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

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.