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Positron Position Operators. I. A Natural Option

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 Psi_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.