Force law in material media, hidden momentum and quantum phases

Accepted Manuscript Force law in material media, hidden momentum and quantum phases Alexander L. Kholmetskii, Oleg V. Missevitch, T. Yarman PII: DOI: Reference:

S0003-4916(16)00069-5 http://dx.doi.org/10.1016/j.aop.2016.03.004 YAPHY 67067

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Annals of Physics

Received date: 16 November 2015 Accepted date: 20 March 2016 Please cite this article as: A.L. Kholmetskii, O.V. Missevitch, T. Yarman, Force law in material media, hidden momentum and quantum phases, Annals of Physics (2016), http://dx.doi.org/10.1016/j.aop.2016.03.004 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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Force law in material media, hidden momentum and quantum phases Alexander L Kholmetskii1, Oleg V. Missevitch2 and T Yarman3,4 1

Belarusian State University, Minsk, Belarus, e-mail: [email protected] Institute for Nuclear Problems, Belarusian State University, Minsk, Belarus 3 Okan University, Akfirat, Istanbul, Turkey 4 Savronik, Eskisehir, Turkey 2

Abstract We address to the force law in classical electrodynamics of material media, paying attention on the force term due to time variation of hidden momentum of magnetic dipoles. We highlight that the emergence of this force component is required by the general theorem, deriving zero total momentum for any static configuration of charges/currents. At the same time, we disclose the impossibility to add this force term covariantly to the Lorentz force law in material media. We further show that the adoption of the Einstein-Laub force law does not resolve the issue, because for a small electric/magnetic dipole, the density of Einstein-Laub force integrates exactly to the same equation, like the Lorentz force with the inclusion of hidden momentum contribution. Thus, none of the available expressions for the force on a moving dipole is compatible with the relativistic transformation of force, and we support this statement with a number of particular examples. In this respect, we suggest applying the Lagrangian approach to the derivation of the force law in a magnetized/polarized medium. In the framework of this approach we obtain the novel expression for the force on a small electric/magnetic dipole, with the novel expression for its generalized momentum. The latter expression implies two novel quantum effects with nontopological phases, when an electric dipole is moving in an electric field, and when a magnetic dipole is moving in a magnetic field. These phases, in general, are not related to dynamical effects, because they are not equal to zero, when the classical force on a dipole is vanishing. The implications of the obtained results are discussed. 1. Introduction The force law in classical electrodynamics (CED) of material media remains a subject of discussions up to the modern time [1-23]. Nowadays, there are two alternative ways to this law. One of them is based on the Einstein-Laub force density, which, in the case of absence of free charged, adopted hereinafter, is postulated in the form [10, 11] 1 P 1 M f EL   P   E  M   B  B E , (1) c t c t where P, M are the polarization and magnetization, and E, B are the electric and magnetic fields, correspondingly. The second approach implies the extension of the Lorentz force law for free charges to the case of bound charges in material media, where the density of four-force f is defined in a usual way as f   F   j  (, =0…3). (2)   Here F  is the tensor of electromagnetic (EM) field, and j is the four-current density, which is determined via the relationship j    M  , (3) and M  is the magnetization-polarization tensor [12, 13]. The spatial components of eq. (2) give the usual Lorentz force law f L  E  j  B , where     P

(4) (5)

2 is the charge density, and (6) j    M  P t is the current density. In general, eqs. (1) and (4) are not equivalent to each other. At the same time, at the second half of the past century it was pointed out [14] that eq. (4) is anyway incomplete, because it does not contain the force component, resulting from the time variation of so called “hidden” momentum of magnetic dipoles in an electric field d 1 Fh   mE , (7) dt c where the hidden momentum is defined as 1 Ph  m  E , (8) c and m being the magnetic dipole moment. As the authors of ref. [14] demonstrate, the existence of the force component (7) is strongly required to resolve the Shockley-James paradox, presented in the mentioned paper. Later, it was recognized that the existence of the force component (7) stems from the general theorem, demanding zero total for any static configurations of charges/currents [15, 16]. One should stress that the recent discussion with respect to the resolution of the paradox by Mansuripur [1] did not require considering the most interesting case of time-dependent hidden momentum of magnetic dipole, which makes incomplete the entire discussion about the force law in material media. Thus, in section 2 we focus our attention to the case, where the hidden momentum of a dipole can vary with time, which, according to the common interpretation of its physical meaning provided long ago (e.g., [14, 16, 19-23]), induces the appearance of corresponding force component (7). We show, however, that the addition of the known expression for this force component to the usual Lorentz force (2) cannot be done covariantly, so that the issuing expression for force on a dipole (named by us as the expanded Lorentz force) does not obey the relativistic transformation of force. The adoption of the Einstein-Laub force law does not resolve the issue, because for a small dipole, the density of Einstein-Laub force integrates exactly to the same equation, like the expanded Lorentz force. In section 3 we present a number of particular examples, which illustrate the validity of our general conclusions. In particular, in sub-section 3.1 we consider the motion of point-like charge with respect to a point-like magnetic dipole and demonstrate that the force component due to time variation of hidden momentum of magnetic dipole is strongly required, in order to maintain the energy-momentum balance for the isolated system of concern. In sub-sections 3.2 we consider the motion of an electric dipole near a homogenously charged insulating line, and in sub-section 3.3 we deal with the motion of a magnetic dipole in a vicinity of a long straight wire carrying a steady current, and show that in both cases the Lorentz force law and Einstein-Laub force law yields relativistically non-adequate results. Thus, we conclude that the problem of covariant formulation of force law in material media remains unsolved to the moment, and in section 4 we discuss some physical difficulties of CED in material media, which indicate that the Lorentz force law, being undoubtedly valid for free charged particles, cannot be directly extended to magnetized/polarized media. As the outcome, we suggest applying the Lagrangian approach to the derivation of the force law in material medium. Taking the Lagrangian density for a magnetized/polarized medium in a relativistically invariant form (section 5), we achieve a novel expression for the force on a small dipole in an EM field, with a novel expression for the generalized momentum of the dipole (section 6). The latter expression implies two novel quantum effects, emerging for an electric dipole in an electric field, and for a magnetic dipole in a magnetic field. We discuss the properties of these phases and the possibility of their measurement. We conclude in section 7.

3 2. Force component due to variation of hidden momentum of a magnetic dipole The notion of hidden momentum was originally introduced for a static distribution of charges, whose center of energy rests in an inertial frame of observation. Then, according to the general relativistic theorem [15, 16], the total momentum of this isolated system must be equal to zero. We reproduce the proof of the latter statement from ref. [16], where the authors notice, first of all, that for any isolated system, the four-divergence of total energy-momentum tensor T (which represents the sum of mechanical T  M and EM T  EM parts) is equal to zero, i.e.  T    T  M   T  EM  0 . Further, for a static system, the total energy density T00 is time-independent, i.e. T 00 t  0 , and hence,  iT 0i  0 , where i=1…3.





Introducing the vector W T 01 c , T 02 c , T 03 c , we see from the previous equality that  W  0 . Then, taking into account that the total momentum P of any isolated system is determined by the equality [17] 1 P i   T 0i dV c (where the integration is carried out over the entire space, and dV is the volume element), we further write: P i   W i dV     Wx i dV   x i  WdV   Wx i dS .

