Observables in gauge theories
Yuri M Makeenko · Cambridge University Press eBooks · 2023
Observables in gauge theoriesModern theories of fundamental interactions are gauge theories.The principle of local gauge invariance was introduced by H. Weyl for the electromagnetic interaction in analogy with general covariance in Einstein's theory of gravitation.An extension to non-Abelian gauge groups was given by Yang and Mills [YM54].A crucial role in gauge theories is played by the phase factor which is associated with parallel transport in an external gauge field.The phase factors are observable in quantum theory, in contrast with the classical theory.For the electromagnetic field, this is known as the Aharonov-Bohm effect.In this chapter we initially consider the matrix notation for the non-Abelian gauge fields and introduce proper non-Abelian phase factors.Then we discuss the relation between observables in classical and quantum theories. Gauge invarianceThe principle of local gauge invariance deals with the gauge transformation (g.t.) of a matter field ψ, which is given by ψ(x) g.t.-→ ψ (x) = Ω(x) ψ(x) .(5.1)Here Ω(x) ∈ G with G being a semisimple Lie group which is called the gauge group (G = SU (3) for QCD).Equation (5.1) demonstrates that ψ belongs to the fundamental representation of G.We restrict ourselves to a unitary gauge group when Ω -1 (x) = Ω † (x) , (5.2) 85