Comment on "Classical and quantum interaction of the dipole".

In [1] Anandan has presented a covariant treatment of the interaction of the electric and magnetic dipole moments of a particle with the electromagnetic field. Our aim is to make some important changes of the results from [1]. Instead of dealing with component form of tensors E, .. [1], we shall deal with tensors as four-dimensional (4D) geometric quantities, E, .. . For simplicity, only the standard basis {eμ; 0, 1, 2, 3} of orthonormal 4-vectors, with e0 in the forward light cone, will be used. Anandan states: “In any frame D and D that couple, respectively, to the electric field components F0i and the magnetic field components Fij are called the components of the electric and magnetic dipole moments.” Then he defines that d and m are the components of D , Eq. (2), and similarly that E and B are the components of F , Eq. (4). Several objections can be raised to such treatment. It is proved in [2] that the primary quantity for the whole electromagnetism is F ab (i.e., in [2] the bivector F ). F ab can be decomposed as F ab = (1/c)(Ev −Ev) + εvcBd, whence E a = (1/c)F vb and B a = (1/2c)εFbcvd, with Eva = B va = 0; only three components of E a and B in any basis are independent. The 4-velocity v is interpreted as the velocity of a family of observers who measures E and B fields. E and B depend not only on F ab but on v as well. In the frame of “fiducial” observers, in which the observers who measure E, B are at rest, v = ce0. That frame with the {eμ} basis will be called the e0-frame. In the e0-frame E 0 = B = 0 and E = F , B = (1/2c)εFjk . In any other inertial frame the “fiducial” observers are moving and v a = ce0 = v eμ; under the passive Lorentz transformations (LT) veμ transforms as any other 4-vector transforms. The same holds for E a and B, e.g., E = Eeμ = ((1/c)F v0)ei = E eμ = ((1/c)F v ν)e ′ μ. E μ transform by the LT again to the components E of the same electric field. There is no mixing with the components of the magnetic field. E are not determined only by F ′μν than also by v. Only in the e0-frame, and thus not in any frame, F i0 and F jk are the electric and magnetic field components respectively. The assertion that, e.g., in any inertial frame it holds that, E = E = 0, E = F i0 and E = F , leads to the usual transformations of the 3-vector E, see, e.g., [3], Eq. (11.149). In [4] the fundamental results are achieved that these usual transformations of the 3-vectors E and B are not relativistically correct and have to be replaced by the LT of the electric and magnetic fields as 4D geometric quantities. The electric and magnetic dipole moment 4-vectors d and m respectively can be determined from dipole moment tensor D in the same way as E and B are obtained from F ; D = (1/c)(ud−ud)+(1/c)εucmd, whence d = (1/c)Dub, and m a = (1/2)εDbcud, with d ua = m ua = 0. u a = dx/ds is the 4-velocity of the particle. The whole discussion about E, B and F ab can be repeated for d, m and D. Now, only in the rest frame of the particle and the {eμ} basis, u a = ce0 and d 0 = m = 0, d = D, m = (c/2)εDjk. It is also stated in [1]: “The electric and magnetic fields in the rest frame ... .” But, there is no rest frame for fields. The whole discussion in [1] has to be changed using different 4-velocities v and u. Thus Eqs. (7) and (6) become (1/2)FabD ba = (1/c)Dau a + (1/c2)Mau , and Da = d Fba, Ma = m F ∗ ba. Instead of Eq. (5) we have that (1/2)FabD ba is the sum of two terms (1/c2)[((Ead ) + (Bam ))(vbu )− (Eau )(vbd )− (Bau )(vbm )] and (1/c)[ε(vaEbucmd+ c 2daubvcBd)]; the second term contains the interaction of Ea with m , and Ba with d . This last result significantly influences Eq. (17) and it will give new interpretations for, e.g., the Aharonov-Casher and the Röntgen phase shifts.