Représentation d''une particule quantique de spin 1/2 par le produit de Moyal et une super-variété symplectique plate

The aim of this paper is to show that it is possible to represent a quantum particle with a spin with the help of a star product and a flat supermanifold. We extend to the supermanifolds the Schouten bracket, the exterior differentiation, the Lie derivative. These notions allow us to define the Poisson bracket and the symplectic supermanifolds. It is easy to define the Moyal product: the Grassman algebra of the superfunctions becomes a Clifford algebra. We suppose that the observables of a quantum particle with a spin are superfunctions defined on a flat supermanifold. The study of these star algebras proves the existence of a spectral resolution for some particular elements. Consequently the equation of motion for an observable admits stationary solutions. The examples considered show that the spin splits a given state into two other ones.     

Quantum geometry from 2 + 1 AdS quantum gravity on the torus

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick’s formula for the area of a polygon with integer vertices.     

Covariant description of Hamiltonian form for field dynamics

Hamiltonian form of field dynamics is developed on a space-like hypersurface in space-time. A covariant Poisson bracket on the space-like hypersurface is defined and it plays a key role to describe every algebraic relation into a covariant form. It is shown that the Poisson bracket has the same symplectic structure that was brought in the covariant symplectic approach. An identity invariant under the canonical transformations is obtained. The identity follows a canonical equation in which the interaction Hamiltonian density generates a deformation of the space-like hypersurface. The equation just corresponds to the Yang-Feldman equation in the Heisenberg pictures in quantum field theory. By converting the covariant Poisson bracket on the space-like hypersurface to four-dimensional commutator, we can pass over to quantum field theory in the Heisenberg picture without spoiling the explicit relativistic covariance. As an example the canonical QCD is displayed in a covariant way on a space-like hypersurface.     

Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions

Nonequilibrium Green’s functions provide a powerful tool for computing the dynamical response and particle exchange statistics of coupled quantum systems. We formulate the theory in terms of the density matrix in Liouville space and introduce superoperator algebra that greatly simplifies the derivation and the physical interpretation of all quantities. Expressions for various observables are derived directly in real time in terms of superoperator nonequilibrium Green’s functions (SNGF), rather than the artificial time-loop required in Schwinger’s Hilbert-space formulation. Applications for computing interaction energies, charge densities, average currents, current induced fluorescence, electroluminescence and current fluctuation (electron counting) statistics are discussed.     

Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/?)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/?)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.     

Symplectic Geometry of Quantum Noise

We discuss a quantum counterpart, in the sense of the Berezin–Toeplitz quantization, of certain constraints on Poisson brackets coming from “hard” symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.     

Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

Quantum theory of irreversibility

A generalization of the Gibbs–von Neumann entropy is proposed based on the quantum BBGKY (Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy as the non-equilibrium entropy for an N$N$-body system. By using a generalization of the Liouville–von Neumann equation describing the evolution of a density superoperator, the entropy production for an isolated system is calculated, being non-zero in general. The existence of a non-zero entropy production allows us, following the procedure of non-equilibrium thermodynamics to introduce a master matrix for which a microscopic expression is obtained. After this, as a test of our theory the quantum Boltzmann equation is derived in terms of a transition superoperator related to this master matrix.     

A note on the propensity concept in quantum mechanics

It is shown that the propensity concept of Wódkiewicz is identical to the notion of the probability distribution of results of a measurement of quantum observables covariant with respect to the galilean group as is defined in the statistical model of quantum measurements.     

Nonequilibrium perturbation theory in Liouville-Fock space for inelastic electron transport

We use a superoperator representation of the quantum kinetic equation to develop nonequilibrium perturbation theory for an inelastic electron current through a quantum dot. We derive a Lindblad-type kinetic equation for an embedded quantum dot (i.e. a quantum dot connected to Lindblad dissipators through a buffer zone). The kinetic equation is converted to non-Hermitian field theory in Liouville-Fock space. The general nonequilibrium many-body perturbation theory is developed and applied to the quantum dot with electron-vibronic and electron-electron interactions. Our perturbation theory becomes equivalent to a Keldysh nonequilibrium Green's function perturbative treatment provided that the buffer zone is large enough to alleviate the problems associated with approximations of the Lindblad kinetic equation.     

