Dieter Schuch, Marcos Moshinsky
For classical canonical transformations, one can, using the Wigner transformation, pass from their representation in Hilbert space to a kernel in phase space. In this paper it will be discussed how the time-dependence of the uncertainties of the corresponding time-dependent quantum problems can be incorporated into this formalism.
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The Weyl-Wigner representations for canonical thermal equilibrium quantum states are obtained for the whole class of quadratic Hamiltonians through a Wick rotation of the Weyl-Wigner symbols of Heisenberg and metaplectic operators. The behavior of classical structures inherently associated to these unitaries is described under the Wick mapping, unveiling that a thermal equilibrium state is fully determined by a complex symplectic matrix, which sets all of its thermodynamical properties. The four categories of Hamiltonian dynamics (Parabolic, Elliptic, Hyperbolic, and Loxodromic) are analyzed. …
Time-symmetric quantum mechanics can be described in the usual Weyl--Wigner--Moyal formalism (WWM) by using the properties of the Wigner distribution, and its generalization, the cross-Wigner distribution. The use of the latter makes clear a strongly oscillating interference between the pre- and post-selected states. This approach allows us to give explicit formulas for the state reconstruction problem, thus generalizing known results to the case of arbitrary observables. In a forthcoming paper we will extend these results to other quantization schemes.
It is shown that a $N\times N$ real symmetric [complex hermitian] positive definite matrix $V$ is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in $SO(m,n)$ [ $SU(m,n)$], for any choice of partition $N=m+n$. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson's theorem which states that if $N$ is even then $V$ is congruent also to a diagonal matrix modulo a symplectic matrix in $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. Applications of these results considered include a generalization of …
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j+1, which is known to tend to the canonical oscillator when j tends to in…
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix. The quantization of a Hamiltonian describing such a system has been done in the canonical case. In this paper, we take a more general approach and look at the system as a Wigner quantum system. Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations. Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n). We determine the …
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulati…
In this paper we generalize the concept of Wigner function in the case of quantum mechanics with a minimum length scale arising due to the application of a generalized uncertainty principle (GUP). We present the phase space formulation of such theories following GUP and show that the Weyl transform and the Wigner function does satisfy some of their known properties in standard quantum mechanics. We utilise the generalized Wigner function to calculate the phase space average of the Hamiltonian of a quantum harmonic oscillator satisfying deformed Heisenberg algebra. It is also shown that average…
Uncertainty and Wigner negativity in Hilbert-space classical mechanics
本文在希尔伯特空间经典力学中重现了量子力学的不确定性关系和威格纳负值特征。
The Gibbs canonical state, as a maximum entropy density matrix, represents a quantum system in equilibrium with a thermostat. This state plays an essential role in thermodynamics and serves as the initial condition for nonequilibrium dynamical simulations. We solve a long standing problem for computing the Gibbs state Wigner function with nearly machine accuracy by solving the Bloch equation directly in the phase space. Furthermore, the algorithms are provided yielding high quality Wigner distributions for pure stationary states as well as for Thomas-Fermi and Bose-Einstein distributions. The …
On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
本文将 Feynman 路径积分推广至 Wigner-Dunkl 量子力学，分析高斯波包、谐振子及带跳跃的 Dunkl 过程，并显式计算 Feynman-Kac 路径积分。
Hidden Supersymmetry in Wigner-Yang Quantum Mechanics
该研究探讨维格纳-杨量子力学系统的超对称结构，推广杨的表示并建立类库仑系统与维格纳-杨简谐振子的胡克-牛顿对偶性，揭示其隐藏超对称性。
The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of K…
The Contextual Modal Logic of a Wigner's Friend Generalization
本文通过考虑量子系统的情境性，反驳了弗拉乌奇格-伦纳思想实验中所提出的量子力学逻辑不一致性的矛盾。
Bell's inequalities in the form given by Wigner are derived from the so-called fundamental assumption of statistical mechanics. I then demonstrate the possible relationship between these inequalities and the second law, particularly if assuming the Pauli exclusion principle dictates the expected outcomes.
Semiclassical approximation of the Wigner function for the canonical ensemble
本文通过Wick旋转的经典轨迹半经典近似正则系综的Wigner函数，并开发数值方案，在单双自由度系统中验证了热力学平均值的准确性。
A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.
Recent developments in elementary quantum mechanics have seen a number of extraordinary claims regarding quantum behaviour, and even questioning internal consistency of the theory. These are, we argue, different disguises of what Feynman described as quantum theory's "only mystery".
