Effective interaction force between an electric charge and a magnetic dipole and locality (or nonlocality) in quantum effects of the Aharonov-Bohm type
Gianfranco Spavieri, George T. Gillies, Miguel E. Rodriguez, Maribel Peréz Pirela
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a y Effective interaction force between an electric charge and a magnetic dipole andlocality (or nonlocality) in quantum effects of the Aharonov-Bohm type. Gianfranco Spavieri ∗ , George T. Gillies, Miguel Rodriguez, and Maribel Perez Centro de F´ısica Fundamental, Facultad de Ciencias,Universidad de Los Andes, M´erida, 5101-Venezuela Department of Mechanical and Aerospace Engineering,University of Virginia, Charlottesville, VA 22904 USA Carrera de F´ısica y Matem´atica, Facultad de Educaci´on,Universidad Nacional de Chimborazo, Riobamba,100150-Ecuador and Carrera de Ingeniar´ıa Ambiental, Facultad de Ingenier´ıa,Universidad Nacional de Chimborazo, Riobamba,100150-Ecuador † Classical electrodynamics foresees that the effective interaction force between a moving chargeand a magnetic dipole is modified by the time-varying total momentum of the interaction fields.We derive the equations of motion of the particles from the total stress-energy tensor, assumingthe validity of Maxwell’s equations and the total momentum conservation law. Applications to theeffects of Aharonov-Bohm type show that the observed phase shift may be due to the relative lagbetween interfering particles caused by the effective local force.
PACS numbers: 03.30.+p, 12.20.Ds, 03.65.Bz, 39.20.+qKeywords: electromagnetic interaction, conservation laws, quantum nonlocality, Aharonov-Bohm effect, hid-den momentum
1- Introduction.
The discussions about the nonlocality, or locality, of the quantum effects of the Aharonov-Bohm (AB) [1]-[3] typehave been developing extensively through the decades [4]-[12]. Some authors claim that there are no forces acting onthe particles in the AB effects [4], [5], [7]. However, other authors sustain diverse interpretations of the AB type ofeffects in terms of the local action of forces [8], [9], [10], [11], [12], [6]. Moreover, it has been argued that nonlocalityclaims are inconsistent with Hilbert-space quantum mechanics [13]. Nonlocal quantum effects are relevant in a widecontext of scientific and heuristic scenarios and, according to the supporters of quantum nonlocality, the effects ofthe AB type represent a new departure from classical physics because they cannot be interpreted by means of localforces acting on the particles. Traditional classical physics requires instead that observable physical effects arise as aresult of a cause, generally the action of a force that produces the effect. Thus, supporters of classical physics claimthat the phase shift of the effects of the AB type may well arise as the result of the action of local forces, which, forexample, may produce a relative lag between particles passing on opposite sides of the line sources, resulting in anobservable quantum phase shift ∆ φ [8], [9], [10], [11], [12], [6].As some other authors have pointed out [7], we believe that there is no consensus regarding the nature andinterpretation of the AB effect, despite of the several, ongoing discussions on the subject. Supporting the view of aninterpretation involving the local action of a force, we consider here three effects of the AB type [1]-[3] where a beamof interfering particles interacts with an external electromagnetic (em) potential (or field) in a, supposedly, force-freeregion of space. In the AB effect [1] a charged particle interacts with the vector potential A of a solenoid, in theAharonov-Casher (AC) effect [2] a neutral particle with a magnetic dipole moment m interacts with an electric field E of a line of charges, and in the Spavieri effect [3] a neutral particle with an electric dipole moment d interactswith the vector potential A of a distribution of magnetic dipoles. If Q is the canonical interaction momentum, thequantum phase of these effects is φ = ~ − R Q · d~ℓ . A unitary vision of these effects is given in Ref. [3], where Q isrelated to the momentum of the interaction fields, Q = ± Q em . Some of the phase shifts ∆ φ arising in these effectshave been verified experimentally: for the AB effect see Refs. [14], [15] and, for tests of the AC effect, see Refs. [16],[17].The basic em interaction in the mentioned AB effects is the one between a particle with an electric charge q and aneutral particle possessing a magnetic dipole moment m . As well known, the standard expression for the interactionforce between a magnetic dipole m and a charge q , in motion with relative velocity v , does not comply with theaction and reaction principle. In fact, neglecting higher order relativistic terms, in the reference frame where thedipole is stationary, the em force acting on q is f emq = q E + ( q/c ) v × B = ( q/c ) v × B , where the electric field is ∗ [email protected] E = 0 for a neutral dipole and B is the magnetic induction field produced by m . Instead, the em force acting on m is f emm = ∇ ( m · B q ), where B q = ( v /c ) × E q is the magnetic field produced by the moving charge, E q being