Abstract We develop the covariant phase space formulation of Weyl-transverse gravity (WTG) in the presence of general timelike and spacelike boundaries. WTG is classically equivalent to General Relativity (GR) but possesses a reduced gauge symmetry consisting of Weyl transformations and transverse diffeomorphisms, together with a fixed background volume form. This structure modifies the variational principle and the definition of conserved quantities relative to GR.We derive the symplectic potential, presymplectic current, and Hamiltonian generators associated with transverse diffeomorphisms, and we identify a set of boundary conditions under which the WTG action is differentiable. These include Dirichlet and Neumann conditions for both the auxiliary Weyl-invariant metric and the dynamical metric, as well as a natural implementation of York boundary conditions, for which WTG exhibits a particularly transparent geometric formulation.We obtain the Noether current and surface charge, clarify the role of the Lagrangian ambiguity related to the cosmological constant, and evaluate the Hamiltonian identity on spacetimes containing a bifurcate Killing horizon. The resulting first-law relation shows that variations of the cosmological constant can contribute nontrivially unless additional physical restrictions are imposed.

We develop a unified Koopman-von Neumann (KvN) operator and Weyl-Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed for quantum mechanics (Weyl quantization, Moyal ⋆-products, and Wigner functionals) to a classical fluid. We construct the appropriate KvN generator (including the required Jacobian term for unitarity) and derive the evolution equation for the corresponding Wigner functional. This framework clarifies when the classical Liouville (Vlasov) description is exact-namely, in quadratic or linear regimes where the Moyal bracket reduces to the Poisson bracket-and when higher-order quantum-like corrections become significant in fully nonlinear regimes. As an analytic example, we obtain a closed-form Wigner solution for a one-dimensional Burgers flow (pressureless Euler) and verify, term by term, that it reproduces the expected Liouville transport (with distributional contributions at the shock). We also compare the phase-space approach with a kinetic (Vlasov-monokinetic) formulation and outline the extension of the framework to three-dimensional flows using a Clebsch variable representation.

The Kochen–Specker theorem shows that it is impossible to assign sharp values to all dynamical variables in quantum mechanics in such a way that the algebraic relations among the values of dynamical variables whose self-adjoint operators commute are the same as those among the operators themselves. We point out that, for quantum theories obtained by quantizing some classical theory, this condition – Kochen–Specker non-contextuality – is implausible from the start because quantization usually changes algebraic relations. We explain why this is so, using the formalism of deformation quantization and its conception of star products , and we illustrate the relevance of this point using various examples of dynamical variables quantized via Weyl quantization and coherent state quantization. Our observations suggest that the relevance of the Kochen–Specker theorem to the question of whether one can assign sharp values to all dynamical variables is rather limited.

The electric interaction between two nearby evolving electrons triggers the correlation between their waves and governs the operation of logical devices called Coulomb entanglers. Of technological interest, in the presence of magnetic fields, are multi-spatial evolution scenarios beyond pure state descriptions. The two-electron density matrix becomes eight-dimensional even for two-dimensional spatial cases, and is thus computationally prohibitive. In this work, we present two new approximations of the two-electron Wigner equation that aim at computational feasibility: a BBGKY approach for reducing the number of variables and a field approximation of the Coulomb-Wigner operator. They exhibit different conceptual aspects that illustrate alternative viewpoints on entanglement: only the evolution provided by the latter model satisfies the orthodox definition of entanglement. Our analysis, based on the Fredholm integral representation of the models, allows us to develop an intuitive picture and physical insight into the process. • The computationally prohibitive two-electron problem of Coulomb entanglement is approached with Wigner (phase space) quantum mechanics. • Two approximate methods, which are well-established in the quantum transport theory, are now applied and analyzed with respect to their relevance to describe the process of entanglement. • Projections over subspaces are used to obtain the BBGKY hierarchy of equations, which gives rise to a model violating the orthodox definition of entanglement. However, we show that the evolution is non-linear and non-Markovian, demonstrating correlations between the reduced (single-electron) states. • The field approximation of the Coulomb-Wigner operator gives rise to classical forces that govern the trajectories’ characteristics of the Liouville operator. We show that the latter, being defined in the two-electron phase space, gives rise to entanglement. • The paper focuses on the physical insights provided by the two derived approximate models on different aspects of the process of entanglement, while the mathematical derivations are given in the appendices.

Abstract We consider a reduced phase space quantisation of a model with T^3 Gowdy symmetry in which gravity has been coupled to Gaussian dust. We complete the quantisation programme in reduced loop quantum gravity (LQG) as well as algebraic quantum gravity (AQG) and derive a Schrödinger-like equation with a physical Hamiltonian operator encoding the dynamics. Due to the classical symmetries of the physical Hamiltonian, the operators are quantised in a graph-preserving way in both cases -- a difference to former models available in the literature. As a first step towards applications of the model in AQG, we consider an ansatz that we use to first construct zero volume states as specific solutions of the Schrödiger-like equation. We then also find states with a vanishing action of the Euclidean part of the physical Hamiltonian and investigate the degeneracies these states experience via the action of the Lorentzian part of the physical Hamiltonian. The results presented here can be taken as a starting point for deriving effective models as well as analysing the dynamics numerically in future work.