As a natural extension of Fan's paper (arXiv: 0903.1769vl [quant-ph]) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation we find new two-fold complex integration transformation about the Wigner operator (in its entangled form) in phase space quantum mechanics and its inverse transformation. In this way, some operator ordering problems can be solved and the contents of phase space quantum mechanics can be enriched.
Both classical and quantum damped systems give rise to complex spectra and corresponding resonant states. We investigate how resonant states, which do not belong to the Hilbert space, fit the phase space formulation of quantum mechanics. It turns out that one may construct out of a pair of resonant states an analog of the stationary Wigner function.
Understanding Squeezed States of Light Through Wigner's Phase-Space
本文利用维格纳相空间分布函数，从经典到量子力学的转变出发，探讨了相空间中的相干态、单模和双模压缩态及压缩真空，并指出双模压缩态自然产生纠缠，同时涉及耦合谐振子和对称群的应用。
Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is marked by its economy, naturalness and more importantly, by its potential for extensions and generalisations to situations where the underlying configuration space is non Cartesian.
The extended Wigner's friend problem deals with two Observers each measuring a sealed laboratory in which a friend is making a quantum measurement. We investigate this problem by relying on the basic rules of quantum mechanics as exposed by Feynman in the well-known "Feynman Lectures on Physics". Although recent discussions have suggested that the extended Wigner's friend problem cannot consistently be described by quantum theory, we show here that a straightforward application of these standard rules results in a non-ambiguous and consistent account of the measurement outcomes for all agents …
A relativistic spin operator is to be the difference between the total and orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all desirable commutation relations as a position operator, can give a proper spin operator. Historically important three spin operators respectively proposed by Bogolubov et al., Pryce, and Foldy-Woutheysen are investigated to manifest a corresponding spin operator to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics described b…
We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\bm J$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their pote…
The Fermi g_F(x,p) function provides a phase space description of quantum mechanics conceptually different from that based on the the Wigner function W(x,p). In this paper, we show that for a peaked wave packet the g_F(x,p)=0 curve approximately corresponds to a phase space contour level of the Wigner function and provides a satisfactory description of the wave packet's size and shape. Our results show that the Fermi function is an interesting tool to investigate quantum fluctuations in the semiclassical regime.
The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard op…
It is shown, that for quantum systems the vectorfield associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg pictures. We illustrate these ambiguities in terms of simple examples.
We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.
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…
Deterministic Mechanical Wigner Negativity via Nonlinear Cavity Quantum Optomechanics in the Unresolved-Sideband Regime
本文提出利用腔响应对机械位置的非线性，在未分辨边带区无需单光子强耦合、额外非线性或条件测量即可确定性制备机械维格纳负性，并证明其在连续驱动稳态下持续存在。
We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalised to accommodate new features which arise when the configuration space changes from $n$-dimensional Euclidean space ${\cal R}^n$ to a Lie group $G$. The notion of canonical momentum is carefully analysed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact $G$ an explic…
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function $W(q,p)$, the marginals of which yield the correct quantum probabilities for $q$ and $p$ separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other wo…
Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics
该研究将Dunkl谐振子及各向同性谐振子、库仑势推广为准精确可解势，获得具有$n+1$个已知本征态的势，并用扩展Dunkl导数简化哈密顿量。
Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $Δ(q',p')$ ($q$-number transform) in phase space quantum mechanics, \[ \iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q), \] and its inverse \[\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'), \] where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of op…
We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the $\ast$-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the $\ast$-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the dens…
The purpose of this comment is to show that a reinterpretation of the results from the Letter: "Physics and Metaphysics of Wigner's Friends: Even performed pre-measurements have no results" allows for reaching the conclusion "pre-measurements have no resul" [arXiv:2003.07464] based only on postulates of quantum mechanics without additional assumptions on irreversibility. Additionally, with supplementary reasoning based on Bell's theorem, one can show that unitary decoherence cannot be solely responsible for the quantum-to-classical transition, and an additional irreversibility model is require…
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space $T^*G$ and the Hilbert space of square integrable functions on $G$. General expressions for the star product for Weyl symbols are presented and explicitly worked ou…
This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular clas…
Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the oth…
Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ${\ee Z}_{D} \times {\ee Z}_{D}$ with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-prese…
We show that the supersymmetric algebra of Witten's quantum mechanics is invariant under a given point canonical transformation. It is shown that Witten's supersymmetric quantum mechanics can be isospectral or not to the seed Hamiltonian depending on the space coordinate you work on. We illustrate our results by generating a new class of exactly solvable supersymmetric partner Hamiltonians which are not isospectral to the seed Hamiltonian.
