Wigner Transform Research Articles

Finite charges from the bulk action

Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.

Third quantization of open quantum systems: Dissipative symmetries and connections to phase-space and Keldysh field-theory formulations

Third quantization of open quantum systems: Dissipative symmetries and connections to phase-space and Keldysh field-theory formulations

Revisiting the thermodynamics of the BTZ black hole with a variable gravitational constant* *Supported by the National Natural Science Foundation of China (12275216, 12105222, 12247103)

The thermodynamics of BTZ black holes are revisited with a variable gravitational constant. A new pair of conjugated thermodynamic variables are introduced, including the central charge C and chemical potential μ. The first law of thermodynamics and the Euler relationship, instead of the Smarr relationship in the extended phase space formalism, are matched perfectly in the proposed formalism. Compatible with the standard extensive thermodynamics of an ordinary system, the black hole mass is verified to be a first order homogeneous function of the related extensive variables, and restores the role of internal energy. In addition, the heat capacity has also resulted in a first order homogeneous function using this formalism, and asymptotic behavior is demonstrated at the high temperature limit. The non-negativity of the heat capacity indicates that the rotating and charged BTZ black holes are thermodynamically stable.

Heat distribution in quantum Brownian motion.

We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

Wigner function dynamics with boundaries expressed as convolution

In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.

Restricted phase space thermodynamics of charged AdS black holes in conformal gravity* *Supported by the National Natural Science Foundation of China (12275138)

The thermodynamics of charged spherically symmetric AdS black holes in conformal gravity is revisited using the recently proposed restricted phase space (RPS) formalism. This formalism avoids all the bizarreness arising in the extended phase space formalism for this model. It is found that the charged AdS black holes in this model may belong to a new universality class that is different from all previously studied cases under the RPS formalism. Besides the distinguished isocharge and isothermal behaviors, the absence of the Hawking-Page transition is another notable feature. However, in the high temperature limit, the thermodynamic behavior of the present model becomes exactly the same as that of the Einstein gravity and black hole scan models, which adds further evidence for the universality of the recently reported correspondence between high temperature AdS black holes and low temperature quantum phonon gases in nonmetallic crystals.

Field theories on ρ-deformed Minkowski space-time

We study one-loop perturbative properties of scalar field theories on the ρ-Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties of the convolution algebra of the Euclidean group linked to the coordinate algebra of the ρ-Minkowski space. The natural integration measure linked to the Haar measure of the Euclidean group defines a trace for the star-product. One-loop properties of the 2-point and 4-point functions for families of complex-valued scalar field theories on ρ-Minkowski space are examined. For scalar theories with orientable interaction, the 2-point function is found to receive UV quadratically diverging one-loop corrections in 4 dimensions while no IR singularities generating UV/IR mixing appears. These however occur in the one-loop corrections to the 4-point function. As well, one-loop 2-point functions for theories with non-orientable interaction involve such IR singularities. These results are discussed.

External gauge field-coupled quantum dynamics: Gauge choices, Heisenberg algebra representations and gauge invariance in general, and the Landau problem in particular

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

Geometric uncertainty relations on Wigner–Yanase skew information

We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics.

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.

Nonadiabatic dynamics near metal surface with periodic drivings: A Floquet surface hopping algorithm.

We develop a Floquet surface hopping approach to deal with nonadiabatic dynamics of molecules near metal surfaces subjected to time-periodic drivings from strong light-matter interactions. The method is based on a Floquet classical master equation (FCME) derived from a Floquet quantum master equation (FQME), followed by a Wigner transformation to treat nuclear motion classically. We then propose different trajectory surface hopping algorithms to solve the FCME. We find that a Floquet averaged surface hopping with electron density (FaSH-density) algorithm works the best as benchmarked with the FQME, capturing both the fast oscillations due to the driving and the correct steady-state observables. This method will be very useful to study strong light-matter interactions with a manifold of electronic states.

Hall conductivity as the topological invariant in magnetic Brillouin zone in the presence of interactions

Hall conductivity for the intrinsic anomalous quantum Hall effect in homogeneous systems is given by the topological invariant composed of the Green function depending on momentum of quasiparticle. This expression reveals correspondence with the mathematical notion of the degree of mapping. A more involved situation takes place for the quantum Hall effect in the presence of external magnetic field. In this case, the mentioned expression remains valid if the Green function is taken in a specific representation, where it becomes the infinite-dimensional matrix [N. Imai et al., Phys. Rev. B 42, 10610 (1990)] or if it is replaced by its Wigner transformation while ordinary products are replaced by the Moyal products [M. A. Zubkov and X. Wu, Ann. Phys. 418, 168179 (2020)]. Both these expressions, unfortunately, are much more complicated and might be useless for the practical calculations. Here we represent the alternative representation for the Hall conductivity of a uniform system in the presence of constant magnetic field. The Hall conductivity is expressed through the Green function taken in Harper representation, when its nonhomogeneity is attributed to the matrix structure while functional dependence is on one momentum that belongs to magnetic Brillouin zone. Our consideration for the interacting systems is nonperturbative and is based on the Schwinger-Dyson equations truncated in a reasonable way. We demonstrate that in this approximation the expression for the Hall conductivity in Harper representation remains valid, where the interacting Green function is to be used instead of the noninteracting one. We, therefore, propose that the obtained expression may be used for the topological description of fractional quantum Hall effect.

