4.3 Restoring Gauge Invariance

In the previous subsections we have developed a self-consistent model for membranes with a spacelike boundary, coupled to a massive tensor field. The price for that result is the apparent loss of manifest gauge invariance. As in the case of ``bubble dynamics'' in $(1+1)$ dimensions, this leads us to the question: is there a way of introducing a mass term into the action without spoiling manifest gauge invariance? Once again, we follow the original Stueckelberg proposal [16] of restoring gauge invariance by introducing a mass term together with a compensating scalar field. Presently, however, we need a modification of Stueckelberg's approach that is suitable for our massive tensor theory of bubble dynamics. The procedure is straightforward. The only novel aspect is that the role of compensating field is now played by a two-index Kalb-Ramond potential $B_{\nu\rho}(x)$ [23]. Accordingly, we modify the action (58) as follows:

$$S = \int d^4x\left[ {1\over 2\cdot 4!} F_{\lambda\mu\nu\rho} F^... ..., A_{\mu\nu\rho}+{1\over m} \partial_{ [ \mu}B_{\nu\rho ]} \right) \right.$$

$$-{m^2\over 2\cdot 3!}\left. \left( A_{\mu\nu\rho}+{1\over m} \... ...^{\mu\nu\rho}+{1\over m} \partial^{ [ \mu}B^{\nu\rho ]} \right) \right]$$ (74)

This action is invariant under the extended tensor gauge transformation

$$\delta A_{\mu\nu\rho}=\partial_ {[ \mu}\Lambda_ {\nu\rho ]}$$ (75)

$$\delta B_{\nu\rho}= -m \Lambda_ {\nu\rho} .$$ (76)

Note that the (gauge invariant) kinetic term for $B$ makes $A$ massive: from a dynamical point of view, the presence of a boundary, or a non conserved current, introduces a mass term for the gauge field that the current is coupled to.
The field equations become

$$\partial_\lambda F^{\lambda\mu\nu\rho}+m^2 \left( A^{\mu\nu\rho} +{1\over m} \partial^{ [ \mu}B^{\nu\rho ]} \right)=g J^{\mu\nu\rho}$$ (77)

$$\partial_\mu \left( A^{\mu\nu\rho} +{1\over m} \partial^{ [ \mu}B^{\nu\rho ]} \right) ={g\over m^2} J^{\nu\rho}$$ (78)

Equation (79) assures the self-consistency of Eq. (78). Moreover, using the same field decomposition as in the previous section, we see that equation (79) fixes the divergenceless component $\widetilde A_{\mu\nu\rho}$:

$$\widetilde A^{\mu\nu\rho} + {1\over m} \partial^{ [ \mu}... ...er m^2} \partial^{ [ \mu} {1\over \Box} J^{\nu\rho ]}$$ (79)

while, Eq. (78) leads to the following expression for $F^{\lambda\mu\nu\rho}$:

$$F^{\mu\nu\rho\sigma} = \sqrt\Lambda \epsilon^{\mu\nu\rho\sigma} + g^2 \partial^{ ... ...a ]} -m^2 \partial^{ [ \mu} {1\over \Box} \widehat A^{\nu\rho\sigma ]}$$

$$\equiv F^{\mu\nu\rho\sigma}_0 -m^2 \partial^{ [ \mu} {1\over\Box} {\widehat A}^{\nu\rho\sigma ]} .$$ (80)

Substituting the above expression into the action, we obtain after some rearrangement

$$S = \int d^4x\left[ {1\over 2\cdot 4!} F_ {0 \mu\nu\rho\sigma}... ...rho} \left( {\Box + m^2\over \Box} \right) \widehat A_{\mu\nu\rho}+ \right.$$

$$+ {g m^2\over 3!} \widehat J^{\mu\nu\rho} {1 \over\Box} \wid... ...+\left. {g^2\over 4 m^2} j^ {\mu\nu} {1\over \Box} j_ {\mu\nu} \right]$$ (81)

$$= - {1\over 2} \int d^4x\left[ \Lambda +{g^2\over 2\cdot 3!} \... ...nu\rho} -{g^2\over 4 m^2} j^ {\mu\nu} {1\over\Box} j^ {\mu\nu} \right]$$ (82)

The final form of the action (83) shows how the introduction of a compensating Kalb-Ramond field, which is necessary for restoring gauge invariance, leads to an ``effective closure'' of the membrane in the physical Minkowskian spacetime and is, in fact, an alternative to the Euclidean procedure of closing the membrane in imaginary time.
However, it seems to us that a careful consideration of the boundary effect in the nucleation process of a vacuum bubble in real spacetime has a clear advantage over the Euclidean formulation in that it brings out the existence of a massive pseudo-scalar degree of freedom which is otherwise hidden in the energy background provided by the cosmological field.