GAUGE-THEORY OF RELATIVISTIC MEMBRANES

A relativistic membrane is usually represented by the Dirac-Nambu-Goto action in terms of the extremal area of a three-dimensional timelike submanifold of Minkowski space. In this paper we show that a relativistic membrane admits an equivalent representation in terms of the Kalb-Ramond gauge field F(munurho) = partial derivative[(mu)B(nurho)] encountered in string theory. At first glance this is somewhat surprising, since the Kalb-Ramond field is usually interpreted as the spin-0 radiation field generated by a closed string. By 'equivalence' of the two representations we mean the following. If x = X(xi) is a solution of the classical equations of motion derived from the Dirac-Nambu-Goto action, then it is always possible to find a differential form of rank 3, satisfying Maxwell-type equations, such that, in a coordinate basis F(munurho)(X(xi)) = constant x X(numurho)/square-root -(1.3!)X(alphabetagamma)X(alphabetagamma) where X(munurho) represents the tangent 3-vector to the membrane world-track. The converse proposition is also true. In the first part of the paper, we show that a relativistic membrane, regarded as a mechanical system, admits a Hamilton-Jacobi formulation in which the H-J function describing a family of classical membrane histories is given by F = dB = dS1 AND dS2 AND dS3 where the three scalar functions S(i)(x) are the Clebsch potentials. In the second part of the paper, we introduce a new Lagrangian of the Kalb-Ramond type which provides a first-order formulation for both open and closed membranes. The advantage of the Lagrangian approach is that it shows explicitly the correspondence between the gauge formulation of the membrane in terms of the Kalb-Ramond potential and the geometric formulation of the membrane in terms of the mechanical coordinates X(mu)(xi). Finally, for completeness, we show that such a correspondence can be established in the very general case of a p-brane coupled to gravity in a spacetime of arbitrary dimensionality.