Stokes' theorem, gauge symmetry and the time-dependent Aharonov-Bohm effect

Stokes' theorem is investigated in the context of the time-dependent Aharonov-Bohm effect -- the two-slit quantum interference experiment with a time varying solenoid between the slits. The time varying solenoid produces an electric field which leads to an additional phase shift which is found to exactly cancel the time-dependent part of the usual magnetic Aharonov-Bohm phase shift. This electric field arises from a combination of a non-single valued scalar potential and/or a 3-vector potential. The gauge transformation which leads to the scalar and 3-vector potentials for the electric field is non-single valued. This feature is connected with the non-simply connected topology of the Aharonov-Bohm set-up. The non-single valued nature of the gauge transformation function has interesting consequences for the 4-dimensional Stokes' theorem for the time-dependent Aharonov-Bohm effect. An experimental test of these conclusions is proposed.


I. INTRODUCTION
In this work we investigate the interplay between Stokes' theorem and gauge symmetry in the context of the time varying Aharonov-Bohm (AB) effect [1,2]. Already the static AB effect -placing an infinite magnetic flux carrying solenoid between the slits of a quantum mechanical two-slit experiment -shows a deep interrelation between gauge symmetry and the 3-dimensional Stokes' theorem. If one allows the current and therefore the magnetic flux through the solenoid to be time dependent then one needs to take into account both the electric field (generated from E = −∂ t A) as well as magnetic field (generated from B = ∇ × A). The introduction of time dependence means that one must consider spacetime coordinates and differentials (i.e. x µ = (t, x) and dx µ = (dt, dx)) in doing the relevant integrals rather than simply spatial coordinates and differentials (i.e. x i = x and dx i = dx).
One needs to consider Stokes' theorem in 4-dimensional Minkowski space-time. We will show that the electric field coming from the time-dependent 3-vector potential, associated with the time-dependent magnetic flux in the solenoid, can be written entirely in terms of the 3-vector potential, A, or in terms of a scalar potential, φ, or some combination of the two. The relationship between these different ways of writing the electric field is through a gauge transformation A µ → A µ + ∂ µ χ. However, the gauge transformation function, χ, and the scalar potential, φ, are found to be non-single valued. This has interesting consequences for Stokes' theorem for this case. This analysis also shows that the phase shift coming from the electric field exactly cancels the time varying part of magnetic AB phase during the period when the potential is being varied with respect to time. Thus, one can experimentally test the conclusions in this work. Although there has been ample experimental confirmation of the static Aharonov-Bohm effect [3,4] there has been no definitive experimental test of the time-dependent Aharonov-Bohm effect to date. The one test that has been performed [5,6] supports the conclusions presented here that the AB phase shift does not inherit the time dependence of the magnetic flux.

II. STOKES' THEOREM USING DIFFERENTIAL FORMS
We begin by reviewing Stokes' theorem in 3 and 4-dimensions. In 3-vector notation the 3-dimensional Stokes' theorem is given by where A is a 3-vector field and the first integral is a closed line integral and the second is an area integral over S whose boundary is ∂S. In differential forms notation Stokes' theorem takes the following elegant form (for a brief review of differential forms at the level needed here see reference [7]). In (2) ω is a p-form, dω (the exterior derivative of ω) is a p + 1-form, c is a p + 1 chain and ∂c is the boundary of c, i.e. a p chain. In this work we will apply Stokes' theorem to electromagnetism so for our p-form ω we take the 1-form vector potential We have given the 1-form A in 4-vector and 3-vector notation. Throughout the paper we set c = 1. The exterior derivative of the 1-form A is the Faraday 2-form F [7] where We have written the Faraday 2-form in 4-vector and 3-vector notation. In (4) we have used the wedge products like dx ∧ dt, dy ∧ dz which are anti-symmetric under exchange of the differentials.
