Lost in Normalization

TL;DR

This paper investigates how the normalization factor in 2d U(1) lattice gauge theory affects the energy spectrum and phase structure, revealing that a specific normalization choice aligns with continuum theory predictions.

Contribution

It demonstrates that the gauge-coupling dependent normalization factor influences the energy landscape and phase behavior, advocating for the  normalization as consistent with continuum limits.

Findings

The  normalization leads to a minimum in the lowest energy at a specific coupling.

The energy landscape becomes multi-valued, indicating a first-order phase transition.

The continuum spectrum is compatible only with the  normalization choice.

Abstract

The consequences of the gauge-coupling dependent normalization-factor of $1/g^{\alpha}$ in the transfer-matrix of 2d U(1) lattice gauge theory are explored. It is seen by the $\alpha=1$ choice that the lowest energy develops a minimum at coupling $g_*=1.125$, leading to a \textit{multi-valued} Gibbs energy similar to the systems with the first-order phase transition. It is argued how the $1/g$ normalization may be regarded as a lost normalization in the commonly used change of variable to the dimensionless angle-variables. Based on the continuum limit at the next-leading order and the Ostrogradsky formulation of higher-order time-derivatives theories, it is argued that the spectrum at continuum is compatible only with the $\alpha=1$ choice.