Quantum Mechanics for Mathematicians

coordinatesx. Such phase transformations are called “gauge transformations”
and form an infinite dimensional group under pointwise multiplication:

Definition(Gauge group).The groupGof functions onM^4 with values in the
unit circleU(1), with group law given by point-wise multiplication

eieφ^1 (x)·eieφ^2 (x)=eie(φ^1 (x)+φ^2 (x))

is called theU(1)gauge group, or group ofU(1)gauge transformations.

Herex= (x 0 ,x)∈M^4 ,eis a constant, andφis a real-valued function

φ(x) :M^4 →R

The vector space of such functions is the Lie algebraLieG, with a trivial Lie
bracket. The constantedetermines the normalization of the Lie algebra-valued
φ(x), with its appearance here a standard convention (such that it does not
appear in the Hamiltonian, Poisson brackets, or equations of motion).
The groupGacts on complex functionsψof space-time as

ψ(x)→eieφ(x)ψ(x), ψ(x)→e−ieφ(x)ψ(x)

Note that, in quantum mechanics, this is a group action on the wavefunctions,
and it does not correspond to any group action on the finite dimensional phase
space of coordinates and momenta, so has no classical interpretation. In quan-
tum field theory though, where these wavefunctions make up the phase space
to be quantized, this is a group action on the phase space, preserving the sym-
plectic structure.
Terms in the Hamiltonian that just involve|ψ(x)|^2 =ψ(x)ψ(x) will be in-
variant under the groupG, but terms with derivatives such as

|∇ψ|^2

will not, since when
ψ→eieφ(x)ψ(x)

one has the inhomogeneous behavior

∂ψ(x)

∂xμ

→

∂

∂xμ

(eieφ(x)ψ(x)) =eieφ(x)

(

ie

∂φ(x)

∂xμ

+

∂

∂xμ

)

ψ(x)

To deal with this problem, one introduces a new kind of field:

Definition(Connection or vector potential).AU(1)connection (mathemati-
cian’s terminology) or vector potential (physicist’s terminology) is a functionA
on space-timeM^4 taking values inR^4 , with its components denoted

Aμ(x) = (A 0 (t,x),A(t,x))

The gauge groupGacts on the space ofU(1)connections by

Aμ(t,x)→Aφμ(t,x)≡Aμ(t,x) +

∂φ(t,x)

∂xμ

(45.1)