How does gauge symmetry arise in QFT and its implications?

[CSnowden]

TL;DR

QFT and the Origin of Gauge Invariance

In an earlier question I asked if the EM field was truly a separate field from the matter field in QFT, as it's field structure is naturally complementary to phase changes in the matter field in just the right way to restore gauge invariance (poorly formed question, but hopefully you get the gist). One response gave a succinct answer in that the QFT Lagrangian clearly contains two separate coupled fields and so there is the answer; however, this does seem to beg the question of how gauge symmetry arises. As the field solutions to the QFT Lagrangian clearly do have a tight gauge symmetry, is there some clearly identifiable structure of the QFT Lagrangian that would cause one to expect that only gauge invariant field solutions would result?

 

[vanhees71]

Gauge invariance first occurs when thinking about how to realize the fundamental space-time symmetry of Minkowski space, i.e., how to realize the proper orthochonous Poincare group. Wigner's famous analysis of the unitary irreducible representations, from which any representation can be built, shows that for physically meaningful realization (admitting microcausal realizations leading to a unitary S-matrix) there are the realizations with $m^2>0$ and $m^2=0$. In the latter case for fields with spin $\geq 1$ you need gauge symmetry, because otherwise you get continuous intrinsic quantum numbers, i.e., something like a "continuous polarization degree of freedom" for the corresponding single-particle Fock states, and such a thing has never been observed.

For the electromagnetic field you have a massless spin-1 representation, and to make the continuous part of the "little group" realized trivially, you have to assume gauge invariance, which leads to the usual two polarization degrees of photons (instead of the 3 spin states for massive vector bosons).

If you now consider interacting field theories thus you have to obey local gauge invariance to make sure that no "unphysical degrees of freedom" mix into the dynamics and thus contribute to the S-matrix, because this would lead to a completely inconsistent theory that cannot be physically interpreted at all (particularly the S-matrix wouldn't be unitary).

 

[CSnowden]

I appreciate your feedback - can you comment on how the QFT Lagrangian is structured to implement the requirement you outline to ensure that 'no "unphysical degrees of freedom" mix into the dynamics and thus contribute to the S-matrix' ? As the Lagrangian solves for the final field structure it seems any requirements must be addressed in it's definition.

 

[vanhees71]

The usual heuristic scheme is "minimal coupling", i.e., you start from a theory with a global symmetry, e.g., the Dirac Lagrangian for free Dirac fields,
$$\mathcal{L}_0=\bar{\psi}(\mathrm{i} \gamma^{\mu} \partial_{\mu}-m) \psi.$$
The symmetry is symmetry under multiplication with a phase factor
$$\psi =\rightarrow \exp(-\mathrm{i} \alpha) \psi.$$
Then you want to make the Lagrangian invariant under local transformations, i.e., when $\alpha \rightarrow \alpha(x)$. To that end you introduce the gauge-boson field and use
$$\partial_{\mu} \rightarrow \partial_{\mu} +\mathrm{i} q A_{\mu}=\mathrm{D}_{\mu},$$
leading to
$$\mathcal{L}_0'=\bar{\psi} (\mathrm{i} \gamma^{\mu} \mathrm{D}_{\mu}-m) \psi,$$
which is invariant under the gauge transformation
$$\psi \rightarrow \exp(-\mathrm{i} q \chi) \psi, \quad A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \chi.$$
Finally you also want the gauge field being a dynamical field. So you add the free field Lagrangian, which should be gauge invariant. The one with the least order of derivatives is
$$\mathcal{L}_{0\text{gauge}}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$
with
$$F_{\mu \nu}=\partial_{\mu} A_{\mu}-\partial_{\nu} A_{\mu}.$$
Adding these lagrangians you end up with the Lagrangian for QED describing electrons, positrons and photons (when setting $q=-e$).

 

[CSnowden]

That is a perfect explanation, many thanks!