Quantum Mechanics for Mathematicians

γ(0)

γ(τ)

γ(1)

C

ψ(γ(1))

eie

∫

γAψ(γ(0))

e

∫

γA

ψ(γ(0))

Figure 45.1: Comparing a complex field at two points in a gauge invariant
manner.

In the path integral formalism (see section 35.3), the minimal coupling of
a single particle to a background electromagnetic field described by a vector
potentialAμcan be introduced by weighting the integral over paths by the
path-dependent phase factor. This changes the formal path integral by
∫
Dγ e
~iS[γ]
→

∫

Dγ e

~iS[γ]

e

ie~∫γA

and a path integral with such a weighting of paths then must be sensibly defined.
This method only works for the single-particle theory, with minimal coupling for
a quantum theory of fields given by the replacement of derivatives by covariant
derivatives described earlier.

45.5 The non-Abelian case

We saw in section 38.2.2 that quantum field theories with aU(m) group acting
on the fields can be constructed by takingm-component complex fields and
a Hamiltonian that is the sum of the single complex field Hamiltonians for
each component. The constructions of this chapter generalize from theU(1)
toU(m) case, getting a gauge groupGof maps fromR^4 toU(m), as well as
a generalized notion of connection and curvature. In this section we’ll outline
how this works, without going into full detail. The non-AbelianU(m) case
is a relatively straightforward generalization of theU(1) case, except for the