IVG2 Speculations on Gauge Theory.

In the normal (Yang-Mills) approach to gauge theory, one uses gauge invariance to deduce the fact that gauge transformations generate vector boson states.  In the molecular vacuum approach, however, one gets the vector boson states “for free.”  The question in the molecular vacuum approach is whether one still has to assume gauge invariance to obtain the results of gauge theory—the covariant derivative and zero mass for vector bosons—or whether the already-given generation of vector boson states by gauge transformations alone is a sufficient condition for deriving the consequences of gauge invariance.  I would speculate that the given generation of vector bosons by gauge transformations is indeed sufficient for the derivation, but I don’t yet see how to do this in a satisfactory way.

The Heisenberg Picture.

A proper derivation of the results of gauge invariance would almost certainly be done using field operators in the Heisenberg picture.  And so the first requirement is to fully understand the Heisenberg picture.  More specifically, how does it fit in with the independent variable approach?  It is possible the fit is very natural, since the solution in the independent variable approach contains the state for all times.  But the point I don’t understand is what it means to add a gauge-induced vector boson state to the Heisenberg vector boson state. 

Possible Wave Function Approach.

One possible approach is to work with wave functions rather than the field operators.  If one assumes that

the wave function of the boson field felt by the fermion (wave function) is the sum of the wave function of the gauge induced boson state and the wave function of the already-present boson state,

then one can indeed derive the results of gauge theory—the covariant derivative and zero mass for vector bosons.  (One transforms the original wave function equations using a gauge matrix, and requires that the new equation depend only on the sum of the original and gauge-induced boson wave function.)  But this does not seem sufficient.

Gauge Transformations and Change of Basis.

An idea that may be useful is to consider gauge transformations to simply be a (space-time dependent) change of basis vectors.  Then neither the underlying independent-variable equation nor the independent-variable solutions are changed by the change of basis; only the wave functions are changed.  But it is difficult to see how this fits in with gauge invariance because gauge invariance refers to invariance of the equations for the representatives (the wave functions, the potentials ) of the states, and there is no guarantee, so far as I can see, that the equations for the representatives remain invariant.