In relativistic space-time, Bohmian theories can be formulated by introducing a privileged foliation of space-time. The introduction of such a foliation-as extra absolute space-time structure-would seem to imply a clear violation of Lorentz invariance, and thus a conflict with fundamental relativity. Here, we consider the possibility that, instead of positing it as extra structure, the required foliation could be covariantly determined by the wave function. We argue that this allows for the formulation of Bohmian theories that seem to qualify as fundamentally Lorentz invariant. We conclude with some discussion of whether or not they might also qualify as fundamentally relativistic.

We present a pilot-wave model for quantum field theory in which the Dirac sea is taken seriously. The model ascribes particle trajectories to all the fermions, including the fermions filling the Dirac sea. The model is deterministic and applies to the regime in which fermion number is superselected. This work is a further elaboration of work by Colin, in which a Dirac sea pilot-wave model is presented for quantum electrodynamics. We extend his work to non-electromagnetic interactions, we discuss a cut-off regularization of the pilot-wave model and study how it reproduces the standard quantum predictions. The Dirac sea pilot-wave model can be seen as a possible continuum generalization of a lattice model by Bell. It can also be seen as a development and generalization of the ideas by Bohm, Hiley and Kaloyerou, who also suggested the use of the Dirac sea for the development of a pilot-wave model for quantum electrodynamics.

Pilot-wave theories provide possible solutions to the measurement problem. In such theories, quantum systems are not only described by the state vector, but also by some additional variables. These additional variables, also called beables, can be particle positions, field configurations, strings, etc. In this paper we focus our attention on pilot-wave theories in which the additional variables are field configurations. The first such theory was proposed by Bohm for the free electromagnetic field. Since Bohm, similar pilot-wave theories have been proposed for other quantum fields. The purpose of this paper is to present an overview and further development of these proposals. We discuss various bosonic quantum field theories such as the Schrödinger field, the free electromagnetic field, scalar quantum electrodynamics and the Abelian Higgs model. In particular, we compare the pilot-wave theories proposed by Bohm and by Valentini for the electromagnetic field, finding that they are equivalent. We further discuss the proposals for fermionic fields by Holland and Valentini. In the case of Holland's model we indicate that further work is required in order to show that the model is capable of reproducing the standard quantum predictions. We also consider a similar model, which does not seem to reproduce the standard quantum predictions. In the case of Valentini's model we point out a problem that seems hard to overcome.1 Postdoctoral Fellow FWO. 2 Corresponding address.

In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q) and configuration q. The evolution of the wavefunction and configuration are respectively determined by the Schrödinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density |ψ(q)| 2 . We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrödinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.1 Present address. Currently Postdoctoral Fellow FWO.2 Note, however, that in principle the theory allows 'nonequilibrium' ensemble distributions, that is, distributions that differ from |ψ| 2 [8][9][10][11].3 Here and below we take = 1.