Gauge Covariance of the Aharonov-Bohm Phase in Noncommutative Quantum Mechanics

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 Schr\"odinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.


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 operatorX i is characterized by the relation where i = 1, 2, 3 stands for the three space coordinates and the constant θ ij is the noncommutativity parameter. Here we have taken the time direction to be commutative [X 0 ,X i ] = 0, 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 f (X) corresponds a Weyl symbol f (x), defined on the commutative counterpart of the space. This amounts to replacing the usual commutative product of functions of operators f (X)g(X) by the Moyal star-product of Weyl symbols, f (x) ⋆ g(x), where, and x are the commutative space coordinates. The canonical quantization condition between the quantum mechanical coordinateX i and momentumP i is the same as in ordinary quantum mechanics; but with the additional relations The wave function Ψ(x) now satisfieŝ All the wave functions and any operators which are dependent on the space-time coordinates should be multiplied by the star product defined above.
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 . 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.

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 m and charge e, under the noncommutative U(1) gauge group, in a magnetic field. The corresponding gauge potential is A i (i = 1, 2, 3).
In the following, we consider only the case of a time-independent background A i ( x). The noncommutative Hamiltonian is given by where P i = −i∇ i . The star U(1) gauge field strength is defined by The transition amplitude from the initial state Ψ i to the final state Ψ f , (Ψ f , e − iHt Ψ i ) is invariant under the following noncommutative gauge transformations, 5 The shift of coordinates of NCQM has previously been used in [3,5,15] Here Ψ(x) is the wave function and U(x) is defined by U(x) = e while in the commutative case, H(x) is invariant under the U(1) gauge transformation.
The propagator K t (x, y) is represented by the bi-local kernel [9, 10] Note that the action of H(x) on e iqx 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 K t (x, y) is explicitly given by where we have used the gauge transformations Here ⋆ x , ⋆ y are the star products defined with respect to x i and y i , 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 N-pieces and taking N → ∞, Here ǫ ≡ t/N and we have used the identity e − iHt 1 e − iHt 2 = e − iH (t 1 +t 2 ) . The reason why gauge covariance is lost in [9,10] is that the quantum mechanical Hamiltonian corresponding to (6) should be treated in the Weyl ordered form if we use the midpoint prescription in the pathintegral formulation. This in turn is a consequence of that the Hamiltonian contains a mixing term betweenP i andX i . This means that the short-time propagator has to be evaluated in the midpoint of x and y, and we must use In this case, the propagator is not bi-locally gauge covariant anymore 6 , We would like to stress that the propagator is bi-locally gauge covariant in the commutative case, namely, If one goes ahead with the midpoint prescription in the noncommutative case, one arrives at a phase shift δφ for an electron wave function after moving around the path C in the noncommutative space given by 6 There is another problem with the midpoint prescription in NCQM. There is an ambiguity in how to define the star product between e −i H(x)t and e iqx in the kernel. Here we have simply assumed that it is given by ⋆x.
It could also be given by ⋆ x , but this does not change the outcome. The propagator is still not bi-locally gauge covariant.
Here the component of θ is defined by θ i = ε ijk θ jk . This is the result obtained in the pathintegral 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 Here is an n-th order expansion of A i in the noncommutativity parameter θ ij . 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 = c = m = e = 1.

