Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off

Abstract

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

Another Proof of Born’s Rule on Arbitrary Cauchy Surfaces

“…2.2; they arise especially from multi-time wave functions [9,11,25,33]; see [23] for an introduction and overview. While certain ways of implementing an ultraviolet cutoff [7,26] lead to multi-time wave functions that cannot be evaluated on arbitrary Cauchy surfaces, models without cutoff define a hypersurface evolution, either on the non-rigorous [28,29] or on the rigorous level [6,[19][20][21][22]. As a consequence, our result proves in particular a Born rule for multi-time wave functions, thereby generalizing a result of Bloch [4] (see also Remark 4 in [24]).…”

Another Proof of Born's Rule on Arbitrary Cauchy Surfaces

“…Some examples are described in [24]; they arise especially from multi-time wave functions [9,11,33,25]; see [23] for an introduction and overview. While certain ways of implementing an ultraviolet cutoff [7,26] lead to multi-time wave functions that cannot be evaluated on arbitrary Cauchy surfaces, models without cutoff define a hypersurface evolution, either on the non-rigorous [28,29] or on the rigorous level [20,21,6,22,19]. As a consequence, our result proves in particular a Born rule for multi-time wave functions, thereby generalizing a result of Bloch [4] (see also Remark 4 in [24]).…”

Boundary Conditions that Remove Certain Ultraviolet Divergences

“…is an operator called the jth partial Hamiltonian; it can roughly be thought of as collecting the terms in the Hamiltonian pertaining to x j , and we have set c = 1. Such a system of equations can be inconsistent, but consistency has been verified for relativistic versions of the Lee model analogous to (2) [24,39,40]. An upshot at this point is that the particle-position representation fits nicely together with Lorentz invariance.…”