Gauge Invariance, the Quantum Metric Tensor and the Quantum Fidelity

Abstract

The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. More over, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. In this paper, the gauge dependence is explicitly shown by computing the quantum metric tensor of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependence has not been observed before, the results of this paper might lead to a new study of topics that were believed to be completely understood.

Classical analog of the quantum metric tensor

“…( 22) also depends on the angle variables, this motivated by the work of Ref. [33]. Another interesting and useful future consideration is how to generalize the metric (19) for a classical field theory.…”

“…In Ref [33]. is shown, however, that under a more general gauge transformation, the quantum metric tensor depends on the gauge.…”

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

“…Of course, it is possible to consider a less restrictive and more general gauge transformation (see Ref. [29], for instance). It is also worth noting that only the real part of the phase space function A (n) i contributes to the Berry connection (24), which is a consequence of Eq.…”

“…where α n is an arbitrary real function of the parameters, meaning that we can choose the basis |n up to a U(1) gauge transformation (see also reference [29]). The relevance of this tensor lies in the fact that it provides the fundamental structures underlying the parameter space.…”

“…This is a consequence of the fact that, while the gauge transformation (2) is restricted to the parameter space, A (n) i (q, p; x) is a mixed structure of the parameter space and the physical phase space. Of course, it is possible to consider a less restrictive and more general gauge transformation (see reference [29], for instance). It is also worth noting that only the real part of the phase space function A (n) i contributes to the Berry connection (24), which is a consequence of equation ( 23) and the fact that…”