itselectric field. Thus, for example, in the direction of motion, with v = v b i in the x direction, we find f emqx = 0, while f emmx = ∂ x ( m · B q ) = 0. The action and reaction principle is not conserved even when q and m are stationary but thecurrent in m varies with time ( ˙m = 0), as pointed out by Shockley-James [18], who claim that their paradox indicatesthat even the conservation law for the linear momentum is not conserved.However, it has been shown ([4], [19]) that the effective force acting on a particle, and the corresponding equationof motion, might be modified if, besides the mechanical momentum of the particle, (mass) × (velocity), the physicalsystem carries an additional momentum related to the interaction fields. In our case the fields of our physical systempossesses a nonvanishing electromagnetic interaction momentum Q em and, moreover, it possesses also a momentum Q h that is due either to the internal stresses [4], [19] or to the charges induced by the field E q [19]. The purpose of ourletter is to derive within classical electrodynamics the equations of motion of q and m after determining the effectiveinteraction force between them, assuming the validity of conservation laws, the action and reaction principle, andthe contribution of the momentum of interaction fields, as also required for the solution of the Shockley-James [18]paradox. The role of the momentum Q h has been taken into account in the nonrelativistic interpretation of the atomicspin-orbit coupling [20] and we show that the (effective) interaction force here derived leads to an interpretation of theAB effects in terms of classical local forces. In fact, the derived interaction force is in agreement with the experimentalevidence of the observed phase shift ∆ φ in the effects of the AB type [14]-[17]. Concerning the test of dispersionlessforces through a long toroid [7], we show that our effective force agrees with the result of the test. Other possibletests for the interaction force, realizable with present technology [21], are discussed.
2- The interaction force between a charge q and a magnetic dipole moment m .The relevant em interaction, taking place in the effects of the AB type, is the one between a charged particle q and a neutral particle with a magnetic dipole moment m . The appropriate tensor describing the system composedby a charge q and a nonconducting dipole m is [4], [19], T µν = θ µν + S µν + δ U µ U ν , complemented by the continuityequation, ∂ µ T µν = 0, where θ µν is the em tensor, S µν is the stress tensor, and δ is the proper density of the propermass. Relevant quantities are the em momentum Q em and the momentum Q h due to stresses (referred to as the hidden momentum) [4], [19], their components being given by, Q iem = 14 πc Z ( E × B ) i dτ = Z θ i dτ ; Q ih = Z S i dτ. (1)Let p q = M q v be the linear momentum of the charge of mass M q and p m = M m v m that of the magnetic dipole m of mass M m . Integration over the volume of the continuity equation, ∂ µ T µν = ∂ µ ( θ µν + S µν + δ U µ U ν ), leads to, ddt ( Q em + Q h ) + Z ( ∂ j θ ij + ∂ j S ij ) dτ + ddt ( p q + p m ) = 0 , (2) ddt ( Q em + Q h ) + ddt ( p q + p m ) = 0 , (3)where, in (3), for a closed isolated system the volume integral of the divergences ∂ j θ ij and ∂ j S ij vanishes, andexpression (2) provides the linear momentum conservation law (3). In the case of the interaction between q and m ,in the dipole approximation Q em and Q h can be expressed [8], [22], [4], [19], as, Q em = Z ρc A ( x ) dτ = qc A ; Q h = − qc A = m c × E q . (4)Moreover, for a finite stationary configuration, expression (3) implies [4], [19], Q em + Q h = qc A + m c × E q = 0 . (5)Since the interaction momentum Q h is a nonlocal quantity, it can be expressed indifferently in terms of A or E q in(4). In the dipole approximation A ( x − x m ) = m × ( x − x m ) / | x − x m | is the vector potential of the magnetic dipole m and E q = q ( x − x q ) / | x − x m | is the electric field of the charge q . In (4) q A has to be evaluated at the positionof the charge x q and c − m × E q at x m .Making use of the relation ∂ β θ αβ = − c − F αλ Jλ , where F αλ is the em field-strength tensor, after volume integration,for our closed system the em and stress force density can be expressed respectively as, f em = − ( d/dt ) Q em and f h = − ( d/dt ) Q h . Then, f em + f h = d ( p q + p m ) /dt or, f em − ddt Q h = ddt ( p q + p m ) = f q + f m , (6)where in (6) f q = d p q /dt and f m = d p m /dt are the effective forces acting on q and m respectively. We can seefrom (6) that the effective forces, f q and f m , and the corresponding equations of motion are affected by the term d Q h /dt . When the magnetic dipole is made of conducting material, for the stationary system the electric field E q induces charges on the magnetic dipole creating an electric field E ind that provides zero total electric field insidethe dipole, E ind + E q = 0 . In this case, if the magnetic dipole is completely shielded, there are no internal stressesand no momentum due to stresses related to m [19]. Still, the original Q em is modified now by the presence of thenonvanishing E ind . In order