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Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field
Accepted Manuscript Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field A.L. Kholmetskii, O.V. Missevitch, T. Yarman
PII: DOI: Reference:
S0003-4916(18)30060-5 https://doi.org/10.1016/j.aop.2018.03.005 YAPHY 67612
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Annals of Physics
Received date : 3 January 2018 Accepted date : 5 March 2018 Please cite this article as: A.L. Kholmetskii, O.V. Missevitch, T. Yarman, Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field, Annals of Physics (2018), https://doi.org/10.1016/j.aop.2018.03.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field A.L. Kholmetskii1, O.V. Missevitch2 and T. Yarman3 1
Department of Physics, Belarus State University, 4 Nezavisimosti Avenue, 220030 Minsk, Belarus Institute for Nuclear Problems, Belarus State University, 11 Bobruiskaya Str., 220030 Minsk, Belarus 3 Department of Engineering, Okan University, Akfirat, Istanbul, Turkey & Savronik, Eskisehir, Turkey 2
Abstract We point out that the known quantum phases for an electric/magnetic dipole moving in an electromagnetic (EM) field must be presented as the superposition of more fundamental quantum phases emerging for elementary charges. Using this idea, we find two new fundamental quantum phases for point-like charges, next to the known electric and magnetic Aharonov-Bohm (A-B) phases, named by us as the complementary electric and magnetic phases, correspondingly. We further demonstrate that these new phases can indeed be derived via the Schrödinger equation for a particle in an EM field, where however the operator of momentum is re-defined via the replacement of the canonical momentum of particle by the sum of its mechanical momentum and interactional field momentum for a system “charged particle and a macroscopic source of EM field”. The implications of the obtained results are discussed. 1. Introduction To the end of past century, four quantum phase effects had been found, which are characterized correspondingly by the electric Aharonov-Bohm (A-B) phase [1] e dt , (1) the magnetic A-B phase [1] e A A ds , (2) c the Aharonov-Casher (A-C) phase [2] 1 mE m0 E ds , (3) c and the He-McKellar-Wilkens (HMW) phase [3, 4] 1 pB p0 B ds , (4) c where ds=vdt is the path element of a charged particle e, moving with the velocity v. The electric (1) and magnetic (2) A-B effects emerge for charged particle in the field of scalar potential and vector potential A, correspondingly; the A-C phase (3) is associated with the hidden magnetic momentum [5-7] of a point-like magnetic dipole m0, moving in an electric field E, while the HMW phase (4) is defined via the hidden electric momentum (according to the terminology introduced in refs. [8, 9]) of a point-like electric dipole p0, moving in a magnetic field B. The vectors m0 and p0 are measured in the rest frame of a dipole, and hereinafter we adopt that the dipoles are electrically neutral.
2
In addition to the phases (1)-(4), recently we have found two new quantum phase effects [8, 9], emerging at the motion of an electric dipole in an electric field, 1 pE 2 p0 // E v ds , (5) c and at the motion of a magnetic dipole in a magnetic field, 1 mB 2 m0 // B v ds , (6) c where is the Lorentz factor, and p0 // , m0 // stand for the component of electric (magnetic) dipole moment collinear with the vector v. We notice that A-C (mE) and HMW (pB) phases, as well as the new phases pE, mB are related to each other via electric-magnetic duality transformations EB, pm [10], and it is comfortable to recognize that the four phases (3)-(6) correspond to all possible combinations of the pair p, m with the pair E, B. The revealing of four quantum phase effects (3)-(6) for moving dipoles, took place during past decades, definitely requires clarifying their physical meaning. Analyzing this problem, we will consider the electric/magnetic dipole as a compact electrically neutral bunch of finite size, composed from elementary charges. This representation suggests that each of the phases (3)-(6) should be explained at the fundamental level via an appropriate phase effect, emerging for every elementary charge of the bunch. As is known, this idea actually works well with respect to the HMW phase (4). Indeed, using the simplest model of the electric dipole p0 (two elementary charges –e and e, separated by a small proper distance d), one can show that the HMW phase for such a dipole can be presented as an algebraic sum of magnetic A-B phases (2) for each charge, composing the dipole, and moving in the presence of vector potential A. In other words,
pB
e A r d d s A r d s L c L
(7)
(r is the radial coordinate), where the path of positive charge is designated as L+, while the path of negative charge is designated as L-. Indeed, introducing the proper polarization P0 of the bunch (so that p0 P0 dV , V being the proper volume of the bunch), we can further write:
pB
1 e A r d d s A r d s A P0 dsdV L c V c L
1 P0 A dsdV 1 P0 B dsdV 1 p0 B ds , c V c V c
which coincides with eq. (4). Here we have used the vector identity [11]
A P dS P A dS AP dS A P dV P AdV 0
0
S
0
S
S
A P dV P AdV ; 0
V
0
V
0
V
0
V
3
then, we have taken into account that the polarization P0 is vanishing on the surface of dipole (so that all integrals in the lhs of this identity are equal to zero), and also used the equalities P0 0 , A B , applying the Coulomb gauge ( A 0 ). At the same time, one can easily realize that the A-C phase (3), as well as the new phase effects (5), (6) cannot be reduced to either electric (1), or magnetic (2) A-B effects for point-like charges composing a dipole. Thus, the problem of physical interpretation of quantum phases (3), (5), (6) remains unsolved. In section 2, we analyze the origin of the phases (3), (5), (6) and show that they indeed can be presented as the composition of new phases for point-like charges composing the dipoles. We named these new quantum phases as the complementary magnetic and electric phases and simultaneously emphasized that these phases cannot be derived either from the Schrödinger equation, or from the Dirac equation for a particle in an electromagnetic (EM) field, which actually may indicate the presence of some missed points in our understanding of quantum phase effects. In section 3 we assert that such missed points actually exist; moreover, they have the fundamental character and are related to the presently adopted procedure for the transition from the classical to quantum description of point-like charges in an EM field. It is related to the definition of operator of momentum, which is commonly related with the canonical momentum of particle in an EM field. However, we show that this relation is, in fact, meaningless from the physical viewpoint. Hence, we suggest redefining the operator of momentum as the sum of mechanical momentum of particle and interactional field momentum for a system “particle and a source of EM field”. Further, we apply the Hamiltonian with the modified definition of the momentum operator to the solution of the Schrödinger equation and actually derive the complementary magnetic and electric phases. By such a way we close the logical chain of our approach, showing that each of the phase effects for dipoles can be explained via the corresponding fundamental phase effect for elementary charge. We discuss and conclude in section 4. 2. Fundamental quantum phases for point-like charges and derivative quantum phases for electric/magnetic dipoles In order to proceed further, we first present the general expression for the quantum phase of an electrically neutral dipole in an EM field
1 1 1 1 1 1 p0 // E v ds 2 m0 // B v ds m0 E ds p0 B ds p E dt m B dt , c 2 c c c
(8) obtained in refs. [8, 9] on the basis of the Lagrangian approach with the interactional Lagrangian 1 density L M F integrated to point-like dipole. (Here M is the magnetization2 polarization tensor, F is the tensor of EM field, and , =0…3). The last two terms on rhs of eq. (24) define the Stark phase [12] and Zeeman phase [13], correspondingly, while the other terms are associated with the moving dipole and presented separately by eqs. (3)-(6). In the present contribution, we omit the last two terms on the rhs of eq. (8), which do not explicitly depend on the velocity of dipole, and deal with the weak relativistic limit, where the remaining four
4
terms in this equation will be presented in the convenient form to the accuracy of calculations c-3 sufficient for our analysis. In this limit, we apply the approximate relationships [10] vm pv p p0 , m m0 (9a-b) c c between the electric p and magnetic m dipole moments for a moving dipole with the corresponding proper values p0, m0. Combining eqs. (9a-b) and (8), we obtain
1 c 2
p
//
E v ds
1 c 2
m
//
B v ds
1 1 1 1 m E p v E ds p B v m B ds. c c c c
(10) Using the equalities p v E E v p vE p p E v , B v m vB m m B v
(where we applied the vector identity a b c ba c ca b ), and taking into account that
p p// p , m m// m , p ds 0 , m ds 0 , we derive 1 p E v ds 1 2 m B v ds 1 m E ds 1 p B ds. (11) 2 c c c c Thus, in the weak relativistic limit, we present the phase (11) via the values measured in a laboratory frame, which will be used below in the analysis of physical meaning of its terms. With respect to the last term of eq. (11) (the HMW phase), we already have shown that this phase is defined as the sum of magnetic A-B phases for each elementary charge composing the electric dipole, eq. (7). Next, we consider one more phase effect for a moving electric dipole, which is presented by the first term on the rhs of eq. (11), i.e. 1 pE 2 p E v ds . (12) c Hereinafter we assume that the inductive component of electric field is equal to zero, and apply the equality E . Substituting this equality into eq. (12), and using the definition
p PdV , we derive: V
pE
1 P v dsdV 1 2 P v dsdV 1 2 v dsdV , 2 c V c V c V
(13)
where we have used the vector identity P P P . The first integral on rhs of eq. (13) is vanishing by the Gauss theorem; in the second integral we have used the equality P , where is the density of bound charges. Introducing again the model of the electric dipole defined above (two elementary charges –e and +e separated by a small distance d), we thus derive 1 1 (14) pE 2 v dsdV 2 e r d e r v ds. c V c Eq. (14) indicates that the quantum phase (12) for an electric dipole, moving in an electric field, originates from a more fundamental quantum phase effect, emerging for the elementary charge e, moving in the field of scalar potential at velocity v, i.e.
