# Nonlocal Quantum Field Theory

## Nonlocal Quantum Field Theory

the common name for the extensions of quantum field theory based on the assumption of the nonpoint nature (nonlocality) of an interaction.

According to traditional quantum field theory, the quantities that describe a physical field can be specified at all points in space-time, and the interaction of the fields is local (that is, it is determined by their values at the coincident points of space-time). Local quantum field theory leads to the appearance of divergences, or physically meaningless, infinitely large values for some physical quantities. The problem of eliminating divergences from the theory is the immediate goal of nonlocal quantum field theory. In addition, some versions of the theory are already being used in planning experiments for verification of the predictions of the existing theory regarding elementary particles and for processing the results of such experiments. The results of the experiments show that the dimensions of the region where the effects of nonlocality could be manifested are in any case smaller than 10^{-15} cm.

The concept of nonlocal interaction had already arisen in classical electrodynamics during attempts to construct a theory of extended charged particles; the action of an electromagnetic field on such particles is determined by the values of the field intensity in the entire region over which the charge is “spread.” The problems observed in the classical model are also typical for nonlocal quantum field theory. For an extended particle to react to any external effects as a unit (this corresponds to the concept of an “elementary,” indivisible particle), the physical interactions (“signals”) must be assumed to propagate instantaneously throughout the particle. At the same time it follows from the theory of relativity that a stipulation regarding the existence of signals that propagate faster than the velocity of light contradicts the principle of causality: the moment of recording of such signals may precede the moment of their emission. Thus, the requirements for integrity of a particle, relativistic invariance, and causality appear to be contradictory.

Nonlocal quantum theory may be created either by direct introduction of factors that “spread” the interaction (called the relativistic formfactors) or in a more radical fashion—for example, by extensions of the theory in which it proves impossible accurately to define a physical quantity “at a point.”

The problems arising in nonlocal quantum field theory, including the problem of reconciliation of the requirements of the theory of relativity and the conditions of causality, touch on the fundamental aspects of physical theory, particularly the concepts of space and time. The introduction of a scale that defines the “spreading” of the particles (more accurately, a scale that is a measure of the violation of the locality of the interaction) may also require a revision of the geometry for very small space-time intervals. It is significant that many attempts at space-time quantization have produced results very close to nonlocal quantum field theory and can even be considered as a physical basis for the introduction of form factors.

A program for the construction of a noncontradictory and physically sound nonlocal quantum field theory has not yet been realized; its consistent development should go more deeply into the concepts of space, time, and matter.

### REFERENCES

Markov, M. A.*Giperony i K-mezony*. Moscow, 1958.

Kirzhnits, D. A. “Nelokal’naia kvantovaia teoriia polia.”

*Uspekhi fizicheskikh nauk*, 1966, vol. 90, fasc. 1, p. 129.

*Trudy mezhdunarodnogo soveshchaniia po nelokal’noi kvantovoi teorii*

**Preprint R2–3590 of the Joint Institute for Nuclear Research. [Dubna, 1967.]**

*polia.*Efimov, G. V. “Nelokal’naia kvantovaia teoriia skaliarnogo polia.”

*Problemy fiziki elementarnykh chastits i atomnogo iadra*, 1970, vol. 1, fasc. 1, p. 256.

V. I. GRIGOR’EV