The dBB theory is often presented as a theory which works only for non-relativistic particles.

This is very unfortunate, because it is completely wrong. The first example of a relativistic dBB field theory, for the electromagnetic field, has been already given in Bohm's original 1952 paper. This misrepresentation may be caused by the fact that the standard form of the Hamiltonian in dBB theory is quadratic in the momentum variables, thus, looks like \(H = \sum_i \frac{1}{2m}p_i^2 + V(q_1,\ldots, q_i,\ldots)\), and the quadratic character is essential to get the nice formula for the guiding equation, so that this does not look like one could easily modify it to obtain a relativistic formula for particle energy.

But the situation is completely unproblematic if one considers relativistic fields.

The scalar field

For a simple scalar field theory, the configuration is defined by the field values \(\varphi(x)\), the momentum is \(\pi(x) = \dot{\varphi}(x)\), and the Hamiltonian is

\[H = \int \mathcal{H} d^3x = \int \frac12 \pi^2(x) + \frac12(\partial_i\varphi(x))^2 + m^2\varphi^2(x) d^3x. \]

So, this Hamiltonian has the same quadratic dependence on the momentum variables as one needs for developing dBB theory in its standard form. It is also important to note that one can easily define a regularization of such a field theory which fits into the same scheme: All one needs to get rid of all (UV as IR) infinities is a spatial lattice with a finite number of nodes \(n_i\) and periodic boundary conditions \(\varphi_{N+1}=\varphi_1\). The resulting lattice theory would have the Hamiltonian

\[H = \sum_i \frac12\pi_i^2 + \frac12 \frac{(\varphi_{i+1}-\varphi_i)^2}{h^2} + m^2 \varphi_i^2,\]

which also fits into the standard dBB form, now even for the simple, unproblematic case of a finite dimensional phase space. Note also that interaction terms with other fields usually do not contain time derivatives, but are simply products of the fields. Thus, they are simply part of the "potential term", which can be, in principle, an arbitrary function of the \(\varphi_i\).

Gauge fields

The situation with gauge fields is unproblematic too. If there is a problem, it is not how to handle relativistic symmetry, but how to handle gauge symmetry. There may be various approaches, but given that Bohm 1952 already gives a solution, it is clear that these are solvable problems.

Fermions

What appears to be problematic are fermion fields. My proposal gives only pairs of Dirac fermions, and, moreover, only in combination with some massive scalar field. This may be sufficient for the standard model, because in the standard model all fermions are part of electroweak pairs, and the massive scalar field may give a nice candidate for dark matter. But this requires that there have to be right-handed neutrinos. If they do not exist, this proposal would not be sufficient.

Are there others? First of all, I want to mention the emergency exit if everything goes wrong with fermions. It has been proposed by Struyve and Westman:

The idea is simply to ignore fermions. Once we have a field theory for bosons, this is sufficient to distinguish macroscopic objects. And this is essentially all we need to recover the equivalence with quantum theory. This simple ignorance of the fermionic sector is in no way a beautiful idea. But it shows one thing: Whatever the problems with fermion fields, this is not a fatal threat to dBB theory.

Thus, a proponent of dBB theory can have a quite relaxed position about fermion fields. So, let's see what we have. There is an interesting proposal made by Holland, in section 10.6.2 ("Fermionic analogue of the oscillator picture of boson fields"), p. 451ff of his book

It uses Euler angles to describe fermion fields. Another field-theoretic proposal for fermions has been proposed in

It uses directly Grassmann variables, and applies a variant of the Dirac field, the van der Waerden field, which consists of two complex components and fulfills a second order equation.

Above proposals have been criticized by Struyve:

I do not want to evaluate here if this criticism is justified or not - given that I have made an own proposal, I could be considered prejudiced here.

Summary

The claim that dBB theory works only for non-relativistic particles is completely wrong. There are proposals for relativistic field theories too. They are unproblematic for bosonic field theories. There are also proposals for fermionic field theories. It is less clear if they are sufficient. But it is clear that a "minimalist approach", which does not introduce any dBB structure to the fermionic part, is viable too.

This is very unfortunate, because it is completely wrong. The first example of a relativistic dBB field theory, for the electromagnetic field, has been already given in Bohm's original 1952 paper. This misrepresentation may be caused by the fact that the standard form of the Hamiltonian in dBB theory is quadratic in the momentum variables, thus, looks like \(H = \sum_i \frac{1}{2m}p_i^2 + V(q_1,\ldots, q_i,\ldots)\), and the quadratic character is essential to get the nice formula for the guiding equation, so that this does not look like one could easily modify it to obtain a relativistic formula for particle energy.

