The connection and Maxwell's theory
We know from quantum mechanics that if we replace the wave-function, , describing the electron field by
that it leaves physical predictions unchanged. We consider the imposition of local invariance on the phase of the electron field,
The problem is that derivatives of are not covariant under this transformation:
.
In order to cancel out the second unwanted term, one introduces a new derivative operator that is covariant. To construct , one introduces a new field, the connection :
.
Then
The term is precisely cancelled out by requiring the connection field transforms as
.
We then have that
.
Note that is equivalent to
which looks the same as a gauge transformation of the gauge potential of Maxwell's theory. It is possible to construct an invariant action for the connection field itself. We want an action that only has two derivatives (since actions with higher derivatives are not unitary). Define the quantity:
.
The unique action with only two derivatives is given by:
.
Therefore, one can derive electromagnetic theory from arguments based solely on symmetry.
The connection and Yang-Mills gauge theory
We now generalize the above reasoning to general gauge groups. One begins with the generators of some Lie algebra:
Let there be a fermion field that transforms as
Again the derivatives of are not covariant under this transformation. We introduce a covariant derivative
with connection field given by
We require that transforms as:
- .
We define the field strength operator
- .
As is covariant, this means that the tensor is also covariant:
Note that is only invariant under gauge transformations if is a scalar, that is, only in the case of electromagnetism.
We can now construct an invariant action out of this tensor. Again we want an action that only has two derivatives. The simplest choice is the trace of the commutator:
The unique action with only two derivatives is given by:
This is the action for Yang-mills theory.