Gauge Covariance of the Aharonov-Bohm Phase

in Noncommutative Quantum Mechanics

Masud Chaichian^{1}^{1}1,
Miklos Långvik^{2}^{2}2,
Shin Sasaki^{3}^{3}3 and Anca Tureanu^{4}^{4}4

Department of Physics, University of Helsinki,

and Helsinki Institute of Physics

P.O. Box 64, FIN-00014 Helsinki, Finland

Abstract

The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov-Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov-Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schrödinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.

## 1 Introduction

In the recent decade, there has been a lot of interest in the study of physics on a noncommutative space-time due to the fact that space-time may exhibit its noncommutativity at the scale of quantum gravity. Especially, string theory, which is considered as the most promising candidate for a theory of quantum gravity, gives rise to space-time noncommutativity [1]. Apart from the string theory motivation, it is interesting to investigate the space-time noncommutativity in a more familiar set-up, like quantum mechanics. Especially, since the result [2], combining Heisenberg’s uncertainty principle with Einstein’s theory of classical gravity, is quantum mechanical in spirit, the purely quantum mechanical treatment of a noncommutative space-time becomes interesting. In [2] one considers a gedanken experiment at very high energy where the high density of the energy-momentum tensor would result in the formation of black holes through the Einstein equations. In this case it would no longer be possible to measure lengths up to arbitrary precision, but space-time would become noncommutative in a similar way as phase-space becomes noncommutative in quantum mechanics.

Various approaches to quantum mechanics on noncommutative space-time have been proposed in [3, 4, 5, 6]. Its space coordinate operator is characterized by the relation

(1) |

where stands for the three space coordinates and the constant is the noncommutativity parameter. Here we have taken the time direction to be commutative , due to the problems with unitarity [7] and causality [8] for a noncommuting time direction. We represent the noncommutativity of space coordinates through the Weyl-Moyal correspondence, in which to each function of operators corresponds a Weyl symbol , defined on the commutative counterpart of the space. This amounts to replacing the usual commutative product of functions of operators by the Moyal star-product of Weyl symbols, , where,

(2) |

and are the commutative space coordinates. The canonical quantization condition between the quantum mechanical coordinate and momentum is the same as in ordinary quantum mechanics;

(3) |

but with the additional relations

(4) |

The wave function now satisfies

(5) |

All the wave functions and any operators which are dependent on the space-time coordinates should be multiplied by the star product defined above.

In the context of noncommutative quantum mechanics (NCQM), many observable quantities have been studied. They include the Aharonov-Bohm (AB) effect [9, 10, 11], the hydrogen atom spectrum and the Lamb shift [5, 12], the Hall effect [13], the Aharonov-Casher effect [14] and so on.

Since all the observables in quantum mechanics should be gauge invariant quantities, it is important to examine the gauge invariance of physical quantities in NCQM. For instance, the gauge invariance (or covariance) of the phase factor of a wave function is directly related to many of the physical observables, such as, the Aharonov-Bohm effect, the Aharonov-Casher effect and the Berry phase.

In this letter, we show that the naive
path integral formulation of NCQM and an approach
where one shifts the coordinates of NCQM [11]
lead neither to a gauge invariant nor to a gauge covariant Aharonov-Bohm phase factor^{5}^{5}5The
shift of coordinates of NCQM has previously been used in [3, 5, 15].
Instead, we propose a gauge covariant formulation of the AB phase
which is consistent with the noncommutative Schrödinger equation.

The organization of this letter is as follows. In section 2, we introduce the path integral formulation of NCQM following the result of [9, 10] especially focusing on the gauge covariance of the formulation. We shall stress the difference between the commutative and noncommutative cases and point out how gauge covariance is broken in the noncommutative case. Section 3 is devoted to another approach to NCQM where one shifts the coordinates to satisfy the usual commutation relations of ordinary quantum mechanics. This approach also breaks gauge invariance but preserves some exotic kind of gauge invariance. In section 4, we propose a gauge covariant AB phase factor which is represented by the path-ordered exponential and is consistent with the Schrödinger equation. Section 5 contains summary and discussion.

## 2 Path integral formulation of NCQM

In this section, we introduce the path integral formulation of NCQM following the derivation of [9, 10]. We consider a particle with mass and charge , under the noncommutative gauge group, in a magnetic field. The corresponding gauge potential is . In the following, we consider only the case of a time-independent background . The noncommutative Hamiltonian is given by

(6) |

where . The star gauge field strength is defined by

(7) |

The transition amplitude from the initial state to the final state , is invariant under the following noncommutative gauge transformations,

(8) |

Here is the wave function and is defined by with a real function . The star element satisfies . The Hamiltonian transforms covariantly under the gauge transformation,

(9) |

while in the commutative case, is invariant under the gauge transformation.

The propagator is represented by the bi-local kernel [9, 10]

(10) |

Note that the action of on is via the star-product defined in (2). This propagator is bi-locally gauge covariant provided the Hamiltonian transforms as in (9). The naive gauge transformation of is explicitly given by

(11) | |||||

where we have used the gauge transformations

Here are the star products defined with respect to and , respectively. This bi-local covariance guarantees the gauge invariance of the probabilities, and should provide the gauge covariant AB phase in the path-integral formulation of NCQM.