With this differential form notation one can write down the expression for the usual static, magnetic AB phase shift. In the case where one has a solenoid with a static current and magnetic field, the 4-vector potential becomes A µ = (0, A), i.e. the scalar potential is zero. Thus the phase AB shift, δα AB , for this case [1, 2] becomes where e is the charge of the particle. In arriving at (5) we have used Stokes' theorem (2) and the fact that B x dy ∧ dz + B y dz ∧ dx + B z dx ∧ dy = B · dS. Note that the AB phase shift, δα AB , can be written equivalently as either the closed path integral of A or the surface integral of the magnetic field, i.e. the curl of A.

III. TIME-DEPENDENT AHARONOV-BOHM EFFECT
Now we use the results of the previous section to address the case when the current and magnetic flux through the solenoid vary with time. To this end we need to write down the 4-vector potential for the solenoid. The 3-vector potential inside and outside the solenoid (which is taken to have a radius ρ = R) is and the scalar potential is normally taken as zero everywhere i.e. φ = 0. We will come back the only difference from the static case is now the magnetic field inside the solenoid is time varying. The magnetic field outside is still zero. The new feature resulting from allowing the magnetic flux to vary with time is that there is an electric field coming from E = −∂ t A.
Explicitly using (6) we find where the overdots are derivatives with respect to time. One point to note is that while (6)  (7) and (8) if the flux is linearly dependent on time i.e.
where B 0 , B 1 are constants. Later in section (IV) we will show that it is possible to consider other variations of flux other than the linear dependence of (9). This is accomplished by changing the spatial dependence from ρ ( 1 ρ ) for the fields inside (outside) the solenoid. For the sinusoidal dependence considered in section (IV) we find that the spatial dependence will be ordinary Bessel functions. However for this section it is good to keep in mind that strictly the time dependence of the flux is that given in equation (9). It turns out that this linear time dependence, although unphysical for arbitrary times (there is some practical limit to how large of a B-field one can make) it is good for illuminating the unique features of the time dependent Aharonov-Bohm effect.
We now evaluate the AB phase shift using the surface area integral of the fields i.e. c F where F is the Faraday 2-form (4). The difference from the static case, aside from the time variation of the magnetic field in (7), is that there is a time dependent electric field (8) which contributes to the phase. The AB phase in terms of the Faraday 2-form (i.e. in terms of the electric and magnetic fields) is In the second line we have written the three electric field terms from the first line as − A·dx using E = −∂ t A and performing the dt integration. Next the three magnetic terms of the first line were converted from differential form notation to 3-vector notation, B · dS. Then using B = ∇ × A and the 3-vector form of Stokes' theorem we end up with + A · dx.
Thus we find that the electric and magnetic contributions in (10) exactly cancel giving the prediction of no time-dependent phase shift for the time-dependent Aharonov-Bohm effect.
We should clarify that this cancellation is only in effect for the time-dependent part of the magnetic field. If the vector potential can be split into time-independent and time-dependent then the magnetic field also splits into time-independent and The electric field only comes from A 1 (x, t) and thus only the time-dependent magnetic field B 1 (x, t) part is canceled by the electric field. Initially one might expect that the AB phase in (10) would be time-dependent with the time-dependence coming from the time-varying magnetic flux. Instead, we find that, at least according to the Faraday 2-form/E& M field expression for the AB phase, that the AB phase is time-independent due to the cancellation of the electric and magnetic field parts. This is exactly what was found in [8] where it was shown that the standard time-  (10). To address these questions we look at the change in phase along some infinitesimal path length ∆x and some associated infinitesimal area ∆S, and show that for these infinitesimal temporal and geometrical quantities that one gets cancellation between the electric contribution in (10) and the time dependent contribution of the magnetic field. Thus adding/integrating these infinitesimal quantities up leads to the result in equation (10). For an infinitesimal time interval, ∆t, length interval ∆ and area interval, ∆S one can write (10) as In (11) we have Taylor expanded B(t) = B 0 +Ḃ∆t + O(∆t) 2 . The B 0 term gives the usual, static part of the Aharonov-Bohm phase shift; the focus of this paper is the time-dependent contribution which to first order in ∆t is given by the second term in the Taylor expansion for B(t). In (11) we have also used ∆x = (ρ∆ϕ)φ, ∆S = 1 2 (R∆ϕ)Rẑ = 1 2 ∆ϕR 2ẑ , and the expression for E out from (8). Adding up/integrating all these infinitesimal phase shifts, with ∆(δα AB ) = 0, gives the result δα AB = 0 in equation (10) i.e. the time-dependent part of the phase shift cancels. The cancellation between the electric and magnetic contributions to the phase in (11) is made more transparent in the case when the flux varies linearly as in (9). In this case one can replace ∆t by a finite time interval T and the electric contribution The magnetic contribution takes the form Comparing these two expressions one finds that the time dependent pieces (i.e. the terms ∝ B 1 T ) cancel leaving only the static contribution (i.e. the term ∝ B 0 ).
