4. Dynamics of the membrane vacuum and the formation of bags

The action (8) leads to the pair of coupled field equations

  

$${\langle} \Big \vert \left( \frac{\delta}{\delta \sigma ^{\mu \nu \rho} (s)} - i g A _{\mu \nu \rho} \right) \Big \vert ^{2} \Psi[S] {\rangle}$$ = 0 (19)

$$\partial _{\lambda} F ^{\lambda \mu \nu \rho} (x) g J ^{\mu \nu \rho} (x)$$ = (20)

which describe the interaction between the membrane field and the $A _{\mu \nu \rho}$ gauge potential. Now, we wish to show that the superconducting membrane condensate contains regions of spacetime, or bags of non superconducting vacuum.
Recall that in a type-II superconductor the magnetic field is confined by the superconducting vacuum pressure within a string-like flux tube. Similarly, the membrane condensate confines the gauge field strength within a membrane-like boundary layer surrounding a region of ordinary vacuum. Indeed, in analogy with the superconducting solution of scalar QED, we assume the following asymptotic boundary condition for $\Psi [S]$

$$\Psi [S] \equiv \frac{\phi}{g ^{2}} e ^{i \displaystyle{\... ...} d x ^{\mu} \wedge d x ^{\nu} \theta _{\mu \nu} (x) \right)$$ (21)

where $\phi$ is a constant. This is the form of the membrane field when the three volume enclosed by S is much larger than the three volume of the vortex spacetime image. Then, from equation (19) we obtain the corresponding asymptotic form of $A _{\mu \nu \rho}$:

$$\left( \frac{\delta}{\delta \sigma ^{\mu \nu \rho} (s)} - ... ... \rho} = \frac{1}{g} \partial _{[ \mu} \theta _{\mu \nu ]}.$$ (22)

Therefore, the flux of $F _{\lambda \mu \nu \rho}$ across a large four dimensional region $\Omega$ enclosing B is given by

 

$$\equiv \frac{1}{4!} \int _{\Omega} F _{\lambda \mu \nu \rho} \, d x ^{\lambda} \wedge d x ^{\mu} \wedge d x ^{\nu} \wedge d x ^{\rho}$$ q _n

$$\frac{1}{3!} \oint _{\Gamma} d x ^{\mu} \wedge d x ^{\nu} \wedge d x ^{\rho} A _{\mu \nu \rho}$$ =

$$\frac{1}{3! g} \oint _{\Gamma} d x ^{\mu} \wedge d x ^{\nu} \wedg... ...tial _{[ \mu} \theta _{\nu \rho ]} = \frac{2 \pi n}{g} \qquad n = 1 , 2 , \dots$$ = (23)

Thus, the physical consequence of the monodromy of $\Psi [S]$ is that the flux of $F _{\lambda \mu \nu \rho}$ through a region enclosing B is quantized in units of $2\pi/g$.
In the superconducting phase, equation (20) becomes

$$\partial _{\lambda} F ^{\lambda \mu \nu \rho} (x) = - \ph... ...a ^{\nu \rho ]} (x) \right) \equiv - j ^{\mu \nu \rho} (x)$$ (24)

in which we have introduced the supercurrent density $j ^{\mu \nu \rho} (x)$. Equation (24) holds only where $\theta ^{\nu \rho} (x)$ is a regular function. In the domain of singularity, where the partial derivatives of $\theta ^{\nu \rho} (x)$ do not commute, the covariant curl of $\partial ^{[ \, \mu} \theta ^{\nu \rho]} (x)$ should be interpreted in the sense of distribution theory. Indeed, if we apply the covariant curl operator to both sides of equation (24), we obtain

$$\partial ^{[ \, \lambda} j ^{\mu \nu \rho ]} = - \phi ^{2... ...[ \, \lambda} \partial ^{\mu} \theta ^{\nu \rho ]} \right)$$ (25)

The last term in (25) may not be disregarded without violating (7). Therefore in order to match (23) with (7), we define

 

$$\partial ^{[ \, \lambda} \partial ^{\mu} \theta ^{\nu \rho]} (x) \equiv \frac{q _{n} g}{\Omega _{B}} \epsilon ^{\lambda \mu \nu \rho} \int _{B} d ^{4} \xi \, \delta ^{4)} \left[ x - z( \xi ) \right]$$

$$\equiv \frac{q _{n} g}{\Omega _{B}} J _{B} ^{\lambda \mu \nu \rho} (x)$$ (26)

where, $\Omega _{B}$ and $J _{B} ^{\lambda \mu \nu \rho} (x)$ are, respectively, the bag four-volume and the bag current. Thus, the supercurrent can be determined from the equation

$$\partial _{\lambda} \partial ^{[ \, \lambda} j ^{\mu \nu \r... ... \partial _{\lambda} J _{B} ^{\lambda \mu \nu \rho} \right)$$ (27)

by means of the Green function method:

$$j ^{\mu \nu \rho} (x) = - \frac{\phi ^{2} q _{n}}{\Omega _... ...B} d ^{4} z \, \partial _{\sigma} \, G ( x - z ; \phi ^{2} )$$ (28)

where G ( x - z ) is the scalar Green function

$$\left[ \partial ^{2} + \phi ^{2} \right] G ( x - z ; \phi ^{2}) = \delta ^{4)} ( x - z ) .$$ (29)

Then, from equation (20), (25) and (28), we find the form of the confined gauge field

$$F ^{\lambda \mu \nu \rho} (x) = - \frac{\phi ^{2} q _{n}}{... ...\rho \sigma} \int _{B} d ^{4} z \, G ( x - z ; \phi ^{2}) .$$ (30)

The analogy between the membrane vacuum and a type-II superconductor now seems manifest: in the ordinary vacuum $A _{\mu \nu \rho}$ does not propagate any degree of freedom. Rather, it corresponds to a uniform energy background. However, in the superconducting phase $A _{\mu \nu \rho}$ becomes a dynamical field describing a massive, spin-0 particle [10]. The source for the massive field is the bag current (26). In a boson particle condensate, the magnetic field is confined to a thin flux tube surrounding the vortex line; in the membrane condensate, the $F _{\lambda \mu \nu \rho}$-field is confined within the membrane which encloses the ordinary vacuum bag. The gauge field provides the ``skin'' of the bag. To complete the analogy, in the next section we show that the thickness of the membrane is given by the inverse of the dynamically generated mass of $F _{\lambda \mu \nu \rho}$.