Interpolating Between the Gauge and Schr\"odinger Pictures of Quantum Dynamics

TL;DR

This paper introduces a method to interpolate between the gauge and Schr"odinger pictures of quantum dynamics by adding a local term, enabling a smooth transition that combines locality with the traditional wavefunction approach.

Contribution

The work develops an interpolation framework that connects the gauge and Schr"odinger pictures, enhancing understanding of quantum dynamics representations.

Findings

Adding a local term allows smooth interpolation between pictures.

Large coefficients recover the Schr"odinger picture.

The gauge picture's locality is explicitly maintained during interpolation.

Abstract

Although spatial locality is explicit in the Heisenberg picture of quantum dynamics, spatial locality is not explicit in the Schr\"odinger picture equations of motion. The gauge picture is a modification of Schr\"odinger's picture such that locality is explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture utilizes (1) a distinct wavefunction associated with each patch of space, and (2) time-dependent unitary connections to relate the Hilbert spaces associated with nearby patches. In this work, we show that by adding an additional spatially-local term to the gauge picture equations of motion, we can effectively interpolate between the gauge and Schr\"odinger pictures, such that when this additional term has a large coefficient, all of the gauge picture wavefunctions approach the Schr\"odginer picture wavefunction (and the connections approach the identity).