Quantum statistical-gauge geometry

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Statistical Mechanics

Background:

• Exact equilibrium constraints like sum rules and Ward identities are difficult to analyze in quantum many-body systems due to distributional local currents and anomaly-like contact terms in commutators.
• Existing methods struggle to systematically expose these constraints, hindering a complete understanding of system dynamics.

Purpose of the Study:

• To present a novel geometric organization of quantum shift symmetry in many-body systems.
• To reformulate hyperforces as covariant derivatives and force balance as a thermal Ward identity using the Kubo-Mori inner product.
• To establish a framework for deriving iterative, all-orders constraints and positivity bounds.

Main Methods:

• Viewing the local shifting superoperator as a statistical-gauge connection.
• Proving bulk flatness for compact smearings, showing generators close on a Lie bracket of vector fields without bulk Schwinger terms.
• Quantizing the diffeomorphism moment map to fix the microscopic generator, ensuring agreement between Weyl and half-density quantization.

Main Results:

• A Ward-Bianchi hierarchy of equal-time constraints derived from the closure of smeared shift generators.
• Demonstration of bulk connection flatness, leading to antisymmetry relations and positivity bounds.
• Analysis of how background electromagnetic fields deform the algebra via density-weighted curvature insertions, not bulk central extensions.

Conclusions:

• The geometric framework provides a powerful tool for understanding equilibrium constraints in quantum many-body systems.
• Global topology influences the system through holonomy, connecting to twisted boundary conditions and flux threading.
• The framework is successfully applied to fractional quantum Hall fluids, linking to the GMP-W_{∞} algebra and verified through exact diagonalization.