Functional Time Evolution, Anomaly Potentials, and the Geometric Phase

TL;DR

This paper provides a geometric interpretation of the anomaly potential in the functional Schrödinger equation for quantum fields, linking it to a gauge connection and geometric phase in the space of spacetime embeddings.

Contribution

It introduces a quantum geometric perspective on the anomaly potential, connecting it to a gauge connection and geometric phase in the space of Cauchy curve embeddings.

Findings

The anomaly potential corresponds to a gauge connection on the vacuum bundle.

The geometric phase (holonomy) relates to the evolution of quantum states along embeddings.

The anomaly potential cancels the geometric phase, ensuring phase consistency in quantum evolution.

Abstract

A free quantum field in 1+1 dimensions admits unitary Schrodinger picture dynamics along any foliation of spacetime by Cauchy curves. Kuchar showed that the Schrodinger picture state vectors, viewed as functionals of spacelike embeddings, satisfy a functional Schrodinger equation in which the generators of time evolution are the scalar field energy-momentum densities with a particular normal-ordering and with a (non-unique) c-number contribution. The c-number contribution to the Schrodinger equation, called the ``anomaly potential'', is needed to make the equation integrable in light of the Schwinger terms present in the commutators of the normal-ordered energy-momentum densities. Here we give a quantum geometric interpretation of the anomaly potential. In particular, we show the anomaly potential corresponds to the expression in a gauge of the natural connection on the bundle of vacuum states over the space of embeddings of Cauchy curves into the spacetime. The holonomy of this connection is the geometric phase associated with dynamical evolution along a closed path in the space of embeddings generated by the normal-ordered energy-momentum densities. The presence of the anomaly potential in the functional Schrodinger equation provides a dynamical phase which removes this holonomy, so that there is no net phase change for quantum transport around closed loops in the space of embeddings.