However the above analysis, as well as that in [8], leaves open the question as to the status of Stokes' theorem in the time-dependent case in regard to the one-form side of equation (2). At first glance it would seem Stokes' theorem is violated in this case which would then call into question the above analysis. In the above we have calculated the AB phase δα AB , through the right hand side of Stokes' theorem as given in (2)  The 4-vector potential in this case is given by A µ = (φ, A) = (0, A(t, x)) where the 3-vector potential is given by (6) and the scalar potential is zero. This gauge choice for A µ does give the correct magnetic (7) and electric fields (8) for this situation. But we will see that there are other gauges for which the value of c ω = A will depend on the gauge due to the non-single valued nature of the gauge transformation function which is connected with the fact that the space in this case is non-simply connected (see page 102-103 of [7] for a discussion on this point). Using φ = 0 and A(t, x) from (6) we apparently find that the AB phase shift in terms of A µ is where in going from the end of the first line to the second line we have used the Stokes' theorem in the form (1) and then used ∇ × A = B. The time dependence of the AB phase shift in (12) is exactly same as the magnetic part of the AB phase shift given in (10) where the phase shift was calculated using the Faraday 2-form. Thus from the 4-vector potential calculation in (12) it appears that we should get the usual magnetic AB phase shift but with a time dependence coming from the time dependence of the magnetic field. This is what was predicted in earlier work on the time-dependent AB effect [9]. But in this way one does not take into account the effect of the electric field. In the work [10] time dependent Berry Phases were briefly considered and it was noted that in the case of time-dependent geometric fluxes that there would be a "motive force" similar to the electromotive force in Faraday's law. However in this paper we emphasize that our analysis shows an exact cancellation of the "magnetic" and "electric" contributions to the time-dependent part of the AB phase shift.
To get an "electric" and "magnetic" cancellation of the potentials, φ , A one would need a non-zero scalar potential. In fact one can find a gauge transformation which does give scalar and 3-vector potentials which give the magnetic and electric fields from (7) (8) but for which φ = 0. We begin by noting that one can get the outside magnetic and electric fields from (7) (8) by taking the scalar potential φ out = R 2Ḃ (t)ϕ/2 and A out = 0. Taking and taking B out = ∇ × A out does give B out = 0. It will be noticed that the scalar potential is non-single valued due to the presence of the angular coordinate ϕ-dependence. Such non-single valued functions can not exist (or are pathological) in simply connected spaces, but the Aharonov-Bohm setup is non-simply connected. Because of this non-single value functions can be considered (see the discussion on page 102-103 of [7]). The non-single valued form of the potentials is connected to the form of the gauge potentials given in (6) by the following gauge transformation where K is some constant in the range 0 ≤ K ≤ 1. Note that the gauge function χ is also non-single valued and this is again connected with the fact that the space in the Aharonov-Bohm setup has non-simply connected topology. With this gauge transformation (13) the outside gauge potentials from (6) become When K = 0 the outside electric field is given purely in terms of the non-zero, single-valued 3-vector potential, and when K = 1 the outside electric field is given purely in terms of the non-zero, non-single-valued scalar potential. For K at intermediate values the outside electric field E = −∇φ out − ∂ t A out comes from a combination of the scalar and 3-vector potential. In all cases the outside magnetic field is zero. One important thing to point out is that the quantity A µ dx µ is no longer gauge invariant due to the non-single valued character of the gauge function χ from (13) which is related to the non-simply connected topology of the Aharonov-Bohm set-up. Explicitly under the gauge transformation (13) we where χ(f ) − χ(i) is the difference between the final and initial point of the path. For a closed path and a single value χ this will be zero and A µ dx µ will be gauge invariant.
