Quantum Mechanics for Mathematicians

The first two equations are for non-relativistic theories, and one can interpret
these equations as describing a single quantum particle (with spin^12 in the
second case) moving in a background electromagnetic field. In the relativistic
Klein-Gordon case, here we are in the case of a complex Klein-Gordon field,
as discussed in section 44.1.2. In all three cases, in principle a quantum field
theory can be defined by taking the space of solutions of the equation as phase
space, and applying the Bargmann-Fock quantization method (in practice this
is difficult, since in general there is no translation invariance and no plane-wave
basis of solutions).

45.4 The geometric significance of the connec-

tion

The information contained in a connectionAμ(x) can be put in a different form,
using it to define a phase for any curveγbetween two points inM^4 :

Definition(Path-dependent phase factor).Given a connectionAμ(x), one can
define for any curveγparametrized byτ∈[0,1], with position at timeτgiven
byx(τ), the path-dependent phase factor

∫

γ

A≡

∫ 1

0

∑^4

μ=1

Aμ(x)

dxμ

dτ

dτ

The effect of a gauge transformationφis

∫

γ

A→

∫ 1

0

∑^4

μ=1

(

Aμ(x) +

∂φ

∂xμ

)

dxμ

dτ

dτ=

∫

γ

A+φ(γ(1))−φ(γ(0))

Note that ifγis a closed curve, withγ(1) =γ(0), then the path-dependent
phase factor

∫

γAis gauge invariant.

Digression.For readers familiar with differential forms,Acan be thought of
as an element ofΩ^1 (M^4 ), the space of 1 -forms on space-timeM^4. The path-
dependent phase

∫

γAis then the standard integral of a^1 -form along a curve
γ. The curvature ofAis simply the 2 -formF =dA, wheredis the de Rham
differential. The gauge group acts on connections by

A→A+dφ

and the curvature is gauge invariant since

F→F+d(dφ)

anddsatisfiesd^2 = 0.