IVF2. The Vacuum State.

The second step in converting from an independent variable theory to the conventional form of the theory is to express the usual states of quantum mechanics—the vacuum, vector boson states and physical spin ½ states—in terms of the first, “bare,” spin ½,  basis.  The form for the vacuum state, , is chosen with an eye to providing an explanation for why a gauge transformation produces a vector boson field proportional to the derivative of the transformation.¹ 

Gauge Transformations.

A gauge transformation is a space-time dependent transformation from the internal symmetry group, assumed here to be SU(6).  An infinitesimal gauge transformation can be written as

(IVF2-1)  

with a sum over ,where is the th generator of SU(6) and  tells how the transformation varies in space and time.  Suppose we apply this to a state containing a single fermion state superimposed on the vacuum.

(IVF2-2)  

where  is the rotated fermion state and  is operator for the vector boson field at x, with q being the charge.  The last line comes from gauge theory, which says that a gauge transformation produces a vector boson field proportional to .  Thus we must have

(IVF2-3)  

How can we achieve this?  The only way is to have  not be invariant under gauge transformations.  Instead, a slowly varying gauge transformation must alter the vacuum in such a way as to produce a vector boson field proportional to . 

Molecular Structure of the Vacuum.

There is a way to do this.  We can construct the vacuum out of “molecules” that are invariant under global internal symmetry transformations but not under local gauge transformations. 

(IVF2-4)         

where  represents a molecule with center of mass at position r, F represents the motion of the center of mass, and f represents the binding of each spin ½ wave function to the molecule (with a force provided by the four-fermion interaction).

The ’s are antisymmetrized four-component spin ½ operators that obeys the Dirac equation.   consists of a creation operator for particles and an annihilation operator for antiparticles while consists of an annihilation operator for particles and a creation operator for antiparticles. Only the particle and charge conjugate particle creation operators survive in Eq. (IVF2-4).  The , and therefore ,  are Lorentz spin invariants.

Because of the antisymmetrization and the product from m=1 to 6, each molecule is invariant under global SU(6) transformations.  But if a gauge transformation varies slowly on a scale defined by the binding function f, then the gauge transformation will produce an alteration of the vacuum state proportional to .  That is, the vector boson fields produced by gauge transformations come from a change in the vacuum state induced by the gauge transformation..

Comments on the Molecular Vacuum State.

A very small, tightly bound vacuum molecule introduces problems into the theory.  A large binding energy is needed, and it is difficult to account for creation and annihilation of particle-antiparticle pairs.  So one could assume that the molecules are not so tightly bound (by letting the   of Eq. (IVF2-4) have a longer range) and that the different molecules can interpenetrate.  In the limit of a very long-range f, this could conceivably go over into something like a Dirac vacuum.

A problem with the interpenetrating-molecule approach is that one runs into trouble with the exclusion principle.  So it is probably best to use some sort of “localized Dirac” vacuum.  For a broken symmetry state, the density of states will be different for different internal symmetry states.

Notes:

1. Blood, F. A., Interaction-mediating vector bosons as collective oscillations of the vacuum. Il Nuovo Cim., 102 1059-1081 (1988).