However for a non-single value gauge function χ(f ) − χ(i) will not be zero for a closed path and A µ dx µ is not gauge invariant. Note that the AB phase shift as given in terms of the Faraday 2-form (10) is still gauge invariant since the electric and magnetic fields as well as the "area" dx µ ∧ dx ν are the same regardless of whether the fields come from a single valued 3-vector potential as in (6), a non-single valued scalar potential φ out = R 2 B(t)ϕ/2 or some combination of the two as in (14). The question arises is there a gauge (i.e. a choice of K in (13)) for which A µ dx µ = − 1 2 F µν dx µ ∧ dx ν ? The answer is "yes" for K = 1 2 . For this choice we have For this gauge choice of the potentials the electric field is seen to come equally from φ out and A out (16) For the gauge choice, K = 1/2, one can see that the contributions of φ out and A out to A µ dx µ cancel thus bringing the gauge potential expression for the AB phase shift into agreement with the Faraday 2-form expression for the AB gauge. For the interval ∆t the infinitesimal contributions from the scalar and vector potentials in (16) become where we have used dx → (ρ∆ϕ)φ. Adding up (i.e. integrating) these infinitesimal contributions from (17) gives A µ dx µ = 0 and we find that the 4-dimensional Stokes' theorem (i.e. A µ dx µ = − 1 2 F µν dx µ ∧ dx ν ) is now satisfied but only for the gauge K = 1 2 . In the arguments above, this gauge dependence of the 4-dimensional Stokes' theorem can be traced to the gauge dependence of A µ dx µ which arises from the non-single valued character of the gauge transformation function χ in (13). This in turn is connected with the non-simply connected topology of the Aharonov-Bohm set-up. We note that the Faraday 2-form side of the 4-dimensional Stokes' theorem, − 1 2 F µν dx µ ∧ dx ν , is gauge invariant even for the non-single valued gauge function χ from (13). Thus, one can experimentally test the correctness (or not) of the above arguments. If one performs the time-dependent AB experiment and finds that the phase shift, δα AB , is not time-dependent then the above arguments are correct; if one performs the time-dependent AB experiment and finds that the phase shift, δα AB , inherits the time dependence of the magnetic field/magnetic flux then the above arguments are not correct. However if this last case is the one selected by experiment one needs to understand why the electric field in the time-dependent case has no influence on the phase shift as given by (10).

IV. SINUSOIDAL FLUX VARIATION
Although up to now we have assumed an arbitrary time variation for the flux, B(t), as already mentioned, due to the restriction coming from Maxwell-Ampere's Law (i.e. ∇×B = ∂ t E) one is implicitly dealing with only a linearly increasing flux as given in equation (9).
While this linearly increasing flux is good for illustrating the basic features of the time dependent Aharonov-Bohm effect (in particular the claimed cancellation between the electric and magnetic contributions to the phase shift given in (10) or (11)) one might ask if more general time variations can be considered, and if so does one still have the same cancellations of the electric and magnetic contributions to the phase shift. In this section we show this is possible for the physically realistic case of sinusoidally varying fields. The reason to focus on sinusoidally varying fields is that they would be the ones most likely used experimentally to test the predictions made in this paper. To begin we will assume a more general ρ dependence for the vector potential. Previously in (6) we had taken the ρ dependence of A as ∝ ρ and ∝ 1/ρ for inside and outside the solenoid respectively. Here we assume a vector potential of the form where F (ρ) is some function of ρ and we have already put in the assumed sinusoidal time dependence with frequency ω. The direction of A is still taken to be in theφ direction.