5
c
1 ev ds , c 2
(15)
and the phase pE represents the sum of elementary phases (15) for each charge. We see that the phase (15), like the electric A-B phase (1), depends on the scalar potential , though in the adopted weak relativistic limit, the phase (15) is about (v/c)2 smaller than the phase (1). We suggest naming the phase (15) as the complementary electric phase, introducing for it the subscript “c”. Next, we consider the A-C phase, presented by the third term on rhs of eq. (11). Analyzing this term, we introduce the magnetization M of a compact bunch of charges, and use the equality m MdV . As before, we adopt that the external electric field E does not contain an inductive component, and is defined as E . Substituting this equality into the term of A-C phase in eq. (11), we present it as the function of scalar potential: 1 1 1 mE M dsdV M dsdV M dsdV , (16) c V c V c V
where we have used the vector identity M M M . We see that the first integral on rhs of eq. (16) is vanishing under integration over infinite volume. In the evaluation of the second integral we will use the equality [10] (17) j c M P t , and assume that the polarization is stationary, i.e. dP dt P t v P 0 . Hence,
P t v P . Substituting this equality into eq. (16), we derive
c M j v P . Further substitution of eq. (18) into eq. (16) yields: 1 mE 2 j v P dsdV . c V
(18)
(19)
Evaluating this integral, we adopt for simplicity the case of a pure magnetic dipole, where the proper polarization P0=0. In this case, the polarization P in eq. (19) has a relativistic origin, i.e. P v M 0 c , and is orthogonal to the vector ds. Hence, eq. (19) takes the form
mE
1 1 j dsdV 2 u dsdV , 2 c V c V
(20)
where we have used the definition of current density j=u, being the charge density of carriers of current, and u their flow velocity. Eq. (20) shows that the A-C phase for a moving magnetic dipole represents the sum of complementary electric phases (15) for carriers of current of the dipole, characterized by the flow velocity u. We can clarify better the physical meaning of eq. (20), if we imagine a magnetic dipole as a small electrically neutral rectangular conducting loop, lying in the plane xy, and moving with the velocity v along the axis x in the presence of electric field E, lying along the y-axis. One can see that the contributions of charges of the left and right sides of the loop to the total complementary electric phase mutually cancel each other, while for the upper and lower sides we obtain from eq. (20):
6
1 upQup lowQlow udx . (20a) c 2 Here up, low are the scalar potentials on the upper and lower segments of the loop, correspondingly, and Qup, Qlow are the charges of carriers of current on these segments. We also used the Galilean law of velocity composition for the carriers of current (where their total velocity u=v+u), which is sufficient within the adopted accuracy of calculations c-3, while for positive ions, immovable in the rest frame of the loop, the velocity is equal to v in all its segments. We also notice that in the adopted accuracy of calculations, it is sufficient to put Qup=QlowQ. Taking also into account that for a small rectangular loop with the length l of each segment, E=-(up-low)/l, we obtain from eq. (20a) 1 1 1 mE 2 QEludx 2 EIl 2 dx mEdx c c c (where I is the current in the loop), which is just the A-C phase for the particular problem in question. Finally, we address to the second term on rhs of eq. (11), which defines the phase 1 mB 2 m B v ds . (21) c
mE
Using the equalities B A , m MdV , we derive:
mB
1 M Av dsdV 1 2 M Av dsdV 2 c V c V
1 A M v dsdV . c 2 V
(22)
The first integral on the rhs of eq. (22) is vanishing due to the Gauss theorem, while in the second integral we use the equality (18). As a result, we obtain 1 (23) mB 3 A j A v P v dsdV . c V We again assume that the external electric field does not contain inductive component (i.e., A t 0 ), and the vector potential A does not explicitly depend on time, i.e. dA dt v A . Adopting also that the polarization P is stationary (when dP dt v P ), we transform the second integral to the form d dA 2 2 2 (24) V A v P v dtdV v V dt A P dtdV v p dt dt . Assuming that at the initial and final time moments the magnetic dipole is located outside the fields, where A is vanishing, we see that the integral (24) is equal to zero, and we finally obtain 1 1 (25) mB 3 j Av dsdV 3 u Av dsdV , c V c V where we have used the equality j=u. In order to clarify the origin of the phase effect (25), we notice that for elementary charge e, moving at some velocity v, the current density is j r r0 ev . Substituting this equality into eq. (25), we obtain for the elementary charge
7
e v Av ds . (26) c 3 Eq. (26) discloses one more fundamental quantum phase effect, emerging for a charge moving in the field of vector potential A. We thus have shown that the phase mB (eq. (21)) for a magnetic dipole, moving in a magnetic field, represents the superposition of fundamental phases (26), which we name as the complementary magnetic phase, attributing to it the subscript “cA”. We can better clarify the relationship between the phase mB for a magnetic dipole in a magnetic field, and the complementary magnetic phase (26), considering again the magnetic dipole as a small electrically neutral rectangular conducting loop, lying in the plane xy, and moving with the velocity v along the axis x. We assume that the magnetic field B is orthogonal to the plane of the loop, and is directed along the axis z. Then, the contribution of charges of the upper and lower sides of the loop to the total complementary magnetic phase (26), by analogy with eq. (20a), can be written as mB lower upper v 3 Ax up Qupu Ax lowQlowu dx v 3 Q Ax ludx , (27) c c y
cA
where we have taken for a small loop Ax up Ax low
Ax l. y
Analogously, the contribution of the left and right sides of the loop reads as A (28) mB left right v 3 Q y ludx . c x Summing up eqs. (27) and (28), we obtain: A A v v mB mB lower upper mB left right 3 Q x y ludx 3 mz Bz dx , c x c y which is just the second term on rhs of eq. (11) for the particular problem in question. By such a way we confirmed that each of the four phases disclosed for a moving dipole (see eq. (11)) finds its physical interpretation via the corresponding fundamental phase for a moving charge. This result is visually demonstrated in Fig. 1. Here we point out that all of the derivative quantum phases disclosed for dipoles are path-dependent and thus, the electric A-B phase (1), which does not explicitly depend on a velocity of particle, cannot contribute to the derivative phase effects. This brings asymmetry into Fig. 1. 3. Hamiltonian of charged particle in an electromagnetic field and quantum phase effects It is known that the standard Lagrangian for a charges particle in an EM field and the corresponding Hamiltonian (see, e.g. [14]) yield only the magnetic (1) and electric (2) A-B phases, so that the origin of the complementary electric (15) and magnetic (26) phases seems unclear. One should mention that these phases look like some relativistic corrections to the A-B phases. In this case, the complementary phases (15) and (26) could be derived via the solution of the Dirac equation for an electron in an EM field. However, as is known, this is also not the case. These observations allow us to assume the presence of missed points in our understanding of quantum phase effects.
