3.3 Massive, Gauge Invariant Phase

The answer to the question raised in the previous subsection was suggested by Stueckelberg a long time ago [16]. The original Stueckelberg proposal was to recover gauge invariance by introducing a compensating scalar field $\theta$ so that the resulting action

$$S_A=\int d^2x \left[ {1\over 4} F_{\mu\nu} F^{\mu\nu}... ...\mu +{1\over m} \partial_\mu \theta \right) \right]$$ (20)

is invariant under the extended gauge transformation

$$A_\mu \longrightarrow A_\mu^\prime = A_\mu + \partial_\mu \lambda$$ (21)

$$\theta \longrightarrow \theta^\prime=\theta-m \lambda .$$ (22)

In this case, the vector field equation

$$\partial_\mu F^{\mu\nu}+m^2 \left( A^\nu + {1\over m} \partial^\nu \theta \right) = e J^\nu$$ (23)

is self-consistent because of the $\theta$-field equation

$$m^2 \partial_\nu \left( A^\nu + {1\over m} \partial^\nu \theta \right) = e \partial_\nu J^\nu .$$ (24)

Evidently, the role of the constraint (24) is to combine the $\widetilde A_\mu$ component of the vector potential with the compensator field in such a way that symmetry is restored with respect to the extended gauge transformation. In our geometric interpretation, this is equivalent to ``closing the world history'' by compensating for the leakage of symmetry through the boundary. In this sense, the generation of mass is the consequence of ``mixing'' two gauge fields, namely the $\widetilde A_\mu$ component of the vector potential with the $\theta$-field. As a matter of fact, Eq. (24) determines the mixed, gauge invariant field to be

$$\widetilde A_\mu +{1\over m} \partial_\mu \theta = {e\over m^2} \partial_\mu {1\over\Box} J$$ (25)

Once the above equation (25) is inserted into the action (20) we obtain

$$S=\int d^2 x \left[ {1\over 4} F_{\mu\nu} F^{\mu\nu}... ...widehat A_\mu -{e^2\over 2m^2} J {1\over \Box} J \right]$$ (26)

which represents an ``effective action'' for the only physical degree of freedom represented by $\widehat A_\mu$.