to take into account the presence of E ind , the relation Q em + Q h = 0 of (5) is replacedby the relation Q em + Q em − ind = 0 where Q em − ind = Q h is the em momentum related to the induced charges andtheir field E ind = − E q inside the dipole. Thus, we assume here the validity of expression (5) with Q h representing,depending on the case, either the momentum due to the stresses or the momentum due to the induced charges. Thephysical results that can be derived from (3) and (6) are supposed to be model-independent and are shown to beholding for both the cases of conducting and nonconducting magnetic dipole [19] in solving the Shokley-James paradox[18]. Then, what (6) implies is that the action of the time-variation of Q h (or Q em − ind = Q h ) has to be taken intoaccount in (6) for determining the effective f q and f m that comply with the equilibrium condition and the momentumconservation law.
3- Action and reaction principle.
For our closed system we are left to split equation (6) into two equations of motion, one for q and one for m . Forour purposes, we consider the case where m is stationary and q is moving relative to m . The time derivative of Q em can be expressed as, f em = − ddt Q em = Z ( ρ E + j × B ) dτ = q E + qc v × B + Z ( j × B q ) dτ (7)= − qc ∂ t A + qc v × B + ∇ ( m · B q ) , where in (7), j × B = j q × B + j × B q . In (7) the terms − qc ∂ t A and qc v × B represent the standard force on q andthe last term stands for the standard force [23], [8], [9] on the magnetic dipole m in the presence of the magneticinduction field B q = c − v × E q of the moving charge. Q em depends on the interacting fields of both of q and m and its time derivative (7) contains terms representing forces localized and acting on q as well as forces localized andacting on m . Being a nonlocal quantity, the momentum Q em is not localized on any of the field sources and thusits time derivative, − ( d/dt )( Q em ), cannot be taken to represent uniquely the forces acting on one of the interactingparticles, e.g., q (or m ). Since similar considerations hold for Q h , the variation − ( d/dt )( Q h ) represents force terms,some acting on q and others on m (and not uniquely on q or on m ).When j q = ( ρ/c ) v forms part of a current loop in a neutral wire, after integrating over the closed loop, the action andreaction principle holds for interaction forces between the loop and the dipole m even for time-varying fields. However,for our system the term ( q/c ) v represents a non-neutral ”open current” element and, due to its nonvanishing electricfield E q , the em momentum Q em = 0. As mentioned above, in this case we have ( q/c ) v × B + ∇ ( m · B q ) = 0, becausethe force ( q/c ) v × B is always perpendicular to the direction of motion, while ∇ ( m · B q ) has also a nonvanishinglongitudinal component in the direction of v . Moreover, the effective interaction force (between q and m ) must besuch as to solve the Shokley-James paradox [18], where the radiation force on q , given by − c − q∂ t ( A ), requires to bebalanced by an equal and opposite force on m . As well known, this paradox can be solved by taking into account themomentum Q h [4], [19] with the related force f h = − ( d/dt ) Q h , which, on account of expression (4), may convenientlywritten as, f h = − ddt Q h = − ddt ( m c × E q ) = ddt ( qc A ) (8)= − ˙m c × E q − m c × ( ∂ t E q ) = − ˙m c × E q + qc ( v · ∇ ) A . When v = 0, the force f h in (8) is − ∂ t ( Q h ) = − c − ( ˙m × E q ) = c − q∂ t ( A ), effective when the current of the magneticdipole is varying with time and em radiation fields are involved. Then, the Shokley-James paradox is solved becausethe force − c − q∂ t ( A ), acting on q in (7), is balanced by the equal and opposite force − c − ( ˙m × E q ) = c − q∂ t ( A )acting on m .Neglecting higher order relativistic terms, expression (5) holds even when q moves with velocity v relative to m .Thus, going beyond the effects of pure radiation fields, we assume that the idea behind the Shokley-James paradox(that em interaction must comply with the action and reaction principle and momentum conservation) can be extendedto include the velocity dependent terms of (7) and (8). For this purpose it is sufficient that the equal and oppositeinteraction forces be related to the gradients of the same interaction energy. The quantity − m · B q represents theinteraction potential energy between m and B q and its negative ∇ leads to the force ∇ ( m · B q ) acting on m . In orderfor the action and reaction principle to be holding, the same potential energy must lead also to an equal and oppositeforce acting on q . In fact, with the help of (4) and (5), we may write, − m · B q = − m · ( v c × E q ) = v · ( m c × E q ) = − qc ( v · A ) , (9)where − ( q/c ) v · A represents the interaction potential energy − R ( j q · A ) dτ [23] of the open current element ( q/c ) v in the presence of the vector potential A . Then, expression (9) implies that, in correspondence to the force term ∇ ( m · B q ) acting on the dipole, there is an equal and opposite force ( q/c ) ∇ ( v · A ) = ( q/c ) v × B + ( q/c )( v · ∇ ) A acting on the charge, as shown below.