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We assert that such missed points actually exist; moreover, they have the fundamental character and are related to the presently adopted procedure for the transition from the classical to quantum description of point-like charges in an EM field. To be more specific, we first remind the known expression for the classical nonrelativistic Hamilton function for a charged particle in an electric/magnetic field
p2 (29) e , 2M where p is the mechanical momentum of particle, and M is its rest mass. As is known, for a free particle, the transition from the classical Hamilton function to the corresponding Hamiltonian is carried out via the replacement of momentum p by the operator pˆ i . However, it is obviH
ous that the replacement p i becomes unsuitable in the presence of EM field, since in this way we lose the possibility to describe the magnetic interaction. In these conditions, it was postulated that the operator i should be associated rather with the canonical momentum Pc of particle in an EM field, i.e. eA Pc p Pˆc i , (30) c than with its mechanical momentum pˆ . In this case, presenting eq. (29) in the equivalent form 2 Pc eA c H e ,
2M we obtain the corresponding Hamiltonian
i eA c e , Hˆ (31) 2M which is adopted up to now in the non-relativistic quantum mechanics for the description of particles in an EM field (see, e.g. [14]). As is well-known, the Schrödinger equation with the Hamiltonian (31) allows us to derive the electric (1) and magnetic (2) A-B phases [1, 15]. At the same time, now we emphasize that the canonical momentum (30) represents, in fact, a formal variable, which emerges in the description of classical charges in the presence of EM field via the Lagrangian approach, and the problem of determination of physical meaning of the canonical momentum is usually skipped in modern representation of classical electrodynamics and quantum mechanics. Obviously, this problem is reduced to the determination of physical meaning of the term eA/c, when we consider a charged particle in an external EM field. Let us show that eA/c describes the momentum of EM field, representing a superposition of the external field and particle’s field (hereinafter referred as the interactional field momentum PEM) in the particular case, where the particle is at rest in the frame of observation. Indeed, in this frame the particle produces only the electric field Ee and thus, designating through E, B the external electric and magnetic fields, correspondingly, we obtain the following expression for the interactional field momentum, when the velocity of the particle v is equal to zero: 1 Ee B dV 1 Ee AdV . PEM v 0 (32) 4c V 4c V 2
Here we carry out the integration over the entire space V, and also have used the equality B A . Further, we involve the vector identity [11]
9
E AdV A E dV E AdV A E dV 0 , e
e
V
e
V
e
V
V
which in the Coulomb gauge ( A 0 ) yields
eA E AdV A E dV 4 AdV c . e
e
V
V
e
(33)
V
Here we have taken into account that E e 0 for a resting particle, and used the Maxwell equation Ee 4e , e being the charge density of particle. Hence, combining eqs. (32) and (33), we obtain (34) PEM v 0 eA c . This equation allows us to disclose the meaning of the canonical momentum Pc in eq. (30): the latter represents the sum of the mechanical momentum of moving particle p and the interactional field momentum eA/c in the particular situation, where the particle would be at rest. Thus, this sum does not have any real physical meaning, and this fact looks bothering, especially with the recognition of the key role of eq. (30) in the transition from the classical to quantum description of particles in an EM field. In these conditions, it seems more reasonable to re-define the operator of momentum as the sum of mechanical momentum of particle p and the interactional field momentum PEM in the case of moving particle. In other words, instead of eq. (30), we postulate p P Pˆ i (35) EM
under the transition from the classical to quantum description of charged particles. Therefore, instead of the Hamiltonian (31), we obtain
i PEM e . Hˆ (36) 2M We emphasize that, in general, for a moving particle PEM eA c , so that the solution of 2
the Schrödinger equation with the Hamiltonian (36) should differ from the corresponding solution with the Hamiltonian (31). We will show below that the difference ( PEM eA c ) can emerge in the order of magnitude (v/c)2. Thus, the mentioned difference could be considered as a relativistic correction to the solution of the Schrödinger equation in the indicated order of magnitude and thus, in a true nonrelativistic case, which is relevant for a considerable part of problems of non-relativistic quantum mechanics, the Hamiltonians (31) and (36) yields identical results. We point out that the standard Hamiltonian (31) already contains the term proportional to -2 2 2 c ( e A 2Mc 2 ), which, however, is negligible in the majority of practical situations. In contrast, the difference ( PEM eA c ) remains significant even for slowly moving particles. In order to analyze the implications of the Schrödinger equation with the Hamiltonian (36), below we consider a charged particle in an external electric E and magnetic B fields, and explicitly express the interactional field momentum PEM, entering into eq. (36), via the field potentials at the location of charge e. Designating the electric Ee and magnetic Be fields generated by the charge, we present the interaction field momentum in the form 1 E Be dV 1 Ee B dV . PEM (37) 4c V 4c V
10
Via vector calculus, the field momentum can be expressed as a function of the scalar and vector A potentials of the external EM field at the location of point-like charge, which reads as eA ve ev A v PEM 2 (38) c c c3 (see the Appendix A). We emphasize that for a spinless particle this equation is exact in the case, where the external field is produced by a macroscopic source. Moreover, it remains exact for a particle with spin in the important practical case (which is used for observation of fundamental quantum phases) of the vanishing electric and magnetic fields at its location. Thus, substituting eq. (38) into eq. (36), we obtain 1 eA ve ev A v 2 (39) P e , 2M c c c3 where all variables are considered as operators. Presenting P Mv and assuming the Coulomb gauge, where the operators v and A commutate with each other (see, e.g. [14]), we derive to the accuracy of calculations c-3: 2
H
2 eA v ev 2 ev 2 A v H e 2 2M c c c3
(40)
e 2 A2 in comparison with other terms of eq. (40), which in any 2Mc 2 practical situation is quite warranted, and stands for the Laplacian. The quantum phase for a charged particle in the presence of EM field is defined via the relationship (see, e.g. [15]) 1 H H 0 dt , (41) where H0 is the Hamiltonian of a particle in the absence of EM field. Thus, eqs. (40), (41) yield the following expression for the quantum phase of a charged particle in an EM field: 1 1 1 1 edt eA ds 2 ev ds 3 e A v v ds , (42) c c c where we designated ds vdt the element of the path of particle. The first two terms on rhs of eq. (42) give the electric (1) and magnetic (2) A-B phases, correspondingly, whereas the third and fourth terms yields the complementary electric (15) and complementary magnetic (26) phase, correspondingly, which we have already disclosed in section 2 via the analysis of quantum phase effects for moving dipoles. Now we emphasize that the complementary electric and magnetic phases cannot, in general, be considered only as some formal relativistic corrections respectively to the electric A-B phase (1) and magnetic A-B phase (2) in the lowest order in (1/c). In particular, we point out that under the absence of electric and magnetic fields at the location of particle, eq. (38) remains exact even for a particle with spin, and thus, being used in the Hamiltonian (29), it determines three separate path-dependent quantum phases (2), (15) and (26), so that all of them should be classified as fundamental. where we neglected the term
11
4. Discussion and conclusion The main result of the present paper is the revealing of two novel fundamental quantum phase effects for a point-like charge: the complementary electric phase (15) and the complementary magnetic phase (26). Their finding represents a direct consequence of the principal step we suggested. i.e. the re-definition of the operator of momentum for charged particle in an EM field according to eq. (35). We have shown above that eq. (35) looks more reasonable from the physical viewpoint, than the standard definition (30) used to the moment. One can see that both quantum phases (15) and (26) depend on the velocity of particle under integration over a closed path and thus, they are not topological. Concurrently we notice that these phases are not dynamical, because they exist in the case, when the classical force on the particle is vanishing. In particular, for the complementary electric phase c, this situation is realized in the case, where the scalar potential is constant along the particle’s path. One more principal result of our analysis is the disclosure of the physical meaning of the quantum phase effects for moving electric/magnetic dipoles as a superposition of quantum phase effects for elementary charges, composing these dipoles. Earlier, when the complementary electric (15) and magnetic (26) phases were not known, the problem of physical interpretation of quantum phases for neutral particles remained unsolved. For example, the common recognition of the A-C effect as the dual to A-B effect [2], in fact, added nothing to the understanding of its physical meaning. Whereas, we have seen that at the fundamental level, the A-C effect represents the manifestation of complementary electric phase effect (15), and thus indicates the reality of the field of scalar potential in quantum mechanics. This result seems especially important, because the A-C effect has already been confirmed experimentally [16-18], whereas other experiments aimed to confirm the reality of scalar potential via the electric A-B effect (1) (e.g., [19]), remain to be a subject of discussions [20]. We add that a direct experimental observation of complementary electric (15) and magnetic (26) phases for electrons moving in the field of scalar and vector potentials seems very difficult, due to the dominant contribution of electric (1) and magnetic (2) phases, correspondingly. However, for electrically neutral dipoles the situation drastically changes: that is, the electric (1) and magnetic (2) AB-phases themselves are suppressed, which opens a principal possibility to measure the phases (3)-(6) alone. Amongst these phases, the attempt of detection of the phases pE (5) and mB (6) seems especially interesting: they do not depend on the known electric (1) and magnetic (2) AB-phases (see Fig. 1), so that their experimental confirmation will serve as the indirect demonstration of the reality of complementary electric phase (15) (next to the available demonstration via the A-C effect) and magnetic phase (26) (for the first time), correspondingly. Possible ways to measure these phases had already been discussed in ref. [9]. Further, it is important to remind that the Klein-Gordon and Dirac equations for a charge in an EM field also imply the formal validity of equation (30) and thus, the latter equation should also be replaced by the physically meaningful equation (35). In this respect it is important to notice that the sum of mechanical momentum of particle p and interactional field momentum PEM represents the spatial components of the four-vector, whose time component is defined as the sum of the energy of particle and the energy of interactional EM field. Thus, the replacement (35) keeps the invariance of the Klein-Gordon and Dirac equations.
