Positron position operators. I. A natural option

Abstract

By “position operators,” we mean here a POVM (positive- operator-valued measure) on a suitable configuration space acting on a suitable Hilbert space that serves as defining the position observable of a quantum theory, and by “positron position operators,” we mean a joint treatment of positrons and electrons. We consider the standard free second-quantized Dirac field in Minkowski space–time or in a box. On the associated Fock space (i.e., the tensor product of the positron Fock space and the electron Fock space), there acts an obvious POVM Pobv, but we propose a different one that we call the natural POVM, Pnat. In fact, it is a PVM (projection-valued measure); it captures the sense of locality corresponding to the field operators Ψs(x) and to the algebra of local observables. The existence of Pnat depends on a mathematical conjecture which at present we can neither prove nor disprove; here we explore consequences of the conjecture. We put up for consideration the possibility that Pnat, and not Pobv, is the physically correct position observable and defines the Born rule for the joint distribution of electron and positron positions. We describe properties of Pnat, including a strict no-superluminal-signaling property, and how it avoids the Hegerfeldt–Malament no-go theorem. We also point out how to define Bohmian trajectories that fit together with Pnat, and how to generalize Pnat to curved space–time.