In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present a proof of this theorem which is both elementary and economical. Central to our proof is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis involves a judicious i…
A fluid analog of the information flux in the phase-space associated to purity and von Neumann entropy are identified in the Weyl-Wigner formalism of quantum mechanics. Once constrained by symmetry and positiveness, the encountered continuity equations provide novel quantifiers for non-classicality (non-Liouvillian fluidity) given in terms of quantum decoherence, purity and von Neumann entropy fluxes. Through definitions in the Weyl-Wigner formalism, one can identify the quantum fluctuations that distort the classical-quantum coincidence regime, and the corresponding quantum information profil…
The measurement problem is seen as an ambiguity of quantum mechanics, or, beyond that, as a contradiction within the theory: Quantum mechanics offers two conflicting descriptions of the Wigner's-friend experiment. As we argue in this note there are, however, obstacles from within quantum mechanics and regarding our perspective onto doing physics towards fully describing a measurement. We conclude that the ability to exhaustively describe a measurement is an assumption necessary for the common framing of the measurement problem and ensuing suggested solutions.
Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton-Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space …
A comprehensive theory of the Weyl-Wigner formalism for the canonical pair angle-angular momentum is presented, with special emphasis in the implications of rotational periodicity and angular-momentum discreteness.
We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum-mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.
We show that the cross Wigner function can be written in the form $W(ψ, ϕ)= \hat S (ψ\otimes \overline{\hatϕ})$ where ${\hatϕ}$ is the Fourier transform of $ϕ$ and $\hat S$ is a metaplectic operator that projects onto a linear symplectomorphism $S$ consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl-Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from def…
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary ope…
"Wigner's friend" refers to a quantum process of which different observers, following the rules of quantum mechanics, give contradictory descriptions. Lostaglio and Bowles have recently claimed to describe a classical system showing the same effect. It is argued that this claim is not justified. it fails to take into account the different meanings of probability in classical and quantum mechanics.
We first review the usefulness of the Wigner distribution functions (WDF), associated with Lindblad and pre-master equations, for analyzing a host of problems in Quantum Optics where dissipation plays a major role, an arena where weak coupling and long-time approximations are valid. However, we also show their limitations for the discussion of decoherence, which is generally a short-time phenomenon with decay rates typically much smaller than typical dissipative decay rates. We discuss two approaches to the problem both of which use a quantum Langevin equation (QLE) as a starting-point: (a) us…
The aim of this paper is to give a simple, geometric proof of Wigner's theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner's theorem is not fully appreciated in general. It is Wigner's theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigne…
Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bou…
A general argument is presented against relativistic, unitary, single-outcome quantum mechanics. This is achieved by combining the Wigner's Friend thought experiment with measurements on a Greenberger-Horne-Zeilinger (GHZ) state, and describing the evolution of the quantum state in various inertial frames. Assuming unitary quantum mechanics and single outcomes, the result is that the Born rule must be violated in some inertial frame: in that frame, outcomes are obtained for which no corresponding term exists in the pre-measurement wavefunction.
We present a discussion of the extended Wigner's friend thought experiment proposed by Frauchiger and Renner in [1]. We show by using various arguments, including textbook quantum mechanics and the ontological approach of Contexts, Systems, Modalities (CSM), that no contradiction arises if one admits that agents must agree on what is considered as a system and what is not. In such a contextually objective approach of quantum mechanics, the apparent contradiction is automatically removed. We also discuss why this mutual agreement between agents is already implicit in the standard formulations o…
An n-particle 3-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p\geq 3n but atypical for p<3n. The branching rules for th…
We provide an introduction into the formulation of non-relativistic quantum mechanics using the Wigner phase-space distribution function and apply this concept to two physical situations at the interface of quantum theory and general relativity: (i) the motion of an ensemble of cold atoms relevant to tests of the weak equivalence principle, and (ii) the Kasevich-Chu interferometer. In order to lay the foundations for this analysis we first present a representation-free description of the Kasevich-Chu interferometer based on unitary operators.
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail an…
Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces S^1 x R, i.e. for the canonical pair angle and angular momentum, was presented (arXiv:1601.02520), main properties of those functions derived, discussed and their usefulness illustrated by examples. The present paper is a continuation which compares properties of the new Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions on planar ones in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase space function…
Efficient Approximation of the Wigner Kernel in Phase-Space Quantum Mechanics
针对符号粒子公式中Wigner核计算复杂的问题，提出基于势函数级数表示的解析近似方法，显著降低计算成本并保持主要特性。
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