We formulate stationary-density-preserving nonreversible perturbations of Fokker--Planck dynamics as gauge fields that deform relaxation spectra while leaving the invariant state fixed. When detailed balance holds, a similarity transformation maps the reversible Fokker--Planck operator to a Witten-Laplacian-type supersymmetric Hamiltonian; nonreversible gauges then appear as non-Hermitian perturbations that preserve the zero mode but modify the excited spectrum. This operator viewpoint gives a common language for relaxation gaps, circulating probability currents, hypocoercive acceleration, and finite control costs. We represent admissible gauge currents by antisymmetric tensor fields and identify the detailed-balance-violating Ohzeki--Ichiki force as a constant symplectic example whose infinite-strength limit is Hamiltonian dynamics. The continuous-time spectral gap alone does not select a finite gauge strength, so we introduce a finite-time regularized objective and an actor--critic procedure for learning the gauge. An exactly solvable anisotropic Gaussian Ornstein--Uhlenbeck benchmark separates the spectral transition from the finite-time optimum and shows that the learned gauge recovers the Lyapunov-equation optimum. A double-well benchmark then illustrates the same constrained selection in a nonconvex metastable landscape. Stochastic gradient methods enter this framework as physically relevant Fokker--Planck systems: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter
ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the pratical use of the formalism, we will use it to study the effects of a heat bath on the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.
We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the practical use of the formalism, the effects of a heat bath in the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath (RISB) model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.
This record contains the manuscript “Canonical Invariants, Quantum Noise, and Memory in Gaussian Open Systems.” We develop an invariant-based formulation of Gaussian open-system dynamics for a time-dependent quantum harmonic oscillator subject to dissipation and noise. The construction combines canonical invariants with stochastic and retarded-mode descriptions, allowing Markovian thermal relaxation and finite-memory non-Markovian dynamics to be treated within a common framework. A central result is that, away from equilibrium, the instantaneous energy is not determined by the entropy alone. For local relaxation, the energy admits a differential state representation of the form[E=E!\left(S,\dot S;\frac{\omega(t)}{\nu_B(t)},\gamma_{\rm eff}(t)\right),]so that the entropy rate (\dot S) carries essential nonequilibrium information. Finite-memory dynamics enlarges the required state data while preserving this basic structure. The formulation also makes explicit the consistency relation between dissipation and the asymmetric quantum-noise spectrum, connects the Markovian limit with thermal GKLS dynamics, and provides a finite-dimensional localization of colored-noise memory kernels. The framework is intended as a basis for studying driven Gaussian open systems and for possible extensions to quantum-field and electromagnetic settings. This version has been submitted to Physical Review Research.
Gauge theories form the foundation of modern physics, with applications ranging from elementary particle physics and early-universe cosmology to condensed matter systems.We perform quantum simulations of the unitary dynamics of a U(1) symmetric gauge field theory and demonstrate emergent irreversible behavior.The highly constrained gauge theory dynamics is encoded in a one-dimensional Bose-Hubbard simulator, which couples fermionic matter fields through dynamical gauge fields.We investigate global quantum quenches and the equilibration to a steady state well approximated by a thermal ensemble.Our work may enable the investigation of elusive phenomena, such as Schwinger pair production and string-breaking, and paves the way for simulating more complex higher-dimensional gauge theories on quantum synthetic matter devices.
We work in the universal information hydrodynamics (UIH) framework developed in two companion papers, where reversible quantum dynamics and irreversible response are organised in a local metriplectic geometry generated by a convex free energy, a Fisher metric, and an antisymmetric no work channel. In this setting the irreversible drift is always a Fisher gradient flow in a weighted
H−1 geometry, equipped with a cost entropy inequality and a curvature coercivity bound that are invariant under the addition of any compatible reversible dynamics. In the present paper we show that this structure has a canonical realisation for Fokker Planck equations, finite Markov chains and GKLS semigroups, and that it can be reconstructed operationally from process and state tomography on real quantum hardware. First, starting from overdamped Langevin dynamics and its Fokker Planck generator, we identify a canonical Fisher Dirichlet operator
G=Qdiag(π) that plays the role of the symmetric mobility on a periodic lattice and saturates the cost entropy inequality mode by mode. We then demonstrate that in a wide class of thermal and driven GKLS models the symmetric part of the real generator in the stationary BKM metric has a density block that coincides exactly with this classical Fisher operator, including nonreversible chains and random qubit and qutrit ensembles, and that the global decay of entropy and Fisher energy is governed by the Markov spectral gap while the Fisher gap controls only local curvature. Finally, we implement a suite of IBM Quantum experiments on a superconducting qubit that realises the full UIH picture on hardware. Process tomography of idle circuits yields an effective real generator
K in the Pauli basis whose metric adjoint split in the BKM metric at the device stationary state produces a clean metriplectic decomposition
K=G+J, with
G metric symmetric and
J metric skew. Independent idle depths are consistent with a single time homogeneous generator, the dissipative spectrum of
G sets an information theoretic speed limit for BKM Fisher decay, and a curvature test shows that the same BKM metric is the local Hessian of quantum relative entropy at the stationary state. All claims are backed by a reproducible Python archive that includes constructive Fokker Planck to Markov to GKLS lifts, hypocoercive decay diagnostics, and hardware K tomography. Taken together, the results in this paper show that this programme is not only mathematically coherent, but experimentally visible, showing strong signatures of universality.