4- The equations of motion for the charge q and the dipole m .As a criterion for identifying which are the effective forces on either q or m , we assume the validity of the actionand reaction principle. With the help of Maxwell’s equation ∇ × B q = c ∂ t E q , equations (7), (8), and the identity ∇ ( m · B q ) = ( m · ∇ ) B q + c m × ( ∂ t E q ), we may write (6) as, f em + f h = − qc ∂ t A + qc v × B + qc ( v · ∇ ) A − ˙m c × E q + m c × ( ∂ t E q ) + ( m · ∇ ) B q = f q + f m (10)The two terms of (10), − qc ∂ t A and − ˙m c × E , are equal and opposite and, algebraically, may cancel. However, from aphysical point of view, we may not suppress them if they represent, as they do in this context, the equal and oppositeaction and reaction forces on q and m , respectively. About the terms qc ( v · ∇ ) A and m c × ( ∂ t E q ) of (10) (which areequal and opposite and, algebraically, may cancel) we consider the following possible interpretations. a ) As done by Aharonov et al. [4], we may assume that the momentum Q h is localized on the dipole m . Then,in this case the force (8) − ( d/dt ) Q h = − ˙m c × E + qc ( v · ∇ ) A = − ˙m c × E − m c × ( ∂ t E q ) is entirely acting on m and qc ( v · ∇ ) A and m c × ( ∂ t E q ) in (10) are equal and opposite force terms that cancel because both act on m . This way,the longitudinal components of the forces disappear and expression (10) becomes, f q + f m = − qc ∂ t A + qc v × B − ˙m c × E + ( m · ∇ ) B q , (11)where the first two terms on the rhs represent f q and the last two represent f m . b ) Nevertheless, as mentioned above, the momentum is a nonlocal quantity and, as such, theoretically and experi-mentally, neither Q em or Q h can be localized on any of the source of the interaction fields. What can be assumed aslocalized are the forces that are related to the time (or space) variations of Q em or Q h . Therefore, a priori, we cannotexclude that the term qc ( v · ∇ ) A may represent a force acting on the current element ( q/c ) v . In this case, we maynot cancel algebraically the two terms qc ( v · ∇ ) A and m c × ( ∂ t E q ) in (10) because they represent, respectively, a forceacting on q and a force on m . Thus, being v × B + ( v · ∇ ) A = ∇ ( v · A ), in the reference frame where the magneticdipole is stationary, expression (10) leads to the following effective forces and corresponding equations of motion, f q = − qc ∂ t A + qc ∇ ( v · A ) = ddt ( p q ) (12) f m = − ˙m c × E + ∇ ( m · B q ) = ddt ( p m ) . (13)The Lagrangian formulation for deriving (12) and (13) will be given in a future contribution.
5- Experimental evidence.