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Then, considering the motion of a free charge in an external EM field produced by a macroscopic source, we arrive again at eq. (38), if we assume the absence of electric and magnetic fields at the location of the charge. Hence, the complementary electric (15) and magnetic (26) phases can be also derived via the Dirac equation for a free electron moving in the field of scalar and vector potentials. At the same time, we have to stress that eq. (38) is no longer valid for a bound electron due to two reasons. The first reason is related to the adopted condition of the constancy of external fields and potentials in a vicinity of point-like charge (see, e.g., eq. (A4)), which is well fulfilled at the motion of electron in the field of macroscopic source, but obviously is violated, for example, for the bound system of “electron plus proton”. The second reason is connected to the fact that a quantum bound system in a stationary energy state does not radiate and thus, any direct classical analogy in the description of such a system (e.g., the application of Maxwell equations, as it is done in the derivation of eq. (38)) becomes, in general, incorrect. In ref. [21] we conjectured the field equations for an imaginary classical charge with the prohibited radiating EM field, which, in particular, indicate that the term eA/c in eq. (30), unlike to the classical case, could be physically meaningful. The detailed analysis of this problem anyway falls outside the scope of the present paper. APPENDIX A. Momentum of interactional EM field for the system “point-like charge in an external field of macroscopic source” via the scalar and vector potentials We define the momentum of interactional EM field according to the standard equation (37) 1 (A1) E Be dV 1 Ee B dV , PEM 4c V 4c V and our immediate goal is to express it via the scalar and vector A potentials of the external EM field at the location of point-like charge. Evaluating the first term on rhs of eq. (A1), we use the equation Be v Ee c for a non-radiating spinless charged particle; hence we obtain 1 E Be dV 1 2 E v Ee dV 1 2 vE Ee dV 1 2 Ee E v dV , (A2) 4c V 4c V 4c V 4c V
where we applied the vector identity a b c ba c ca b . Calculating the first integral on rhs of eq. (A2), we adopt for simplicity that the external electric field E represents the Coulomb field, i.e. E . Thus, we obtain
1 vE Ee dV v 2 Ee dV . 2 4c V 4c V
Using the vector identity Ee Ee Ee , we further derive
1 4c 2
v v ve vE E dV 4c E dV 4c E dV c . e
V
e
2
V
e
2
V
2
(A3)
13
Here we have taken into account that the first integral on rhs if eq. (A3) is vanishing under the integration over the entire space due to the Gauss theorem, while in the second integral we used the Maxwell equation Ee 4e , where e is the charge density of moving particle. Considering the second integral on rhs of eq. (A2), we point out that for the system in question, composed from macroscopic source of field and point-like charge, we can adopt the external field E as the constant value in a vicinity of any point-like charge. Therefore, we can always choose such a volume of integration V, where the vector E remains practically constant, while the electric field of charge Ee becomes negligible on the boundary of V. Hence, we can write 1 Ee E v dV E v 1 2 Ee dV . (A4) 2 4c V 4c V ' Introducing the scalar e and vector Ae ve c potentials for a moving charge, and taking into account that the variation of A in any fixed spatial point happens due to the motion of v ), we present its electric field as charge (i.e., t 1 Ae 1 1 E e e e v Ae e 2 v v e . c t c c Hence, eq. (A4) takes the form 1 Ee E v dV E v 1 2 edV E v v 4 v edV . (A5) 2 4c V 4c V ' 4c V ' The first term on rhs of this equation can be transformed into a surface integral, where the scalar potential of charge is vanishing; let us show that the second term is vanishing, too. For this purpose, we involve the vector identity [11]
v dV v dS v dV 0 , e
V'
e
(A6)
e
S'
V'
where the first integral is vanishing due to the adopted requirement that the scalar potential e is equal to zero on the boundary of volume V, while in the second integral we used the equality v 0 . Thus, combining eqs. (A2), (A3), (A5) and (A6), we get the first term on rhs of eq. (A1) as the function of the scalar potential of the external field: 1 E Be dV ve2 (A7) 4c V c Next, we evaluate the second integral on rhs of eq. (A1), 1 Ee B dV 1 Ee AdV . 4c V 4c V
(A8)
Using the identity [11]
E AdV A E dV E AdV A E dV 0 e
V
e
V
e
V
e
(A9)
V
in the Coulomb gauge A 0 , and applying again the Maxwell equation Ee 4e , we further derive
14
E AdV A E dV 4 AdV , e
e
V
(A10)
e
V
V
Substituting eq. (A10) into eq. (A8), we obtain 1 Ee B dV 1 A Ee dV eA . 4c V 4c V c
(A11)
Eq. (A11) demonstrates that even in the case, where the external electric field is equal to zero (like, for example, in observations of the magnetic A-B effect), the interactional field momentum, in general, is not equal to eA/c, and additionally includes the first term on the rhs, which, as we will see below, is not vanishing. For its evaluation, we involve the Maxwell equation 1 Be 1 Ee v Be , (A12) c t c where we have taken into account that the time variation of the magnetic field of particle hap v .Thus, the first term on rhs of eq. (A11) reads as pens due to its motion only, when t 1 A Ee dV 1 2 A v Be dV . (A13) 4c V 4c V Further, we use the vector identity v Be v Be v Be v Be ,
(A14)
where we have taken into account that v does not depend on r, and Be 0 . Substituting eq. (A14) into eq. (A13), we derive: 1 A Ee dV 1 2 A v Be dV . 4c V 4c V
(A15)
Applying now the identity (A9) to the rhs of eq. (A15), we obtain
A v B dV v B AdV A v B dV v B AdV e
e
V
e
V
V
v Be B dV A v Be dV . V
e
V
(A16)
V
In the derivation of this equation, we have used the equalities A B and A 0 (the Coulomb gauge). Now we emphasize that for the considered system “point-like charged particle and macroscopic source of EM field”, we can adopt the vector potential A as the constant value in a vicinity of any point-like charge. Hence, by analogy with eq. (A4), we present the second term on rhs of eq. (A16) as
A v B dV A v B dV 0 , e
V
e
V'
which thus disappears due to the Gauss theorem. Therefore, we get from eqs. (A15) and (A16) 1 A Ee dV 1 2 B v Be dV 4c V 4c V (A17) 1 1 v B Be dV Be B v dV . 4c 2 V 4c 2 V For the considered system “macroscopic source of EM field and point-like charge”, we can again adopt the external magnetic field to be practically constant in a vicinity of point-like charge, and
15
to move the product B v outside the integral in the second term on the rhs of this equation by analogy with eq. (A4). Therefore, due to the equality Be Ae (Ae being the vector potential produced by the charge), the second integral is transformed into a surface integral, where the vector potential Ae is equal to zero. Thus, this integral is vanishing, and eq. (A17) acquires the form 1 A Ee dV 1 2 vB Be dV v 2 A Be dV 4c V 4c V 4c V
v v A Be dV A Be dV . 2 4c V 4c 2 V
(A18)
where we have used the vector identity A Be Be A A Be . The first integral on rhs of eq. (A18) disappears owing to the Gauss theorem, while for further transformation of the second integral, we use the Maxwell equations 1 Ee 4 Be ev , Ee 4e . (A19) c t c Substituting eqs. (A19) into eq. (A18), we obtain
1 A Ee dV 1 2 v A 1 Ee 4 ev dV 4c V 4c V c t c 1 1 3 ev A v dV v A v Ee dV c V 4c 3 V
(A20)
ev A v v A v Ee dV . 3 c 4c 2 V
Let us show that the second term on rhs of eq. (A20) is vanishing for the considered system of macroscopic source of EM field and point-like charge, where we can adopt the vector potential A as the constant value in a vicinity of charge. Thus, by analogy with eq. (A4), we write
A v E dV A v E dV . e
e
V
V'
Next, we use once again the identity (A6), this time for the vectors A and E:
A v E dV A v E dV A E v dS A v E dV . e
V
e
V
e
S
e
(A21)
V
The first integral on rhs of this equation is equal to zero, since the electric field of charge Ee disappears on the surface S, while the second integral is vanishing due to the equality v =0. Thus, combining eqs. (A20), (A21) and (A11), we obtain 1 Ee B dV eA ev A3 v (A22) 4c V c c Finally, taking the sum of eqs. (A7) and (A22), we arrive at the expression for the interactional EM field momentum via the potentials of the external field: 1 E Be dV 1 Ee B dV eA ve2 ev A3 v , (A23) PEM 4c V 4c V c c c which is eq. (38) used in section 3.
16
References [1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [2] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319. [3] X.-G. He and B.H.J. McKellar, Phys. Rev. A47 (1993) 3424. [4] M. Wilkens, Phys. Rev. Lett. 72 (1994) 5. [5] W. Shockley and R. P. James, Phys. Rev. Lett. 18 (1967) 876. [6] S. Goleman and J.H. Van Vleck, Phys. Rev. 171 (1968) 1370. [7] Y. Aharonov, P. Pearle and L. Vaidman, Phys. Rev. A 37 (1988) 4052. [8] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, EPL 113 (2016) 14003. [9] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Ann. Phys. 368 (2016) 139. [10] J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998). [11] O.D. Jefimenko, Electromagnetic Retardation and Theory of Relativity, 2nd ed. (Electret Scientific Company, Star City, 2004), Appendix 1. [12] A. Miffre, M. Jacquey, M. Büchner, et al., Eur. Phys. J. D38 (2006) 353. [13] S. Lepoutre, A. Gauguet, G. Trnec, et al., Phys. Rev. Lett. 109 (2012) 120404. [14] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, New York, 1965). [15] Y. Aharonov and D. Bohm, Phys. Rev. 123 (1961) 1511. [16] A. Cimmino, G.I. Opat, A.G. Klein, et al. Phys. Rev. Lett. 63 (1989) 380. [17] W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E. Mooij, Phys. Rev. Lett. 71 (1993) 2311. [18] M. König, A. Tschetschetkin, E. M. Hankiewicz, et al., Phys. Rev. Lett. 96 (2006) 06804. [19] A. Van Oudenaarden, M.H. Devoret, Yu.V. Nazarov, J.E. Mooij, Nature 391 (1998) 768. [20] A. Walstad, Int. J. Theor. Phys. 49 (2010) 2929. [21] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Phys. Scr. 82 (2010) 045301.
17
Electric A-B phase
HMW phase pB
Magnetic A-B phase A
A-C phase
mE
Complementary electric phase c
Complementary magnetic phase cA
phase pE
phase mB
Fundamental phases
Derivative phases
Fig. 1. Relationship between fundamental quantum phases for a charged particle and corresponding derivative phases for a moving dipole.
