Observables in Relativistic Quantum Mechanics

Abstract

The conventional statement of statistical determinism is that “the expectation values of all (Heisenberg) observables are determined by the expectation values of the observables at one time.” This requires that a full algebra of self-adjoint operators be in one-to-one correspondence with measurement procedures performed at one time. For instance, it requires that if two noncommuting observables p and q are defined at $\text{t\hspace{0.17em}=\hspace{0.17em}0}$ , there should exist a measurement procedure at $\text{t\hspace{0.17em}=\hspace{0.17em}0}$ corresponding to $\mathrm{p+q}$ . No such procedure is known. The contrast between the positive assertion of the existence of certain laboratory procedures and the inability to describe them constitutes perhaps the weakest point of quantum mechanics. However, the conventional statement of statistical causality is shown to be untenable in a relativistic theory. This paper proposes a weaker form of causality which (1) uses measurements made within a truncated light cone rather than at one time for predictive purposes, and (2) which involves only strictly localized states, i.e., states which are vacuumlike outside a finite volume. Failure of the conventional causality statement implies that the set of quasilocal observables is not necessarily linear, i.e., if A and B are in a set, $\mathrm{A+B}$ is not necessarily in it. This remark may open the way to a systematic inquiry into the problems of associating laboratory procedures to self-adjoint operators.