Second Quantization and Lorentz Invariance

The problem of Lorentz invariance is treated in the conventional quantum field theory as follows: I) Field variables are transformed as classical quantities. II) Next, the procedure of the second quantization is performed in the new coordinate system. Then the commutation relations are retained to be invariant. However, as Wignerl) has pointed out, the invariance of the commutation relations does not mean the existence of unitary operators which transform the field quantities as <pt = U<p UIn fact these operators have never been constructed in the conventional theory. Moreover, the above procedure produces the invariance of the Feynman amplitudes, but the connection between the Feynman amplitudes and the scattering amplitudes does not seem to be Lorentz invariant. In spite of these iflcompletenesses, however, the present field theory has given the correct results in a rather wide range of problems. Therefore, if the present field theory dOEs not contain any inconsistency in itself the above procedure may be considered to be satisfactory, and the problem of Lorentz invariance need not be pursued any more. But, if one considers the local field theory to be unsatisfactory and intends to find out the possibility to construct new theories, this problem must be reinvestigated as a guiding principle, and the following two problems may become important: I) \Vhether it is possible to construct any theory which can describe the creation and annihilation of particles and possesses the unitary representation of Lorentz group. II) In the theory which does not need the unitary representation of Lorentz gruup, how are the Lorentz transformations treated in a consistent way? As for the latter problem, SegaP) has recently proposed an interesting attempt. This paper will Le devoted to the investigation of the former probem. Then we