3. The action

After the preparatory discussion of the previous section, we assign to the field $\Psi [S]$ the following action

 

$$- \frac{1}{2 \cdot 4!} \int d ^{4} x F ^{\lambda \mu \nu \rho}F _... ... \frac{1}{3!} \oint [DS] \langle \vert {\mathcal{D}}\Psi [S] \vert ^{2} \rangle$$ S =

$${\mathcal{D}}_{\mu \nu \rho} \Psi [S] \equiv \left( \frac{\delta}{\delta \sigma ^{\mu \nu \rho} (s)} - i g A _{\mu \nu \rho} \right) \Psi[S]$$ (8)

where, at present, $\oint [DS] \dots$ is a formal way to write the functional sum over equivalence classes of closed surfaces with respect to reparametrization invariance, and $F _{\lambda \mu \nu \rho}= \partial _{ [ \mu} A _{\nu \rho \sigma]}$ is the gauge field strength of the rank-three tensor gauge potential $A _{\mu \nu \rho}(x)$. The shorthand notation used in (8) is convenient but hides some essential features of the action functional which are worth discussing at this point. From our vantage point, the key property of the action (8) is its invariance under the extended gauge transformation

$$\Psi ' [S] \Psi [S] \exp \left( i \frac{g}{2} \oint _{S} d y ^{\mu} \wedge d y ^{\nu} \Lambda _{\mu \nu} (y) \right)$$ = (9)

$$A _{\mu \nu \rho}' A _{\mu \nu \rho} + \partial _{[ \mu} \Lambda _{\nu \rho]}$$ = (10)

This transformation consists of an ``ordinary'' gauge term for $A _{\mu \nu \rho}(x)$, which is defined over spacetime, and a non local term for the phase of the membrane functional $\Psi [S]$.
The second term in the action contains a spacetime integral which is not explicitly shown in the expression (8). The reason is that a free theory of surfaces is invariant under translations of the center of mass

$$x ^{\mu} = \big{\langle} x ^{\mu} (s) \big{\rangle}$$ (11)

so that the membrane action functional in (8) contains the spacetime four-volume as the corresponding zero-mode contribution. This translational invariance is broken by the coupling to an ``external'' field $A _{\mu \nu \rho}(x)$, in which case the four dimensional zero-mode integral is no longer trivial [8]. It can be factored out by inserting the ``unity operator'' [9]

$$\int d ^{4} x \, \delta ^{4)} \left[ { \textstyle \oint ... ...extstyle \oint _{S} d ^{2} s \sqrt{\gamma} } \right) = 1$$ (12)

into the functional integral. Then, we define the sum over surfaces as a sum over all the surfaces with the center of mass in x, and then we integrate over x:

$$\oint [D S] ( \dots ) \int d ^{4} x \oint [ D x ^{\mu} (s) ] \delta ^{4)} \left[ x - \langle x(s) \rangle \right] ( \dots ) \ ,$$ =

$$\equiv \int d ^{4} x \oint [ D S _{x} ] ( \dots )$$ (13)

All of the above applies to the quantum mechanical formulation of surfaces interpreted as geometric objects. On more physical grounds, membranes represent energy layers characterized by a typical thickness, say $1 / \Lambda$, which will be determined later on. To take into account the finite thickness of a physical membrane, the singular delta-function which corresponds to the ``thin film approximation'', has to be smeared into a regular function sharply peaked around $\langle x ^{\mu} (s) \rangle$. The simplest representation for such a function is given by a momentum space gaussian

$$\delta ^{4)} \left[ x - \langle x(s) \rangle \right] \right... ...angle x ^{\mu} (s) \rangle) - k ^{2} / ( 2 \Lambda) } } .$$ (14)

However, as long as we work at a distance scale much larger than the membrane transverse dimension, we can approximate the physical extended object with a geometrical surface. In what follows we shall refer to the regularized delta-function only when it is strictly necessary. With the above prescriptions, the action (8) can be written as the spacetime integral of a lagrangian density

$${\mathcal{S}} [ \Psi ^{*} , \Psi ; A _{\mu \nu \rho}] = \in... ...ert {\mathcal{D}} \Psi [S] \Big \vert ^{2} \rangle \right\}$$ (15)

and the interaction between the membrane field current and the $A _{\mu \nu \rho}$ potential is described by

 

$${\mathcal{S}} _{\mathrm{int.}} [ S ; A _{\mu \nu \rho}] \frac{g}{2 \cdot 3!} \int d ^{4} x \, \oint [ D S _{x} ] \, \left... ...!\!\! \strut \longleftrightarrow} \Psi [S] \rangle \right] A ^{\mu \nu \rho}(x)$$ =

$$\frac{g}{2 \cdot 3!} \int d ^{4} x \oint [ D S _{x} ] \cdot$$ =

$$\qquad \cdot \left[ \langle i \Psi ^{*} [S] \frac{{\strut \longle... ...vert \Psi [S] \vert ^{2} A ^{\mu \nu \rho} \rangle \right] A _{\mu \nu \rho}(x)$$

$$\equiv \frac{g}{3!} \int d ^{4} x \, J ^{\mu \nu \rho} (x) A _{\mu \nu \rho}(x)$$ (16)

where g is the gauge coupling constant of dimension two in energy units. As a classical ``charge'', it describes the strength of the interaction among volume elements of the world-tube swept in spacetime by the membrane evolution. In our functional field theory g enters as the interaction constant between the membrane field current and the gauge potential. Equation (16) exhibits a characteristic London form which alerts us about the occurrence of non-trivial vacuum phases. Indeed, the current implicitly defined in the last step in equation (16), can be rewritten in the following form

 

$$J ^{\mu \nu \rho} (x) \oint [ D S _{x} ] \langle i \Psi ^{*} [S] \frac{{\strut \longlef... ...S] \rangle + g \oint [ D S _{x} ] \vert \Psi [S] \vert ^{2} A ^{\mu\nu\rho} (x)$$ =

$$\equiv I ^{\mu \nu \rho} (x) + \frac{\varphi ^{2} (x)}{g} A ^{\mu \nu \rho}(x)$$ (17)

where we have introduced the scalar field $\varphi (x)$

$$\varphi ^{2} (x) \equiv g ^{2} \oint [ D x ^{\mu} (s) ] \... ...le ) - k ^{2} / (2 \Lambda) } } \vert \Psi[S] \vert ^{2}$$ (18)

which we interpret as the order parameter associated with membrane condensation in the same way that the Higgs field is the order parameter associated with the boson condensation of point-like objects. In the ordinary vacuum $\varphi (x) = 0$, i.e. there are no centers of mass, and therefore no membranes. Alternatively, we define a vacuum characterized by a constant ``density of centers of mass'', $\varphi(x) = \mathrm{const.} \ne 0$, as a membrane condensate.
We will show in Section 4 that the membrane condensate acts as a superconductor upon the gauge potential, turning $A _{\mu \nu \rho}(x)$ into a massive scalar field. The problem of surface condensation is thus reduced to studying the distribution of their representative, pointlike, centers of mass. Conversely, we show in Section 5 that membrane condensation can be driven by the $A _{\mu \nu \rho}(x)$-field quantum corrections alone, and is accounted for by an effective potential ascribed to the extended object. This is Umezawa's self-consistency condition transplanted in our own formalism.