LOCALITY OF QUANTIZED SCALAR FIELDS FOR GENERATIONS OF PARTICLES

Yu V Kulish,E V Rybachuk

doi:10.26565/2312-4334-2017-4-01

Abstract

It is shown that an integral corresponding to the contribution of one particle to equal-time commutator of quantized scalar fields diverges in a reality, contrary to usual assumption that this integral vanishes. It means that commutator of scalar fields does not vanish for space-like intervals between the field coordinates. In relation with this divergence the generalization of the Klein-Gordon equation is considered. The generalized equation is presented as products of the operators for the Klein-Gordon equation with different masses. The solutions of derived homogeneous equations are sums of fields, corresponding to particles with the same values of the spin, the electric charge, the parities, but with different masses. Such particles are grouped into the kinds (or families, or dynasties) with members which are the particle generations. The commutator of fields for the kinds of particles can be presented as sum of the products of the commutators for one particle and the definite coefficients. The sums of these coefficients for all the generation equal zero. The sums of the products of these coefficients and the particle masses to some powers equal zero too, i.e., for these coefficients some relations exist. In consequence of these relations the commutators of the fields for the particle generations vanish on space-like intervals. Thus, the locality (the microcausality) is valid for the fields of the particle kinds. It is possible if the number of the generations is greater than two.

Highlights

Показано,що інтеграл відповідний внеску однієї частинки в одночасний коммутатор квантованих скалярних полів в дійсності розбігається, в протилежність звичайному припущенню, що цей інтеграл дорівнює нулю
It is confirmed by explicit calculations. In consequence of this divergence, the commutator of scalar fields for one particle can be non-zero value. It means that the locality principle can be violated for these fields
We believe that the causality principle must be valid for the commutator of the scalar fields

Summary

LOCALITY OF QUANTIZED SCALAR FIELDS FOR GENERATIONS OF PARTICLES

It is shown that an integral corresponding to the contribution of one particle to equal-time commutator of quantized scalar fields diverges in a reality, contrary to usual assumption that this integral vanishes. It means that commutator of scalar fields does not vanish for space-like intervals between the field coordinates In relation with this divergence the generalization of the Klein-Gordon equation is considered. The sums of the products of these coefficients and the particle masses to some powers equal zero too, i.e., for these coefficients some relations exist In consequence of these relations the commutators of the fields for the particle generations vanish on space-like intervals. ЛОКАЛЬНІСТЬ КВАНТОВАНИХ СКАЛЯРНИХ ПОЛІВ ДЛЯ ПОКОЛІНЬ ЧАСТИНОК Ю.В. In the present paper the divergence of the integral (4) is studied in details As this divergence leads to non-zero value of the commutator (1) on space-like intervals, it means that the locality principle can be violated. To confirm non-zero value of the commutator (1) for the space-like intervals we explicitly derive the integral (4).

Br sin

ELIMINATION OF DIVERGENCES IN COMMUTATORS OF TOTAL FIELDS

Ak m

CONCLUSIONS