 

 

S

In the latter integral, the integration is carried out over the surface S enclosing the volume V. Thus, if the vector W vanishes at infinity faster than 1/r3 (which is certainly the case for bound field components generated in static system), the latter integral vanishes at infinity, and we obtain1 Pi=0. At the same time, the momentum of EM field Pf of this configuration is not equal to zero. Since the mechanical momentum of the system, related to its motion (hereinafter designated as PM), is equal to zero in the frame of observation, then one more component of mechanical momentum Ph (is not directly related to a motion of the system and named as “hidden”) must exist, in order to counterbalance the momentum of EM field Pf of the considered charge distribution. In other words, for the isolated system in question we have the equality PM  Ph  Pf  0 , or, which is the same at PM=0, Ph  Pf  0 . (9) One should notice that any system of mechanically free charges cannot compose a static (or stationary) configuration and thus, the notion of hidden momentum can be introduced only for bound charges in material media, though the inclusion of free charges to such configuration is also possible. In fact, the hidden momentum is related to bound currents, and its emergence for various models of magnetic dipoles is analyzed in numerous publications (see, e.g. [16, 20, 21]). Next we assume that the hidden momentum in eq. (9) becomes time-dependent, and its variation happens without applying any external forces to the considered isolated system of charges. In this case, the center of energy of the system remains immovable, and PM=0 at any time moment, when the radiation of the system is negligible. Hence, in the center-of-energy frame, the time evolution of this system is described by the equalities

1

We mention the paper by Franklin [18], who questioned the validity of this theorem. However, one can see that the inequality Pi0 derived in [18] for a static configuration “charge plus magnetic dipole” is valid only for EM component of the total momentum, as shown in [19]. Thus, the conclusion by Franklin about the incorrectness of the mentioned theorem is erroneous.

4 Ph t  0    Pf t  0 ,

(10a)

dPh (t ) dt   dPf (t ) dt ,

(10b) which means that the hidden momentum is equal to the field momentum with the reverse sign at any time moment. The most popular representative of such static system is the charge e and magnetic dipole m, separated by some distance r and resting in a frame of observation. This system has been also used by Mansuripur in the formulation of his paradox [1], where, however, the magnetic dipole moment m was assumed to be time-independent. The hidden momentum of this configuration is defined by eq. (8), while the field momentum, as is shown in refs. [16, 20], is equal to Pf   m  E  c , in a full agreement with eqs. (8) and (10a). Now, instead of considering a uniform motion of this system (as in [1]), we assume that both the charge and the dipole remain at rest in a laboratory frame, but the magnetic dipole moment slowly decreases with time (e.g., due to thermal losses of carriers of current in the magnetic dipole; hereinafter we assume that the radiation of the system is negligible). This is just a configuration of the Shockley-James paradox [14], where the notion of hidden momentum had been introduced for the first time, in order to get its consistent resolution. The paradox is formulated in the following way. First, we notice that the magnetic dipole generates the vector potential A  m  r r3 , which for the time-dependent m leads to the appearance of inductive electric field and the related force e A e m t  rme (11) eEind    3 c t c rme at the location of the charge e, and rme is the radius-vector, joining the dipole m and charge e. Due to the force (11), the charge begins to accelerate, so that in the absence of reactive force on magnetic dipole, the center of mass of this configuration also accelerates, thus violating the relativistic theorem [15, 16] reproduced above. This paradox is resolved via the introduction of a reactive force on the magnetic dipole due to time variation of its hidden momentum (8), i.e. dP 1 e m t  rem  , (12) Fh   h   m t  E    3 dt c c rem where we have used the expression E  erem rem for the electric field of charge e at the location of magnetic dipole m; rem is the radius-vector, joining the charge e and dipole m. Hence, taking into account that rem=-rme, we see that the forces (11) and (12) mutually balance each other, so that the center of energy of this configuration remains at rest in a laboratory frame. By such a way the Shockley-James paradox is fully resolved. A point of trouble is that the density of four-force (2) does not contain any term, representing the force component due to time variation of hidden momentum and thus, the only way out, presumably, is to introduce “manually” (in Mansuripur terminology [22]) this force term into the rhs of eq. (2). However, doing so, we face a number of serious difficulties, which are discussed below. First of all, one should emphasize that the expression for hidden moment (8) has been derived for a static (or stationary) situation, and, in general, it is not obvious that it remains valid for dynamical problems of CED. Nevertheless, after the publication [14], the expression (8) was commonly used in all problems of CED, dealing with the motion of magnetic dipoles in an electric field, without rigorous proof of its validity in the general case, where both the mechanical and EM momenta of interacting charges/dipoles vary with time. Correspondingly, the force term due to time variation of hidden momentum is commonly written in the form (7). 3

5 Even so, one should demand that the introduction of the force density due to hidden momentum contribution into eq. (2) should be done in a way, which does not destroy the covariance of this equation. This is possible only in the case, where the components of four-force density due to the variation of hidden momentum compose a four-vector. Thus, for further progress, we have to conjecture the time-time component of four-vector “hidden energy-momentum density”. In ref. [23], the author suggested to write the energy component of the magnetic dipole, associated with its hidden momentum, in the form (13) U h  v  m  E  c . However, the hidden energy (13), and hidden momentum (8) cannot compose a fourvector, because in the general case, the electric field, as the component of the tensor of EM field, can be a non-zero quantity in one frame K, and can become zero in some other frame K0. In this case, both the hidden energy and the hidden momentum do vanish in the frame K0, whereas in the frame K they are not vanishing. Obviously, this situation is impossible for four-vectors. When E does not vanish in all frames of observation, the same line of reasoning can be applied to the magnetic dipole moment m: it can be equal to zero in one reference frame (e.g., in the rest frame of electric dipole K0, where Uh, Ph=0), but can be a finite value in another reference frame K, wherein the dipole is moving (and where, in general, Uh, Ph0). These observations also indicate that the quantities Uh, Ph cannot compose a four-vector. One more option for covariant description of the force component due to hidden momen tum contribution is to relate the time variation of hidden energy-momentum density ph with the field energy-momentum density p f

dph



d   dp f

where p f







via the generalization of eq. (10b), i.e.

d ,

(14)

is the momentum density of EM field ( p f  u c , p f  S c (i=1…3); u being the 0

i

i

2

EM energy density, and S the Poynting vector), and  is the proper time. Hence, instead of eq. (2) we obtain the force density in the form  (15) f   F   j   dph d , or, due to the equality (15),  f   F   j   dp f d . (16) We will not discuss here the general applicability of eq. (14) to non-stationary systems, containing magnetized/polarized media. Yet we should stress that eqs. (15), (16) imply the case of negligible radiation, which is often adopted in the analysis of systems, possessing hidden momentum. Thus, the field momentum density, entering into this equation, stands for the bound (or velocity-dependent) field solely. At the same time, for such a bound field, the energy density u and energy flux density S do not compose a four-vector, because the four-divergence of EM energy-momentum tensor for the bound EM field is not equal to zero. In these conditions, the  momentum density p f is not four-vector, too. This leads to the known difficulties, e.g., in the definition of total mass of the classical electron, to the appearance of “4/3 problem for the electron, and some others, which we do not consider in the present paper, and address to the relevant publications (e.g., [12-13, 17, 24-27]). It is important to point out that in these conditions, the force density f defined by eqs. (15) or (16), in general, is not a four-vector. Therefore, the relativistic transformation of force, which contains the term of hidden momentum contribution, can be incompatible with the Lorentz transformation for force. The validity of this statement is illustrated in the next section, where we continue our analysis in 3d-form.

