[19] The question of interest now arises: “Is there a gauge choice for which the external scalar and vector potentials produced by a localized NR source vanish?”, or, in contrast, “Can the potentials influence the physics outside of a NR source region?”. Aharonov and Bohm taught us to be especially careful when addressing these questions. For this purpose, we consider next two distinct possibilities.

#### 4.1. A Communication Scenario

[20] Figure 2 illustrates schematically the first possibility. We picture a transmitting station T consisting of a hypothetical time-varying NR source that generates vanishing external fields and nonvanishing external potentials, along with a receiving station R consisting of a potential-measuring device based on the A-B effect. The question of interest is whether the receiving station R can (or cannot) acquire signals contained in the potentials produced by the NR source at the transmitting station T. It turns out (we shall show this in Section 4.2) that this question can be addressed in the usual way, i.e., by asking whether a gauge transformation of the form (refer to Equations (A1, A5, A6, A7))

exists that suppresses the potentials. The answer to the latter question is found to be “Yes”, i.e., a NR gauge exists that eliminates the potentials. In particular, for a localized NR source, the choice Φ′ = 0 for the scalar potential in Equation (12) yields from Equations (A3,10)

In this NR gauge, both potentials **A**′(**r**) and Φ′(**r**) are thus seen to vanish everywhere outside the NR source region.

[21] Note that in this context the localized NR source is very unique because of the particular form of the associated transverse vector potential **A**(**r**) in Equation (10). This feature of the NR source played a key role in the above cancellation of the associated external potentials. In contrast, for a localized radiating source, **H**(**r**) = ∇ × **A**(**r**) ≠ 0 outside the source's support. Consequently, **A**(**r**) cannot be of the form Equation (10) for a radiating source (recall that ∇ × **J**_{T}(**r**) = 0 if **r** ∉ σ).

[22] Summarizing, the above result, Equations (12,13), rules out any possibility of using the strategy illustrated in Figure 2 for new secure communications by NR potentials. This conclusion will become more evident after investigating next the more general remote sensing scenario depicted in Figure 3.

#### 4.2. A Remote Sensing Scenario

[23] Figure 3 depicts a yet more tricky scenario. Here one considers the same NR source as shown in Figure 2. However, unlike the situation in Figure 2, the electron path integrals are now allowed to “chain” the NR source. Thus they can cross potentially nonzero magnetic fluxes created in the interior of the NR source. The question of practical interest is whether one can extract NR source information contained in the source's internal magnetic fluxes by measuring the perhaps nontrivial, external NR potentials.

[24] To address this problem, one is forced to consider the path integrals (co-chains) that determine the electron phase shift associated with the A-B effect [*Lee et al.*, 1992; *Peshkin and Tonomura*, 1989; *Carron*, 1995; *Aharonov and Bohm*, 1959]. It is not hard to show from Equation (1) that for A-B experiments in the exterior of a spatially localized NR source, the A-B phase shift is determined by the closed-path integral ∮_{C}*d***x** · **A**(**x**, *t*) where *C* is a path that surrounds the NR source. The other, Φ-path integral, vanishes identically for electron paths in the exterior of a NR source. We note from Equations (A1, A5, A7) that, in general, if *C* is an A-B path that surrounds (with no loss of generality) only one NR source and *S*_{i} is an interior surface of that NR source through which one can measure the total magnetic flux Ψ_{i}(*t*), where *C*_{i} = ∂*S*_{i} is the boundary of that interior surface, then the path integral

This connects the electric field co-chain directly with the time derivative of the corresponding NR source vector potential co-chain. Since the fields vanish everywhere outside the NR source, the enclosed magnetic flux is contributed only by the internal fluxes inside the source. Figures 3 and 4 illustrate schematically the relevant A-B path and its associated enclosed magnetic flux Ψ_{i}.

[25] For a NR source, **E**(**r**, *t*) = 0 everywhere outside the source support for all times *t*. Then for any path taken outside a NR source, expression (14) reduces to the fundamental result

i.e., the co-chain of the NR source vector potential must be a constant for all time. This “static” condition is necessary and applies to the most general, time-varying, localized NR source. In particular, note that if the co-chain value is zero at *t* = 0 as it would be for the general electrodynamic case, i.e., for those cases that can vary with time, then it will be zero for all time. It is only for a static NR source that this co-chain value can be nonzero for *t* = 0, hence, for all time. Thus, only for static NR fields, i.e., those that cannot vary with time, the total internal fluxes available for the relevant noninvasive A-B experiments can be nonzero constants.

[26] The fundamental result Equation (15), therefore, tells us that it is not possible to measure time-dependent, information-carrying aspects of the external NR potentials. Consequently, no electrodynamic information about the source can be detected in the external potentials.

[27] The triviality of the NR potentials described by Equations (12, 13) is now evident in the usual A-B terms. In particular, we note that the formulation leading to Equations (12,13) implicitly assumes that the magnetic flux crossing the A-B electron paths is exactly zero. Such paths do not “invade” the vicinity of the source region (see Figure 5a). Now, since the fields vanish everywhere outside the NR source, it follows at once that the relevant A-B magnetic fluxes are zero for the situation in Figure 5a. This situation is also perfectly addressed locally, i.e., in differential form, by enforcing ∇ × **A**(**r**) = **H**(**r**) = 0 for **r** ∉ σ, as we required, in fact, in the formulation leading to Equations (12, 13).

[28] On the other hand, the situation depicted in Figures 3, 4 and 5b shows that for the remote sensing application the A-B path essentially “enters” a simply connected region σ′ enclosing the NR source support σ. In this case, the A-B measurements can involve internal fluxes of the NR source. Therefore, care must be exercised in evaluating the possible physical significance of the potentials as has been known since the time of *Aharonov and Bohm*'s [1959] original paper. In this case the approach employed in connection with Equations (12,13) is incomplete. Instead, one must investigate the A-B path integrals. By using this general approach, the possibility of observing quantum-mechanical effects of the potentials was found in the present paper (see the discussion in Equation (15)) to be very limited. In particular, only static effects were found to be potentially measurable. One concludes that the A-B effect cannot be used for communications or imaging applications, both of which require dynamic information.

[29] Finally, a connection is worth making to a paper [*Afanasiev and Stepanovsky*, 1995] that presents the opposite view. *Afanasiev and Stepanovsky* [1995] provides a number of examples of time-dependent NR sources with supposedly nonvanishing external potentials. The NR sources in *Afanasiev and Stepanovsky* [1995] are infinitesimally small, and are confined to the origin. They do not involve multiply connected regions and, therefore, cannot induce A-B effects. In other words, the question of measurability of potentials associated with such sources can be addressed directly with the gauge transformation approach presented in Equations (12,13). After some manipulations, one finds that the external potentials of the examples of *Afanasiev and Stepanovsky* [1995] vanish trivially with the NR gauge transformation in Equations (12,13). Finally, the authors of that study argue that perhaps the finite counterparts of their infinitesimal NR sources can exhibit time-dependent A-B effects. However, this contradicts the necessary static condition derived here, Equation (15). This result establishes in the most general case that A-B effects associated with NR potentials are possible only in static situations.