2. The formalism

The picture of membrane superconductivity, as opposed to vortex superconductivity, can be visualized as ``islands'' of normal vacuum surrounded by a ``sea'' of massive $A _{\mu \nu \rho}(x)$-quanta. In general, the emergence of different vacuum phases in the ground state of a physical system is accompanied by the formation of boundary layers between the various vacuum domains. In our approach, these boundaries are approximated by geometrical manifolds of various dimensionality (p-branes). This is the point where we depart from Umezawa's approach: p-branes are introduced at the outset with their own action functional, and therefore possess their own dynamics independently of an underlying local field theory. Classical bubble-dynamics has been studied in detail [4], and this paper represents a tentative step toward the quantum formulation. The paradigm of the quantum approach is a line field theory introduced several years ago by Marshall and Ramond as a basis for a second quantized formulation of closed string electrodynamics [5]. We are interested in the case of relativistic, spatially closed membranes whose history in spacetime is represented by infinitely thin (1+2)-dimensional Lorentzian submanifolds of Minkowski space (M). In a first quantized approach, membrane coordinates and momenta become operators acting over an appropriate space of states. However, the non linearity of the theory and the invariance under reparametrizations introduce severe problems in the first quantized formulation, e.g. operator anomalies in the algebra of constraints. At least for closed membranes, one can bypass these difficulties by considering a field theory of geometric surfaces [6]. If we consider the abstract space F of all possible bubble configurations, then we are led to consider a field theory of quantum membranes in which the membrane field is a reparametrization invariant, complex, functional of the two-surface S which we assume to be the only boundary of the membrane history. Our objective, then, is to introduce and discuss the action which governs the evolution of quantum membranes regarded as 3-dimensional timelike submanifolds of Minkowski space. To this end, our first step is to introduce the 3-volume derivative $\delta / \delta \sigma ^{\mu \nu \rho} (s)$, which extends the notion of ``loop derivative'' introduced, some years ago, in the framework of the loop formulation of gauge theories [7]. The underlying idea is this: suppose we attach at a given point s of the surface S, an infinitesimal, closed surface $\delta S$. This procedure is equivalent to a deformation of the initial shape of S in the neighborhood of s, thereby changing the enclosed volume by an infinitesimal amount $\delta V$. Then, we define the volume derivative of $\Psi [S]$ through the relation

$$\delta \Psi [S] \equiv \Psi [S \oplus \delta S] - \Psi [S... ...rho} (s)} \, d x ^{\mu} \wedge d x ^{\nu} \wedge d x ^{\rho}$$ (1)

in the limit of vanishing $\delta V$. This definition is ``local'' to the extent that it involves a single point on the surface. For the whole S , an averaging procedure is required

$$\langle \dots \rangle \equiv \left( \oint _{S} d ^{2} s \... ...{-1} \oint _{S} d ^{2} s \sqrt{\gamma} \left( \dots \right)$$ (2)

where, $\gamma = x _{\nu \rho}x ^{\nu \rho}$ is the determinant of the metric induced over S by the embedding $y ^{\mu} = x ^{\mu} ( s ^{i} )$ and $x ^{\nu \rho}= \partial ( x ^{\nu} , x ^{\rho} ) / \partial ( s ^{1} , s ^{2})$ is the surface tangent bi-vector. The volume derivative is related to the more familiar functional variation $\delta / \delta x ^{\mu} (s)$ by the relation

$$\frac{\delta}{\delta x ^{\mu} (s)} = \frac{1}{2} \, x ^{\nu \rho} \frac{\delta}{\delta \sigma ^{\mu \nu \rho} (s)} .$$ (3)