Continuous majorization in quantum phase space

We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

Time evolution of density matrices as a theory of random surfaces

In the operatorial formulation of quantum statistics, the time evolution of density matrices is governed by von Neumann’s equation. Within the phase space formulation of quantum mechanics it translates into Moyal’s equation, and a formal solution of the latter is provided by Marinov’s path integral. In this paper we uncover a hidden property of the Marinov path integral, demonstrating that it describes a theory of ruled random surfaces in phase space.

Matrix quantization of gravitational edge modes

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra mathfrak{sdiff} (S2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra mathfrak{su} (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}{mathbb{R}}^{S^2} arises as the large-N limit of mathfrak{sl}left(N,mathbb{C}right)oplus mathbb{R} and that the full area-preserving corner symmetry algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}mathfrak{sl}{left(2,mathbb{R}right)}^{S^2} is the large-N limit of the pseudo-unitary group mathfrak{su} (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the mathfrak{su} (N) structure constants, as well as an explicit computation of the full mathfrak{diff} (S2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N.

Statistics of phase space localization measures and quantum chaos in the kicked top model.

Quantum chaos plays a significant role in understanding several important questions of recent theoretical and experimental studies. Here, by focusing on the localization properties of eigenstates in phase space (by means of Husimi functions), we explore the characterizations of quantum chaos using the statistics of the localization measures, that is the inverse participation ratio and the Wehrl entropy. We consider the paradigmatic kicked top model, which shows a transition to chaos with increasing the kicking strength. We demonstrate that the distributions of the localization measures exhibit a drastic change as the system undergoes the crossover from integrability to chaos. We also show how to identify the signatures of quantum chaos from the central moments of the distributions of localization measures. Moreover, we find that the localization measures in the fully chaotic regime apparently universally exhibit the beta distribution, in agreement with previous studies in the billiard systems and the Dicke model. Our results contribute to a further understanding of quantum chaos and shed light on the usefulness of the statistics of phase space localization measures in diagnosing the presence of quantum chaos, as well as the localization properties of eigenstates in quantum chaotic systems.

$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators

In this note, we establish L^p -bounds for the semigroup e^{-tq^w(x,D)} , t \ge 0 , generated by a quadratic differential operator q^w(x,D) on \mathbb{R}^n that is the Weyl quantization of a complex-valued quadratic form q defined on the phase space \mathbb{R}^{2n} with non-negative real part \operatorname{Re} q \ge 0 and trivial singular space. Specifically, we show that e^{-tq^w(x,D)} is bounded from L^p(\mathbb{R}^n) to L^q(\mathbb{R}^n) for all t > 0 whenever 1 \le p \le q \le \infty , and we prove bounds on \parallel { e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel in both the large t \gg 1 and small 0 < t \ll 1 time regimes.Regardin g L^p \rightarrow L^q bounds for the evolution semigroup at large times, we show that \parallel{e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel is exponentially decaying as t \rightarrow \infty , and we determine the precise rate of exponential decay, which is independent of (p,q) . At small times 0 < t \ll 1 , we establish bounds on \parallel {e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel for (p,q) with 1 \le p \le q \le \infty that are polynomial in t^{-1} .

Filter functions for the Glauber-Sudarshan P -function regularization

The phase-space quasiprobability distribution formalism for representing quantum states provides practical tools for various applications in quantum optics such as identifying the nonclassicality of quantum states. We study filter functions that are introduced to regularize the Glauber-Sudarshan $P$ function. We show that the quantum map associated with a filter function is completely positive and trace preserving and hence physically realizable if and only if the Fourier transform of this function is a probability density distribution. We also derive a lower bound on the fidelity between the input and output states of a physical quantum filtering map. Therefore, based on these results, we show that any quantum state can be approximated, to arbitrary accuracy, by a quantum state with a regular Glauber-Sudarshan $P$ function. We propose applications of our results for estimating the output state of an unknown quantum process and estimating the outcome probabilities of quantum measurements.

Making sense of transmission resonances and Smith lifetimes in one-dimensional scattering: The extended phase space quantum trajectory picture

Resonances are ubiquitous in a wide range of physical and chemical phenomena. Their impact on quantum scattering processes renders their study as important as it can be puzzling. In this paper, we illustrate the accuracy of a fully quantum, purely trajectory based reformulation of quantum mechanics proposed by one of the authors (Poirier) to acquire insights on shape resonances through direct and accurate computation of the diagonal elements of Smith’s lifetime matrix. This study also generalizes the relationship between the quantum trajectory propagation time and the Eisenbud–Wigner time delay – introduced in our previous publication (Dupuy et al., 2022) for symmetric potentials – to the general case of asymmetric potential profiles. In addition, we show how the complex amplitudes of the scattering matrix can be extracted from left- and right-incident quantum trajectories. Finally, we demonstrate that extended phase space quantum trajectories not only recover S-matrix and quantum time quantities, but they also provide their own picture of resonant phenomena, as dynamically distinct events characterized by an integer number of closed orbits in the quantum phase space.

Systematics of boundary actions in gauge theory and gravity

We undertake a general study of the boundary (or edge) modes that arise in gauge and gravitational theories defined on a space with boundary, either asymptotic or at finite distance, focusing on efficient techniques for computing the corresponding boundary action. Such actions capture all the dynamics of the system that are implied by its asymptotic symmetry group, such as correlation functions of the corresponding conserved currents. Working in the covariant phase space formalism, we develop a collection of approaches for isolating the boundary modes and their dynamics, and illustrate with various examples, notably AdS3 gravity (with and without a gravitational Chern-Simons terms) subject to assorted boundary conditions.