The correct choice, a ) or b ), for the effective force on the charge q in motion, needs to be corroborated experimentally.In the experiment by Becker and Batelaan [7] the long macroscopic toroid, adopted for testing the time of flight of q , has not been used to observe the AB effect. Moreover, it is reasonable to assume that the vector potential A is nearly uniform inside the long toroid where, in the direction of motion, ( v · ∇ ) A ≃
0. Thus, no action on themoving charge is exerted inside the toroid by the force term qc ( v · ∇ ) A of (12) and, consequently, no variation of itstime of flight is foreseen, in agreement with the experimental results [7]. In fact, we expect our force to be actingbriefly on the moving q at the beginning and end of the toroid only. The ideal experiment to more easily detect thelongitudinal component of the force must adopt an arrangement where the perpendicular component v × B vanishesand, moreover, A is not uniform in the direction of motion, so that ( v · ∇ ) A q = 0. Such an arrangement is obtainedin the case of the AB effect with a standard toroid, where the force (12) is nonvanishing. Actually, in the experimentperformed by Tomonura et al. [15], which detects the Aharonov-Bohm phase shift ∆ φ AB , a microscopic toroid hasbeen used and the result corroborates the existence of a longitudinal force.The effect of a local force on the phase shift of the AB system has been derived in Refs. [8], [9] and we reconsiderit here starting from the free-force phase φ = ~ − ( p · x − ¯ Et ) of the interfering wave function of the electronsin the AB effect. In the presence of the small force f q , the particles moving on opposite sides of the solenoidacquire a relative lag that produces the phase shift ∆ φ , either because of the particles relative variation δ v or δ x [8], [9], [19]. Assuming that the particle of mass M q slightly changes its momentum p q = M q v under theaction of the force f q = M q d v /dt , we have δ v = M − q R t f q dt . If x ( t ) is the position of the particle when p q isconstant, with vdt = dx along the path of the particle (from −∞ to + ∞ ), the force has the effect to change it by∆ x = R ( δv ) dt = M − q R dt R t ( f q ) x dt = ( vM q ) − R dt R t [ ∇ ( qc A · v )] x dx . Then, ∆ x = ( vM q ) − R ∞−∞ qc A · v dt , leadingto the phase shift δφ = ~ − ( p q · ∆ x ) = ~ − qc R ∞−∞ A · d x . Because of the topological properties of the system and thesymmetry of vector potential A , the resulting relative phase shift between particles moving along the opposite sidesof the solenoid is, ∆ φ = − ~ − qc Z ∞−∞ A · d x = ~ − qc H C A · d x = ∆ φ AB , (14)where ∆ φ AB is the observable Aharonov-Bohm phase shift.Similar conclusions may be drawn for the Aharonov-Casher [2] and Spavieri [3] effects. In the case of the ACeffect, m is moving with velocity v m = − v relative to a static electric charge distribution. Then, in (13) the term ∇ ( m · B ) = − c − ∇ ( m · v m × E ) = ∇ ( P m · E ) = ( P m · ∇ ) E represents the force on the electric dipole moment P m = c − v m × m of the moving m . Thus, the force on the moving magnetic dipole m turns out to be given by thesame expression as derived by Boyer in his classical interpretation of the AC effect [8], [9]. The experimental evidencefor the existence of the local force is given by the experiments cited in [16] and [17].
6- Concluding remarks
We have derived within classical electrodynamics the expressions (12) and (13) for the effective interacting forceon the charge q and on the dipole m , respectively. Starting from first principles, we describe the isolated system bymeans of the total tensor T µν assuming the validity of the continuity equation ∂ µ T µν = 0 and the conservation law ofthe total linear momentum. Our effective force expressions, which solve the Shockley-James paradox, account for aforce term acting in the direction of motion. This term foresees that an observable phase shift must take place betweeninterfering particles encircling the magnetic flux in the AB effect. The phase shift occurs because of the relative lagof the particles produced by the local action of the force. The best experimental evidence for the existence of thelongitudinal force term in the AB and AC effects, is given by the observed phase shift in the tests of Refs. [14]-[17].The tradition in physics, based on cause and effect, requires observed effects to be explained by means of the localcauses that produce them, be their origin due to electromagnetic interaction or even interacting quantum systems.With our approach, the observed AB phase shifts can be interpreted in terms of the action of local em effectiveforces, reinforcing the view of the classical origin of the effects of the AB type. For a conclusive interpretation of theAharonov-Bohm effect it is essential, as suggested in Ref. [7], to close the loopholes that exist in the tests of the eminteraction. Ideally, tests of the em forces acting in the q − m interaction (some are discussed in Ref. ?? ) should aimat verifying the validity of the action and reaction principle, the conservation laws, and the conclusive expressions ofboth forces f q and f m .
7- Acknowledgments