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dBB field theories in the relativistic domain
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.  


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.  


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.
I didn't realize the situation with fermions was so uncertain.

Right-handed neutrinos: as theory, I don't think this is a big stumbling block. Everybody wants the existence of right-handed neutrinos, since neutrinos have been shown to have mass. In the standard model, for Higgs to give mass to neutrinos requires right-handed neutrinos of one sort or another. It may be a problem for your theory but it's also a problem for everyone else. You can calmly wait until they solve it somehow.

Practically however it helps illustrate that relativistic fermions are a problem for dBB. After all fermions are the most common, best understood of particles. If you require speculative right-handed neutrinos, just to explain a simple electron moving fast, it's not encouraging.

Then there's this "emergency exit". In that "solution" we simply give up on fermions as beables. In QED the only way we know about them is their interaction with bosons, photons. So we ignore fermions as such but explain how photon field is affected by them; and, photon field is all we can observe anyway; so, good enough.

Maybe it works but it's not very satisfactory.

It's surprising LM (or other dBB opponents) don't bang on this question more. Perhaps it's because they don't understand it. In that case your publicizing it argues well for the honesty of yourself and dBB'ers in general. You're not interested in "winning" but getting at the truth. If there's a problem you want it well-known, so everyone can help solve it. If someone can prove it kills dBB - fine, if that's the truth let's hear it! This is the scientific attitude in a nut-shell and I wish everyone would follow your example.

As far as I know your fermion theory works well enough to cover this base, given the current incomplete state of the Standard Model (due to massive neutrinos). They won't object to your right-handed neutrinos since they need them also.

However - intuition tells me this relativistic-fermion problem may point to a really fundamental issue. Perhaps the problem is NOT with dBB, but with relativity! dBB already says that Lorentz invariance can't be maintained forever. Sure it's true at the levels we've tested so far but (according to Bohm) at the very fundamental level beyond our current ability to test, it breaks down.

Maybe dBB isn't too comfortable with relativistic fermions because they don't actually exist! They're only an approximation of the truth, because relativity is not correct, fundamentally. Maybe dBB is revealing this to us. Assuming it's the correct theory (the point of view you take for this work), when you try to force it to accept completely relativistic fermions it tells you: no, those are logically inconsistent.

In that case the correct approach would be to develop a fermion theory which only approximates relativistic requirements at the level we can experiment on at this time.

Does that seem like a possibility?
Ward Struyve is indeed a good example of a dBB supporter looking for serious problems in dBB theory.  In arxiv:0904.0764 I had to make some computations to solve another problem he has proposed.  

Struyve's "minimalist model" I see just as a useful tool for discussions, to meet claims that there exist no viable variant of dBB theory for RQFT, in discussions where the opponents want to reject realism together with causality to preserve fundamental relativity, a sort of second defense line or so.  It is clearly not a satisfactory model.  

Regarding the right-handed neutrinos I'm quite optimistic, I would have included them into my model even without any evidence about neutrino masses. Anyway, my theory predicts that they do not have any interactions with any gauge fields, so they would be unobservable. 

In my opinion it is clear that relativistic symmetry is only a large distance approximation.  The Lorentz group is nothing but the symmetry group of the standard wave equation 
\[ \square u(t,x) = (\partial_t^2 - \partial_i^2) u(t,x) = 0.\]
Of course, there will be a lot of more complex wave equations of similar form for various vector and tensor fields, which will have the same symmetry group too.  Now, this wave equation is the natural large distance limit of a normal lattice theory.  But for the lattice theory, the Lorentz group is no longer a symmetry group.  

The idea that this symmetry group, which appears in such a natural way as an approximation in a natural equation we already know from standard acoustics, is, instead, some fundamental theory, is imho quite artificial.

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