PII: DOI: Reference:
S0003-4916(18)30060-5 https://doi.org/10.1016/j.aop.2018.03.005 YAPHY 67612
To appear in:
Annals of Physics
Received date : 3 January 2018 Accepted date : 5 March 2018 Please cite this article as: A.L. Kholmetskii, O.V. Missevitch, T. Yarman, Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field, Annals of Physics (2018), https://doi.org/10.1016/j.aop.2018.03.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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1
Quantum phases for point-like charged particles and for electrically neutral dipoles in an electromagnetic field A.L. Kholmetskii1, O.V. Missevitch2 and T. Yarman3 1
Department of Physics, Belarus State University, 4 Nezavisimosti Avenue, 220030 Minsk, Belarus Institute for Nuclear Problems, Belarus State University, 11 Bobruiskaya Str., 220030 Minsk, Belarus 3 Department of Engineering, Okan University, Akfirat, Istanbul, Turkey & Savronik, Eskisehir, Turkey 2
Abstract We point out that the known quantum phases for an electric/magnetic dipole moving in an electromagnetic (EM) field must be presented as the superposition of more fundamental quantum phases emerging for elementary charges. Using this idea, we find two new fundamental quantum phases for point-like charges, next to the known electric and magnetic Aharonov-Bohm (A-B) phases, named by us as the complementary electric and magnetic phases, correspondingly. We further demonstrate that these new phases can indeed be derived via the Schrödinger equation for a particle in an EM field, where however the operator of momentum is re-defined via the replacement of the canonical momentum of particle by the sum of its mechanical momentum and interactional field momentum for a system “charged particle and a macroscopic source of EM field”. The implications of the obtained results are discussed. 1. Introduction To the end of past century, four quantum phase effects had been found, which are characterized correspondingly by the electric Aharonov-Bohm (A-B) phase [1] e dt , (1) the magnetic A-B phase [1] e A A ds , (2) c the Aharonov-Casher (A-C) phase [2] 1 mE m0 E ds , (3) c and the He-McKellar-Wilkens (HMW) phase [3, 4] 1 pB p0 B ds , (4) c where ds=vdt is the path element of a charged particle e, moving with the velocity v. The electric (1) and magnetic (2) A-B effects emerge for charged particle in the field of scalar potential and vector potential A, correspondingly; the A-C phase (3) is associated with the hidden magnetic momentum [5-7] of a point-like magnetic dipole m0, moving in an electric field E, while the HMW phase (4) is defined via the hidden electric momentum (according to the terminology introduced in refs. [8, 9]) of a point-like electric dipole p0, moving in a magnetic field B. The vectors m0 and p0 are measured in the rest frame of a dipole, and hereinafter we adopt that the dipoles are electrically neutral.
2
In addition to the phases (1)-(4), recently we have found two new quantum phase effects [8, 9], emerging at the motion of an electric dipole in an electric field, 1 pE 2 p0 // E v ds , (5) c and at the motion of a magnetic dipole in a magnetic field, 1 mB 2 m0 // B v ds , (6) c where is the Lorentz factor, and p0 // , m0 // stand for the component of electric (magnetic) dipole moment collinear with the vector v. We notice that A-C (mE) and HMW (pB) phases, as well as the new phases pE, mB are related to each other via electric-magnetic duality transformations EB, pm [10], and it is comfortable to recognize that the four phases (3)-(6) correspond to all possible combinations of the pair p, m with the pair E, B. The revealing of four quantum phase effects (3)-(6) for moving dipoles, took place during past decades, definitely requires clarifying their physical meaning. Analyzing this problem, we will consider the electric/magnetic dipole as a compact electrically neutral bunch of finite size, composed from elementary charges. This representation suggests that each of the phases (3)-(6) should be explained at the fundamental level via an appropriate phase effect, emerging for every elementary charge of the bunch. As is known, this idea actually works well with respect to the HMW phase (4). Indeed, using the simplest model of the electric dipole p0 (two elementary charges –e and e, separated by a small proper distance d), one can show that the HMW phase for such a dipole can be presented as an algebraic sum of magnetic A-B phases (2) for each charge, composing the dipole, and moving in the presence of vector potential A. In other words,
pB
e A r d d s A r d s L c L
(7)
(r is the radial coordinate), where the path of positive charge is designated as L+, while the path of negative charge is designated as L-. Indeed, introducing the proper polarization P0 of the bunch (so that p0 P0 dV , V being the proper volume of the bunch), we can further write:
pB
1 e A r d d s A r d s A P0 dsdV L c V c L
1 P0 A dsdV 1 P0 B dsdV 1 p0 B ds , c V c V c
which coincides with eq. (4). Here we have used the vector identity [11]
A P dS P A dS AP dS A P dV P AdV 0
0
S
0
S
S
A P dV P AdV ; 0
V
0
V
0
V
0
V
3
then, we have taken into account that the polarization P0 is vanishing on the surface of dipole (so that all integrals in the lhs of this identity are equal to zero), and also used the equalities P0 0 , A B , applying the Coulomb gauge ( A 0 ). At the same time, one can easily realize that the A-C phase (3), as well as the new phase effects (5), (6) cannot be reduced to either electric (1), or magnetic (2) A-B effects for point-like charges composing a dipole. Thus, the problem of physical interpretation of quantum phases (3), (5), (6) remains unsolved. In section 2, we analyze the origin of the phases (3), (5), (6) and show that they indeed can be presented as the composition of new phases for point-like charges composing the dipoles. We named these new quantum phases as the complementary magnetic and electric phases and simultaneously emphasized that these phases cannot be derived either from the Schrödinger equation, or from the Dirac equation for a particle in an electromagnetic (EM) field, which actually may indicate the presence of some missed points in our understanding of quantum phase effects. In section 3 we assert that such missed points actually exist; moreover, they have the fundamental character and are related to the presently adopted procedure for the transition from the classical to quantum description of point-like charges in an EM field. It is related to the definition of operator of momentum, which is commonly related with the canonical momentum of particle in an EM field. However, we show that this relation is, in fact, meaningless from the physical viewpoint. Hence, we suggest redefining the operator of momentum as the sum of mechanical momentum of particle and interactional field momentum for a system “particle and a source of EM field”. Further, we apply the Hamiltonian with the modified definition of the momentum operator to the solution of the Schrödinger equation and actually derive the complementary magnetic and electric phases. By such a way we close the logical chain of our approach, showing that each of the phase effects for dipoles can be explained via the corresponding fundamental phase effect for elementary charge. We discuss and conclude in section 4. 2. Fundamental quantum phases for point-like charges and derivative quantum phases for electric/magnetic dipoles In order to proceed further, we first present the general expression for the quantum phase of an electrically neutral dipole in an EM field
1 1 1 1 1 1 p0 // E v ds 2 m0 // B v ds m0 E ds p0 B ds p E dt m B dt , c 2 c c c
(8) obtained in refs. [8, 9] on the basis of the Lagrangian approach with the interactional Lagrangian 1 density L M F integrated to point-like dipole. (Here M is the magnetization2 polarization tensor, F is the tensor of EM field, and , =0…3). The last two terms on rhs of eq. (24) define the Stark phase [12] and Zeeman phase [13], correspondingly, while the other terms are associated with the moving dipole and presented separately by eqs. (3)-(6). In the present contribution, we omit the last two terms on the rhs of eq. (8), which do not explicitly depend on the velocity of dipole, and deal with the weak relativistic limit, where the remaining four
4
terms in this equation will be presented in the convenient form to the accuracy of calculations c-3 sufficient for our analysis. In this limit, we apply the approximate relationships [10] vm pv p p0 , m m0 (9a-b) c c between the electric p and magnetic m dipole moments for a moving dipole with the corresponding proper values p0, m0. Combining eqs. (9a-b) and (8), we obtain
1 c 2
p
//
E v ds
1 c 2
m
//
B v ds
1 1 1 1 m E p v E ds p B v m B ds. c c c c
(10) Using the equalities p v E E v p vE p p E v , B v m vB m m B v
(where we applied the vector identity a b c ba c ca b ), and taking into account that
p p// p , m m// m , p ds 0 , m ds 0 , we derive 1 p E v ds 1 2 m B v ds 1 m E ds 1 p B ds. (11) 2 c c c c Thus, in the weak relativistic limit, we present the phase (11) via the values measured in a laboratory frame, which will be used below in the analysis of physical meaning of its terms. With respect to the last term of eq. (11) (the HMW phase), we already have shown that this phase is defined as the sum of magnetic A-B phases for each elementary charge composing the electric dipole, eq. (7). Next, we consider one more phase effect for a moving electric dipole, which is presented by the first term on the rhs of eq. (11), i.e. 1 pE 2 p E v ds . (12) c Hereinafter we assume that the inductive component of electric field is equal to zero, and apply the equality E . Substituting this equality into eq. (12), and using the definition
p PdV , we derive: V
pE
1 P v dsdV 1 2 P v dsdV 1 2 v dsdV , 2 c V c V c V
(13)
where we have used the vector identity P P P . The first integral on rhs of eq. (13) is vanishing by the Gauss theorem; in the second integral we have used the equality P , where is the density of bound charges. Introducing again the model of the electric dipole defined above (two elementary charges –e and +e separated by a small distance d), we thus derive 1 1 (14) pE 2 v dsdV 2 e r d e r v ds. c V c Eq. (14) indicates that the quantum phase (12) for an electric dipole, moving in an electric field, originates from a more fundamental quantum phase effect, emerging for the elementary charge e, moving in the field of scalar potential at velocity v, i.e.
