Wigner Transform Research Articles

Development in Phase Space Quantum Mechancis for Constructing Different Masses’ Entangled State Representation

For developing the phase space quantum mechanics we explore how to construct different masses’ entangled state representation with the use of tthe Weyl-Wigner correspondence. We shall adopt a new approach, which uses the Radon transform of two-mode Wigner operator in normally ordering form to realize our goal. As an application of the entangled state representation the new normally ordered generalized squeezing operator is derived. The technique of integration within ordered product of operators is fully used.

On the quantization of a neutron star

This paper deals with quantum gravitation applied to a simple neutron star model. For the quantization process, we use the Weyl prescription that can be used for functions that are not defined in the phase space. The Weyl quantization is applied to the expression of gravitational energy defined in the context of Teleparallelism Equivalent to General Relativity (TEGR). From this, a quantum equation is obtained whose observable is the classical energy. As a consequence, we obtained discretizations for the mass of the star and its angular momentum.

Quantum corrections to the entropy and its application in the study of quantum Carnot engines.

Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ℏ by using the phase-space formulation, and the zeroth-order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ℏ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights into the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.

Brane dynamics from the first law of entanglement

In this note, we study the first law of entanglement in a boundary conformal field theory (BCFT) dual to warped AdS cut off by a brane. Exploiting the symmetry of boundary-centered half-balls in the BCFT, and using Wald’s covariant phase space formalism in the presence of boundaries, we derive constraints from the first law for a broad range of covariant bulk Lagrangians. We explicitly evaluate these constraints for Einstein gravity, and find a local equation on the brane which is precisely the Neumann condition of Takayanagi [6] at linear order in metric perturbations. This is analogous to the derivation of Einstein’s equations from the first law of entanglement entropy. This machinery should generalize to give local linearized equations of motion for higher-derivative bulk gravity with additional fields.

Pseudodifferential Weyl Calculus on (Pseudo-)Riemannian Manifolds

One can argue that on flat space {mathbb {R}}^d, the Weyl quantization is the most natural choice and that it has the best properties (e.g., symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization—we call it the balanced geodesic Weyl quantization. Among other things, we prove that it maps square-integrable symbols to Hilbert–Schmidt operators, and that even (resp. odd) polynomials are mapped to even (resp. odd) differential operators. We also present a formula for the corresponding star product and give its asymptotic expansion up to the fourth order in Planck’s constant.

Quantum effects in two-qubit systems interacting with two-mode fields: Dissipation and dipole-dipole interplay effects

We explore two coupled dipole qubits interacting with two-mode cavity fields through non-degenerate two-photon processes and under the influence of intrinsic noise. Our investigation is based on the influence of the coupling constants between different constituents of the system on the quantum coherence and entropy squeezing. We investigate the phase space information, entanglement, and purity via the Husimi function. It is found that the robustness of the positivity and the negativity of the superposition of generalized coherent state is highly sensitive to the coupling constants. The phase space quantum information, the matter-light entanglement, and the stationary mixture state can be controlled by the two-qubit coupling as well as the coupling to the environment. The effect of the intrinsic dissipation and the two-qubit coupling on the generated squeezing via the qubit-system variable squeezing or the entropy squeezing is investigated. They lead to revoke the squeezing phenomenon.

Infrared behavior of Weyl gravity: Functional renormalization group approach

Starting from an ultraviolet fixed point, we study the infrared behavior of quantum Weyl gravity in terms of a functional renormalization group (RG) flow equation. To do so, we employ two classes of Bach-flat backgrounds, namely maximally symmetric spacetimes and Ricci-flat backgrounds in the improved one-loop scheme. We show that in the absence of matter fields and with a topological term included, the effective action exhibits dynamical breaking of scale symmetry. In particular, it is shown that apart from a genuine IR fixed point that is reached at a zero-value of the running scale, the RG flow also exhibits bouncing behavior in the IR regime. We demonstrate that both $\beta_C$ and $\beta_E$ reach the RG turning point (almost) simultaneously at the same finite energy scale, irrespectively of the chosen background. The IR fixed point itself is found to be IR-stable in the space of the considered couplings. Ensuing scaling dimensions of both operators are also computed. Salient issues, including the connection of the observed bouncing RG flow behavior with holography and prospective implications in early Universe cosmology, are also briefly discussed.

XAct Implementation of the Theory of Cosmological Perturbation in Bianchi I Spacetimes

This paper presents a computational algorithm to derive the theory of linear gauge invariant perturbations on anisotropic cosmological spacetimes of the Bianchi I type. Our code is based on the tensor algebra packages xTensor and xPert, within the computational infrastructure of xAct written in Mathematica. The algorithm is based on a Hamiltonian, or phase space formulation, and it provides an efficient and transparent way of isolating the gauge invariant degrees of freedom in the perturbation fields and to obtain the Hamiltonian generating their dynamics. The restriction to Friedmann–Lemaître–Robertson–Walker spacetimes is straightforward.

Effective dynamics from coherent state path integral of full loop quantum gravity

A new routine is proposed to relate Loop Quant Cosmology (LQC) to Loop Quantum Gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path integral formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path integral formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in $\mu_0$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\bar{\mu}$-scheme of effective dynamics and the continuum limit of the path integral formula.

Time-frequency analysis of the Dirac equation

The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in [40]. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis [56]; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.

An algebraic approach to Koopman classical mechanics

Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving a noncommutative operator algebraic variant of the Koopman–von Neumann approach. In this form, the measurement theory for unary classical mechanics can be the same and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical signal analysis approach that can also be successfully applied to the measurement problem of quantum mechanics. The development offers elementary mathematics that allows a formal reconciliation of “collapse” and “no-collapse” interpretations of quantum mechanics.

