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International Journal of Modern Physics B

GAUGE INVARIANCE IN NON-RELATIVISTIC MANY-BODY THEORY

Jürg Martin Fröhlich, Urban M. Studer

Published Jun 1, 1992• 14 citations
Publisher

Abstract

We review some recent results on the physics of two-dimensional, incompressible electron and spin liquids. These results follow from Ward identities reflecting the U(1) em × SU(2) spin -gauge invariance of non-relativistic quantum mechanics. They describe a variety of generalized quantized Hall effects.
DOI
10.1142/s0217979292001092

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1. Introduction 2. The Pauli Equation and its Symmetries {2.1} Gauge-Invariant Form of the Pauli Equation {2.2} Aharonov-Bohm Effect {2.3} Aharonov-Casher Effect 3. Gauge Invariance in Non-Relativistic Quantum Many-Particle Systems {3.1} Differential Geometry of the Background {3.2} Systems of Spinning Particles Coupled to External Electromagnetic and Geometric Fields {3.3} Moving Coordinates and Quantum-Mechanical Larmor Theorem 4. Some Key Effects Related to the U(1)×SU(2)U(1) \times SU(2)U(1)×SU(2) Gauge Invariance of Non-Relativistic Quantum Mechanics {4.1} ``Tidal'' Aharonov-Bohm and ``Geometric'' Aharonov-Casher Effects {4.2} Flux Quantization {4.3} Barnett and Einstein-deHaas Effects {4.4} Meissner-Ochsenfeld Effect and London Theory of Superconductivity {4.5} Quantum Hall Effect 5. Scaling Limit of the Effective Action of Fermi Systems, and Classification of States of Non-Relativistic Matter 6. Scaling Limit of the Effective Action of a Two-Dimensional, Incompressible Quantum Fluid {6.1} Scaling Limit of the Effective Action {6.2} Linear Response Theory and Current Sum Rules {6.3} Quasi-Particle Excitations and a Spin-Singlet Electron Pairing Mechanism 7. Anomaly Cancellation and Algebras of Chiral Edge Currents in Two- Dimensional, Incompressible Quantum Fluids {7.1} Integer Quantum Hall Effect and Edge Currents {7.2} Edge Excitations in Spin-Polarized Quantum Hall Fluids 8. Classification of Incompressible Quantum Hall Fluids {8.1} QH Fluids and QH Lattices: Basic Concepts {8.2} A Dictionary Between the Physics of QH Fluids and the Mathematics of QH Lattices {8.3} Basic Invariants of Chiral QH Lattices (CQHLs) and their Physical Interpretations {8.4} General Theorems and Classification Results for CQHLs {8.5} Maximally Symmetric CQHLs {8.6} Summary and Physical Implications of the Classification Results
I point out two of the subtleties referred to in the title. The first is that gauge-invariant magnetic systems may realized under general circumstances, as suggested by a simple theorem. The second subtlety is that care is needed to identify the field theory simulated by a cold-atomic lattice gauge system. Though the simplest such model confines in 2+1 dimensions, it has non-relativistic ``gluon" excitations. Time-reversal invariance is spontaneously broken in this system. The confinement mechanism is related to an extra U(1) gauge invariance.There is a model, suggested long ago by D. Rohlich and me, which is known to have relativistic spin waves. One of the outstanding theoretical problems is a better determination of the energy-momentum relation of spin waves in different magnetic gauge systems.
In a Lagrange formulation of nonrelativistic quantum mechanics gauge invariance corresponds to a local U(1) symmetry of the Lagrangian density. We review how this gauge symmetry yields the electromagnetic interaction of the free and the locally interacting N-body system and why nonlocal interactions require additional exchange contributions. A nonrelativistic Ward-Takahashi identity is then derived. This relates electromagnetic operators to the full propagator of the particles in question, which may be off their energy shell.