On the Retardation Effects in the Non-local Field Theory

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Though recently many papers have been published about the non-local field theory of elementary particles such as by Yukawa), yet at present these theories are too abstract to obtain definite results. On the other hand Heisenberg's new monistic theory) of field is considered to tell us our objects to reach, but it is also far separated from the present concrete theory. Then on our way to reach this theory, there must be many questions to discuss. At present one of the ways to proceed on our theoretical studies about elementary particles, may be to start, firstly, to discuss the relation between the following two theories about the electron, which are founded on the valid experimental evidences, and secondly, to unite these theories into one formulation. This is the idea of the present paper. The first theory of the electron is Tomonaga-Schwiniger's quantum electro-dynamics in a relativistically covariant formulation on a dualistic standpoint, (hereafter designated as T-S theory) which explains experimentally anomalous magnetic moment of the electron and many radiative reaction effects. Contrary to T-S theory there is another important theory of the electron proposed by Bopp, which follows the school of Dirac's and Mie-Born's classical theories of the electron founded on a monistic standpoint, and explains the experimental mass spectrum of elementary particles, though it is qualitative. So theoretically the latter theory is rather interesting. Apparently it looks as if these two theories were based on fundamentally different standpoints and do not fuse into one formulation. In the present report the author introduces in the first place the outline of Bopp's theory, which is necessary to develop our theory. The different points of the above two theories are discussed and we bring to light how the unified formulation of two theories is accomplished. Following these discussions we shall propose a field theory for the above aim, which is nothing but a simple non-local field theory assuming non-locality only in commutation relations of field quantities, and considering the expansion by a retardation parameter, but with different interpretation about Lagrange function contrary to T-S theory. In our formulation the 4-dimensional functionals constructed from various dynamical quantities (as electromagnetic potential, spinor fields and especially Lagrange function), vary following generalized Schrodinger equations of motion. Then we shall show that Dirac's wave equation with matter field and Maxwell's wave equation with electromagnetic field are derived from the 1st order approximation about retardation parameter of our simple Lagrange function, in which case it is derived also that the instantaneous velocity of a particle is equal to light velocity. In the 2nd approximation of retardation parameter the wave equation of matter field takes a generalized form of Dirac's equation, which is slightly different from the equations treated by Honl-Papapetrou and Bopp. Thus it is shown that Bopp's theory is transformed into a dualistic non-local field theory with a simple formulation. But their full discussions will be postponed till the following paper. Though we have assumed non-local commutation relations with respect to field quantities, and studied the non-local effects generated from these. commutation relations, we did not determine concretely these commutative functions in this paper. As in the generalized Dirac's equation the moments of these quantities with respect to retardation parameter appear, these quantities will be determined by comparing them with experimental results. In this point our theory is based on phenomenological standpoint with respect to commutation. relations. Also in the present paper the Heisenberg representation is used to make clear the relation between Bopp's theory and ours.
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On the Retardation Effects in the Non-local Field Theory

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