6 3. Expanded Lorentz force and Einstein-Laub force on a moving dipole: general presentation and particular examples First of all, we present the expression for the force F, acting on a point-like dipole with the proper electric dipole moment p0 and magnetic dipole moment m0, derived, e.g., in ref. [6] on the basis of eqs. (4)-(6): 1 F   p   E  m  B   v    p  B , (17) c where   1  p  v v  v  m0 , p  p0  (18) 0 c v 2   1 m  v v  p0  v m  m0  (19) 0 c v 2 are electric and magnetic dipole moments, correspondingly, in an inertial frame of observation,





1 / 2

wherein the dipole moves with the velocity v, and   1  v 2 c 2 is the Lorentz factor. We remind that eq. (17) implies the constancy of p0, m0, and it has been obtained long ago by Vekstein [28]. Recently, we generalized eq. (17) to the case of variable electric dipole moment p0 and magnetic dipole moment m0 and added the force term due to the time variation of hidden momentum of the dipole. As a result, we obtained the expression for force in a more general form as [29] 1 d  p  B   1 d m  E  . F   p  E   m  B   (20) c dt c dt The first three terms on rhs of this equation originates from the Lorentz force law in material media (eqs. (4)-(6)), with the addition of the hidden momentum contribution (the last term). Due to this reason, we can name eq. (20) as the expanded Lorentz force (ELF) on a dipole. Now it is interesting to compare eq. (20) with expression for force on a dipole, derived via the Einstein and Laub force law (1), which for stationary polarization/magnetization, and point-like dipoles integrates to (e.g., [6]): 1 1 FEL   p   E  m   B  p  v   B  m  v   E , (21) c c where v is the velocity of dipole in a frame of observation. In the general case, where both P and M are time-dependent, the straightforward calculations yield two additional terms in eq. (21), i.e. 1 1 1 dp 1 dm FEL   p   E  m   B  p  v   B  m  v   E  B E. (22) c c c dt c dt For further transformation of eq. (22), we use the vector identities m    B  m  B  m  B , p    E    p  E    p  E , as well as the Maxwell equations 1 E 1 B  B  ,  E   , (23a-b) c t c t assuming the absence of source charges at the location of dipole. Hence, we get 1 B 1 E 1 FEL   p  E   m  B   p   m  p  v   B  c t c t c 1 1 dp 1 dm m  v   E  B E  c c dt c dt 1 dp 1 1  B  1 dm  E   p  E   m  B    B  p  v   B    E  m   v   E  , or c dt c c  t  c dt  t 

7 FEL   p  E   m  B  

1d  p  B   1 d m  E  . (24) c dt c dt We reveal that the Einstein-Laub force on a dipole (24) exactly coincides with eq. (20), obtained via the Lorentz force law with the addition of term due to hidden momentum contribution. Thus, in contrast to the existing opinion (see, e.g. [1, 22]), the Einstein-Laub force law and the expanded Lorentz force law occur equivalent to each other with respect to the force acting on a point-like dipole, moving in an EM field. Hence, the introduced abbreviation ELF for the force (20) can be equally applied to eq. (24), too, which, according to the reader’s desire, can be also decoded as the Einstein-Laub force. Therefore, our further analysis implemented in sections 3 and 4 is equally applicable to both approaches, based on the Lorentz force and the Einstein-Laub force, correspondingly. With respect to other known force laws in material media (the Minkowskian and Abraham force laws [30]) we notice that modern experiments seem invalidate the Minkowskian force [31], while the Abraham force density, being equivalent to Einstein-Laub force density for homogenous media [30], also yields eq. (24) for a point-like dipole. In sub-section 3.1 we consider a motion of point-like charge with respect to magnetic dipole and demonstrate that the last term on rhs of eq. (20) actually plays the important role in the fulfillment of the law of conservation of total energy-momentum for this isolated system. At the same time, in sub-sections 3.2 and 3.3 we present examples, which disclose the incompatibility of ELF (20) with the relativistic transformation of force2.

3.1. Point-like charge and magnetic dipole in a relative motion Consider a point-like charge e, moving at the velocity v along the axis x towards a point-like magnetic dipole located in the origin of coordinates and possessing the constant proper magnetic dipole moment m0 along the axis z. For this configuration, we want to determine mutual forces between the charge e and the magnetic dipole, using eq. (17), or, alternatively, eq. (20), in order to clarify the role of the last term on its rhs (the hidden momentum contribution). One can see that the mutual forces between the charge and the dipole have non-vanishing y-components only, and we further suppose that the dipole is fixed on a massive platform, while the charge is moving inside a thin insulating tube oriented along the axis x and attached to the same platform (see Fig. 1). In this case, the mechanical forces on the charge and the dipole due to the platform exactly counteract the EM forces in this system, and the velocity of charge has only the x-component and does not vary with time. First, we calculate the force acting on the dipole due to the moving charge, which, according to eq. (17) applied to the configuration of Fig. 1, is presented by the single term Fm  m0  Be  , where Be stands for the magnetic field of the moving charge. We find that the non-vanishing ycomponent of this force has the value 2 E   Fm y  m0  Be z  m0v y  m03ve 1  v 2  , (25) y c y cx t   c  where we have used the known expression for the electric field of a moving charge [12, 13] e 1 v2 c2 r (26) Ee  32 1  v 2 c 2 sin 2  r 3 with =0 for the configuration of Fig. 1.

 

2









We point out that this transformation is applicable to any particles with non-variable internal structure (e.g., pointlike particles) and, in general, it can be violated, when an internal structure varies with time, giving rise to the corresponding variation of the energy-momentum components, lying beyond the approximation of point-like particles. The latter situation can be realized, where dp0/dt, dm0/dt0 (see section 5). However, at p0, m0=const (i.e., the case considered in this section), the violation of relativistic transformation of force would indicates the incorrectness of EFL.

8 Determining the reactive force on the charge, we involve the expression for the zcomponent of magnetic field of constant magnetic dipole [32] Bm z   m0 x 3 t  . Hence the EM force on the charge reads as Fe y  e v  B y  m03ve . (27) c cx t  Thus, the force on the dipole due to the moving charge (25), and the reactive force on the charge due to the dipole (27) both lie in the positive y-direction, and the resultant force, lying in the same direction, m ve  v2  (28) Fy  Fm y  Fe y  03  2  2  , cx t   c  induces the accelerated motion of the platform along the axis y. This result already looks doubtful. What is worse, both of the forces (25) and (27) keep their sign at the time moments, when the charge passes the dipole3 and continues to move away from the dipole. This result obviously contradicts the energy-momentum conservation law for the isolated system “charge and magnetic dipole on a platform”. Indeed, taking the initial time moment, when the platform is at rest in a frame of observation, and the x-coordinate of the charge x-, and taking the final time moment, when x+, we see that at these time moments, the energy and momentum of interactional EM field between the charge and the dipole is equal to zero. However, during the motion of charge between these time moments, the platform is continuously accelerated due to the force (28) and finally acquires a final portion of momentum and, correspondingly, a final portion of motional energy. This result drastically contradicts the total energy-momentum conservation law for the system in question, and clearly shows the non-adequacy of the force law (17). The only way to recover the conservation of total energy-momentum for the configuration of Fig. 1 is to abandon eq. (17) in the favor of eq. (20), which includes the force term due to time variation of hidden momentum. Considering again the force on magnetic dipole due to moving charge, we have two nonvanishing terms in eq. (20): E 1 Fm  m0  Be   m0  e , (29a) c t where Ee is the electric field of the moving charge at the location of magnetic dipole. Using the Maxwell equation (23a) and the vector identity m0    Be   m0  Be   m0  Be , we can present eq. (29a) in the equivalent form v  Ee Fm  m0   Be  m0    . (29b) c Combining eqs. (26) and (29b), we obtain a single non-vanishing y-component of force: 2   Fm y   1  m0  vEz    m03ve 1  v 2  . (30) c  z  cx t   c  The reactive force on the moving charge due to magnetic dipole, as before, is defined by eq. (27). Hence, we see that, at least to the accuracy of calculations (v/c)2, the forces (27) and (30) mutually counteract each other, and the net force on the platform vanishes. Thus, no violation of the total energy-momentum conservation law happens with application of eq. (20). We add that in a higher accuracy of calculations, the force on the platform is not equal to zero, and is proportional to (v/c)3, i.e. m e v3 (31) Fy  3 0 3 x t  c (the sum of eqs. (27) and (30)). 3

Here we assume that the charge and dipole are point-like, and this event occurs without their collision.

