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Stress-Gauge Invariance and Quotient-Space Structure of Projected Momentum Dynamics in Many-Particle Quantum Hydrodynamics

Kawaguchi Masanori

Published Oct 2, 2026• 0 citations
Publisher

Abstract

This preprint develops a quotient-space formulation of stress-gauge invariance in many-particle quantum hydrodynamics (MPQHD) within the Mori–Zwanzig projection formalism. For a fixed Hamiltonian, equilibrium inner product, slow-variable subspace, and Mori projector, the paper derives a necessary and sufficient condition under which two microscopic momentum-flux tensors generate the same orthogonal momentum force. This criterion is then used to define a projector-dependent equivalence relation among stress tensors and to construct the corresponding quotient space. By the first isomorphism theorem, this quotient space is naturally identified with the image of the projected divergence operator. The framework is applied to the Wyatt and Kuzmenkov stress representations in exact MPQHD. Since their difference is divergence-free, the two tensors belong to the same projected-stress equivalence class and therefore generate identical orthogonal forces and identical full Mori–Zwanzig memory kernels. The paper also clarifies the relation to the Bohm–Madelung formulation. The many-body Bohm quantum potential contributes to the quantum-force density and quantum stress, which in turn enter the total projected force and memory kernel. However, a separately defined “quantum viscosity” based only on a representation-dependent quantum-stress subkernel is not generally invariant. The resulting framework distinguishes microscopic stress representations from the projected transport dynamics retained by a chosen coarse-graining and provides a mathematical basis for analyzing generalized hydrodynamics, nonlocal transport, memory effects, and Bohmian many-body correlations. This manuscript is a preprint and has not yet undergone peer review.

DOI
10.5281/zenodo.23098655

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This preprint develops a quotient-space formulation of stress-gauge invariance in many-particle quantum hydrodynamics (MPQHD) within the Mori–Zwanzig projection formalism. For a fixed Hamiltonian, equilibrium inner product, slow-variable subspace, and Mori projector, the paper derives a necessary and sufficient condition under which two microscopic momentum-flux tensors generate the same orthogonal momentum force. This criterion is then used to define a projector-dependent equivalence relation among stress tensors and to construct the corresponding quotient space. By the first isomorphism theorem, this quotient space is naturally identified with the image of the projected divergence operator. The framework is applied to the Wyatt and Kuzmenkov stress representations in exact MPQHD. Since their difference is divergence-free, the two tensors belong to the same projected-stress equivalence class and therefore generate identical orthogonal forces and identical full Mori–Zwanzig memory kernels. The paper also clarifies the relation to the Bohm–Madelung formulation. The many-body Bohm quantum potential contributes to the quantum-force density and quantum stress, which in turn enter the total projected force and memory kernel. However, a separately defined “quantum viscosity” based only on a representation-dependent quantum-stress subkernel is not generally invariant. The resulting framework distinguishes microscopic stress representations from the projected transport dynamics retained by a chosen coarse-graining and provides a mathematical basis for analyzing generalized hydrodynamics, nonlocal transport, memory effects, and Bohmian many-body correlations. This manuscript is a preprint and has not yet undergone peer review.
In the first part of this paper, the many-particle quantum hydrodynamics (MPQHD) equations for a system containing many particles of different sorts are derived exactly from the many-particle Schrödinger equation. It includes the derivation of the many-particle continuity equations (MPCE), many-particle Ehrenfest equations of motion (MPEEM), and many-particle quantum Cauchy equations (MPQCE) for any of the different particle sorts and for the total particle ensemble. The new point in our analysis is that we consider a set of arbitrary particles of different sorts in the system. In MPQCEs, there appears a quantity called pressure tensor. In the second part of this paper, we analyze two versions of this tensor in depth -- the Wyatt pressure tensor and the Kuzmenkov pressure tensor. There are different versions because there is a gauge freedom for the pressure tensor similar to that for potentials. We find that the interpretation of all quantities contributing to the Wyatt pressure tensor is understandable but for the Kuzmenkov tensor, it is difficult. Furthermore, the transformation from Cartesian coordinates to cylindrical coordinates for the Wyatt tensor can be done in a clear way, but for the Kuzmenkov tensor, it is rather cumbersome.
Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract
Abstract Bohm developed the Bohmian mechanics (BM), in which the Schrödinger equation is transformed into two differential equations: a continuity equation and an equation of motion similar to the Newtonian equation of motion. This transformation can be executed both for single-particle systems and for many-particle systems. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including one differential equation for the mass balance and two differential equations for the momentum balance, and we extended their analysis in a prework (K. Renziehausen, I. Barth in Prog. Theor. Exp. Phys. 2018:013A05, 2018) for the case that the particle ensemble consists of different particle sorts. The purpose of this paper is to show how the differential equations of MPQHD can be derived for such a particle ensemble with the differential equations of BM as a starting point. Moreover, our discussion clarifies that the differential equations of MPQHD are more suitable for an analysis of many-particle systems than the differential equations of BM because the differential equations of MPQHD depend on a single position vector only while the differential equations of BM depend on the complete set of all particle coordinates.
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.
The conserved probability current of the Schrodinger field admits a dimension-dependent gauge representation that provides a convenient setting for comparing hydrodynamic, topological and infrared structures. We use this representation to develop a unified treatment of several deformations of the Madelung momentum. BF couplings incorporate electromagnetic interactions, Berry and spin connections, projected non-abelian adiabatic data, and intrinsic phase holonomy within the same framework. In (2+1)(2+1)(2+1) dimensions, the Chern-Simons deformation can be reduced to a nonlocal functional of the wavefunction. Its Coulomb-gauge form separates into a phase-sensitive geometric contribution and a density-dependent dynamical contribution; we show explicitly how this separation changes under a general gauge transformation and recover the topological braiding phase in the well-separated adiabatic regime. For systems with a spatial boundary, we derive the equal-time symplectic structure, the associated surface charges, and their boundary algebra. Finally, after a nonlinear interaction produces the Bogoliubov acoustic branch, the radiative memory can be expressed in the dual two-form variables. Its zero-average component is generated by a residual large gauge transformation, whereas the monopole component remains independent global data. This provides a gauge-theoretic description of the memory/asymptotic-symmetry sector of the acoustic infrared structure.
We present a constructive derivation of a time-dependent deformation functional theory---a collective variable approach to the nonequilibrium quantum many-body problem. It is shown that the motion of infinitesimal fluid elements (i.e., a set of Lagrangian trajectories) in an interacting quantum system is governed by a closed hydrodynamics equation with the stress force being a universal functional of Green's deformation tensor gij{g}_{ij}gij​. Since the Lagrangian trajectories uniquely determine the current density, this approach can be also viewed as a representation of the time-dependent current-density functional theory. To derive the above theory, we separate a ``convective'' and a ``relative'' motions of particles by reformulating the many-body problem in a comoving Lagrangian frame. Then, we prove that a properly defined many-body wave function (and thus any observable) in the comoving frame is a universal functional of the deformation tensor. Both the hydrodynamic and the Kohn-Sham formulations of the theory are presented. In the Kohn-Sham formulation, we derive a few exact representations of the exchange-correlation potentials, and discuss their implication for construction of nonadiabatic approximations. We also discuss a relation of the present approach to a recent continuum mechanics of the incompressible quantum Hall liquids.