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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. Moreover, 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. A related concept is the quantum fidelity, which is also shown to depend on the gauge in this paper. The gauge dependences are explicitly shown by computing the quantum metric tensor and the quantum fidelity 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 dependences have not been observed before, the results of this paper might lead to a new study of topics that are believed to be completely understood.

The main purpose for constructing the quantum metric tensor (QMT) was to define a distance in the system’s parameter space [

In general, there has been much interest in the geometrical properties of quantum systems. In [

The QMT was constructed by looking for a gauge independence [

In this paper, we use the Landau problem to show the gauge dependence of the QMT and the QF. For this reason, in Section 2 we describe the Landau problem in the symmetric gauge. Section 3 shows the QMT for one of the ground states in different gauges. While Section 4 introduces a gauge independent definition of the QMT, Section 5 shows the calculation of this new definition for the Landau problem. On the other hand, Section 6 shows the gauge dependence of the QF and explains its origin. Finally, a discussion and our conclusions are written in Section 6.

The Landau problem [

where

these

we can select the angular momentum in the z direction,

where m is a label for the angular momentum in the z direction, such that

In this case, we see that the wavefunction depends on the parameters space and the physical space x.

The QMT,

where

with this definition, the corresponding distance will be [

It is proved that the QMT is gauge invariant [

For the purpose of this paper, it is sufficient to consider only the variation of B, therefore the parameter space will be 1-dimensional, with

whereas

therefore

In order to prove the gauge dependence of the QMT, we make the calculation in different gauges. It is known [

the corresponding wave functions obey

According to the theory [

This particular

and with

Now, in Equation (13), we set

From Equation (16) and the definition of the QMT, we compute that

The presence of g in Equation (17) clearly implies gauge dependence. This gauge dependence is inherited by the distance in the parameter space. This means that we do not have a gauge independent distance in the parameter space. In the specific case of Equation (17) the distance is minimum when we work in the symmetric gauge (

If we perform a gauge transformation in the parameter space, given by

then

It has been assumed that the phase

when Equation (20) is valid, the tensor presented in Equation (6) is gauge invariant. This means that the QMT is gauge invariant when

However, some phases, and its derivatives, may depend on the physical space,

Before constructing the real gauge invariant QMT, we note that Equation (6) can be written as

or, in the representation of coordinates

because

For constructing the gauge invariant QMT, we need a function

when we perform a change of gauge given by Equation (18). With this new connection, the gauge invariant QMT will be

or

In Equations (24) and (25), we recognize the covariant derivative,

which transforms like

under a change of gauge. Using the covariant derivative, the QMT takes the form

Equation (28) defines a gauge invariant QMT. Since Equation (27) is valid, then Equation (28) will always be gauge independent. Here we can see that if, instead of Equation (27), we perform a no Abelian gauge transformation we would generalize the QMT to a no abelian QMT.

However, we need to find the correct connection that transforms like it is shown in Equation (23). The form of the new connection

Continuing with the example presented in Section 3.2, the new QMT is given by

The fact that

therefore, according to Equation (23), and using that

thus, under the transformation of Equation (18), we get

Applying Equation (32) to the state given by Equation (16), we obtain

for any gauge. That is, the QMT proposed in this paper is gauge independent.

As it was mentioned in the introduction, the QF is also useful to measure the distance between states. If the quantum system depends on n parameters

where

In Equation (35) we can see dependence with the parameter g, therefore the QF also depends on the gauge chosen. This dependence occurs for the same reason that it appears in the QMT i.e. the phase difference is not independent of the internal product. If we start with a gauge whose state vector is

if we now perform the gauge transformation given by Equation (18), the QF fidelity will take the form

again, when the phase can be taken outside of the internal product Equation (37) simplifies to

and the fidelities in different gauges coincide. However, for more general gauges, like the presented in the example studied here, we cannot take the phase outside and, therefore, the fidelities do not coincide.

We explicitly showed that the QMT and the QF depend on the gauge. This dependence is directly related to the phase difference between the wave functions in different gauges: when the change of gauge introduces a phase whose derivatives

We also proposed a real gauge invariant QMT by defining a new connection

This work was partially supported by DGAPA-PAPIIT grant IN103716; CONACyT project 237503, and scholarship 419420. We also wish to acknowledge Unidad de Posgrado, UNAM for the support and the workshop “Academic Writing” during the preparation of this paper.

Alvarez-Jiménez, J. and Vergara, J.D. (2016) Gauge Invariance, the Quantum Metric Tensor and the Quantum Fidelity. Journal of Modern Physics, 7, 1627-1634. http://dx.doi.org/10.4236/jmp.2016.713147