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Physical review. E

Quantum statistical-gauge geometry

Pham Van Hai

Published Jun 1, 2026• 4 citations
Publisher

Abstract

Exact equilibrium constraints in quantum many-body systems, such as sum rules and Ward identities, can be hard to expose because local currents are distributional and equal-time commutators may contain anomaly-like contact terms. We present a simple geometric organization of the recently identified quantum shift symmetry. The local shifting superoperator is viewed as a statistical-gauge connection acting on observables, so hyperforces are covariant derivatives and force balance becomes a thermal Ward identity expressed with the Kubo-Mori inner product. For compact smearings we prove bulk flatness: smeared shift generators close on the Lie bracket of vector fields with no bulk Schwinger term. This closure yields an iterative, all-orders Ward-Bianchi hierarchy of equal-time constraints that closes on a finite set of force insertions and a bilinear force-gradient kernel, leading to general antisymmetry relations and positivity bounds. We fix the microscopic generator by quantizing the diffeomorphism moment map, where Weyl and half-density quantization agree and give a self-adjoint operator without bulk ordering anomalies. Although the bulk connection is flat, global topology remains through holonomy, which links naturally to twisted boundary conditions and flux threading. Background electromagnetic fields deform the algebra through explicit density-weighted curvature insertions rather than a bulk central extension. As an application we treat fractional quantum Hall fluids on a torus, connect the deformation to the guiding-center GMP-W_{∞} algebra, and verify the key identities with exact diagonalization.