Using this form of the vector potential in (17) to calculate the magnetic and electric fields via B = ∇ × A, E = −∂ t A and then inserting these in ∇ × B = ∂ t E we arrive at the following equation for F (ρ) where primes denote differentiation with respect to ρ. Changing variables to x = ωρ the equation (18) becomes where now the primes denote differentiation with respect to x. Equation (19) is solved by the ordinary Bessel functions of order 1 namely J 1 (x) and Y 1 (x). The solutions for the vector potentials inside and outside the solenoid now take the form where A 1 , C 1 , D 1 are constants to be determined from boundary conditions. Note that A in One final point to make in this case of a sinusoidally varying flux is that now there is a non-zero magnetic field outside the solenoid. In the previous case using A out from (6) one got B out = ∇ × A out = 0. However using A out from (20) gives B out = ∇ × A out = 0.
Thus for a sinusoidally varying field there are both E and B fields outside the solenoid which makes this conceptually different from the usual time independent Aharonov-Bohm effect. However as mentioned above the time dependent parts of the electric and magnetic contributions from (10) or (11) still cancel.

V. MULTIVALUED MONOPOLE POTENTIAL
There is another situation where one encounters non-single valued potentials and gauge transformation functions -the potentials for a magnetic monopole. The following 3-vector potential (now using spherical polar coordinates r, θ, ϕ, rather than the cylindrical coordinates, ρ, ϕ, z used in the previous section) yields a monopole magnetic field B = ∇ × A = gr/r 2 . The vector potential in (21) is single valued, but it has the usual Dirac string singularity pathology along the negative z-axis i.e. θ = π. One can also obtain a magnetic monopole field from A monopole = − g(1+cos θ) r sin θφ which has a Dirac string singularity along the positive z-axis i.e. θ = 0. These two forms of the monopole 3-vector potential are related by the gauge transformation A → A − ∇χ with χ = 2gϕ. In this case the gauge function χ is non-single valued but the two forms of the gauge potential A monopole are single valued. Thus, this is not exactly like the time-dependent AB effect of the previous section.
It is easy to see that one can also get a magnetic monopole field from the following, alternative 3-vector potential [11] [12] A monopole = − gϕ sin θ rθ , which does not have Dirac string singularity of (21) but it is non-single valued due to the ϕ dependence of A θ . The two vector potentials in (21) and (22) are related by a gauge transformation of the form Here we see that both the gauge transformation function χ in (23) and the 3-vector gauge potential (22) are non-single valued, which then is similar to the time-dependent AB effect of the previous section. In this work we just point out the similarity between the timedependent AB effect and magnetic monopoles. We will return to a detailed analysis of the monopole case in future work.

VI. SUMMARY
In this paper we have investigated the time-dependent AB effect and related issues connected with Stokes' theorem and gauge symmetry. We found that the 4-dimensional Strokes' theorem as given by (2) with ω = A and dω = dA = F is gauge dependent since ∂c A is gauge dependent. This comes about due to the non-single valued character of scalar potential φ from (14) and the non-single valued character of the gauge function χ from (13). This non-single valued character of φ and χ are the result of the topology of the Aharonov-Bohm set-up being non-simply connected [7]. Since the quantity ∂c A is gauge dependent the equality of the left and right hand sides of Stokes' theorem given in (2) will only be true for a specific gauge. In terms of the scalar and 3-vector potentials given in (14) the gauge where the left hand side and right hand side of Stokes' theorem are equal is K = 1/2. This analysis can be tested experimentally by performing the time-dependent Aharonov-Bohm experiment. If one finds that δα AB does not inherit the time-variation of the magnetic flux then the above analysis is correct; if δα AB inherits the time dependence of the magnetic field then the above analysis is not correct.
We concluded by showing that some of these issues of non-single valuedness of the gauge potentials and gauge transformation function also appear in the analysis of magnetic monopoles. We will return to a more detailed examination of this in future work.