The Vacuum Polarization tensor in 1+1 dimensional space-time

TL;DR

This paper investigates the gauge invariance of the vacuum polarization tensor in 1+1 dimensional quantum field theory, revealing that regularization restores gauge invariance but affects the free field energy spectrum.

Contribution

It demonstrates that the non-gauge invariant result arises from a lower bound on free field energy and shows how regularization impacts the energy spectrum in 1+1D QFT.

Findings

Vacuum polarization tensor in 1+1D is not gauge invariant without regularization.

Regularization restores gauge invariance but introduces states with negative free field energy.

Lower bound on free field energy explains non-gauge invariant results.

Abstract

Quantum field theory (QFT) is supposed to be gauge invariant. However it has been well established that a direct calculation of the vacuum polarization tensor produces a non-gauge invariant result. In this paper it will be shown that this problem is due to the fact that there is a lower bound to the free field energy in QFT. The vacuum polarization tensor will be calculated in 1+1 dimensional space-time and is shown not to be gauge invariant. The gauge invariance of the theory can be restored through a regularization procedure which eliminates the non-gauge invariant terms. However it will be shown that this will impact the free field energy. If the free field energy is defined so that the vacuum state has an energy of zero then the impact of regularization is to introduce states whose free field energy is less than zero.