# Causality Principle

## Causality Principle

in physics, a general principle establishing the permissible limits of the influence of physical events on each other. According to the principle, a given event cannot influence events that have already occurred, a notion reflected in such statements as “the future does not influence the past” and “the cause event precedes the effect event in time.” The causality principle also requires the absence of any mutual influence between events for which the application of the concepts “earlier” and “later” has no meaning—an event, for example, that is earlier to one observer but appears later to another observer. According to the special theory of relativity, such a situation arises when the spatial distance between events is so great and the time interval between the events is so small that the events can be connected only by a signal propagating faster than light. Since the causal relationship can be realized only by a signal connecting the events, the requirement of the absence of a causal relationship leads to the famous result that motion cannot occur at a speed exceeding the speed of light in a vacuum.

In physical theory, the causality principle is used primarily to choose boundary conditions for the corresponding dynamics equations so as to ensure the uniqueness of the solution of the equations. Thus, in the solution of Maxwell’s equations of electrodynamics, the causality principle chooses between advanced and retarded potentials in favor of the latter. Similarly, in quantum field theory the causality principle imparts uniqueness to the results of the Feynman-diagram technique, which is an important instrument in the theoretical description of interacting fields or particles. Moreover, the causality principle permits the general properties of quantities that describe the response of a physical system to external influences to be established. An example is the analytic properties of the dielectric constant of a system as a function of frequency (the Kramers-Kronig dispersion relations). Another important example is the dispersion relations in the theory of scattering of strongly interacting particles, or hadrons. These relations are a unique instance of an exact dependence between directly observable quantities—the amplitude of forward elastic scattering and the total cross section—that is derived without the use of any model conceptions of elementary particles. The role of the causality principle in the theory of elementary particles has grown with the development of a special axiomatic approach whose goal is to describe the interactions of particles directly on the basis of the general principles (postulates) of the theory. In the axiomatic approach, whose accomplishments include the derivation of the dispersion relations, the causality principle is assigned the constructive role of one of the principal postulates—along with the requirements of relativity theory and quantum theory.

The causality principle is verified without question by experiment in the macroscopic domain and by general human experience. Its validity, however, on the subnuclear level studied by the physics of elementary particles is not obvious. This is because in the formulation of the causality principle an event is understood to be a “point” event, which occurs at a given point in space at a given time. The causality principle we have been discussing up to now is accordingly also called the principle of microscopic causality. It should be noted, however, that the constraints resulting from quantum theory and the theory of relativity make the physical realization of a point event impossible. Any event —that is, any act involving the interaction of particles—unavoidably has a finite extent in space and time. Therefore, on the microscopic level the causality principle loses its direct physical meaning and becomes a formal requirement. We thus can speak of the possible violation of the causality principle “in the small”; its validity is, of course, preserved on the macroscopic level. Such a weakened causality principle is called the principle of macroscopic causality. A quantitative formulation of it that adequately reflects the constraints indicated above has not yet been developed. The numerous attempts to generalize quantum field theory that are associated with nonlocal quantum field theory are based on this principle.

The causality principle with which modern physics deals is a specific physical statement that is substantially narrower than the general philosophical notion, which sees causality as the interdependence and determinacy of a sequence of events. The causality problem acquired great importance in the formulation of quantum mechanics, when the question of whether or not determinism is contradicted by a probabilistic description of microphenomena was widely discussed. The understanding of the necessity of rejecting the straight determinism of classical mechanics in considering the statistical regularities of the microworld led to a negative answer to this question. The apparent contradiction with the general causality principle can be explained by the unsuitability of classical physics for the description of microobjects. The transition to an adequate description in the language of wave functions leads to a situation in which even in quantum mechanics the initial state of the system completely defines its entire subsequent evolution—for known interactions of the system.

The problem of the observance of the general causality principle—that is, causality in the philosophical sense—still retains its importance today in the analysis of possible forms of violation of the physical causality principle “in the small.” Such analysis has been fostered by the development of nonlocal field theory, by the investigation of the problem of motion at speeds higher than the speed of light, and by special experiments for the purpose of verifying the causality principle. The analysis must explain what forms of violation of the causality principle lead to an unfamiliar and what forms to an inadmissible (from the viewpoint of the general causality principle) situation. For example, the replacement of the original causality principle by the opposite statement (“the past does not influence the future”) does not contradict the general causality principle even though it leads to highly unfamiliar consequences. In this case, the chain of cause and effect relationships is not broken but appears in time-inverted form. A contradiction with the general causality principle arises if it is assumed that the causal relationship can be directed both forward and backward in time. In such a case, a closed cycle of cause and effect relationships would be possible that would lead to violation of the principle that an effect event does not influence the cause event that produced it. This principle is substantially broader and more adequate to the general causality principle than the original causality principle. If the effect were capable of influencing its own cause, this influence could be expressed in the vanishing of the cause event, a situation that would obviously entail a break in the connection between cause and effect. For example, if a wave emitted by a radiator were capable, after reflection, of returning at an earlier time, it could explode the radiator even before the radiator began operating. The fundamental impossibility of traveling into the past in a time machine follows from the same considerations.

A number of complicated and profound problems that still await solution is associated with the causality principle in modern physics.

### REFERENCE

“Sverkhsvetovye dvizheniia i spetsial’naia teoriia otnositel’nosti.” In*Einshteinovskii sbornik 1973.*(Translated from English.) Preface by D. A. Kirzhnits and V. N. Sazonov. Moscow, 1974.

D. A. KIRZHNITS