5
c
1 ev ds , c 2
(15)
and the phase pE represents the sum of elementary phases (15) for each charge. We see that the phase (15), like the electric A-B phase (1), depends on the scalar potential , though in the adopted weak relativistic limit, the phase (15) is about (v/c)2 smaller than the phase (1). We suggest naming the phase (15) as the complementary electric phase, introducing for it the subscript “c”. Next, we consider the A-C phase, presented by the third term on rhs of eq. (11). Analyzing this term, we introduce the magnetization M of a compact bunch of charges, and use the equality m MdV . As before, we adopt that the external electric field E does not contain an inductive component, and is defined as E . Substituting this equality into the term of A-C phase in eq. (11), we present it as the function of scalar potential: 1 1 1 mE M dsdV M dsdV M dsdV , (16) c V c V c V
where we have used the vector identity M M M . We see that the first integral on rhs of eq. (16) is vanishing under integration over infinite volume. In the evaluation of the second integral we will use the equality [10] (17) j c M P t , and assume that the polarization is stationary, i.e. dP dt P t v P 0 . Hence,
P t v P . Substituting this equality into eq. (16), we derive
c M j v P . Further substitution of eq. (18) into eq. (16) yields: 1 mE 2 j v P dsdV . c V
(18)
(19)
Evaluating this integral, we adopt for simplicity the case of a pure magnetic dipole, where the proper polarization P0=0. In this case, the polarization P in eq. (19) has a relativistic origin, i.e. P v M 0 c , and is orthogonal to the vector ds. Hence, eq. (19) takes the form
mE
1 1 j dsdV 2 u dsdV , 2 c V c V
(20)
where we have used the definition of current density j=u, being the charge density of carriers of current, and u their flow velocity. Eq. (20) shows that the A-C phase for a moving magnetic dipole represents the sum of complementary electric phases (15) for carriers of current of the dipole, characterized by the flow velocity u. We can clarify better the physical meaning of eq. (20), if we imagine a magnetic dipole as a small electrically neutral rectangular conducting loop, lying in the plane xy, and moving with the velocity v along the axis x in the presence of electric field E, lying along the y-axis. One can see that the contributions of charges of the left and right sides of the loop to the total complementary electric phase mutually cancel each other, while for the upper and lower sides we obtain from eq. (20):
6
1 upQup lowQlow udx . (20a) c 2 Here up, low are the scalar potentials on the upper and lower segments of the loop, correspondingly, and Qup, Qlow are the charges of carriers of current on these segments. We also used the Galilean law of velocity composition for the carriers of current (where their total velocity u=v+u), which is sufficient within the adopted accuracy of calculations c-3, while for positive ions, immovable in the rest frame of the loop, the velocity is equal to v in all its segments. We also notice that in the adopted accuracy of calculations, it is sufficient to put Qup=QlowQ. Taking also into account that for a small rectangular loop with the length l of each segment, E=-(up-low)/l, we obtain from eq. (20a) 1 1 1 mE 2 QEludx 2 EIl 2 dx mEdx c c c (where I is the current in the loop), which is just the A-C phase for the particular problem in question. Finally, we address to the second term on rhs of eq. (11), which defines the phase 1 mB 2 m B v ds . (21) c
mE
Using the equalities B A , m MdV , we derive:
mB
1 M Av dsdV 1 2 M Av dsdV 2 c V c V
1 A M v dsdV . c 2 V
(22)
The first integral on the rhs of eq. (22) is vanishing due to the Gauss theorem, while in the second integral we use the equality (18). As a result, we obtain 1 (23) mB 3 A j A v P v dsdV . c V We again assume that the external electric field does not contain inductive component (i.e., A t 0 ), and the vector potential A does not explicitly depend on time, i.e. dA dt v A . Adopting also that the polarization P is stationary (when dP dt v P ), we transform the second integral to the form d dA 2 2 2 (24) V A v P v dtdV v V dt A P dtdV v p dt dt . Assuming that at the initial and final time moments the magnetic dipole is located outside the fields, where A is vanishing, we see that the integral (24) is equal to zero, and we finally obtain 1 1 (25) mB 3 j Av dsdV 3 u Av dsdV , c V c V where we have used the equality j=u. In order to clarify the origin of the phase effect (25), we notice that for elementary charge e, moving at some velocity v, the current density is j r r0 ev . Substituting this equality into eq. (25), we obtain for the elementary charge
7
e v Av ds . (26) c 3 Eq. (26) discloses one more fundamental quantum phase effect, emerging for a charge moving in the field of vector potential A. We thus have shown that the phase mB (eq. (21)) for a magnetic dipole, moving in a magnetic field, represents the superposition of fundamental phases (26), which we name as the complementary magnetic phase, attributing to it the subscript “cA”. We can better clarify the relationship between the phase mB for a magnetic dipole in a magnetic field, and the complementary magnetic phase (26), considering again the magnetic dipole as a small electrically neutral rectangular conducting loop, lying in the plane xy, and moving with the velocity v along the axis x. We assume that the magnetic field B is orthogonal to the plane of the loop, and is directed along the axis z. Then, the contribution of charges of the upper and lower sides of the loop to the total complementary magnetic phase (26), by analogy with eq. (20a), can be written as mB lower upper v 3 Ax up Qupu Ax lowQlowu dx v 3 Q Ax ludx , (27) c c y
cA
where we have taken for a small loop Ax up Ax low
Ax l. y
Analogously, the contribution of the left and right sides of the loop reads as A (28) mB left right v 3 Q y ludx . c x Summing up eqs. (27) and (28), we obtain: A A v v mB mB lower upper mB left right 3 Q x y ludx 3 mz Bz dx , c x c y which is just the second term on rhs of eq. (11) for the particular problem in question. By such a way we confirmed that each of the four phases disclosed for a moving dipole (see eq. (11)) finds its physical interpretation via the corresponding fundamental phase for a moving charge. This result is visually demonstrated in Fig. 1. Here we point out that all of the derivative quantum phases disclosed for dipoles are path-dependent and thus, the electric A-B phase (1), which does not explicitly depend on a velocity of particle, cannot contribute to the derivative phase effects. This brings asymmetry into Fig. 1. 3. Hamiltonian of charged particle in an electromagnetic field and quantum phase effects It is known that the standard Lagrangian for a charges particle in an EM field and the corresponding Hamiltonian (see, e.g. [14]) yield only the magnetic (1) and electric (2) A-B phases, so that the origin of the complementary electric (15) and magnetic (26) phases seems unclear. One should mention that these phases look like some relativistic corrections to the A-B phases. In this case, the complementary phases (15) and (26) could be derived via the solution of the Dirac equation for an electron in an EM field. However, as is known, this is also not the case. These observations allow us to assume the presence of missed points in our understanding of quantum phase effects.
