearity existing in Kerr media. Nonlinear coherent states and the thermodynamics of nonlinear oscillators are studied.

On the Moyal deformation of Nahm equations in seven dimensions

We show how the reduced (anti-)self-dual Yang–Mills equations to seven dimensions described by the Nahm equations can be carried over to the Weyl–Wigner–Moyal formalism. In the process some new solutions for the cases of gauge groups SU(2) and SL(2,R) are explicitly obtained.

DEFORMATION QUANTIZATION OF RELATIVISTIC PARTICLES IN ELECTROMAGNETIC FIELDS

The Weyl–Wigner–Moyal formalism for Dirac second-class constrained systems has been proposed recently as the deformation quantization of Dirac bracket. In this paper, after a brief review of this formalism, it is applied to the case of the relativistic free particle. Within this context, the Stratonovich–Weyl quantizer, Weyl correspondence, Moyal ⋆-product and Wigner function in the constrained phase-space are obtained. The recent Hamiltonian treatment for constrained systems, whose constraints depend explicitly on time, are used to perform the deformation quantization of the relativistic free charged particle in an arbitrary electromagnetic background. Finally, the system consisting of a charged particle interacting with a dynamical Maxwell field is quantized in this context.

F-oscillators deformation for Moyal algebras

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 deformation of Groenewold kernel related to general kinds of nonlinear vibrations described by f-oscillators are considered.

Deformation quantization of fermi fields

Deformation quantization for any Grassmann scalar free field is described via the Weyl–Wigner–Moyal formalism. The Stratonovich–Weyl quantizer, the Moyal ★-product and the Wigner functional are obtained by extending the formalism proposed recently in [I. Galaviz, H. García-Compeán, M. Przanowski, F.J. Turrubiates, Weyl–Wigner–Moyal Formalism for Fermi Classical Systems, arXiv:hep-th/0612245] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich–Weyl quantizer, the Moyal ★-product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.

Weyl–Wigner–Moyal formalism for Fermi classical systems

The Weyl–Wigner–Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. The Weyl correspondence is studied by computing the relevant Stratonovich–Weyl quantizer. The Moyal ★-product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.

Star-quantization of an infinite wall

In deformation quantization (a.k.a. the Wigner–Weyl–Moyal formulation of quantum mechanics), we consider a single quantum particle moving freely in one dimension, except for the presence of one infinite potential wall. Dias and Prata pointed out that, surprisingly, its stationary-state Wigner function does not obey the naive equation of motion, i.e., the naive stargenvalue (*-genvalue) equation. We review our recent work on this problem that treats the infinite wall as the limit of a Liouville potential. Also included are some new results: (i) we show explicitly that the Wigner-Weyl transform of the usual density matrix is the physical solution, (ii) we prove that an effective-mass treatment of the problem is equivalent to the Liouville one, and (iii) we point out that self-adjointness of the operator Hamiltonian requires a boundary potential, but one apparently different from that proposed by Dias and Prata. PACS Nos.: 03.65.–w, 03.65.Ca, 03.65.Ge

Phase space distributions and a duality symmetry for star products

A duality property for star products is exhibited. In view of it, known star-product schemes, like the Weyl–Wigner–Moyal formalism, the Husimi and the Glauber–Sudarshan maps are revisited. The tomographic map, which has been recently described as yet another star product scheme, is considered. It yields a noncommutative algebra of operator symbols which are positive definite probability distributions. Through the duality symmetry a new noncommutative algebra of operator symbols is found, equipped with a new star product. The kernel of the star product is established in explicit form and examples are considered.

Entropy and Wigner Distribution Functions Revisited

Several expressions for quantum entropy proposed in the literature are evaluated within the Weyl—Wigner—Moyal phase-space representation of quantum mechanics, with emphasis on some important subtle points in this approach. It has been found that the Renyi—Susmann entropy and its linearization are distinguished because of their properties.

Scalar Charged Particle in Weyl–Wigner–Moyal Phase Space. Constant Magnetic Field

A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables) is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with the peculiarities of the constraints on the initial conditions. An effective increase in coherence between eigenstates of the Hamiltonian is found and possibilities of its experimental observation are discussed.

On phase-space representations of quantum mechanics

We discuss a class of representations of quantum mechanics which uses functions defined on a parameter space to represent observable quantities. We show that infinitesimal canonical transformations could be used to introduce a phase-space-like structure consistent with the requirements of quantum mechanics. The resulting family of phase-space representations of quantum mechanics contains many well-known representations as special cases, e.g., the Weyl–Wigner–Moyal, normal and antinormal one. It is also flexible enough to represent, e.g., PT -symmetric theories, introduced recently within the context of non-Hermitian quantum mechanics.

Chern–Simons hadronic bag from quenched large- N QCD

SU( N) reduced, quenched, gauge theories have been shown to be related to string theories. We extend this result and show how a 4-dimensional, reduced, quenched, Yang–Mills theory, supplemented by the topological term, can be related through the Wigner–Weyl–Moyal correspondence to an open 3- brane model. The boundary of the 3-brane is described by a Chern–Simons 2-brane. We identify the bulk of the 3-brane with the interior of a hadronic bag and the world-volume of the Chern–Simons 2-brane with the dynamical boundary of the bag. We estimate the value of the induced bag constant to be a little less than 200 MeV.

Weyl–Wigner–Moyal formulation of a Dirac quantized constrained system

An extension of the Weyl–Wigner–Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl product of these equivalent classes is defined. The new Moyal bracket is shown to be compatible with the Dirac bracket for constrained systems.

Moyal brackets, star products and the generalised Wigner function

The Wigner—Weyl—Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal—Nahm equation.

ON THE REDUCED SU(N) GAUGE THEORY IN THE WEYL–WIGNER–MOYAL FORMALISM

Weyl–Wigner–Moyal formalism is used to describe the large N limit of reduced SU(N) quenching gauge theory. Moyal deformation of Schild–Eguchi action is obtained.