9 This result indicates that for the problem sketched in Fig. 1, the hidden momentum of magnetic dipole does not exactly counterbalance the momentum of interactional EM field between the charge and the dipole. This means, in particular, that the equality (14) assumed above is not exactly fulfilled in dynamical situations. At the same time, the time variation of difference between field momentum and hidden momentum for the problem in question is counterbalanced by the time variation of mechanical momentum of platform due to the force (31), in full agreement with the law of conservation of total energy-momentum for the problem in Fig. 1. 3.2. Electric dipole and homogenously charged line Consider the motion of an electric dipole p0 near a homogeneously charged straight line, as shown in Fig. 2. In the frame K0, the line produces the electric field E0 y  20 y0 at the location of dipole, and, according to eq. (20), there is a single non-vanishing force component, acting on the electric dipole along the axis y 1 2 2 p  (32) F0 y   p  E   20 0 (  u  1  u 2 c 2  ). y y0  u Next, we determine the force on the dipole for an observer in the frame K, wherein the charged line moves at the constant velocity v along the axis x, while the dipole has the velocity V{v, u/v, 0}, where  v  1  v 2 c 2  . In this frame, due to relativistic transformation of fields (see, e.g. [33]), the y-component of electric field at the location of dipole is equal to E y  20 v y0 . (33) Besides, the moving line generates a magnetic field with a non-vanishing z-component in the plane xy: (34) Bz  20v v cy0 . According to eqs. (18), (19), the electric dipole moment of a moving dipole has a single non-vanishing y-component p y   p0  u , (35) and the magnetic dipole moment m has a single non-vanishing z-component mz  p0v c u . (36) Then, in the frame K, eq. (20) along with eqs. (33)-(36), yield the following y- and xcomponents of force on the dipole: F   p E y p0v Bz 20 p0 Fy   p  E   m  B    0    0y , (37) 2 y y  u y  u c y  v y0  u  v 1 2

1 d  p0 Bz  1 d  p0vEy       0. (38) c dt   u  c dt  c u  Comparing the obtained results (37), (38) with the relativistic transformation of force [33]  F F  v v  1 1  v F0  u    F   0  0 2 1    1  v  u  c 2  , (39) 2  v c  v   v  we find that for the configuration of Fig. 2, this transformation yields Fy  F0 y  v , which agrees with eq. (37). However, the x-component of force should be equal to Fx  F0 yuv c 2 , which is clearly at odds with eq. (38). The non-adequacy of eq. (38) can be seen from the fact that at the vanishing x-component of force, the x-component of momentum of dipole should be conserved in K, i.e. Fx  

10 d  V Mv   0 , dt

where  V  1  V 2 c 2  , and M is the rest mass of the dipole. Hence, due to the constancy of v, we derive (40) d V dt  0 , which is obviously wrong, because the dipole is accelerated along the y-axis of the frame K due to the force (37). Looking closer at the structure of eq. (38), we can find that the compatibility with the relativistic transformation of force is restored, when the second term on its rhs (the force due to variation of hidden momentum) is excluded. For the problem considered, this force component emerges in the frame K due to the non-vanishing value of magnetic dipole moment (36) of a moving electric dipole. Thus, we have found that for the problem in question, the inclusion of the term due to hidden momentum variation into eq. (20) leads to non-adequate results, whereas without this term, we formally get a relativistically consistent solution. 1 2

3.3. Magnetic dipole moving with respect to a long straight conducting wire Now we consider the modification of the previous problem, where the straight charged line is replaced by a conducting line, carrying a steady current I0, while the electric dipole p0 is replaced by the magnetic dipole m0{0,0,-m0}, see Fig. 3. In the frame K0, the magnetic field of the line exists at the location of dipole, which has a single non-vanishing z-component B0 z  2I 0 cy . Then eq. (20) tells us that the force on the dipole has a single non-vanishing y-component, i.e. F0 x  0 , (41) Fm0 y   m0  B0   12 d u  m0  B y . (42) y c dt In order to simplify further calculations, we assume that the mass of the dipole M is large enough, and its acceleration along the axis y (du/dt) due to the force (41) is negligible. Hence, during the time intervals near t=0, we can adopt the velocity of dipole u as the constant value. With this simplification, eq. (42) yields 2 F0 y  2 I 0 m0 cy 2 u . (43) Evaluating the components of force on the dipole in the frame K, we omit detailed calculations, which can be done straightforwardly, and present the final results. In particular, one can find that the y-component of force on the dipole is determined in the frame K via the equation F 2I m Fy  2 0 0 2  0 y (44) v cy  v u in a full agreement with the relativistic transformation of force (39). The x-component of force on the dipole in the frame K is calculated as the sum of the third and fourth terms of eq. (20), i.e. 1d  p  B x  1 d m  E x  0 , Fx  (45) c dt c dt where we have taken into account that p  v  m0 c ; the magnetic field B has the components 0,0,  v B0 z , the electric field E has the components 0,  v v  B0 y c ,0 , and m=m0 for the con-





figuration of Fig. 3. However, the result (45) is in strong contradiction with the transformation (39), which, for the force components (41) and (42), observed in K0, gives the non-vanishing value

11

uv 2I 0 m0 uv . (46)  c 2 cy 2 u 2 c 2 By analogy with the problem considered in sub-section 4.2, we can show that eq. (45) leads to the wrong inequality (40). The compatibility with the relativistic transformation of force is restored, when the third term of eq. (20), emerging due to the electric dipole moment of a moving magnetic dipole, is eliminated. This result once more illustrates the non-adequacy of ELF (20). Fx  F0 y

Having considered the problems in Figs. 1-3, we have shown in sub-section 3.1 (Fig. 1) that the force term due to time variation of hidden momentum (the last term of eq. (20)) can play the important role to prevent the violation of the total energy-momentum conservation law for an isolated system. At the same time, for the problem in sub-section 3.2 (Fig. 2), the presence of this force component in eq. (20) yields a relativistically non-adequate solution (40), when the magnetic dipole moment originates from the motion of electric dipole. What is more, the prob1 d  p  B  , where lem considered in sub-section 3.3 (Fig. 3) demonstrates that the force term c dt the electric dipole moment is developed by a moving magnetic dipole, also gives the same nonadequate solution (40), although its existence is directly implied in the Lorentz force law (2). These results reflect the impossibility of covariant description of the force law in material media via the Lorentz force, or the Einstein-Laub force, as we revealed above. In our opinion, this situation is related to the existence of some difficulties in physical interpretation of CED in material media, which were not pointed out in a due extent (see the next section), and which seems making inapplicable a direct extension of the Lorentz force law for freely moving charges (where it is undoubtedly correct) to a magnetized/ polarized medium. 4. Difficulties in the physical interpretation of the Lorentz and Einstein-Laub force laws in material media Rigorously speaking, the incompleteness of the Lorentz force law in material media in its standard form (2) is already seen from the fact that we have to add “manually” the force component due to hidden momentum contribution (here we even do not discuss the failure to make such an addition in a covariant form, as we disclosed in section 2). Hence, the four-force density (2) itself cannot, in general, be directly related to the derivative of total four-momentum density on proper time due to the missing of hidden momentum component and thus, it might be incompatible with the total energy-momentum conservation law and relativistic transformation of force (39). This observation explains the failure of the Lorentz force law (2) to provide relativistically adequate solutions, in particular, for the problem in Fig. 3. Below we discuss some other difficulties in the physical interpretation of CED in material media, which indeed make questionable the extension of the Lorentz force law for free particles to magnetized/polarized media. 4.1 Magnetization and bound currents in material media First, we address to eq. (3), whose structure guarantees that the four-current density j represents a four-vector. This means, in particular, that its time-like component (5) stands for the total charge density, and the space-like components (6) corresponds to total current density in a medium. When the medium is at rest, there are no difficulties in physical interpretation of these components. In particular, the first term on rhs of eq. (6) describes the contribution of circulating currents into the total current density, while the second term of this equation stands for the currents, emerging due to time variation of polarization. At the same time, one should remember that in a frame of observation, where the medium is moving, the total current density does also include the convective current of charges, resting with respect to the medium (e.g., positive ions of the frame of a current loop), and solely aimed to counterbalance the charge of conduction

