A Electrodynamical and inertial gauge fields

A possible set of additional assumptions selecting Maxwell’s theory from the class of all possible gauge fields is given by
                ∫                 ′   ′                           ∫                    ′   ′
                          ′ ϱ (x   , t )                        1            ′ J   (x   , t )
Φ (x,   t)  =       d3x     ------------,    A    (x,   t) =    --     d3x     --k-----------,
                                    ⃗ ′         k                                       ⃗ ′
                            |⃗x  -   x  |                        c               |⃗x  -   x  |

Here, the potentials are written in terms of four localized functions ϱ and Jk , where   ′                   ⃗ ′
t   =   t -   |⃗x  -   x  |∕c and ϱ and J
   k obey a continuity equation. The quantities (110) are well known as retarded potentials in Lorentz gauge [16]. Using the representation (110) it may easily be shown that the corresponding fields ⃗E and B⃗ obey the full set of Maxwell’s equations, including the two inhomogeneous equations with sources ϱ and J⃗,
∂ E⃗                      ∂                1 ∂ E⃗       4 π
------ =   4 π ϱ,         -----×   ⃗B   -   -------- =   ----J⃗  .
 ∂ ⃗r                      ∂ ⃗r              c  ∂ t        c

An alternative possibility is, of course, to postulate the validity of Eqs. (111) for the fields directly, without making use of potentials. The interesting problem of the coexistence of the Galilei-invariant Schrödinger equation and the Lorentz-invariant Maxwell equations has been discussed recently [9].

It is less well-known that inertial forces and linearized gravitational forces provide an alternative realization, besides electrodynamics, for the fields  ⃗    ⃗
E  ,  B  [15]. We restrict ourselves here to a discussion of inertial forces, whose description in a quantum-theoretical context seems particularly interesting. Let us consider a particle in an arbitrary accelerated reference frame, whose movement relative to an inertial frame is given by a translation vector ⃗x  (t )
   0 and a rotation velocity ⃗ω (t ) . As is well known such a particle experiences an inertial force field
⃗                                                    ˙                                        ˙
FI  (x,  x˙,  t ) =   -  m  ⃗a (t) -  2m   ⃗ω  (t) ×  ⃗x -  m   ⃗ω (t )×  ( ⃗ω (t ) ×   ⃗x  )-  m  ⃗ω  (t )×  ⃗x,

where ⃗a (t ) =   ¨⃗x   (t )
              0 . In order to use Eq. (112) in the present field theory, the particle velocity ⃗x˙ has to be replaced by the velocity field (34) and the electric charge e by the ’gravitational charge’ m . Then, comparison of Eqs. (112) and (35) shows that inertial forces are described by a field
F⃗  (x,   t) =   m   ⃗E   (x,   t) +   ----˜ (x⃗,   tv) ×   B⃗   (x,  t),
  I                     I                                   I

                        B⃗I   (x,  t ) =   2  c ⃗ω  (t)

E⃗   (x,  t)  =   -  ⃗a (t ) -   ⃗ω (t )  ×  [⃗ω  (t ) ×   ⃗x ] -   ⃗ω˙ (t)  ×   ⃗x.

Both inertial fields have the dimension of an acceleration. It is easy to see that they obey the homogeneous Maxwell equations (31) and that appropriate potentials, as defined by (30), are given by
                              1-                 2
φI  (x,  t ) =   ⃗x ⃗a (t ) -      [⃗x  ×   ⃗ω  (t )]

        A⃗   (x,  t)  =   c ⃗ω  (t)  ×   ⃗x.


We see that inertial forces do also fit into the above gauge scheme although fundamental differences exist. In contrast to the electrodynamic field there are no additional field equations for the inertial field. Rather, its space-time dependence is more or less ’rigid’ and can only be influenced by means of the input parameters ⃗x   (t )
  0 and ⃗ω (t ) . Inertial fields do in contrast to electrodynamical fields not fit into the mathematical scheme gauge theory.

The fact that electrodynamical and inertial (gravitational) fields share a common (gauge) constraint, is sometimes interpreted as an indication of a common origin of both theories. We do not want to discuss this fascinating hypothesis here but mention only that these two gauge fields may also occur simultaneously; such a situation may simply be described by means of a linear combination of fields and potentials. Of course, the electrodynamic fields E⃗ and B⃗ are now defined with respect to the accelerated coordinate frame. (The homogeneous Maxwell equations hold in the accelerated frame as well; this is a condition for the potentials to exist. On the other hand the two inhomogeneous Maxwell equations change their form in accelerated coordinate frames [8]). In order to obtain Schrödinger’s equation for a statistical ensemble of charged particles in arbitrary accelerated reference frames, the replacements
e             e         m
--A⃗  -  →    --A⃗  +   ---A⃗I
c             c          c

  e φ  - →    e φ  +   m  φI

have to be performed in Eq. (61). The resulting theory is invariant with respect to the gauge transformation
                      S    =   S  +   χ

    ′            1 ∂  χ1          ′                1  ∂ χ2
  φ   =   φ  +   ---------,     φ    =   φ    +   -----------
                 e   ∂ t          I         I     m    ∂  t

   ′             c ∂  χ1           ′                c  ∂  χ2
A⃗   =   A⃗  -   ---------,     A⃗    =   A⃗I   -    ----------
                 e  ∂  ⃗x           I                m   ∂ ⃗x

                 χ   =  χ1   +   χ2   +   C  .



If the gauge for the inertial potentials is fixed according to (116), (117) and the above replacements are performed, the present approach leads to a very simple derivation of the quantum-theoretical Hamiltonian for an ensemble of charged particles under the influence of an electrodynamic field in an non-inertial reference frame [8].