搜索标题: Unitarity, Gauge Invariance, and Renormalizability
搜索作者: sage

Unitarity is a requirement of the consistency of any theory. In gauge theories, this is not entirely obvious. From Lorentz invariance, we will write a gauge field as A_mu. However, we know that there are degrees of freedoms in this gauge field that is not physical. Therefore, for unitarity, we must require those degrees of freedom does not contribute to any physical observable. This is ensured by gauge symmetry.

A confusing point is that we must choose a gauge to do our calculation. It is not trivial to verify any calculation is in the end gauge invariant. However, Fadeev-Popov procedure shows that our theory preserve gauge symmetry with any gauge fixing, since F-P is manifestly gauge invariant (a rather technical way to see this is the BRST symmetry).

Another confusing point is that, naively, the classical gauge symmetry is not necessary translating into a full symmetry of every observable, since the full theory is only defined by a regularization+renormalization procedure. For example, the naive cut-off regularization breaks the manifest gauge symmetry (it is still secret gauge invariant in the end). However, it is possible to regulate the theory in a manifestly gauge invariant way. One of the famous example is dimensional regularization.

Yet another seemingly confusing point is that whether all these nice property is still true in a theory where gauge symmetry is spontaneously broken. The answer is that the symmetry is only broken by the vacuum (Or ther physical states does manifest gauge symmetry). However, gauge symmetry is still a good symmetry for the Lagrangian and provides all the services it suppose to do. another way of saying this is that the theory still has a gauge symmetry, although non-linearly realized.

Now, there is also the issue of renormalizability. It turns out that gauge theories are renormalizable. Gauge symmetry played an important role here in that it constrains the type of divergences we can have and relate different divergences. The definite proof is provided by 'tHooft. On the other hand, in the modern point view of effective field theory, renormalizability is not a requirement for a consistent theory.