Positron Position Operators. I. A Natural Option

Abstract

By "position operators," I 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," I mean a joint treatment of positrons and electrons. I 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 P obv , but I propose a different one that I call the natural POVM, P nat . In fact, it is a PVM (projection-valued measure); it captures the sense of locality corresponding to the field operators Ψ s (x) and to the algebra of local observables. The existence of P nat depends on a mathematical conjecture which at present I can neither prove nor disprove; here I explore consequences of the conjecture. I put up for consideration the possibility that P nat , and not P obv , is the physically correct position observable and defines the Born rule for the joint distribution of electron and positron positions. I describe properties of P nat , including a strict no-superluminal-signaling property, and how it avoids the Hegerfeldt-Malament no-go theorem. I also point out how to define Bohmian trajectories that fit together with P nat , and how to generalize P nat to curved space-time.