Gauge fields

Gauge fields are a special class of field theories based on the principles of the theory of symmetry groups. Einstein's general relativity is one example but up to now nobody has made a quantum theory of it.

QED is the standard example of a quantum field theory based on gauge symmetry but called abelian, whereas QCD uses so-called non-abelian symmetry characterised by the non-equivalence after permutation of the action of two symmetry operations on the state of a system described by these quantum theories; this is called non-commutativity.

It is suspected that all coherent relativistic quantum field theories are based on the existence of so-called gauge invariance symmetries which should allow them to be united.