We study the metaplectic representation of the Jacobi group (the semi-direct product of the Heisenberg group by SU(1, 1)) by using the complex Weyl correspondence. In particular, we give explicit formulas for the complexWeyl symbols of the metaplectic representation operators and we prove that the complex Weyl correspondence is a Stratonovich–Weyl correspondence for the metaplectic representation.

Classical physics and visible matter are insufficient to account for cosmic phenomena such as flat galactic rotation curves or the apparent acceleration of the Universe, leading to the conventional introduction of dark matter and dark energy. However, their persistent invisibility suggests that these effects may stem not from hidden components but from deeper physical principles. Here, we propose that the dark-sector phenomenology emerges from nonlinear quantum forces absent in classical dynamics, originating from the infinite hierarchy of quantum corrections in the Wigner–Moyal phase-space formulation. Previous studies have shown that these higher-order corrections vanish under coarse resolution yet grow exponentially when the system is resolved more finely. We reinterpret this resolution-dependent quantum complexity as a relativistic effect amplified by gravitational potential, whose normalization reproduces spacetime curvature in the weak-field limit and generates additional forces on cosmic scales. When the mass distribution can be described as a macroscopic wave packet, the resulting quantum-corrected force naturally reproduces galactic rotation curves without invoking dark matter. Conversely, when the background potential defining the quantum corrections is tied to the observer’s causal horizon, it weakens with distance, causing the quantum terms to diminish and the dynamics to gradually converge toward classical behavior — making the faraway Universe appear to contain less dark matter. This horizon-dependent suppression naturally reduces the inferred luminosity distances of distant galaxies, accounting for the Pantheon[Formula: see text] Type Ia supernova data without invoking dark energy and giving the impression of an accelerating Universe. Our results suggest that the combined phenomenology of relativity, dark matter, and dark energy may arise from gravitationally regulated quantum statistical dynamics, offering a unified and observationally consistent alternative to the standard dark-sector paradigm.

We establish an intuitive connection between transformation optics (TO) and the classical invariants of étendue and radiance. Through explicit application of the optical-metric formulation of TO, we demonstrate that any smooth, passive, impedance-matched transformation performs as a canonical (symplectic) mapping on optical phase space. Combined with Hamiltonian ray dynamics, this implies that Liouville's theorem applies as well to transformation-optical media, enforcing phase-space volume preservation and fundamental constraints on radiance and étendue under passive mappings. To our knowledge, this explicit phase-space formulation has not been systematically developed within the transformation-optics framework. We derive strict analytical bounds on achievable field enhancement. In particular, we show that the maximum average intensity attainable in any passive TO concentrator is limited solely by the geometric area-compression ratio of the underlying coordinate transformation, independent of the specific material realization. We apply this framework to zero-index media, optical-null media, and illusion devices and find the same rules. Consequently, our findings demonstrate that TO redistributes optical intensity without increasing radiance, consistent with a Liouville-type constraint. This result provides a consistent, metric-based explanation for fundamental concentration limits in passive, impedance-matched metamaterials.

Abstract Quantum phases in low-dimensional mixed-spin systems continue to attract significant interest due to their rich correlation structure and relevance to modern quantum technologies. Here, we investigate the one-dimensional mixed spin-(1/2, S ) Heisenberg model in an external magnetic field using a phase-space formulation based on the generalized Wigner function. Focusing on mixed-spin (1/2, 1) and (1/2, 3/2) systems, we show that phase-space indicators, including Rényi entropies and Wigner negativity, provide a complementary characterization of quantum phases and correlations, consistent with phase structures obtained from conventional observables. Our analysis reveals three robust regions in the phase diagram: ferrimagnetic, quantum–spin–liquid-like, and fully polarized. Ground-state level crossings emerge near the corresponding boundaries, marking finite-size precursors of thermodynamic phase transitions. By combining phase-space diagnostics with finite-size scaling, we identify how quantum correlations evolve toward the thermodynamic limit. These results deepen the understanding of correlation-driven phases in mixed-spin chains and establish phase-space methods as powerful tools for analyzing complex quantum magnetic systems.

In this work, we study the regularity of the Cauchy problem for the free transport equation, and by using the inverse Wigner transformation, we reduce this problem to the Cauchy problem of a class of linear homogeneous hyperbolic Schrödinger equation. We prove firstly the analytical smoothing effect of Cauchy problem for Schrödinger type equation if the initial datum is exponential decay. Finally we prove the directional propagation of the exponential decay and also analytic regularity for free transport equation.

It has been argued that the Feynman path integral formalism leads to a quantization rule, and that this is the Born–Jordan rule which gives the unique quantization rule consistent with the correct short-time propagator behavior of the propagator for nonrelativistic systems. We examine this short-time approximation and conclude, contrary to prevailing views, that the asymptotic expansion applies only to Hamiltonian functions that are at most quadratic in the momentum and with constant mass. While the Born–Jordan rule suggests the appropriate quantization of functions in this class, there are other rules which give the same answer, most notably the Weyl quantization scheme.

