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A dBB theory for fermion fields - Printable Version

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A dBB theory for fermion fields - Schmelzer - 05-12-2016

Fermion fields are a difficult issue for dBB theory.  Lubos Motl is certainly not the only one who thinks that to incorporate them into a dBB theory is impossible. In  Bohmian mechanics is incompatible with loop corrections he writes: 

Quote:A problem is that 
the fermionic fields can't have any classical values at all.
There can't be any (nonzero) classical values of the fermionic fields because those would be Grassmannian numbers which anticommute with each other. But no two nonzero numbers  a,b  obey \(ab=-ba\).  So there doesn't even exist a mathematically possible configuration space for these "fermionic field beables". This strategy is completely failing for fermions!


The starting point: A lattice theory for a real-valued scalar field

Let's see how this problem can be solved.  First of all, we want to have a field theory.  And we already know how to define a dBB theory for a lattice regularization of a relativistic scalar field.  The mass term can be, of course, replaced by an arbitrary potential \(V(\varphi)\), so that what we have to care about is the Lagrangian
\[ S = \int \frac12 \eta^{\mu\nu}\partial_\mu \varphi \partial_\nu \varphi -V(\varphi)  d^4x,\]
with Hamiltonian
\[ H = \int \frac12 \pi^2 + \frac12 (\nabla\varphi)^2 + V(\varphi)  d^3 x.\]
Let's define the corresponding lattice regularization, for simplicity for an 1D space only and setting \(h=1\).
\[ H = \sum_n \frac12 \pi_n^2 +  \frac12 (\varphi_n-\varphi_{n+1})^2 + V(\varphi).\]
This lattice Hamiltonian is of the standard form necessary for defining a dBB theory in the usual way, and this does not depend on the potential \(V(\varphi)\).  

First step:  From a real field to a \(\mathbb{Z}_2\)-valued field

As a consequence, we obtain a possibility to define a dBB theory also for a potential with a degenerated vacuum state, say, \(V(\varphi) = |\varphi|^2 + e^{-|\varphi|^2}\).  So we have two minima, where each taken alone would look approximately like a standard massive scalar particle.  But there would be also an energy split between the symmetric and the anti-symmetric combinations.  So, the theory will be approximately a theory of some massive scalar particle, with a mass which we can make as large as we like, and a \(\mathbb{Z}_2\)-valued field, something usually named a spin field.  

Thus, to describe a \(\mathbb{Z}_2\)-valued lattice field theory is not a problem at all for dBB theory.  

Second step: From a spin fields to a fermion field

A \(\mathbb{Z}_2\)-valued lattice field theory is not yet a fermion theory.  We can define lattice operators \(\sigma^i_n\) which look nicely in each node:
\[\sigma_n^i\sigma_n^j = \delta_{ij}+i\varepsilon_{ijk}\sigma_n^k.\]

But the operators on different nodes commute, they have to commute by construction:
\[ \left[\sigma^i_m, \sigma^j_n\right]  = 2i\delta_{mn}      \varepsilon_{ijk}\sigma^k_n. \]
Instead, the operators of a fermionic field theory would have to anticommute:
\[\{\psi_m,\psi^*_n\} = \delta_{mn},\;\{\psi^*_m,\psi^*_n\}=\{\psi_m,\psi_n\}=0.\]
With
\[\psi_n^1 = \psi_n + \psi_n^*,\; \psi_n^2 = -i(\psi_n - \psi_n^*),\;\psi_n^3 = -i\psi_n^1\psi_n^2 \]
we can make them locally similar, but they nonetheless anticommute.  

But there is a solution:  Let's choose some ordering \(<\) between the nodes. Then, define
\[ \psi^{1/2}_n = \sigma^{1/2}_n \prod_{m>n}{\sigma^3_m}, \qquad  \psi^3_n =\sigma^3_n,\]
or, reverse,
\[ \sigma^{1/2}_n  = \psi^{1/2}_n \prod_{m>n}{\psi^3_m}, \qquad \sigma^3_n =\psi^3_n.\]
This defines an isomorphism between the two algebras of operators.  A quite well-known isomorphism, known from Clifford algebra theory,  where it is used to show that \(\textit{Cl}^{N,N}(\mathbb{R})\cong M_2(\textit{Cl}^{N-1,N-1}(\mathbb{R}))\cong M_{2^N}(\mathbb{R})\).

This is, of course, not yet all one has to care about - the local operators \(\sigma^{1/2}_n\) look global in terms of the \(\psi^{1/2}_n\).  But in 1D, everything appears completely nice, and the natural operator becomes the lattice Dirac operator.  Higher dimensions are more tricky, and there is no exact equivalence between the natural energy operators and the lattice Dirac operator.  Nonetheless, they appear sufficiently close.  For the details, see sect. 5 of Schmelzer, I.: A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Found. Phys. vol. 39, 1, p. 73-107  (2009), arXiv:0908.0591.

An interesting point of this construction is that the Dirac operator which is defined on the spatial lattice is that of a staggered discretization of the Dirac-Kähler equation on the exterior bundle \(\Lambda(\mathbb{R}^3)\).  It corresponds to the Dirac equation in its original form \(i \partial_t \psi = -i\alpha^i\partial_i \psi + m \beta \psi\).  The fermion doubling gives in this case two Dirac fermions.  So, the theory does not allow to describe a single Dirac fermion, only pairs of Dirac fermions.  

Fortunately, this is not really a problem, given that in Nature (at least in the SM) fermions appear only in electroweak pairs.