IVF1. Basis Vectors in the Underlying Theory 

We cannot directly convert the underlying, pre-representational theory to the conventional representational form of quantum mechanics because we don’t know the specific form of Eq. (D-1).  Nevertheless, under reasonable assumptions, we can still gain valuable insight by using general principles.  The conversion from independent variable theory to the current, conventional form of quantum mechanics will be done in two steps.  The first step is to assume there is a spin ½ basis.  And the second is to assume the conventional vacuum, fermion and boson states can be expressed in terms of the spin ½ basis.

The Spin ½ Basis.

No matter what the specific form of the underlying equation is, we must convert it to representational form by the use of some set of basis vectors.  We would like to have the antisymmetry of spin ½ states built into the theory.  To achieve this, we assume that the initial complete set of states consists solely of functions that form a particlelike spin ½ representation of the inhomogeneous Lorentz group.  These might, for example, be solutions to the (single-bare-particle) eigenvector equation

(IVF1-1)  

It is assumed these basis vectors belong to the n representation of SU(n), the set of all nxn unitary matrices with determinant 1, with m denoting the internal symmetry label.  One can Fourier transform to obtain states labeled by position.  Solutions will then be sums of antisymmetrized products of these states. 

Restrictions on the States.

It is possible that we may want to restrict the spin ½ basis to zero mass, left-handed states.  This depends on work to be done in the as-yet-incomplete Sec. IVH.

Antisymmetrized Fermion Field Operators.

To build in antisymmetry, we assume the complete set of states belongs to the one-dimensional, antisymmetrized representation of the permutation group.  This implies that all physical solutions are to be represented by sums of antisymmetrized products of spin ½ functions.  The effect of this is that one can convert the whole problem into a form where states and operators are expressed in terms of anticommuting spin ½ creation and annihilation operators.

Note that the vector boson operators, bilinear in the fermion operators, will obey symmetric statistics.  See Sec. IVF4.

The Spin ½ Basis Equation of Motion.

As we remarked in Sec. IVD2, the spin ½ basis vectors can always be chosen so the wave functions obey the Dirac equation.  We assume this is done.  The equations of motion for the spin ½ field operators will then be the Dirac equation, modified to include an interaction term.  If we give the Lagrangian density from which the equations of motion are derived, one might have, as an example,

(IVF1-2)  

 is a vector with n components in the SU(n) space and the ’s are the  nxn matrix generators of SU(n).  .  (Nambu and Jona-Lasinio¹ proposed a four-fermion theory, but their treatment of the vacuum and boson states is different from ours.) 

Renormalization.

One might object that four-fermion theories are not in general renormalizable²—that is, an expansion in terms of Feynman diagrams gives non-cancelable infinities—and so such theories are suspect.  However, the use of fermion states alone as a basis is only an intermediate stage in this theory.  The final basis will be in terms of vector boson states and the vacuum, as well as fermion states.  It is presumably only in this final basis that a Feynman diagram expansion is appropriate.  So the renormalization problems encountered when attempting to use a Feynman diagram expansion at an intermediate stage, where the expansion does not seem appropriate, are not, I think, sufficient reason to negate the insights offered by this approach.

Notes:

1. Nambu, Y. and Jona-Lasinio, G., Dynamical model of elementary particles based on an analogy with superconductivity. Phys. Rev. 122, 345-358 (1961). 

2. Marshak, R. E.  Conceptual Foundations of Modern Particle Physics Ch. 4 (World Scientific, Singapore, 1993).