12 electrons. However, the charges, immovable with respect to the medium, obviously do not contribute to its magnetization, even if the medium moves, and the neglect of this fact often leads to confusing results. As example, we remind the paper by Fischer [34], where he defined the magnetic dipole moment 1 (47) m   r  j dV 2V via the total current density j in any frame of references. Then, applying the Lorentz transformations to r and j, he derived the transformation m   1 v  v  m   1  p  v  , m 0  0 0  2  1v 2 2 in the mathematically correct way, but in contradiction with the definition of tensor M, and with a common sense, at least in the last term on rhs. In order to obtain the correct general expression for magnetization, we introduced in ref. [35] a notion of the proper current density jpr for a moving medium, and its difference from the total current density j is expressed by the fact that the latter is measured with a unit area, resting in a laboratory (which, in general, corresponds to a correct measurement procedure in special relativity), while the former is measured with a unit area, co-moving with the medium. One can see that in this case the contribution of immovable (in the rest frame of the medium) charges is excluded, and, as shown in ref. [36], the expression for magnetization (47) keeps its form both for a resting and moving medium, when the total charge density j is replaced by jpr, i.e. 1 (48) m   r  j pr dV . 2V With the latter expression, the relativistic transformation of magnetic dipole moment becomes compatible with the common definition of the magnetization-polarization tensor [36]. We thus conclude that in eq. (6), the total current density j and magnetization M should be determined via different operational procedures (with the involvement of a resting and a comoving unit areas, correspondingly, in a frame of observation), which, in general, is at odds with the spirit of special relativity. In the present analysis, we do not comment further on the revealed difficulty and only notice that, in general, this could be one of the reasons, restricting the applicability of ELF to magnetized/polarized media. 4.2. Electromagnetic field of a moving dipole and relativistic transformation of electric/magnetic dipole moment Next, we present one more physical difficulty in CED of material media, emerging in the analysis of relativistic transformation of electric (18) and magnetic (19) dipole moments. In this way, we first introduce the following definitions: - we name a small electrically neutral bunch of charges as a “pure electric dipole”, if its proper electric dipole moment p0 (measured in the rest frame of the bunch) differs from zero, while the proper magnetic dipole moment m0, as well as the multipole moments of higher orders, are equal to zero (e.g., two point-like charges –q and +q separated by a small distance l); - we name a small electrically neutral bunch of charges as a “pure magnetic dipole”, if its proper magnetic dipole moment m0 differs from zero, while the proper electric dipole moment, as well as the multipole moments of higher orders, are equal to zero (e.g., a tiny closed conducting loop with a steady current). Further, we designate pm  v  m0 c (49) the electric dipole moment, developed by a moving magnetic dipole m0, and m p  p0  v c (50) the magnetic dipole moment, developed by a moving electric dipole p0.

13 It is known that eq. (49) describes a “relativistic polarization” of a moving magnetic dipole (see, e.g. [13, 35]), while the effect (50), being non-relativistic in its origin, arises due to convective currents of moving charges of opposite sign, composing an electric dipole [13]. We emphasize an important property of the electric dipole moment pm: as we will see below, the electric field generated by this dipole in a frame of observation, is similar to the electric field of a pure electric dipole, moving at velocity v and having the proper electric dipole moment equal to pm. In other words, the expression for the electric field generated by the electric dipole moment pm, has the same structure, as that of a moving pure electric dipole. At the same time, we demonstrate below that the expression for the magnetic field, generated by a magnetic dipole mp, in no way can be reduced to the expression for the magnetic field of a pure magnetic dipole, which has the proper magnetic dipole moment mp, and moving with velocity v in a frame of observation. However, in ELF (20) no distinction is made with respect to the force on a pure electric/magnetic dipole, and on the electric/magnetic dipole components (49), (50). As is known, this force contributes into the balance between a mechanical momentum of an isolated system of moving charges/currents, and a momentum of their EM field, and it is not obvious that the neglect of different nature of the pure electric/magnetic dipoles, and the electric/magnetic dipoles (49), (60), is warranted in the analysis of this problem. So, let us calculate the field potentials in the Lorenz gauge for a pure magnetic dipole (m00, p0=0), and for a pure electric dipole (p00, m0=0) in an inertial frame of observation, where a dipole moves with the constant velocity v. Hereinafter we imply the constancy of vectors p0, m0, and we pay special attention to the field potentials, generated due to the electric dipole moment pm (49), as well as due to the magnetic dipole moment mp (50). First, consider a pure magnetic dipole m0 in its rest frame K0, where it produces the vector potential [32]) 3 (51) A0m  m0  r0 r0 , and the scalar potential  0 m  0 . Here r0 is the radial coordinate of the frame K0, and the subscript “m” reminds that we calculate the potentials of a pure magnetic dipole. Next, we determine the field potentials of the magnetic dipole in a laboratory frame K, wherein the dipole has the constant velocity v. Applying the Lorentz transformations for vector and scalar potentials between K0 and K, we obtain:  A  v v   1  m0  r0  m0  r0  v v   1 , (52) Am  A0m  0 m 2 3 3 v r0 r0 v 2 m    A0 m  v  c . (53) Involving eq. (51), we present eq. (53) in the equivalent form m  r  v   v  m0  r0   pm  r0 , (54) m   0 3 0 3 3 r0 c r0 c r0 where we have used the vector identity, a  b c  c  a  b , and then eq. (49). Eq. (54) indicates that the scalar potential of a moving magnetic dipole is defined merely by the electric dipole moment pm. We also notice that eq. (54) can be expressed via the radial coordinate r of the frame K, using the Lorentz transformation for length [33]. Now let us determine the field potentials for a pure electric dipole, which has the electric dipole moment p0 in its rest frame K0. In this frame, the dipole produces the scalar potential [32] 0 p  p0  r0 r0 3 , (55) and vector potential A0 p  0 , where the subscript “p” stands for the potentials, generated by a pure electric dipole. Using the Lorentz transformation for the potentials, we find in the frame K: p r (56)  p  0 p   0 3 0 , r0