Wigner distribution and Boltzmann-Shannon entropy for a diatomic molecule under polarized electric field interaction

We study the classical chaos appearing in a diatomic molecules BeO, CO and CN due to the interaction with a circularly polarized electric field, and its signature in Quantum Mechanics through the Wigner distribution function and the Boltzmann-Shannon entropy. We found a motion out of the center of the quantum phase space defined by Wigner function when the classical system becomes chaotic, and we found a jumping behavior of the average Boltzmann-Shannon entropy with respect the electric field strength when the classical system becomes chaotic, indicating a sudden increasing in the disorder (or sudden lost of information) in the quantum system.

Localization Operators Associated with the Hypergeometric Wigner Transform Related to the Cherednik Operators in the Case of the Root System $$BC_{d}$$

Localization operators have a relatively recent development in pure and applied mathematics. Motivated by Wong’s approach, we will study in this paper the time–frequency analysis associated with the hypergeometric Wigner transform related to the Cherednik operators in the case of the root system $$BC_{d}$$.

On Integrals Over a Convex Set of the Wigner Distribution

We provide an example of a normalized $$L^{2}({\mathbb {R}})$$ function u such that its Wigner distribution $${\mathcal {W}}(u,u)$$ has an integral $$>1$$ on the square $$[0,a]\times [0,a]$$ for a suitable choice of a. This provides a negative answer to a question raised by Flandrin (Proc IEEE Int Conf Acoustics 4(1):2176–2179, 1988). Our arguments are based upon the study of the Weyl quantization of the characteristic function of $${{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}}$$ along with a precise numerical analysis of its discretization.

Application of Grassmann phase space theory to Cooper pair model

This paper concerns the application of Grassmann phase space theory (GSPT) to treat the dynamical evolution of systems of identical fermions, such as ultracold gases of fermionic atoms. Phase space theory (which originated from quantum optics) is increasing in importance since it overcomes certain issues associated with other theoretical methods, such as Greens functions, variational methods, quantum-Monte-Carlo equations, etc. In phase-space theory quantum states are represented by quasi-probability distribution functions of phase space variables associated with canonical system operators—such as annihilation, creation operators. Evolution is described via a Fokker-Planck equation for the distribution function, which is equivalent to Ito stochastic equations for (time dependent) stochastic phase space variables. Quantum correlation functions given as averages of products of phase space variables over the quasi-probability distributions then become stochastic averages of products of stochastic phase space variables. In GSPT, the phase space variables are Grassmann numbers, but as computer representation of g-numbers is difficult, Grassmann phase space methods were regarded as being computationally inaccessible. However, previous work using the un-normalised B distribution shows that computer representation of Grassmann variables is unnecessary. Stochastic averages of products for quantum correlation functions at later times are related linearly to stochastic averages at earlier times via stochastic matrices only involving c-numbers. Thus, GSPT calculations of quantum correlation functions now only involve c-number computations. This paper presents the first correct numerical calculation of a quantum correlation function for a fermionic system using stochastic methods based on Grassmann phase space theory, namely the time dependence of the coherence between two Cooper pair states in a four-mode fermion system, where the short and finite time solutions can be compared to known exact results. Good agreement between the stochastic and exact results is found, showing that GPST is a valid approach for treating fermionic systems. The treatment of time evolution involves a novel use of the eigenvalues and biorthogonal column eigenvectors of a stochastically determined c-number matrix M and its transpose. Other topics of interest in ultra-cold fermi gases for which the GSPT could be applied are highlighted, such as the strong interaction regime for the BEC/BCS crossover achieved using magnetically tuned Feshbach resonance techniques.

On the Stratonovich approach for a satellite dynamics

In contrast to a vector non-linear stochastic differential equation (SDE) describing the satellite dynamics under the ‘fluctuating aerodynamic torque’, this paper analyses a second-order fluctuation equation for the radial perturbation about the given orbit. The second-order fluctuation equation for the radial perturbation has found its application for the satellite orbital stability. After accomplishing a phase space formulation, we arrive at the two-dimensional SDE. Most notably, the inaccurate choice of stochastic integral describing the satellite stochastic dynamics will have influence on their estimation, stability and control. For this reason, we develop a noise equation of the satellite dynamics in the Stratonovich setting. The satellite dynamics in the Stratonovich sense can be expressed equivalently in the Itô setting by accounting additional correction terms in the system non-linearity term of the SDE. This paper develops the estimation theory of satellite dynamics via the Stratonovich calculus. The analytic findings are useful to the trajectory estimation of the orbiting satellite under the influence of atmospheric dust perturbations, where the observations are not available.

Time evolution law of Wigner operator in diffusion channel

As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the <i>P</i> representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.

Dissipation and Semigroup on $H^k_n$: Non-cutoff Linearized Boltzmann Operator with Soft Potential

In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudo-differential calculus and in general, for a specific class of Weyl quantization, the $L^2$ dissipation implies $H^k_n$ dissipation. This kind of estimate is also known as the G{\aa}rding's inequality.

On stringy de Sitter spacetimes

We reexamine a family of models with a 3+1-dimensional de Sitter spacetime obtained in the standard tree-level low-energy limit of string theory with a non-trivial anisotropic axion-dilaton background. While such limiting approximations are encouraging but incomplete, our analysis reveals a host of novel features, and shows these models to relate standard and well understood supersymmetric string theory solutions. Finally, we conjecture that this de Sitter spacetime naturally arises by including more of the stringy degrees of freedom, such as a recently advanced variant of the non-commutative phase-space formalism, as well as the analytic continuation of a complex two-dimensional Fano variety arising as a small resolution in a Calabi-Yau 5-fold.

Compositions of states and observables in Fock spaces

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [ 29 , 13 ]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.