IVG1. Gauge Theory 

Yang-Mills Gauge Theory.

The pivotal paper on gauge transformations was that of Yang and Mills,¹ in which the internal symmetry group was taken to be SU(2) (isotopic spin).  In that paper, they say “We define isotopic gauge as an arbitrary way of choosing the orientation of the isotopic spin axes at all space-time points…. We then propose that all physical processes be invariant under an isotopic gauge transformation  where S represents a space-time dependent isotopic spin rotation.” 

We can infer more detailed, specific assumptions from this:

● At each point in space, there is a set of n axes related to the internal symmetry group SU(n).  There is no real understanding, in my opinion, of what the axes refer to. 

● The internal symmetry states of all particle states—all fermions, all bosons—are measured with respect to the same set of axes.

● The equations of quantum mechanics are assumed to be invariant under arbitrary rotations of the axes at each point in space.  This does not follow from any other mathematically established principle; it is a separate assumption.

The principle of gauge invariance almost completely determines the form of the interactions in elementary particle physics, from the replacement of the usual derivative with the covariant derivative to the zero mass of vector bosons associated with non-broken symmetries.  We know from its far-reaching, verified consequences that the results of the Yang-Mills proposal on gauge invariance are correct (when applied to the appropriate internal symmetry group).  However, since a gauge transformation is not part of the invariance group, I do not see any convincing group theoretic (or any other) rationale for the basic assumption of gauge theory—that the forms of the equations remain invariant under gauge transformations. 

In Sec. IVG2, we will lay down principles that may be useful in deriving the same consequences that follow from the assumption (currently justified only because its consequences are correct) of gauge invariance.

Notes:

1. Yang, C. N. and Mills, R. I. Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev., 96, 191-195 (1954).