14 v  p0  r0  . (57) 3 c cr0 Rigorously speaking, these expressions are valid for point-like electric and magnetic dipoles. Hereinafter we also imply their applicability to small dipoles of a macroscopic size, providing from the requirement that in these expressions the radial coordinates r0 and r are much larger than the sizes of the dipoles. Now let us compare the pair of equations (54), (56) for m, p, as well as the pair of equations (52), (57) for Am, Ap. In particular, in the comparison of eqs. (54), (58), we reveal that the electric dipole pm, emerging due to relativistic polarization of the moving magnetic dipole, generates the scalar potential, like a pure electric dipole, moving at velocity v and having the proper electric dipole moment equal to pm. Due to this reason, the total electric field Ep produced by the electric dipole moment of a moving dipole, is proportional to p (i.e., the sum of components containing the proper electric dipole moment (the first and second terms of eq. (18)) and the electric dipole moment pm (the third term of eq. (18)). Thus, when the dipole is moving in the external electric field E and magnetic field B, the expression for the component of field momentum, being defined by the product EpB, is proportional to p, too. Since the time variation of field momentum in many cases is related to the force on a dipole, we infer that this force could be also explicitly proportional to p. This result formally agrees with the structure of the first and third terms of eq. (20), though, as we have seen with the problem in Fig. 3, this occurs insufficient to justify the correctness of the third term of eq. (20). Next, we compare eqs. (52), (57) and find that the vector potentials Am and Ap occur quite different: The lines of Am circulate in space, whereas Ap is always parallel to the velocity v, and is presented by open lines. In these conditions, the components of the magnetic field Bm produced by a moving dipole, should have a different structure in the terms containing m0 and mp, correspondingly. This means that the expression for the field Bm cannot be proportional to the magnetic dipole moment m of a moving dipole defined by eq. (19), but should represent a more complicated function of m0, mp. Hence, when a dipole is moving in an external EM field, the fraction of interactional field momentum defined by the product EBm, is not proportional to the magnetic dipole moment m, either. Due to a closed link of force on dipole with the time variation of field momentum, this force also should represent some explicit function of m0, mp, which is not realized in eq. (20). At the same time, in the presence of an external magnetic field B solely (when the interaction between magnetic dipole moment and B-field happens with the vanishing field momentum), the difference between magnetic fields generated by the magnetic dipole m0 and the magnetic dipole mp, correspondingly, does not affect the field momentum and occurs not essential. Therefore, in the force component m  B  (the second term on rhs of eq. (20)), the application of relativistic transformation for magnetic dipole moment (19) might be allowed. However, when we consider the interaction of a magnetic dipole moment with an external electric field E (i.e., the hidden momentum contribution), the interactional momentum of EM field essentially depends on the structure of magnetic field of a moving dipole and thus, it cannot be proportional to m, but should be a more complicated function of m0, mp, as we have mentioned above. Correspondingly, in the general case, for the force component due to time variation of hidden momentum, the transformation (19) becomes inapplicable. This implies that this force component should have different structure for a true magnetic dipole (defined by the first and second terms of eq. (19)) and for magnetic dipole moment developed by a moving electric dipole (the third term in eq. (19)). In this respect, we confess that the particular problems considered in section 3, have not been selected occasionally: regarding the the configuration in Fig. 1, where the time variation of hidden momentum happens for a true magnetic dipole, the related force term plays an important role in the implementation of total energy-momentum conservation law for the isolated system in question. At the same time, for the configuration of Fig. 2, where the hidden momentum contriAp  

vop



15 bution fully originates from the magnetic dipole moment (50) developed by a moving electric dipole, eq. (20) yields a non-adequate solution. These examples confirm our conclusion that the force component due to hidden momentum contribution should be different for a true magnetic dipole, and for a magnetic dipole moment, associated with the motion of electric dipole. Nevertheless, for the sake of a covariant formulation of CED in material media in its present form, this circumstance is, in fact, totally ignored in eq. (20). Here we do not discuss other physical difficulties of CED in material media; rather we want to demonstrate that the problem of consistent Lorentz-invariant description of the force on a moving dipole (both with the Lorentz force law, where the term due to hidden momentum contribution is included, and with the Einstein-Laub force law) remains, in fact, unsolved. Therefore, in the next section we suggest applying the Lagrangian formalism for the solution of this problem. 5. Force law in material media via the Lagrangian approach We use the invariant Lagrangian density (see e.g. [37]) 1 lint  M αβ Fαβ , (58) 2 describing the interaction of an electrically neutral magnetized/polarized material medium with an external EM field in the absence of free charges. For a compact bunch of charges with the proper electric p0 and magnetic m0 dipole moments, moving in an EM field with velocity v, eq. (58) integrates to the interactional Lagrangian Lint  p  E  m  B , where p, m are defined by eqs. (18) and (19). Hence, designating M the rest mass of the dipole, we obtain the total Lagrangian in the form Mc 2 (59) L  p E  m B ,



which, at the given E, B, can be considered as the function of radial coordinate r and velocity v of a dipole. Therefore, the minimization of action yields the known equation L d L  , (60) r dt v where we obtain L   p  E   m  B  , (61) r   p0 //  E v  m0 //  B v 1 L 1 (62)  Mv    m0  E    p0  B  . 2 2 v c c c c , Deriving the latter equation, we presented the electric and magnetic dipole moments in the form p v  m0 m p v , m  0 //  m0  0 , p  0 //  p0   c  c resulting from eqs. (18), (19) where the subscripts “//”, “” denote the components, being either collinear, or orthogonal to v. Combining eqs. (60)-(62), we obtain the novel expression for the force on a dipole d d   p0 //  E v d  m0 //  B v F  Mv    p  E   m  B     dt dt c2 dt c2 (63) d 1 d 1  p0  B   m0  E , dt c dt c derived with the invariant Lagrangian (59). We add that the energy of dipole in an EM field reads as

16

  p0 //  E v 2  m0 //  B v 2 L  L  Mc 2   p  E   m  B    v c2 c2 (64)   p0 //  E  m0 //  B  1 1  v   p0  B   v  m0  E      p0  E     m0  B , c c   which shows that this energy, being expressed via proper electric/magnetic dipole moments and the fields measured in a laboratory, does not depend on electric hidden momentum   p B  c and magnetic hidden momentum m0  E  c 4, and is solely determined by the interaction of a proper electric dipole moment with an the electric field, and the interaction of a proper magnetic dipole moment with a magnetic field; In the ultra-relativistic limit (where the first and third terms on rhs of eq. (64) are suppressed), the components of the proper electric (magnetic) dipole moments orthogonal to v give a dominating contribution to the energy. Comparing the novel expression for force (63) with EFL (20), we see that the firsts two terms on rhs of these equations are identical to each other, and the difference between these equations emerges in the structure of terms, containing the total time derivative. The physical meaning of these force components in eq. (63) can be understood via the analysis of generalized momentum of electric/magnetic dipole in an EM field (62). The last two terms are well understandable; however, in contrast to analogous terms in eq. (20), the electric and magnetic hidden momenta depend only on the proper dipole moments, and do not include the components v  m0 c for a moving magnetic dipole and p0  v c for a moving electric dipole. We find this result comfortable from the physical viewpoint. Indeed, consider the case, where p0, m0, E and B are time-independent. Then, for example, the force component due to variation of electric hidden momentum reads as  p0  v   B  c for a dipole, moving with velocity v in a frame of observation. This means that in the rest frame of the dipole, the nonvanishing time derivative B t emerges. In the case, where the vector B is orthogonal to p0 (i.e., the case, where the vector product  p0  B  takes the maximum value), the time variation of magnetic field induces the appearance of a circulating electric field (23b), which has opposite signs at its ends (when the origin of coordinate system coincides with the midpoint of electric dipole). Hence, the force at these ends is of the same sign, and its direction is determined by the vector product  p0  B t  . Analogously, we disclose the origin of the force component m0  v  E  c in the rest frame of the dipole. Addressing now to eq. (20), we find that in the considered case, where p0, m0, E and B do not depend on time, this equation additionally predicts the presence of the force components  v  m0  v   B  c 2 and  p0  v  v   E  c 2 in the third and fourth terms, correspondingly. However, both of these force terms disappear in the rest frame of a dipole, so that the reason for their emergence for a laboratory observer remains mysterious. In fact, the problem in Fig. 2 demonstrates the non-adequacy of the force term  p0  v  v   E  c 2 , while the problems sketched in Fig. 3 proves the non-adequacy of the U  v

term  v  m0  v   B  c 2 . Concurrently, the reader can check that with eq. (63), where the mentioned terms do not appear, the relativistic transformation of force (39) for the problems in Figs. 2 and 3 is restored. Next contribution to the generalized momentum is supplied by the second and third terms of eq. (62), and the best way to understand their physical meaning is to analyze the problems, where these terms solely determine the total force on the dipole.