The propagator can be represented by the products of short-time propagators in the infinite time evolution by separating the time interval into -pieces and taking ,

(12) |

Here and we have used the identity

(13) | |||||

We would like to stress that the propagator is bi-locally gauge covariant in the commutative case, namely,

(14) |

If one goes ahead with the midpoint prescription in the noncommutative case, one arrives at a phase shift for an electron wave function after moving around the path in the noncommutative space given by

(15) |

Here the component of is defined by . This is the result obtained in the path-integral formulation in the midpoint prescription [9, 10]. The same result has been obtained by the perturbative analysis of the Schrödinger equation [16].

We can explicitly check that this result is neither gauge invariant nor covariant under the gauge transformations

(16) |

Here is an -th order expansion of in the noncommutativity parameter . As we mentioned, this gauge non-covariance originates from the Weyl ordering of the quantum mechanical Hamiltonian and hence, from the midpoint prescription in the path-integral. In the next section, we will use another approach to derive the the AB phase in NCQM. From here on, for simplicity, we shall use .

## 3 The phase shift in terms of a shift of coordinates

It is known that the noncommutativity of space in quantum mechanics can be interpreted as ordinary quantum mechanics with deformed Hamiltonian. This deformation can be performed via a shift of coordinates [3, 5, 15].

Consider quantum mechanics on a noncommutative space, with the commutation relation among coordinate and momentum operators as

(17) |

Following the procedure adopted in [3, 5], the shifted coordinate and momentum

(18) | |||||

(19) |

satisfy

(20) |

Thus NCQM now reduces to ordinary quantum mechanics but with deformed Hamiltonian . The gauge potential in the Hamiltonian can be expanded as

(21) |

Consequently, the noncommutative Hamiltonian is interpreted as the deformed Hamiltonian

(22) |

in ordinary quantum mechanics. The Hamiltonian (22) is no longer star-gauge covariant as a consequence of shifting the coordinates. This is because the potential is given in the noncommutative space and it transforms as

(23) |

However, the potential is not given in this type of noncommutative space, but the ordinary quantum mechanical one, and consequently does not transform similarly to (23). Therefore, the star gauge covariance of the Hamiltonian is lost in (22).

The Schrödinger equation corresponding to (22) is

(24) |

The solution to this equation is obtained from the commutative solution through the shift of coordinates

(25) |

where is the solution of the equation with vanishing gauge potential and is now the eigenvalue of as only acts on in (24) because of the antisymmetry of . It was shown [11] that the phase shift in this solution is equivalent to the path integral result obtained in [9, 10], i.e. equation (15) in the previous section and thus is neither gauge invariant nor covariant.

A comment is in order about the gauge invariance of this approach. In view of the shifted coordinate, the Hamiltonian and any physical observables are manifestly invariant under the coordinate shifted gauge transformation but not under the ordinary star gauge transformation. Here the coordinate shifted gauge transformation is defined by the commutative gauge transformation evaluated in the shifted coordinate .

## 4 The gauge covariant phase factor: the Wilson loop

In this section we propose a gauge covariant phase factor which can be obtained with the help of the Wilson loop operator. Let us first consider the AB phase in commutative quantum mechanics. The Schrödinger equation in the presence of a time independent vector potential is

(26) |

This equation is solved by

(27) |

Here is the solution of the Schrödinger equation in the absence of the vector potential. The integral is performed along a path which ends in the point .

The phase factor in (27) is clearly gauge invariant under the gauge transformation . The AB phase in the commutative case is evaluated as the gauge invariant magnetic field through Stokes theorem where the boundary of is the closed path . Consequently the observable is gauge invariant (see, e.g., [17, 18]).

On the other hand, the Schrödinger equation in NCQM is

(28) |

where all -dependent terms are evaluated by the star product with respect to . We recall that a gauge invariant quantity in a non-Abelian gauge theory is the Wilson loop. Wilson loops have been previously used in the context of noncommutative gauge field theories for constructing observable quantities, as well as new representations of the noncommutative gauge groups, forbidden by the no-go theorem of noncommutative gauge theories (see e.g. [19, 20, 21] and references therein). They are defined by the gauge trace of the path-ordered exponential. Inspired by this, we consider the Ansatz for the solution to (28) as

(29) |

Here the symbol P stands for path ordering. The parameter parametrizes the path with endpoints and , where and . is the solution of the free Schrödinger equation

(30) |

In the case of the AB experiment, represents the location of the source of electrons and represents the point at which the intensity of the beam is evaluated. The free solution can also be viewed as a wavefunction at the point from which it is taken to by the free propagator, .

The definition of the path-ordered exponential is

(31) | |||||

This is nothing but a Wilson line in noncommutative gauge theory [19] and under NC gauge transformations it transforms as:

(32) |

It can be shown (see Appendix A) that this path ordered exponential satisfies the equation

(33) |

Let us check the Ansatz (29), starting with the r.h.s. of the NC Schrödinger equation (28), which reads:

(34) |

For the evaluation of (34) we shall need:

(35) | |||||

(36) | |||||

where and stands for .