Our second step towards the formulation of the membrane wave equation, is to introduce the concept of monodromy for the $\Psi [S]$ field, since this notion is directly linked to the physical interpretation of the membrane field. Our requirement is that $\Psi [S] \equiv A [S] e ^{i \Theta [S]}$ be a single valued functional of S, i.e. the phase

$$\Theta [S] \equiv \frac{1}{2} \oint _{S} d x ^{\mu} \wedge d x ^{\nu} \theta _{\mu \nu} (x)$$ (4)

can vary only by $2 \pi n$, n=1,2,..., under transport along a ``loop'' in surface space. This condition constitutes the basis of the analogy with a type-II superconductor. In order to illustrate the precise meaning of this analogy, it is convenient to interpret the motion of a bubble in the abstract space F in which each point corresponds to a possible bubble configuration. Then, the 3-volume derivative introduced above represents the spacetime image of the generator of translations in F-space and ``classical motion'' in F-space corresponds to a continuous surface deformation in Minkowski space. With this understanding, we define a ``line'' in F-space as a one-parameter family of ``points'', i.e., surface configurations $\{ S ; t \}$ in physical space. Let each surface in the family be represented by the embedding equation $x ^{\mu} = x ^{\mu} ( s ^{1} , s ^{2} , t = \mathrm{const.} )$, where t is the real parameter labelling in a one-to-one way each surface of the family, so that $x ^{\mu} = x ^{\mu} ( s ^{1} , s ^{2} ; t)$ represents the embedding of the whole family. However, the same relation can be interpreted as the embedding of a single three-surface whose t =const. sections reproduce each surface of the family. In a similar way, we define a ``loop'' of surfaces as a one-parameter family of surfaces in which the first and the last are identified. Then, according to our definition of volume derivative as the spacetime image of the translation generator in surface space, we define the circulation of $\Theta [S]$ as the flux of the covariant curl of $\theta _{\mu \nu} (x)$ :

 

$$\Delta \Theta [S] \equiv \frac{1}{3!} \oint _{\Gamma} d x ^{\mu} \wedge d x ^{\nu} \wedge d x ^{\rho} \frac{\delta \Theta [S]}{\delta \sigma ^{\mu \nu \rho} (s)}$$

$$\frac{1}{3!} \oint _{\Gamma} d x ^{\mu} \wedge d x ^{\nu} \wedge d x ^{\rho} \partial _{[ \mu} \theta _{\nu \rho ]} (x)$$ = (5)

where $\Gamma : x ^{\mu} = x ^{\mu} ( s ^{1} , s ^{2} ; t)$, with $\partial \Gamma = \emptyset$ represents the spacetime image of the integration path in surface space.
Finally, we define a vortex line in surface space, as a one-parameter family of surfaces $\{ V ; t \}$ for which the amplitude of the membrane field vanishes, i.e. $\vert \Psi [ V _{t} ] \vert = 0$. In order to avoid boundary terms and thus simplify calculations, we assume that the vortex line is closed. In other words, the spacetime image of the vortex line is a compact three surface without boundary that we shall denote by $\partial B$.
Suppose the test loop of surfaces $\Gamma \equiv \partial \Omega$ surrounds the vortex line $\partial B$, then the monodromy of $\Psi [S]$ implies the quantization condition :

$$\Delta \Theta [S] = \frac{1}{3!} \oint _{\Gamma} d x ^{\mu... ...\theta _{\nu \rho ]} = 2 \pi n \ , \quad n = 1 , 2 , \dots$$ (6)

If the flux (6) is quantized, then $\theta$ is a singular function within B. Indeed, using Stokes' theorem, we rewrite (6) as

 

$$\frac{1}{4!} \int _{\Omega} \! \! \! d x ^{\mu} \wedge d x ^{\nu}... ...} \wedge d x ^{\sigma} \partial _{[ \mu} \partial _{\nu} \theta _{\rho \sigma]} \frac{1}{4!} \int _{B} \! \! \! d x ^{\mu} \wedge d x ^{\nu} \wed... ...} \wedge d x ^{\sigma} \partial _{[ \mu} \partial _{\nu} \theta _{\rho \sigma]}$$ =

$$2 \pi n \ne 0.$$ = (7)

The ``Bag'' B is the domain of singularity of the phase 2-form $\theta _{\mu \nu} (x)$ and represents the spacetime image of the ``vortex interior''.