  p B  c as the “latent momentum” of electric dipole. At this stage, we find more appropriate to call the product m0  E  c as the magnetic hidden momentum, and the product   p0  B  c as the electric hidden momentum for the reasons, which we will explain elsewhere. 4

In ref. [38], we named the product

17 Consider, for example, a magnetic dipole m0 with the proper mass M, which is oriented along the constant magnetic field B0, and both these vectors are parallel to the axis x of the rest frame of the dipole K0. In this frame, no force and no torque is exerted on the dipole, and its total energy in the magnetic field is equal to U 0  Mc 2  m0  B0  . In another inertial frame K, wherein the dipole is moving with the constant velocity v along the axis x, its total energy is equal to U=U0, and total momentum is Px  Mv  vm0  B0  c 2 . (65) Now we assume that the magnetic dipole represents two small disks with opposite charges, rotating in the opposite directions around a common axis and, due to their friction, the disks slowly decelerate with a negligible radiation [16, 20]. In this process, the conservation of total energy signifies that the rotational energy of disks is transformed into their heating, while the conservation of momentum (65) yields the equality d Mv   d vm0  B0  c 2   d vm0 //  B  c 2 , (66) dt dt dt which is just the force (63) measured in the frame K. (Here B{Bx,0,0}, and Bx=B0x). Further, due to the constancy of v in the frame K, eq. (66) means the decrease of the rest mass of the dipole dM d m0 //  B   , dt dt c2 which is related to the corresponding decrease of mechanical stress energy in the decelerated disks, emerging due to the interaction of moving charges of both disks with the magnetic field 5. Consider the modification of this problem, where the vectors m0 and B0, being parallel to each other, are both orthogonal to velocity v. Then we similarly obtain in the frame K d Mv   d vm0  B0  c 2 . (67) dt dt In order to express eq. (67) via the fields in the frame K, we use the identity a  b  c   ba  c   ca  b , and then the relativistic transformation of fields between K0 and K. Hence, we obtain d Mv    d 1 m0  E  , dt dt c which shows that at m0v, the process of deceleration of disks, composing a magnetic dipole, is related in the frame K to the force component due to time variation of magnetic hidden momentum, as already was found, e.g., in refs. [16, 20]. In a similar way we can grasp the meaning of the third term in eq. (63). Thus, the application of Lagrangian approach to the derivation of the force law in material media, leading to the new expression (63), seems attractive from the physical viewpoint, and provides a clear physical interpretation of all of the terms of eq. (63). 6. Quantum phases of electric/magnetic dipole The Lagrangian approach, we applied to the derivation of the classical force (63), also allows us to determine the corresponding quantum phases. For this purpose we write the Hamilton function H  L v  v  L , which in the quantum limit straightforwardly determines the Hamiltonian Hˆ and the corresponding total phase for a dipole in an EM field

5

We notice that, in general, eq. (66) is not compatible with the relativistic transformation of forces for the reasons mentioned in the footnote 2.

18 1  1 1 Hdt   2    p0 //  E v  ds  2   m 0 //  B v  ds   c c (68) 1 1 1 1 m 0  E   ds    p0  B   ds    p  E dt   m  B dt , c  c   where ds =vdt is the path element. The last two terms on rhs of eq. (68) determine the Stark phase [39] and Zeeman phase [40], correspondingly, the third and the fourth terms stand for the Aharonov-Casher (A-C) phase [41] and He-McKellar-Wilkens (HMW) phases [42, 43], correspondingly, and all these phases had been observed [40, 44]. At the same time, we point out that eq. (68) additionally includes the integrals containing    p0 //  E v c 2 ,   m 0 //  B v c 2 , (the first and the second term on rhs), which thus should be related to the corresponding quantum phases even in the case, where the classical force is vanishing (e.g., at constant fields E, B). Thus, at the motion of an electric dipole p0 in an electric field E, the quantum phase has the form   pE   2   p0 //  E v  ds . (69) c Analogously, at the motion of a magnetic dipole m0 in a magnetic field B, the corresponding quantum phase reads as   mB   2  m0 //  B v  ds . (70) c At constant p0, m0, E and B, we obtain in the non-relativistic case (1): v v  pE   2 p0 // E// L   2 p0 //  , (71) c c and v  mB   2 m0 // B// L , (72) c where L is the length of the path of dipole in the field region, and  is the potential difference between the initial and end points of the path, lying in the field region. We further observe that the phases (69), (70) do depend on the velocity v of a dipole and hence, unlike to Aharonov-Bohm phase [45], A-C and HMW phases, they do not represent topological phases. Besides, they cannot be classified as dynamic phases, since the classical force on a dipole can be equal to zero, while the values of pE and mB are not vanishing. In addition, these phases depend on mutual orientations of vectors p0, E, m0, B and v. These observations allow us to assume that eqs. (69), (70) determine some new kind of quantum phases, which, like A-C and HMW phases, are related to each other via electric-magnetic duality transformations EB, pm [12]. From the experimental viewpoint, both phases (71) and (72) can be reliably detected. The phase (71) can be measured via application of the technique of quantum interference of molecules, having a large proper electric dipole moment (e.g., BaS, where p0310-29 Cm [44]). Then, at v/c10-4, and 105 V, we obtain pE10 mrad. (73) The measurement of phase (72) can be done via the neutron interferometry. For example, for a neutron beam, passing a solenoid along its axis with the velocity v10-4c, we obtain with B=10 T, L=1 m and m0=9.610-24 J/T mB30 mrad. (74) The estimated phases (73), (74) are even larger than the typical phases for the A-C and HMW effects. However, one should notice that the phase (73) can be mixed with the Stark phase [39], while the phase (74) can be mixed with the Zeeman phase [40]. At the same time, the Stark and Zeeman phases, being determined by the last two terms in eq. (68), in a real situation are in-



19 versely proportional to the velocity of dipole v. In contrast, the phases (69), (70) are directly proportional to v, which provides us with experimental tools (not discussed here for brevity) for separation of these phases. One more option for the identification of the phases pE and mB is related to their sensitivity to the direction between vectors p0, E and v for pE, and to the direction between vectors m0, B and v for mB. This property allows us to obtain experimentally important arguments in the favour of the force law (63) versus the ELF (20). Indeed, eq. (20) implies that the generalized momentum of dipole should contain the sum of the terms 1 v  m0  B   p0  v  E  , (75) c2 resulting from the substitution of transformations (18), (19) into third and fourth terms of eq. (20), containing the total time derivatives. Using the vector identity a  b  c   ba  c   ca  b , we obtain in the case, where p0, Ev, and m0, Bv: 1 v  m0  B   p0  v  E    v m02  B   v  p02  E  . 2 c c c At the same time, at p0//v, and m0//v, the contribution (75) to the Hamiltonian disappears. Hence, the corresponding phases resulting from eq. (75) read as 1  pE EFL    2   p0  E v  ds , (76) c 1  mB EFL    2  m0  B v  ds . (77) c The comparison of eqs. (69) (defining pE) and (76) (defining pE(ELF)) shows that at the motion of electric dipole along the normal to a parallel plate charged capacitor (where p0, E//v), the phase pE takes the maximal value, and the phase pE(ELF) is equal to zero, whereas at the motion of electric dipole along the plates of a capacitor (where p0, Ev), pE is vanishing, while pE(ELF) becomes maximal. Thus, this experiment will allow us to make a crucial choice between the force laws (63) and (20). In a similar way, we can compare the magnetic phases (70) and (77), and to get one more possibility to test the force law (63) versus ELF (20). 7. Conclusion In the present paper, first of all, we highlighted the incompleteness of the analysis of the problem of hidden momentum in the discussion with respect to the Mansuripur paradox, where the crucial case of time-varied hidden momentum of a magnetic dipole is, in effect, omitted. At the same time, the presence of the force term due to time variation of hidden momentum unambiguously results from the general theorem about zero total momentum of any stationary configuration of charges and currents. In the support of this statement, in sub-section 3.1 we considered a motion of a point-like charge with respect to a point-like magnetic dipole, where the force component due to time variation of hidden momentum plays an important role in the implementation of the law of conservation of total energy-momentum for the isolated system in question. The absence of this force component in the Lorentz force law (2) indicates that the fourforce density (2), in general, is not directly related to the derivative of total mechanical fourmomentum density on proper time, due to the missing hidden momentum component and thus, the 3d-form of this equation occurs incompatible with the relativistic transformation of force (39). A “manual” addition of the force component due to hidden momentum contribution to the Lorentz force law in material media does not remedy the situation, because the mentioned force term cannot be added covariantly, as we have discussed in section 2. At the same time, such an expanded Lorentz force (ELF) law in 3d-form (20) occurs fully equivalent to the Einstein-Laub force law with respect to the force on a small electric/magnetic dipole, and their incompatibility