The aim of these notes is to examine the propagation of isotropic Sobolev (micro)singularities for an evolution problem whose generator is given by the Weyl quantization of a complex-valued quadratic form on phase space.

A bstract In gravitational theories with boundaries, diffeomorphisms can become physical and acquire a non-vanishing Noether charge. Using the covariant phase space formalism, on shell of the gravitational constraints, the latter localizes on codimension-2 surfaces, the corners. The corner proposal asserts that these charges, and their algebras, must be important ingredients of any quantum gravity theory. In this manuscript, we continue the study of quantum corner symmetries and algebras by computing the entanglement entropy and quantum informational properties of quantum states abiding to the quantum representations of corners in the framework of 2-dimensional gravity. We do so for two classes of states: the vacuum and coherent states, properly defined. We then apply our results to JT gravity, seen as the dimensional reduction of 4d near extremal black holes. There, we demonstrate that the entanglement entropy of some coherent quantum gravity states — states admitting a semiclassical description — scales like the dilaton, reproducing the semiclassical area law behavior and further solidifying the quantum informational nature of entropy of quantum corners. We then study general states and their gluing procedure, finding a formula for the entanglement entropy based entirely on the representation theory of 2d quantum corners.

The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff-Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this article, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree-Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling-Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born-Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.

This paper investigates the quantization of the Schwarzschild–de Sitter (SdS) black hole (BH) using the Misner–Sharp–Hernandez (MSH) mass as the internal energy in a reduced phase space framework. After introducing the canonical variables of the reduced phase space, we derive a discrete spectrum for the surface areas of the BH event horizon (EH) as well as MSH masses. We utilized the MSH mass spectrum to obtain the entropy of the BH. The entropy of the BH and cosmic EHs reveals a logarithmic correction to the Bekenstein–Hawking term. Our results support the robustness of the logarithmic form of quantum corrections in SdS thermodynamics.

We introduce entropic measures to quantify non-classical resource in hybrid spin-boson systems. We discuss the stabiliser Rényi entropy in the framework of phase space quantisation and define an analogous hybrid magic entropy and a mutual magic entropy that capture the distribution of quantum magic across spin and bosonic subsystems. We use these entropic measures to demonstrate two key phenomena: the detection of the superradiant phase transition in the Dicke model and the quantum dynamics of magic in the Jaynes-Cummings model. We develop a Monte Carlo numerical scheme to practically evaluate these entropic measures in interacting many-body systems.

Using a global rotation by θ about the z-axis in the spin sector of the Jordan–Wigner transformation rotates Pauli matrices X^ and Y^ in the x−y-plane, while it adds a global complex phase to fermionic quantum states that have a fixed number of particles. With the right choice of angles, this relates expectation values of Pauli strings containing products of X^ and Y^ to different products, which can be employed to reduce the number of measurements needed when simulating fermionic systems on a quantum computer. Here, we derive this symmetry and show how it can be applied to systems in Physics and Chemistry that involve Hamiltonians with only single-particle (hopping) and two-particle (interaction) terms. We also discuss the consequences of this for finding efficient measurement circuits in variational ground state preparation.

Abstract We provide a systematic construction for local quantum circuits hosting free fermions in disguise (FFD), both with staircase and brickwork architectures. Similar to the original Hamiltonian model introduced by Fendley, these circuits are defined by the fact that the Floquet operator corresponding to a single time step can not be diagonalized by means of any Jordan–Wigner transformation, but still displays a free-fermionic spectrum. Our construction makes use of suitable non-local transfer matrices commuting with the Floquet operator, allowing us to establish the free fermionic spectrum. We also study the dynamics of these circuits after they are initialized in arbitrary product states, proving that the evolution of certain local observables can be simulated efficiently on classical computers. Our work proves recent conjectures in the literature and raises new questions on the classical simulability of FFD.

A bstract In this work, we attempt to construct bit thread configurations for various backgrounds using expressions from the covariant phase space formalism. We find that when the Ryu-Takayanagi surface is same as the horizon, such expressions are sufficient. In other cases, it differs by gradient of a harmonic function. We explore its relation to Wald and differential entropy, and re-express the first law of entanglement entropy in terms of bit threads. Inclusion of quantum effects imposes some constraints on the bulk entanglement via the dominant energy condition. We also apply our method to ascertain a bit thread configuration in a certain dynamical spacetime.

• Definition of a gauge-independent density matrix and Wigner function according to Stratonovic. • Derivation of a Wigner equation that fully includes the electromagnetic fields E and B. • Strong formulation of the gauge-independent Wigner equation using pseudo-differential operators. • Weak formulation of the gauge-independent Wigner equation using integral operators. • Quantum mechanics: standard formulation requires the use of electrodynamic potentials; in the phase space formulation, the fields E and B can be used. Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.