8
We assert that such missed points actually exist; moreover, they have the fundamental character and are related to the presently adopted procedure for the transition from the classical to quantum description of point-like charges in an EM field. To be more specific, we first remind the known expression for the classical nonrelativistic Hamilton function for a charged particle in an electric/magnetic field
p2 (29) e , 2M where p is the mechanical momentum of particle, and M is its rest mass. As is known, for a free particle, the transition from the classical Hamilton function to the corresponding Hamiltonian is carried out via the replacement of momentum p by the operator pˆ i . However, it is obviH
ous that the replacement p i becomes unsuitable in the presence of EM field, since in this way we lose the possibility to describe the magnetic interaction. In these conditions, it was postulated that the operator i should be associated rather with the canonical momentum Pc of particle in an EM field, i.e. eA Pc p Pˆc i , (30) c than with its mechanical momentum pˆ . In this case, presenting eq. (29) in the equivalent form 2 Pc eA c H e ,
2M we obtain the corresponding Hamiltonian
i eA c e , Hˆ (31) 2M which is adopted up to now in the non-relativistic quantum mechanics for the description of particles in an EM field (see, e.g. [14]). As is well-known, the Schrödinger equation with the Hamiltonian (31) allows us to derive the electric (1) and magnetic (2) A-B phases [1, 15]. At the same time, now we emphasize that the canonical momentum (30) represents, in fact, a formal variable, which emerges in the description of classical charges in the presence of EM field via the Lagrangian approach, and the problem of determination of physical meaning of the canonical momentum is usually skipped in modern representation of classical electrodynamics and quantum mechanics. Obviously, this problem is reduced to the determination of physical meaning of the term eA/c, when we consider a charged particle in an external EM field. Let us show that eA/c describes the momentum of EM field, representing a superposition of the external field and particle’s field (hereinafter referred as the interactional field momentum PEM) in the particular case, where the particle is at rest in the frame of observation. Indeed, in this frame the particle produces only the electric field Ee and thus, designating through E, B the external electric and magnetic fields, correspondingly, we obtain the following expression for the interactional field momentum, when the velocity of the particle v is equal to zero: 1 Ee B dV 1 Ee AdV . PEM v 0 (32) 4c V 4c V 2
Here we carry out the integration over the entire space V, and also have used the equality B A . Further, we involve the vector identity [11]
9
E AdV A E dV E AdV A E dV 0 , e
e
V
e
V
e
V
V
which in the Coulomb gauge ( A 0 ) yields
eA E AdV A E dV 4 AdV c . e
e
V
V
e
(33)
V
Here we have taken into account that E e 0 for a resting particle, and used the Maxwell equation Ee 4e , e being the charge density of particle. Hence, combining eqs. (32) and (33), we obtain (34) PEM v 0 eA c . This equation allows us to disclose the meaning of the canonical momentum Pc in eq. (30): the latter represents the sum of the mechanical momentum of moving particle p and the interactional field momentum eA/c in the particular situation, where the particle would be at rest. Thus, this sum does not have any real physical meaning, and this fact looks bothering, especially with the recognition of the key role of eq. (30) in the transition from the classical to quantum description of particles in an EM field. In these conditions, it seems more reasonable to re-define the operator of momentum as the sum of mechanical momentum of particle p and the interactional field momentum PEM in the case of moving particle. In other words, instead of eq. (30), we postulate p P Pˆ i (35) EM
under the transition from the classical to quantum description of charged particles. Therefore, instead of the Hamiltonian (31), we obtain
i PEM e . Hˆ (36) 2M We emphasize that, in general, for a moving particle PEM eA c , so that the solution of 2
the Schrödinger equation with the Hamiltonian (36) should differ from the corresponding solution with the Hamiltonian (31). We will show below that the difference ( PEM eA c ) can emerge in the order of magnitude (v/c)2. Thus, the mentioned difference could be considered as a relativistic correction to the solution of the Schrödinger equation in the indicated order of magnitude and thus, in a true nonrelativistic case, which is relevant for a considerable part of problems of non-relativistic quantum mechanics, the Hamiltonians (31) and (36) yields identical results. We point out that the standard Hamiltonian (31) already contains the term proportional to -2 2 2 c ( e A 2Mc 2 ), which, however, is negligible in the majority of practical situations. In contrast, the difference ( PEM eA c ) remains significant even for slowly moving particles. In order to analyze the implications of the Schrödinger equation with the Hamiltonian (36), below we consider a charged particle in an external electric E and magnetic B fields, and explicitly express the interactional field momentum PEM, entering into eq. (36), via the field potentials at the location of charge e. Designating the electric Ee and magnetic Be fields generated by the charge, we present the interaction field momentum in the form 1 E Be dV 1 Ee B dV . PEM (37) 4c V 4c V
10
Via vector calculus, the field momentum can be expressed as a function of the scalar and vector A potentials of the external EM field at the location of point-like charge, which reads as eA ve ev A v PEM 2 (38) c c c3 (see the Appendix A). We emphasize that for a spinless particle this equation is exact in the case, where the external field is produced by a macroscopic source. Moreover, it remains exact for a particle with spin in the important practical case (which is used for observation of fundamental quantum phases) of the vanishing electric and magnetic fields at its location. Thus, substituting eq. (38) into eq. (36), we obtain 1 eA ve ev A v 2 (39) P e , 2M c c c3 where all variables are considered as operators. Presenting P Mv and assuming the Coulomb gauge, where the operators v and A commutate with each other (see, e.g. [14]), we derive to the accuracy of calculations c-3: 2
H
2 eA v ev 2 ev 2 A v H e 2 2M c c c3
(40)
e 2 A2 in comparison with other terms of eq. (40), which in any 2Mc 2 practical situation is quite warranted, and stands for the Laplacian. The quantum phase for a charged particle in the presence of EM field is defined via the relationship (see, e.g. [15]) 1 H H 0 dt , (41) where H0 is the Hamiltonian of a particle in the absence of EM field. Thus, eqs. (40), (41) yield the following expression for the quantum phase of a charged particle in an EM field: 1 1 1 1 edt eA ds 2 ev ds 3 e A v v ds , (42) c c c where we designated ds vdt the element of the path of particle. The first two terms on rhs of eq. (42) give the electric (1) and magnetic (2) A-B phases, correspondingly, whereas the third and fourth terms yields the complementary electric (15) and complementary magnetic (26) phase, correspondingly, which we have already disclosed in section 2 via the analysis of quantum phase effects for moving dipoles. Now we emphasize that the complementary electric and magnetic phases cannot, in general, be considered only as some formal relativistic corrections respectively to the electric A-B phase (1) and magnetic A-B phase (2) in the lowest order in (1/c). In particular, we point out that under the absence of electric and magnetic fields at the location of particle, eq. (38) remains exact even for a particle with spin, and thus, being used in the Hamiltonian (29), it determines three separate path-dependent quantum phases (2), (15) and (26), so that all of them should be classified as fundamental. where we neglected the term
11
4. Discussion and conclusion The main result of the present paper is the revealing of two novel fundamental quantum phase effects for a point-like charge: the complementary electric phase (15) and the complementary magnetic phase (26). Their finding represents a direct consequence of the principal step we suggested. i.e. the re-definition of the operator of momentum for charged particle in an EM field according to eq. (35). We have shown above that eq. (35) looks more reasonable from the physical viewpoint, than the standard definition (30) used to the moment. One can see that both quantum phases (15) and (26) depend on the velocity of particle under integration over a closed path and thus, they are not topological. Concurrently we notice that these phases are not dynamical, because they exist in the case, when the classical force on the particle is vanishing. In particular, for the complementary electric phase c, this situation is realized in the case, where the scalar potential is constant along the particle’s path. One more principal result of our analysis is the disclosure of the physical meaning of the quantum phase effects for moving electric/magnetic dipoles as a superposition of quantum phase effects for elementary charges, composing these dipoles. Earlier, when the complementary electric (15) and magnetic (26) phases were not known, the problem of physical interpretation of quantum phases for neutral particles remained unsolved. For example, the common recognition of the A-C effect as the dual to A-B effect [2], in fact, added nothing to the understanding of its physical meaning. Whereas, we have seen that at the fundamental level, the A-C effect represents the manifestation of complementary electric phase effect (15), and thus indicates the reality of the field of scalar potential in quantum mechanics. This result seems especially important, because the A-C effect has already been confirmed experimentally [16-18], whereas other experiments aimed to confirm the reality of scalar potential via the electric A-B effect (1) (e.g., [19]), remain to be a subject of discussions [20]. We add that a direct experimental observation of complementary electric (15) and magnetic (26) phases for electrons moving in the field of scalar and vector potentials seems very difficult, due to the dominant contribution of electric (1) and magnetic (2) phases, correspondingly. However, for electrically neutral dipoles the situation drastically changes: that is, the electric (1) and magnetic (2) AB-phases themselves are suppressed, which opens a principal possibility to measure the phases (3)-(6) alone. Amongst these phases, the attempt of detection of the phases pE (5) and mB (6) seems especially interesting: they do not depend on the known electric (1) and magnetic (2) AB-phases (see Fig. 1), so that their experimental confirmation will serve as the indirect demonstration of the reality of complementary electric phase (15) (next to the available demonstration via the A-C effect) and magnetic phase (26) (for the first time), correspondingly. Possible ways to measure these phases had already been discussed in ref. [9]. Further, it is important to remind that the Klein-Gordon and Dirac equations for a charge in an EM field also imply the formal validity of equation (30) and thus, the latter equation should also be replaced by the physically meaningful equation (35). In this respect it is important to notice that the sum of mechanical momentum of particle p and interactional field momentum PEM represents the spatial components of the four-vector, whose time component is defined as the sum of the energy of particle and the energy of interactional EM field. Thus, the replacement (35) keeps the invariance of the Klein-Gordon and Dirac equations.