20 with the relativistic transformation of force has been demonstrated in section 3 with the problems presented in Figs. 2 and 3. In section 4 we discussed the problem of determination of magnetization for a moving media via bound currents, as well as the difference of EM fields, generated by true electric/magnetic dipoles, and electric/magnetic dipoles moments pm (eq. (49), mp (eq. (50)), resulting from the motion of dipoles. In our opinion, the results, which we achieved, explain the failure of ELF to provide relativistically adequate description of interaction of moving dipoles with EM field. In this respect, we suggest an approach to the derivation of force law in material media, based on a relativistically invariant Lagrangian density (58). Thus, we obtained a novel expression for the force on a moving dipole (63), its generalized momentum (62) and energy (64), with a clear clarification of the physical meaning of these equations. We have analyzed the physical meaning of the force components in eq. (63), and provided their clear interpretation. The expression for the Hamiltonian, associated with the motion of a moving dipole (68) indicates that in the addition to the known topological A-C and HMW phases, there are two nontopological phases, related to the motion of an electric dipole in an electric field (eq. (69)), and the motion of magnetic dipole in a magnetic field (eq. (70)). The estimated values of these phases are even larger than the A-C and HMW phases, and the experiments for their detection will allow us to test concurrently the force law (63) versus ELF (20). Finally, one can introduce the common designation for the quantum phases, associated with the motion of dipoles in EM field: pE, mB, pB (HMW phase), mE (A-C phase), which correspond to all possible combinations of the pair p, m with the pair E, B, and all of these phases are commonly derived via the Hamiltonian (68). References 1. M. Mansuripur, Phys. Rev. Lett. 108, 193901 (2012). 2. D.A.T. Vanzella, Phys. Rev. Lett. 110, 089401-1 (2013). 3. S.M. Barnett, Phys. Rev. Lett. 110, 089402-1 (2013). 4. P.L. Saldanha, Phys. Rev. Lett. 110, 089403-1-2 (2013). 5. M. Khorrami, Phys. Rev. Lett. 110, 089404-1 (2013). 6. D. Cross, arXiv:1205:5451. 7. T.M. Boyer, Am. J. Phys. 80, 962 (2012). 8. A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Prog. Electromagnetic Research B45, 83 (2012). 9. D. J. Griffiths and V. Hnizdo, arXiv:1205.4646 (2012). 10. A. Einstein and J. Laub, Ann. Phys. (Leipzig) 331, 532 (1908). 11. A. Einstein and J. Laub, Ann. Phys. (Leipzig) 331, 541 (1908). 12. J.D. Jackson, Classical Electrodynamics, 3rd ed (Wiley, New York, 1998). 13. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism 2nd Edition, (Wiley, New York, 1962). 14. W. Shockley and R. P. James, Phys. Rev. Lett. 18, 876 (1967). 15. S. Goleman and J.H. Van Vleck, Phys. Rev. 171 (1968) 1370. 16. Y. Aharonov, P. Pearle and L. Vaidman, Phys. Rev. A 37, 4052 (1988). 17. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields 2nd edition, (New York, Pergamon Press, New York, 1962). 18. J. Franklin, Am. J. Phys. 82, 869 (2014). 19. V. Hnizdo, Am. J. Phys. 83, 279 (2015). 20. L. Vaidman, Am. J. Phys. 58, 978 (1990). 21. V. Hnizdo, Am. J. Phys. 60, 279 (1992). 22. M. Mansuripur, Proc. SPIE 8455, 845512 (2012). 23. V. Hnizdo. Am. J. Phys. 65, 55 (1997). 24. F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, Mass., 1965).

21 25. C. Teitelboim, D. Villarroel and Ch. G. van Weert. R. Nuovo Cimento 3, 1 (1980). 26. A.L. Kholmetskii, O.V. Missevitch and T. Yarman. Phys. Scr. 83, 055406 (2011). 27. A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Can. J. Phys. 93, 691 (2015). 28. G.E. Vekstein, Eur. J. Phys. 18, 113 (1997). 29. A.L. Kholmetskii, O.V. Missevitch and T. Yarman. Eur. J. Phys. 33, L7 (2012). 30. I. Brevik. Phys. Rep. 52, 133 (1979). 31. G.L.J.A. Rikken and B.A. van Tiggelen. Phys. Rev. Lett. 108, 230402 (2012). 32. R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2, (Addison-Wesley, Reading, MA, 1963). 33. C. Møller, The Theory of Relativity (Clarendon Press, Oxford, 1973). 34. G.P. Fischer, Am. J. Phys. 39, 1528 (1971). 35. A.L. Kholmetskii and T. Yarman, Eur. J. Phys. 31, 1233 (2010). 36. A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Prog. Electromagnetic Research B47, 263 (2013). 37. M. Fabrizio and A. Morri, Electromagnetism of Continuous Media (Oxdord Uviversity Press, Oxford, 2003). 38. A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Eur. Phys. J. Plus 129, 215 (2014). 39. A. Miffre, M. Jacquey, M. Büchner, et. al. Eur. Phys. J. D38, 353 (2006). 40. S. Lepoutre, A. Gauguet, G. Trnec, et al. Phys. Rev. Lett. 109, 120404 (2012). 41. Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984). 42. X.-G. He and B.H.J. McKellar, Phys. Rev. A47, 3424 (1993). 43. M. Wilkens, Phys. Rev. Lett. 72, 5 (1994). 44. A. Cimmino, G.I. Opat, CA.G. Klein, et al. Phys. Rev. Lett. 63, 380 (1989). 45. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).

22

Insulating tube e

.

m0

v

Platform

y x

Fig. 1. Interaction of moving charge e with magnetic dipole m0, oriented along the axis z. The dipole is fixed on a massive platform, while the charge is moving inside a thin insulating tube oriented along the axis x and also fixed on the platform.

23

u(t)

E0 p0

0

y

y K

x

K0

x

v

Fig. 2. An electric dipole p0{0,-p0, 0} is moving with the velocity u{0,u, 0} with respect to homogeneously charged long line oriented along the axis x, and having the linear charge density 0, as viewed in the inertial frame K0 (the rest frame of the line). Using eq. (20), we propose to calculate the force on the dipole in the frame K0, as well as in another inertial frame K, moving with respect to K0 at a constant velocity v in the negative x-direction.

24

u

B0

m0

I0

y

y K

x

K0

x

v

Fig. 3. Motion of magnetic dipole m0 in the field of a long straight conducting line, carrying a steady current I0, as viewed in the inertial frame K0 (the rest frame of the line). We want to determine the force on the dipole in the frame K0, as well as in another inertial frame K, moving with respect to K 0 with constant velocity v in the negative xdirection, using eq. (20).