The l.h.s. of the NC Schrödinger equation (28) is

(37) | |||||

This is exactly as in (34). Thus the Ansatz (29) satisfies

(38) |

The path ordered exponential (31) is hard to evaluate explicitly but it can be done for an infinitesimal closed path in the 1-2 plane depicted in fig.1. We can show that

(39) | |||||

where is the infinitesimal parameter and are unit vectors along the directions 1 and 2. The star product is evaluated at and the field strength is defined by (7). The result is manifestly gauge covariant. A generalization of this result to is possible by replacing by , where are the generators of .

The NCAB phase factor for a path from to is given by

(40) |

where the path is parametrized appropriately in the line integral. In view of the gauge transformation (32), it transforms as

(41) |

under a gauge transformation.

The path-ordered phase factor appearing here is quite similar to the non-Abelian counterpart of the AB phase [22]. This would be related to the topological features of the phase factor which will be studied elsewhere [23].

One important consistency check for the Ansatz (29) is its gauge covariance. The wave function has to transform in the fundamental representation of , and its Hermitian conjugate, correspondingly, in the antifundamental representation,

(42) | |||||

(43) |

in order to insure the gauge covariance of the NC Schrödinger equation. One can show that the gauge transformation (32) of the path ordered exponential is compatible with this gauge covariance requirement. Indeed, since is a solution of the NC Schrödinger equation (28) with the initial condition , it follows that, according to (42), the initial condition will transform under gauge transformations as

(44) |

On the other hand, the formal general solution of (28) can be written using the total propagator :

(45) |

The total propagator factorizes into the free propagator and the gauge-field-dependent phase factor, such that the solution can be written as:

(46) |

By comparing (29) with (46), it is clear that

(47) |

and, in view of the fact that the free propagator does not transform under gauge transformations, while the initial solution transforms as (44), the solution of the free Schrödinger equation will have the peculiar gauge transformation:

(48) |

We should emphasize out that is not actually a genuine solution of a free Schrödinger equation, but an artifact of the factorization of the total propagator as in (46). In other words, from the dynamical point of view satisfies the free Schrödinger equation, while inheriting at the same time the gauge transformation property (44) of the initial solution of (28).

The gauge transformations (32) and (48) provide the consistency check for the gauge covariance of defined by the Ansatz (29). As a result, the noncommutative Schrödinger equation (28) is covariant under a noncommutative gauge transformation. This guarantees that the observable probability density , for the AB-effect of two waves differing by a phase depending on the paths or ,

(49) | |||||

is gauge invariant.

## 5 Summary and discussion

In this letter, we have studied the gauge covariance of the wave function phase factor in the framework of NCQM.

Due to the fact that the phase factor in a wave function is frequently related to a physical observable, it is important to investigate the gauge invariance and covariance of it in NCQM. The AB phase factor is probably the most familiar observable phase factor in quantum mechanics.

The naive path-integral formulation of NCQM violates the star gauge covariance of the AB phase. The origin of this violation comes from the Weyl ordered quantum mechanical Hamiltonian and midpoint prescription in the short-time propagator. This is quite different from the commutative case where the Hamiltonian itself is gauge invariant and hence the propagator is bi-locally gauge covariant.

The same result is obtained by shifting the coordinates of NCQM, whence the star gauge invariance/covariance is broken. However, some exotic gauge invariance, the ”shifted gauge invariance” (See end of section 3) is preserved although the physical meaning of this type of gauge invariance is not clear.

We have found a gauge covariant AB phase factor which is defined by the path-ordered exponential. This resembles the well-known Wilson loop in non-Abelian gauge theory. We have shown that the path-ordered exponential is consistent with the noncommutative Schrödinger equation. We would like to stress that our result is quite similar to the non-Abelian AB phase proposed in [22]. This is very natural because the star gauge symmetry is essentially non-Abelian, which can be seen from eq. (7).

The AB phase factor is related to the Dirac monopole quantization
and topological properties of the theory and it would be
interesting to find the gauge invariant quantization condition
corresponding to the noncommutative Dirac monopole, especially due
to the results in [24] on noncommutative monopoles, dyons and
solitonic solutions. It would also be interesting to investigate the star
gauge invariant path-integral formulation of NCQM [23].

Acknowledgments

We are indebted to Masato Arai, Peter Prešnajder and Sami Saxell for discussions and useful comments.
The work of S. S. is supported by the bilateral program of Japan Society
for the Promotion of Science (JSPS) and Academy of Finland, “Scientist
Exchanges.” A. T. acknowledges the grant no. 121720 of the Academy of Finland.

## Appendix A

In this appendix the relation

(A.1) |

where

(A.2) | |||||

is proven. The parametrization of the path is as follows: and , so that and .

We will begin by considering the path ordered exponential as a continuous function of the parameter in the form

(A.3) | |||||

This can be differentiated with respect to using the result

(A.4) |

It gives

(A.8) | |||||