12
Then, considering the motion of a free charge in an external EM field produced by a macroscopic source, we arrive again at eq. (38), if we assume the absence of electric and magnetic fields at the location of the charge. Hence, the complementary electric (15) and magnetic (26) phases can be also derived via the Dirac equation for a free electron moving in the field of scalar and vector potentials. At the same time, we have to stress that eq. (38) is no longer valid for a bound electron due to two reasons. The first reason is related to the adopted condition of the constancy of external fields and potentials in a vicinity of point-like charge (see, e.g., eq. (A4)), which is well fulfilled at the motion of electron in the field of macroscopic source, but obviously is violated, for example, for the bound system of “electron plus proton”. The second reason is connected to the fact that a quantum bound system in a stationary energy state does not radiate and thus, any direct classical analogy in the description of such a system (e.g., the application of Maxwell equations, as it is done in the derivation of eq. (38)) becomes, in general, incorrect. In ref. [21] we conjectured the field equations for an imaginary classical charge with the prohibited radiating EM field, which, in particular, indicate that the term eA/c in eq. (30), unlike to the classical case, could be physically meaningful. The detailed analysis of this problem anyway falls outside the scope of the present paper. APPENDIX A. Momentum of interactional EM field for the system “point-like charge in an external field of macroscopic source” via the scalar and vector potentials We define the momentum of interactional EM field according to the standard equation (37) 1 (A1) E Be dV 1 Ee B dV , PEM 4c V 4c V and our immediate goal is to express it via the scalar and vector A potentials of the external EM field at the location of point-like charge. Evaluating the first term on rhs of eq. (A1), we use the equation Be v Ee c for a non-radiating spinless charged particle; hence we obtain 1 E Be dV 1 2 E v Ee dV 1 2 vE Ee dV 1 2 Ee E v dV , (A2) 4c V 4c V 4c V 4c V
where we applied the vector identity a b c ba c ca b . Calculating the first integral on rhs of eq. (A2), we adopt for simplicity that the external electric field E represents the Coulomb field, i.e. E . Thus, we obtain
1 vE Ee dV v 2 Ee dV . 2 4c V 4c V
Using the vector identity Ee Ee Ee , we further derive
1 4c 2
v v ve vE E dV 4c E dV 4c E dV c . e
V
e
2
V
e
2
V
2
(A3)
13
Here we have taken into account that the first integral on rhs if eq. (A3) is vanishing under the integration over the entire space due to the Gauss theorem, while in the second integral we used the Maxwell equation Ee 4e , where e is the charge density of moving particle. Considering the second integral on rhs of eq. (A2), we point out that for the system in question, composed from macroscopic source of field and point-like charge, we can adopt the external field E as the constant value in a vicinity of any point-like charge. Therefore, we can always choose such a volume of integration V, where the vector E remains practically constant, while the electric field of charge Ee becomes negligible on the boundary of V. Hence, we can write 1 Ee E v dV E v 1 2 Ee dV . (A4) 2 4c V 4c V ' Introducing the scalar e and vector Ae ve c potentials for a moving charge, and taking into account that the variation of A in any fixed spatial point happens due to the motion of v ), we present its electric field as charge (i.e., t 1 Ae 1 1 E e e e v Ae e 2 v v e . c t c c Hence, eq. (A4) takes the form 1 Ee E v dV E v 1 2 edV E v v 4 v edV . (A5) 2 4c V 4c V ' 4c V ' The first term on rhs of this equation can be transformed into a surface integral, where the scalar potential of charge is vanishing; let us show that the second term is vanishing, too. For this purpose, we involve the vector identity [11]
v dV v dS v dV 0 , e
V'
e
(A6)
e
S'
V'
where the first integral is vanishing due to the adopted requirement that the scalar potential e is equal to zero on the boundary of volume V, while in the second integral we used the equality v 0 . Thus, combining eqs. (A2), (A3), (A5) and (A6), we get the first term on rhs of eq. (A1) as the function of the scalar potential of the external field: 1 E Be dV ve2 (A7) 4c V c Next, we evaluate the second integral on rhs of eq. (A1), 1 Ee B dV 1 Ee AdV . 4c V 4c V
(A8)
Using the identity [11]
E AdV A E dV E AdV A E dV 0 e
V
e
V
e
V
e
(A9)
V
in the Coulomb gauge A 0 , and applying again the Maxwell equation Ee 4e , we further derive
14
E AdV A E dV 4 AdV , e
e
V
(A10)
e
V
V
Substituting eq. (A10) into eq. (A8), we obtain 1 Ee B dV 1 A Ee dV eA . 4c V 4c V c
(A11)
Eq. (A11) demonstrates that even in the case, where the external electric field is equal to zero (like, for example, in observations of the magnetic A-B effect), the interactional field momentum, in general, is not equal to eA/c, and additionally includes the first term on the rhs, which, as we will see below, is not vanishing. For its evaluation, we involve the Maxwell equation 1 Be 1 Ee v Be , (A12) c t c where we have taken into account that the time variation of the magnetic field of particle hap v .Thus, the first term on rhs of eq. (A11) reads as pens due to its motion only, when t 1 A Ee dV 1 2 A v Be dV . (A13) 4c V 4c V Further, we use the vector identity v Be v Be v Be v Be ,
(A14)
where we have taken into account that v does not depend on r, and Be 0 . Substituting eq. (A14) into eq. (A13), we derive: 1 A Ee dV 1 2 A v Be dV . 4c V 4c V
(A15)
Applying now the identity (A9) to the rhs of eq. (A15), we obtain
A v B dV v B AdV A v B dV v B AdV e
e
V
e
V
V
v Be B dV A v Be dV . V
e
V
(A16)
V
In the derivation of this equation, we have used the equalities A B and A 0 (the Coulomb gauge). Now we emphasize that for the considered system “point-like charged particle and macroscopic source of EM field”, we can adopt the vector potential A as the constant value in a vicinity of any point-like charge. Hence, by analogy with eq. (A4), we present the second term on rhs of eq. (A16) as
A v B dV A v B dV 0 , e
V
e
V'
which thus disappears due to the Gauss theorem. Therefore, we get from eqs. (A15) and (A16) 1 A Ee dV 1 2 B v Be dV 4c V 4c V (A17) 1 1 v B Be dV Be B v dV . 4c 2 V 4c 2 V For the considered system “macroscopic source of EM field and point-like charge”, we can again adopt the external magnetic field to be practically constant in a vicinity of point-like charge, and
15
to move the product B v outside the integral in the second term on the rhs of this equation by analogy with eq. (A4). Therefore, due to the equality Be Ae (Ae being the vector potential produced by the charge), the second integral is transformed into a surface integral, where the vector potential Ae is equal to zero. Thus, this integral is vanishing, and eq. (A17) acquires the form 1 A Ee dV 1 2 vB Be dV v 2 A Be dV 4c V 4c V 4c V
v v A Be dV A Be dV . 2 4c V 4c 2 V
(A18)
where we have used the vector identity A Be Be A A Be . The first integral on rhs of eq. (A18) disappears owing to the Gauss theorem, while for further transformation of the second integral, we use the Maxwell equations 1 Ee 4 Be ev , Ee 4e . (A19) c t c Substituting eqs. (A19) into eq. (A18), we obtain
1 A Ee dV 1 2 v A 1 Ee 4 ev dV 4c V 4c V c t c 1 1 3 ev A v dV v A v Ee dV c V 4c 3 V
(A20)
ev A v v A v Ee dV . 3 c 4c 2 V
Let us show that the second term on rhs of eq. (A20) is vanishing for the considered system of macroscopic source of EM field and point-like charge, where we can adopt the vector potential A as the constant value in a vicinity of charge. Thus, by analogy with eq. (A4), we write
A v E dV A v E dV . e
e
V
V'
Next, we use once again the identity (A6), this time for the vectors A and E:
A v E dV A v E dV A E v dS A v E dV . e
V
e
V
e
S
e
(A21)
V
The first integral on rhs of this equation is equal to zero, since the electric field of charge Ee disappears on the surface S, while the second integral is vanishing due to the equality v =0. Thus, combining eqs. (A20), (A21) and (A11), we obtain 1 Ee B dV eA ev A3 v (A22) 4c V c c Finally, taking the sum of eqs. (A7) and (A22), we arrive at the expression for the interactional EM field momentum via the potentials of the external field: 1 E Be dV 1 Ee B dV eA ve2 ev A3 v , (A23) PEM 4c V 4c V c c c which is eq. (38) used in section 3.
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References [1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [2] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319. [3] X.-G. He and B.H.J. McKellar, Phys. Rev. A47 (1993) 3424. [4] M. Wilkens, Phys. Rev. Lett. 72 (1994) 5. [5] W. Shockley and R. P. James, Phys. Rev. Lett. 18 (1967) 876. [6] S. Goleman and J.H. Van Vleck, Phys. Rev. 171 (1968) 1370. [7] Y. Aharonov, P. Pearle and L. Vaidman, Phys. Rev. A 37 (1988) 4052. [8] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, EPL 113 (2016) 14003. [9] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Ann. Phys. 368 (2016) 139. [10] J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998). [11] O.D. Jefimenko, Electromagnetic Retardation and Theory of Relativity, 2nd ed. (Electret Scientific Company, Star City, 2004), Appendix 1. [12] A. Miffre, M. Jacquey, M. Büchner, et al., Eur. Phys. J. D38 (2006) 353. [13] S. Lepoutre, A. Gauguet, G. Trnec, et al., Phys. Rev. Lett. 109 (2012) 120404. [14] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, New York, 1965). [15] Y. Aharonov and D. Bohm, Phys. Rev. 123 (1961) 1511. [16] A. Cimmino, G.I. Opat, A.G. Klein, et al. Phys. Rev. Lett. 63 (1989) 380. [17] W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E. Mooij, Phys. Rev. Lett. 71 (1993) 2311. [18] M. König, A. Tschetschetkin, E. M. Hankiewicz, et al., Phys. Rev. Lett. 96 (2006) 06804. [19] A. Van Oudenaarden, M.H. Devoret, Yu.V. Nazarov, J.E. Mooij, Nature 391 (1998) 768. [20] A. Walstad, Int. J. Theor. Phys. 49 (2010) 2929. [21] A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Phys. Scr. 82 (2010) 045301.
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Electric A-B phase
HMW phase pB
Magnetic A-B phase A
A-C phase
mE
Complementary electric phase c
Complementary magnetic phase cA
phase pE
phase mB
Fundamental phases
Derivative phases
Fig. 1. Relationship between fundamental quantum phases for a charged particle and corresponding derivative phases for a moving dipole.