But the situation is completely unproblematic if one considers relativistic fields.

The scalar field

For a simple scalar field theory, the configuration is defined by the field values \(\varphi(x)\), the momentum is \(\pi(x) = \dot{\varphi}(x)\), and the Hamiltonian is

\[H = \int \mathcal{H} d^3x = \int \frac12 \pi^2(x) + \frac12(\partial_i\varphi(x))^2 + m^2\varphi^2(x) d^3x. \]

So, this Hamiltonian has the same quadratic dependence on the momentum variables as one needs for developing dBB theory in its standard form. It is also important to note that one can easily define a regularization of such a field theory which fits into the same scheme: All one needs to get rid of all (UV as IR) infinities is a spatial lattice with a finite number of nodes \(n_i\) and periodic boundary conditions \(\varphi_{N+1}=\varphi_1\). The resulting lattice theory would have the Hamiltonian

\[H = \sum_i \frac12\pi_i^2 + \frac12 \frac{(\varphi_{i+1}-\varphi_i)^2}{h^2} + m^2 \varphi_i^2,\]

which also fits into the standard dBB form, now even for the simple, unproblematic case of a finite dimensional phase space. Note also that interaction terms with other fields usually do not contain time derivatives, but are simply products of the fields. Thus, they are simply part of the "potential term", which can be, in principle, an arbitrary function of the \(\varphi_i\).

Gauge fields

The situation with gauge fields is unproblematic too. If there is a problem, it is not how to handle relativistic symmetry, but how to handle gauge symmetry. There may be various approaches, but given that Bohm 1952 already gives a solution, it is clear that these are solvable problems.

Fermions

What appears to be problematic are fermion fields. My proposal gives only pairs of Dirac fermions, and, moreover, only in combination with some massive scalar field. This may be sufficient for the standard model, because in the standard model all fermions are part of electroweak pairs, and the massive scalar field may give a nice candidate for dark matter. But this requires that there have to be right-handed neutrinos. If they do not exist, this proposal would not be sufficient.

Are there others? First of all, I want to mention the emergency exit if everything goes wrong with fermions. It has been proposed by Struyve and Westman:

- W. Struyve, H. Westman, A new pilot-wave model for quantum field theory, AIP Conf. Proc. 844, 321-339 (2006), arXiv:[quant-ph]0602229

- W. Struyve, H. Westman, A minimalist pilot-wave model for quantum electrodynamics, Proc. R. Soc. A 463, 3115-3129 (2007), arXiv:0707.3487

The idea is simply to ignore fermions. Once we have a field theory for bosons, this is sufficient to distinguish macroscopic objects. And this is essentially all we need to recover the equivalence with quantum theory. This simple ignorance of the fermionic sector is in no way a beautiful idea. But it shows one thing: Whatever the problems with fermion fields, this is not a fatal threat to dBB theory.

Thus, a proponent of dBB theory can have a quite relaxed position about fermion fields. So, let's see what we have. There is an interesting proposal made by Holland, in section 10.6.2 ("Fermionic analogue of the oscillator picture of boson fields"), p. 451ff of his book

- P.R. Holland, The Quantum Theory of Motion, Cambridge University Press 1993.

It uses Euler angles to describe fermion fields. Another field-theoretic proposal for fermions has been proposed in

- A. Valentini, Pilot wave theory of fields, gravitation and cosmology, in J.T. Cushing, A. Fine, S. Goldstein (eds.), Bohmian mechanics and quantum theory : an appraisal, Springer, Dordrecht 1996.

It uses directly Grassmann variables, and applies a variant of the Dirac field, the van der Waerden field, which consists of two complex components and fulfills a second order equation.

Above proposals have been criticized by Struyve:

- W. Struyve, Pilot-wave theory and quantum fields, Rept.Prog.Phys.73:106001 (2010), arxiv:0707.3685

- W. Struyve, The de Broglie-Bohm pilot-wave interpretation of quantum theory, PhD. Thesis, Ghent University, Ghent (2004), arxiv:quant-ph/0506243

I do not want to evaluate here if this criticism is justified or not - given that I have made an own proposal, I could be considered prejudiced here.

Summary

The claim that dBB theory works only for non-relativistic particles is completely wrong. There are proposals for relativistic field theories too. They are unproblematic for bosonic field theories. There are also proposals for fermionic field theories. It is less clear if they are sufficient. But it is clear that a "minimalist approach", which does not introduce any dBB structure to the fermionic part, is viable too.