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 Following the procedure adopted in [3,5], the shifted coordinate and momentum satisfy Thus NCQM now reduces to ordinary quantum mechanics but with deformed Hamiltonian H(X,P ) →H(x,p). The gauge potential in the Hamiltonian can be expanded as Consequently, the noncommutative Hamiltonian H(X,P ) = 1 2 (P i + A i (X)) 2 is interpreted as the deformed Hamiltoniañ 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 A i (X) is given in the noncommutative space and it transforms as However, the potential A i (x) 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 The solution to this equation is obtained from the commutative solution through the shift of coordinates where ψ is the solution of the equation with vanishing gauge potential and p l is now the eigenvalue ofp l asp l only acts on Ψ in (24) because of the antisymmetry of θ kl . 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) This equation is solved by Here ψ(x, t) is the solution of the Schrödinger equation in the absence of the vector potential.
The integral is performed along a path C which ends in the point x.
The phase factor exp [ − i (27) is clearly gauge invariant under the U(1) gauge transformation δA i = −∂ i λ(x). The AB phase in the commutative case is evaluated as the gauge invariant magnetic field B through Stokes theorem C d ξ · A = S d S · B where the boundary of S is the closed path C. Consequently the observable is gauge invariant (see, e.g., [17,18]).

On the other hand, the Schrödinger equation in NCQM is
where all x-dependent terms are evaluated by the star product with respect to x. 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.
ψ(x, x 0 , t) is the solution of the free Schrödinger equation In the case of the AB experiment, x 0 represents the location of the source of electrons and x represents the point at which the intensity of the beam is evaluated. The free solution ψ(x, x 0 , t) can also be viewed as a wavefunction at the point (x 0 , t 0 ) from which it is taken to (x, t) by the free propagator, K free (x, t; x 0 , t 0 ).
The definition of the path-ordered exponential is This is nothing but a Wilson line in noncommutative gauge theory [19] and under NC gauge transformations it transforms as: It can be shown (see Appendix A) that this path ordered exponential satisfies the equation Let us check the Ansatz (29), starting with the r.h.s. of the NC Schrödinger equation (28), which reads: For the evaluation of (34) we shall need: where Ψ = e P ⋆ x 0 ψ and e P stands for P exp ⋆x 0 −i 1 0 ds dξ i ds A i (x 0 + ξ(s)) . The l.h.s. of the NC Schrödinger equation (28) is This is exactly H ⋆ x Ψ as in (34). Thus the Ansatz (29) satisfies The path ordered exponential (31) is hard to evaluate explicitly but it can be done for an infinitesimal closed path C l in the 1-2 plane depicted in fig.1. We can show that where ǫ ≪ 1 is the infinitesimal parameter and e 1 , e 2 are unit vectors along the directions 1 and 2. The star product is evaluated at x and the field strength is defined by (7). The result is manifestly gauge covariant. A generalization of this result to U ⋆ (N) is possible by replacing where T a are the generators of U(N).
The NCAB phase factor for a path a from x 0 to x is given by where the path a is parametrized appropriately in the line integral. In view of the gauge transformation (32), it transforms as 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 Ψ(x, x 0 , t) has to transform in the fundamental representation of U ⋆ (1), and its Hermitian conjugate, correspondingly, in the antifundamental representation, 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 Ψ(x, x 0 , t) is a solution of the NC Schrödinger equation (28) with the initial condition Ψ(x, x 0 , t 0 ) = Ψ(x 0 , t 0 ), it follows that, according to (42), the initial condition will transform under gauge transformations as On the other hand, the formal general solution of (28) can be written using the total propagator The total propagator factorizes into the free propagator and the gauge-field-dependent phase factor, such that the solution can be written as: By comparing (29) with (45), it is clear that and, in view of the fact that the free propagator does not transform under gauge transformations, while the initial solution Ψ(x 0 , t 0 ) transforms as (43), the solution ψ(x, x 0 , t) of the free Schrödinger equation will have the peculiar gauge transformation: We should emphasize out that ψ(x, x 0 , t) is not actually a genuine solution of a free Schrödinger is gauge invariant.

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 U(1) 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 U(1) 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 U(1) 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].
We will begin by considering the path ordered exponential as a continuous function of the parameter s ′ in the form This can be differentiated with respect to s ′ using the result It gives The newly obtained relation (A.9) can also be written in the form  so that the star product with respect to x 0 in (A.11) can safely be transformed into a star product with respect to x and therefore we finally have which is exactly (A.1) or (33).