^{1}

^{1, 2}

^{1}

^{1}

^{1}

^{2}

We show that a “dynamical” interaction for arbitrary spin can be constructed in a straightforward way if gauge and Lorentz transformations are placed on the same foundation. As Lorentz transformations act on space-time coordinates, gauge transformations are applied to the gauge field. Placing these two transformations on the same ground means that all quantized field like spin-1/2 and spin-3/2 spinors are functions not only of the coordinates but also of the gauge field components. As a consequence, on this stage the (electromagnetic) gauge field has to be considered as classical field. Therefore, standard quantum field theory cannot be applied. Despite this inconvenience, such a common ground is consistent with an old dream of physicists almost a century ago. Our approach, therefore, indicates a straightforward way to realize this dream.

After the formulation of general relativity which explained fources on a geometric ground, physicists and mathematicians tried to incorporate the electromagnetic interaction into this geometric picture. Weyl claimed that the action integral of general relativity is invariant not only under space-time Lorentz transformations but also under the gauge transformation, if this is incorporated consistently [

In Section

Relativistic field theories are based on the invariance under the Poincaré group

For an operator

is commutative, that is,

While the Lorentz transformation

As an operator

The principle of relativity states that a change of the reference frame cannot have implications for the motion of the system. This means that (

It may be reasonable to introduce an external field directly into the Poincaré algebra which can be applied to classically understand the elementary particle. To do so one has to transform the generators of the Poincaré group to be dependent on the external field in such a way that the new, field-dependent generators obey the commutation relations (

is commutative, that is,

At this point we specify

The second claim is that the dynamical transformation operator

Note that the plane-wave solution of the Dirac equation was found more than 70 years ago by Wolkow [

Note that due to the antimutation of the

Finally, as a consequence of the explicit form (

As a consequence of gauge invariance and Lorentz type of

the invariance of the wave function under gauge transformations,

the explicit shape of

the invariance of

the “dynamical” interaction for any spin as given by

as a consequence of (

Let us close again with Weyl In [

The work is supported by the Estonian target financed Project no. 0180056s09 and by the Estonian Science Foundation under grant no. 8769. S. Groote acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under Grant 436 EST 17/1/06.