Commutation superoperator of a state and its applications to the noncommutative statistics

For a normal state on a von Neumann algebra the space of square-integrable operators is introduced. As distinct from the L² space in the classical probability theory, it possesses an additional skew-symmetric form and the associated superoperator, which is a convenient tool to describe commutation properties in L². It is shown that a state on the algebra of canonical commutation relations is Gaussian (quasi-free) iff the space of canonical observables is an invariant subspace of the corresponding commutation superoperator. Basing on these ideas a new approach to some problems in the noncommutative statistic is developed.     

Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

Theory of irreversible processes in nonequilibrium quantum systems. II. Cluster expansions with contraction for homogenous systems

On the basis of a method developed in a previous paper, a systematic rule for obtaining a symmetrized collision superoperator of the Van Hove generalized master equation including an arbitrary number of particles is given. In the formalism, the quantum statistical effect is taken into account through the use of contractions (internal and external contractions) on the basis of the cluster expansion. As an application of this general rule, a symmetrized collision superoperator including the effect of three-particle collisions is obtained.     

State transfer in dissipative and dephasing environments

The European Physical Journal D - By diagonalization of a generalized superoperator for solving the master equation, we investigated effects of dissipative and dephasing environments on quantum...     

Conformal Symmetry and Quantum Relativity

The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincare symmetry of field theory can be extended to the larger conformal symmetry. We use these symmetries to define quantum observables associated with positions in space-time, in the spirit of Einstein theory of relativity. This conception of localization may be applied to massive as well as massless fields. Localization observables are defined as to obey Lorentz covariant commutation relations and in particular include a time observable conjugated to energy. While position components do not commute in the presence of a nonvanishing spin, they still satisfy quantum relations which generalize the differential laws of classical relativity. We also give of these observables a representation in terms of canonical spatial positions, canonical spin components, and a proper time operator conjugated to mass. These results plead for a new representation not only of space-time localization but also of motion.     

Deformation theory applied to quantization and statistical mechanics

After a review of the deformation (star product) approach to quantization, treated in an autonomous manner as a deformation (with parameter ) of the algebraic composition law of classical observables on phase-space, we show how a further deformation (with parameter ) of that algebra is suitable for statistical mechanics. In this case, the phase-space is endowed with what we call a conformal symplectic (or conformal Poisson) structure, for which the bracket is the Poisson bracket modified by terms of order (1, 0) and (0, 1). As an application, one sees that the KMS states (classical or quantum) are those that vanish on the modified (Poisson or Moyal-Vey) bracket of any two observables, multiplied by a conformal factor.     

Quantum conditional probability in one-dimensional scattering

A result of Wan and McLean on the asymptotic separation of states in quantum mechanics is used to analyze the motion of a particle across a one-dimensional potential of finite range in terms of quantum conditional probabilities. It is shown that probabilities conditional on transmission or reflection, defined according to the Lüders rule, yield the results to be expected by intuitive argument. The theorem of total probabilities, based on the events of transmission and reflection, is shown to hold for a class of observables, and examples are given both of observables which belong to this class and of observables which do not.     

Joint distributions of observables on quantum logics

A joint distribution of a set of observables on a quantum logic in a statem is defined and its properties are derived. It is shown that if the joint distribution exists, then the observables can be represented in the statem by a set of commuting operators on a Hilbert space.     

On the observables on quantum logics

Two postulates concerning observables on a quantum logic are formulated. By Postulate 1 compatibility of observables is defined by the strong topology on the set of observables. Postulate 2 requires that the range of the sum of observables ought to be contained in the smallestC-closed sublogic generated by their ranges. It is shown that the Hilbert space logicL(H) satisfies the two postulates. A theorem on the connection between joint distributions of types 1 and 2 on the logic satisfying Postulate 2 is proved.