Relativistic Generalization of Bohm's Theory

In the special relativistic region the Dirac equation replaces the Schrodinger equation for the spinning electron. The Dirac spinor wave function and the electromagnetic four-vector potential are "second-quantized" as local field linear operators on a "Fock space" of states that tell how many particles and antiparticles are in each "field oscillator" attached to a classical spatial mode function which is defined by the symmetry and the boundary conditions.

These local field operators create and destroy the particles and antiparticles. Bohm is able to extend his picture to this case as well. As a practical matter the standard formalism as developed by Tomonaga, Schwinger, Feynman and Dyson is the one that must be used to get numbers out. However, Bohm's picture may show how to get a generalized quantum field theory that violates the microcausality axiom and has strictly finite renormalizations of mass, charge and wavefunction normalization. Such a generalization is not even thinkable in the standard picture because the Lorentz-invariant functional derivatives of Tomonaga's theory, for example, cannot even be defined if microcausality is violated

Chapter 12 of Holland's book is "Relativistic quantum theory".

Can the trajectory concept be retained in the relativistic quantum domain? And how do we interpret relativistic quantum fields?... It is known that Lorentz covariant wave equations can have consequences that conflict with relativity. ... Especially in field theory, the individual processes that build up and explain the covariant statistical predictions of the formalism exhibit features that are in conflict with our entrenched ideas regarding covariance and locality as necessary concomitants of relativity theory. That is, relativity is statistically valid but the individual events do not have an intrinsically relativistic character. p.498
The fourth component of the current density can be negative for relativistic quantum fields. This implies negative probability density which is hard to interpret not only in Bohm’s theory but in orthodox theory.

The effective variable mass squared M^2 has a factor (1 + Q) where Q is the wave operator on the square root of rho divided by rho multiplied by the square of the bare mass m. (p. 499 eq. 12.1.2)

The current density four-vector is not generically timelike everywhere on the particle trajectory. Indeed, the effective variable mass M can become imaginary describing a faster-than-light tachyon.

A theory of material objects in which an initially time-like future pointing trajectory may pass through the light cone to become space-like, an even move backwards in time is clearly unacceptable. p.500
I beg to differ. Holland’s poison is my fine wine. I think it was Stephen Weinberg who pointed out “failures of nerve” in the history of physics in which theorists did not take the consequences of their equations seriously enough. The kinds of difficulties Holland points to are, in fact, tantalizing opportunities that may lead to the proper understanding of dark matter, of exotic matter needed to support time-travel wormholes, the and the Alcubierre warp drive and the associated vacuum instability that advanced intelligences might use in their super-technology.

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