DOI
10.1103/jb7b-rsfb

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Exact equilibrium constraints in quantum many-body systems, such as sum rules and Ward identities, can be hard to expose because local currents are distributional and equal-time commutators may contain anomaly-like contact terms. We present a simple geometric organization of the recently identified quantum shift symmetry. The local shifting superoperator is viewed as a statistical-gauge connection acting on observables, so hyperforces are covariant derivatives and force balance becomes a thermal Ward identity expressed with the Kubo-Mori inner product. For compact smearings we prove bulk flatness: smeared shift generators close on the Lie bracket of vector fields with no bulk Schwinger term. This closure yields an iterative, all-orders Ward-Bianchi hierarchy of equal-time constraints that closes on a finite set of force insertions and a bilinear force-gradient kernel, leading to general antisymmetry relations and positivity bounds. We fix the microscopic generator by quantizing the diffeomorphism moment map, where Weyl and half-density quantization agree and give a self-adjoint operator without bulk ordering anomalies. Although the bulk connection is flat, global topology remains through holonomy, which links naturally to twisted boundary conditions and flux threading. Background electromagnetic fields deform the algebra through explicit density-weighted curvature insertions rather than a bulk central extension. As an application we treat fractional quantum Hall fluids on a torus, connect the deformation to the guiding-center GMP-W_{∞} algebra, and verify the key identities with exact diagonalization.
We address gauge invariance in the statistical mechanics of quantum many-body systems. The gauge transformation acts on the position and momentum degrees of freedom, and it is represented by a quantum shifting superoperator that maps quantum observables onto each other. The shifting superoperator is anti-self-adjoint, and it has noncommutative Lie algebra structure. These properties induce exact equilibrium sum rules that connect locally resolved force and hyperforce densities for any given observable. We argue that the framework is amenable to tight integration into quantum hyperdensity functional theory and that it generalizes naturally to nonequilibrium.
We provide an extended account of the recent statistical mechanical theory of gauge invariance against operator shifting in quantum many-body systems [J. Müller and M. Schmidt, Phys. Rev. E 114, L022104 (2026)10.1103/dzz3-m28k]. The gauge transformation is enacted by a shifting superoperator that displaces the fundamental position and momentum degrees of freedom. The shifting superoperator constitutes a map between Hilbert space operators, and it features Lie algebra commutator structure. Averages of general observables remain invariant under the shifting both in and out of thermal equilibrium, as well as in ground states. The gauge invariance induces exact sum rules that interconnect global observables and associated locally resolved correlation functions. In particular we describe the resulting one-body force, hyperforce, product, and two-body sum rules. We relate the shifting superoperator to a previously formulated quantum canonical transformation and present the generalization of quantum shifting to multicomponent systems. The gauge theory respects fundamental fermionic and bosonic particle properties, as we demonstrate by proving the compatibility of operator shifting and exchange symmetry. We formulate the quantum version of hyperdensity functional theory to provide formal access to hyperforces as well as to general averaged quantum observables via universal density functionals. For time-dependent situations, we describe quantum dynamical gauge invariance and prove exact dynamical sum rules for nonequilibrium situations, as generated by Hamiltonian time dependence. We argue for the fundamental status of statistical mechanical gauge invariance based on the compliance of the underlying geometry with canonical quantization according to Dirac's correspondence principle. Analogies and differences of the quantum mechanical sum rules with their classical counterparts remain indicative of the respective levels of description.
Statistical-gauge structures generated by current operators provide a symmetry-based description of forces and hyperforces in closed quantum statistical mechanics, but their status in open quantum dynamics remains largely unexplored. Here we develop a generalized statistical-gauge framework for Markovian open quantum systems governed by Lindblad evolution. The construction formulates statistical shifts in a smeared operator language, thereby avoiding singular local expressions while preserving the force and hyperforce content of the closed theory. Within this framework, the action of the Lindblad generator is analyzed in a jump-resolved representation that separates Hamiltonian transport from dissipative contributions in jump space. The central result is a modular Ward identity that holds for any full-rank quantum state, independently of equilibrium, stationarity, or unitary dynamics. We show that weak covariance is controlled by the full dissipative structure in jump space, rather than by invariance of individual jump operators or by particle-number conservation. This leads to a classification of continuum reservoirs according to density weight and spatial rate profile, and shows that Lindblad dynamics can deform the transported shift algebra through its product defect even when the bath is weakly covariant. Physically, the theory explains why uniform one-body loss or pumping may preserve the statistical-gauge structure, whereas density dephasing, multiparticle loss, and nonuniform bath profiles generically produce covariance-breaking residuals. These residuals provide practical diagnostics for reservoir engineering, approximate master equations, and numerical many-body simulations of open quantum systems.
Anomalies of generalized symmetries are closely related to the statistics of topological excitations. Truncating a symmetry transformation to a finite region produces an operator that creates symmetry defects on its boundary; we call this a symmetry patch operator. Such operators resemble hopping operators that create topological excitations at the boundary of their support, but this geometric similarity does not justify identifying the two. For finite Abelian groups in one spatial dimension, we prove that if the same family of local operators defines both valid hopping operators and compatible symmetry patches, the type-III component of the anomaly is −2-2−2 times that of the statistics. We establish a one-to-one correspondence between statistics and anomaly by treating hopping operators as symmetric operators that commute with the global symmetry, following the description of symmetry through the algebra of local symmetric operators. We explain this correspondence through gauging by coupling boundary matter to a Dijkgraaf-Witten gauge field in one higher spatial dimension. Hopping operators act on the gauge-invariant degrees of freedom. After fixing the background gauge field, the residual gauge transformations become the global symmetry, and the gauge invariance of hopping operators becomes the requirement that they commute with this symmetry.
Abstract We develop a geometric formulation of nonrelativistic quantum mechanics in Randers–type backgrounds based on the effective quantum connection induced by the Randers one-form. Starting from the classical Randers action, we derive the nonrelativistic Hamiltonian through the Legendre transformation and obtain the corresponding Schrödinger equation by canonical quantization. The resulting formalism naturally yields the kinetic momentum, probability current, effective force, and the curvature associated with the induced quantum connection. We show that the nontrivial quantum effects associated with the spatial effective connection are characterized by its local curvature and global holonomy, leading to three complementary geometric regimes. Globally trivial flat connections leave the local dynamics unchanged apart from uniform energy shifts, flat connections with nontrivial holonomy generate geometric phases, twisted boundary conditions, and persistent probability currents, whereas nonvanishing effective curvature gives rise to noncommuting kinetic momenta, Landau-type quantization, a characteristic geometric length, and curvature-controlled degeneracy. The formalism is further extended to nonuniform backgrounds through the first Born approximation, where the vector contribution to elastic scattering is shown to probe the transverse component of the effective connection and, consequently, the Fourier components of its effective curvature. These results provide a unified geometric framework for nonrelativistic quantum mechanics in Randers–type backgrounds and identify holonomy and effective curvature as the fundamental global and local mechanisms through which the induced quantum connection modifies quantum behavior.