Constrained system : U(1) pure gauge theory

U(1) pure gauge theory  : theory of light (without matter)

\(\mathcal{L}_{\text{free}} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\)

 

quantization of the photon field

 

http://www.ecm.ub.es/~espriu/teaching/classes/fae/LECT4.pdf

gives physical and unphysical states

Hilbert space of physical states

 

quantize unconstrained system and then impose the constraint

 

 

remark

if matter exists, we get QED

\(\mathcal{L}_{\text{free}} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\)