Abstract. Recently doubts have been raised about the ability of pilot wave theories with field ontology to recover the predictions of quantum field theory. In particular, Struyve has questioned that the overlap between wave functionals of macroscopically different states with fixed particle number is really non-significant. With numerical computations and some further plausibility arguments we show that the overlap between n-particle states in field theory decreases almost exponentially with the number of particles and becomes non-significant already for small particle numbers. 1.
Abstract. Recently doubts have been raised about the ability of pilot wave theories with field ontology to recover the predictions of quantum field theory. In particular, Struyve has questioned that the overlap between wave functionals of macroscopically different states with fixed particle number is really non-significant. With numerical computations and some further plausibility arguments we show that the overlap between n-particle states in field theory decreases almost exponentially with the number of particles and becomes non-significant already for small particle numbers. 1.
States in quantum field theory (QFT) are represented by many-particle wave functions, such that a state describing n particles depends on n space–time positions. Since a general state is a superposition of states with different numbers of particles, the wave function lives in the configuration space identified with a product of an infinite number of four-dimensional Minkowski space–times. The squared absolute value of the wave function is interpreted as the probability density in the configuration space, from which the standard probabilistic predictions of QFT can be recovered. Such a formulation and probabilistic interpretation of QFT allows one to interpret the wave function as a pilot wave that describes deterministic particle trajectories, which automatically includes a deterministic and continuous description of particle creation and destruction. In particular, when the conditional wave function associated with a quantum measurement ceases to depend on one of the space–time coordinates, then the 4-velocity of the corresponding particle vanishes, describing a trajectory that stops at a particular point in space–time. In a more general situation a dependence on this space–time coordinate is negligibly small but not strictly zero, in which case the trajectory does not stop but the measuring apparatus still behaves as if this particle has been destroyed.
In [2], a detailed argument is presented on a version of pilot-waves, given by at Theory of Exclusively Local Beables. What the author of [2] considers to be his crucial proposal is described in the title of section 3 as 'complicated, ugly, and highly contrived'. The reason for such a proposal is claimed to be the widespread perception among quantum physicists, according to which 'those sympathetic to the pilot-wave ontology no doubt expected that, for a system of N particles moving in three spatial dimensions, the theoretical description would be of N wave-particle pairs - each pair consisting of a point particle guided in some way by an associated wave propagating in 3-space. But Schroedinger's wave function for such an N-particle system was emphatically not a set of N (interacting) waves, each propagating in 3-space. It was, rather, a single wave propagating in the 3N-dimensional configuration space for the system', [2, p. 5]. In the present paper it is argued that the mentioned widespread expectation need not be seen as being betrayed by what actually happens. In particular, the construction in section 3 of [2] may be avoidable.
The purpose of this paper is to present an overview of recent work on pilot-wave approaches to quantum field theory. In such approaches, systems are not only described by their wave function, as in standard quantum theory, but also by some additional variables. In the non-relativistic pilot-wave theory of deBroglie and Bohm those variables are particle positions. In the context of quantum field theory, there are two natural choices, namely particle positions and fields. The incorporation of those variables makes it possible to provide an objective description of nature in which rather ambiguous notions such as 'measurement' and 'observer' play no fundamental role. As such, the theory is free of the conceptual difficulties, such as the measurement problem, that plague standard quantum theory.
We reply to claims (by Deutsch, Zeh, Brown and Wallace) that the pilot-wave theory of de Broglie and Bohm is really a many-worlds theory with a superfluous configuration appended to one of the worlds. Assuming that pilot-wave theory does contain an ontological pilot wave (a complex-valued field in configuration space), we show that such claims arise from not interpreting pilot-wave theory on its own terms. Specifically, the theory has its own ('subquantum') theory of measurement, and in general describes a 'nonequilibrium' state that violates the Born rule. Furthermore, in realistic models of the classical limit, one does not obtain localised pieces of an ontological pilot wave following alternative macroscopic trajectories: from a de Broglie-Bohm viewpoint, alternative trajectories are merely mathematical and not ontological. Thus, from the perspective of pilot-wave theory itself, many worlds are an illusion. It is further argued that, even leaving pilot-wave theory aside, the theory of many worlds is rooted in the intrinsically unlikely assumption that quantum measurements should be modelled on classical measurements, and is therefore unlikely to be true.
Motivated by Wallace's thesis that pilot wave beables should be decoherence-preferred to recover quantum predictions, we consider the relation between pilot wave beables and decoherence. We prove that without any connection between beables and decoherence the overlap between macrcopic states becomes negligible. This is sufficient to recover quantum predictions, so that Wallace's thesis has to be rejected. A natural connection between decoherence and beables appears if the decomposition into systems used by decoherence is based on the beables. While our first result becomes inapplicable in this case, we present evidence that the overlap becomes negligible too.
Abstract This chapter replies to claims that the pilot-wave theory of de Broglie and Bohm is really a many-worlds theory with a superfluous configuration appended to one of the worlds. Assuming that pilot-wave theory does contain an ontological pilot wave (a complex-valued field in configuration space), the chapter shows that such claims arise from not interpreting pilot-wave theory on its own terms. Specifically, the theory has its own (‘subquantum’) theory of measurement, and in general describes a ‘non-equilibrium’ state that violates the Born rule. Furthermore, in realistic models of the classical limit, one does not obtain localised pieces of an ontological pilot wave following alternative macroscopic trajectories: from a de Broglie–Bohm viewpoint, alternative trajectories are merely mathematical and not ontological. Thus, from the perspective of pilot-wave theory itself, many worlds is an illusion. It is further argued that, even leaving pilot-wave theory aside, the theory of many worlds is rooted in the intrinsically unlikely assumption that quantum measurements should be modelled on classical measurements, and is therefore unlikely to be true.