Some Jump Processes in Quantum Field Theory

Abstract

A jump process for the positions of interacting quantum particles on a lattice, with time-dependent transition rates governed by the state vector, was first considered by J.S. Bell. We review this process and its continuum variants involving "minimal" jump rates, describing particles as they get created, move, and get annihilated. In particular, we sketch a recent proof of global existence of Bell's process. As an outlook, we suggest how methods of this proof could be applied to similar global existence questions, and underline the particular usefulness of minimal jump rates on manifolds with boundaries.

Stochasticity and Bell-type quantum field theory

“…which represents the space of all the possible configurations of a variable but finite number of particles on this lattice (see Tumulka and Georgii (2005) for details, here I follow their notation).…”

“…. ] + considers only the positive part of the quantity between the squared brackets, setting the value equal to 0 whenever this quantity is negative; for details see Tumulka and Georgii (2005). Furthermore, the authors show this is a special case of (4), which is the jump rate defined in Dürr et al (2005).…”

“…To complete the picture it should be said that the transitions permitted can only be made to n + 1 or n − 1 particles, where n is the number of particles of a generic configuration q: a particle can appear, so that the transition to n + 1-states correspond to a birth processes, and disappear, meaning that the transition to n − 1-states correspond to death processes (or a particle can be replaced by a pair (pair creation) and vice versa (pair annihilation)). The results on the global existence, coherence and uniqueness of the jump rates in BTQFT can be found in Dürr et al (2005) and Tumulka and Georgii (2005).…”

An abstract framework for interior-boundary conditions

“…These ideas were later extended to more singular models by the second author [21,22]. Other aspects of specific models with interior-boundary conditions were investigated in [42,27,37]. These constructions of self-adjoint operators are related to singular number-preserving interactions that can be described by (generalised) boundary conditions and classified in terms of self-adjoint extensions of certain "minimal" operators (see e.g.…”

A minimalist pilot-wave model for quantum electrodynamics

“…A continuum model put forward by Colin (2003aColin ( ,b, 2004, which was further developed by Colin & Struyve (2007), seems to confirm Bell's expectation. On the other hand, Dürr et al have developed a continuum version of Bell's model which is stochastic (Dürr et al 2003(Dürr et al , 2004a(Dürr et al , 2005aTumulka & Georgii 2005). The fact that there are two generalizations of Bell's lattice model for the continuum originates in a different reading of Bell's work (Colin & Struyve 2007;Tumulka 2007a).…”