New expressions for the Aharonov–Bohm phase and consequences for the fundamentals of quantum mechanics

Appendix A. Hamilton function of electromagnetic field: explicit presentation

Here, we address to the definition of the Hamilton function given by (7)–(9) and first calculate the derivative

Further substitution of (A1) and (8) into (7) yields:

Taking into account the equalities [12]

we further derive

A similar derivation of (A3) can be also found in [23].

Appendix B. Quantum phase effects for point-like charged and dipoles and the principle of superposition of quantum phases

Here, we shortly reproduce our way to a new expression for the quantum phase of an electrically charged particle, which consists of two steps.

The first step is to derive an updated expression for the AB phase of the electric/magnetic dipole based on a relativistically consistent description of a point-like dipole in an EM field [24], [25]: a problem, which, to our surprise, still has not been solved before, though its cornerstone – the covariant expression for the Lagrangian density l of a material medium – looks evident (see, e.g., [26])

where M ^μν is the magnetization–polarization tensor.

Further integration of the Lagrangian density (B1) over the volume of a compact dipole allowed us to obtain a new covariant expression for the Lagrangian, with a subsequent derivation of a new motional equation and a new Hamilton function for the dipole [24], [25]. Its further generalization to the quantum limit with application of equation (6) yields the following new expression for the total quantum phase of a moving dipole [24], [25]

where p _0//, m _0// denote vector components collinear with a velocity of the dipole v.

In order to simplify further calculations, we will use a weak relativistic limit, corresponding to the accuracy of calculations c ⁻³, where equation (B2) can be presented in a convenient approximate form [25]

with all quantities evaluated for a laboratory observer.

Note that the first and second terms on the rhs of equation (B3) correspond to the previously known Aharonov–Casher (AC) phase [27] and the He–McKellar–Wilkens (HMW) phase [28], [29], while the remaining two terms describe new quantum phases, corresponding to the motion of an electric dipole in an electric field

and the motion of a magnetic dipole in a magnetic field

The revealed representation of the total quantum phase of a moving electric/magnetic dipole as the sum of four components (B3) makes topical the problem of determining the physical meaning of each of these components. The importance of this problem is enforced by the fact that Aharonov and Casher presented only a formal derivation of the AC phase (B4) [27] without discussing its physical meaning, whereas attempts to interpret the origin of the HMW phase by its authors [28], [29] were not satisfactory and were not widely recognized.

In our analysis of equation (B3), we assumed that all components of quantum phase of a moving dipole should have a common physical origin and be composed from quantum phases of point-like charges, entering into the dipole. We called this rule as the principle of superposition of quantum phases (SQP), and its validity looks natural due to the linearity of the fundamental equations of quantum physics.

In order to analyze co-jointly quantum phase effects for charges and dipoles, it is convenient to express the phase (B3) through potentials of EM field. Below we solve this problem consequently for each phase component entering into equation (B3).

Aharonov–Casher phase δ _mE (first term on the rhs of equation (B3))

Presenting this phase through the scalar potential ϕ, we assume for simplicity that the inductive component of the electric field is equal to zero (∂ A /∂t = 0), which is always fulfilled in observations of the AB phase, and we obtain

In addition, using the definition

(where M is magnetization), we get

where V stands for the volume of the dipole, ds is the path lement, and we have used the vector identity

Further on, we have taken into account that the volume integral ∫ V ∇ × M ϕ d V can be transformed into a surface integral by the Gauss theorem, where the magnetization M vanishes.

To evaluate the remaining integral in (B6-3), we apply the equality

where P is polarization. Assuming the case of stationary polarization, where ∂ P / ∂ t = − v ⋅ ∇ P , we further derive:

Here, for simplicity, we assume the case of a pure magnetic dipole, when the proper polarization is equal to zero. Then the polarization P in equation (B6-5) has a relativistic origin (i.e., P = v × M ₀/c) and is orthogonal to the vector d s . Hence, equation (B6-5) yields

where we have used the equality j = ρu; ρ being the charge density, and u is the flow velocity of carriers of current.

He–McKellar–Wilkens phase δ _pB (second term on the rhs of equation (B3))

This phase can be expressed in terms of the vector potential A as follows:

In the derivation of (B7), we have used the equalities

(where V is the volume of the dipole), as well as the vector identity (see Appendix A of [20])

further we used the fact that the polarization P vanishes on the surface of the dipole S, so that all integrals on the lhs of this identity are equal to zero. Finally, we applied equality ∇ × P = 0 and the Coulomb gauge (∇ ⋅ A = 0).

The phase δ _pE (the third term on rhs of equation (B3))

Presenting this phase through the scalar potential ϕ, we once again assume that the inductive component of the electric field is equal to zero (∂ A /∂t = 0). Then, using (B6-1) along with equations (B6-1), (B6-2), and (B7-3), we derive:

In the derivation of this equation, we also applied the vector identity ∇ ⋅ P ϕ = ϕ ∇ ⋅ P + P ⋅ ∇ ϕ and have taken into account that the volume integral ∫ V ∇ ⋅ P ϕ d V can be transformed into a surface integral, at which the polarization P is vanishing.

Phase δ _mB (fourth term on the rhs of equation (B3))

Using equalities (B6-2) and (B7-1), we obtain:

where we have used the vector identity ∇ ⋅ M × A = A ⋅ ∇ × M − M ⋅ ∇ × A and taken into account that the first integral on the rhs of (B9-1) is vanishing by the Gauss theorem.

Evaluating the second integral, we again assume stationary polarization, i.e., d P / d t = ∂ P / ∂ t + v ⋅ ∇ P = 0 . Hence, combining equations (B9-1) and (B6-4), we obtain

Let us show that the second integral in (B9-2) is vanishing when choosing such initial t _i and final t _f time moments, when the magnetic dipole is located outside the fields. Therefore, we have:

which is equal to zero at A (t _i ) = A (t _f ) = 0. Thus, (B9-2) yields

where we used the equality j = ρu.

Total AB phase for a moving dipole and for a moving charge

Summing up (B6)B9)–(B9) for the phase components of a moving dipole in the presence of an EM field, we obtain the total AB phase for a moving dipole

Further on, we take into account that for any system of N discrete charges e, the charge density and current density at any point r _i are respectively defined by the equations

Thus, addressing the SQP principle and comparing (B10) and (B11a-b), we obtain the velocity-dependent component of the AB phase of an electrically moving charge as the sum

which reproduces the velocity-dependent component of the AB phase of a charged particle (the last three terms of equation (11)).

In Figure 1, we schematically represent the relationship between the velocity-dependent quantum phases of dipoles and charges established through the SQP principle.

Figure 1:

Adapted from [16]. Relationship between velocity-dependent quantum phases for charged particles and moving dipoles.