On Duality in the Born-Infeld Theory

The duality symmetric action for the Born-Infeld theory in terms of two potentials, coupled with non-trivial backgroud fields in four dimensions is established. This construction is carried out in detail by analysing the hamiltonian structure of the Born-Infeld theory. It is shown the equivalence with the usual Born-Infeld theory. Nowadays the concept of duality is widely recognized by its unifying role in physics. The five known different superstring theories are now unified by duality in the framework of M Theory (see for example [1]). The simplest case where duality appears are the Maxwell’s equations without sources, interchanging the equations of motions and the Bianchi identities. Schwarz and Sen [2] have developed a method to raise duality symmetry at the level of the action but at the price of losing the explicit Lorentz invariance. The classical and quantum equivalence with electromagnetism has been well established [3], [4], [5]. Earlier, Deser and Teitelboin [6] noticed that the Maxwell theory in its hamiltonian formultation is invariant under non-local dualty transformations. Moreover, several attempts have been made at conciliating duality symmetry with Lorentz invariance [7]. On the other hand, the Born-Infeld theory [8], initially conceived as an alternative for electromagnetism, has recently received considerable attention because the world volume action of a D-brane is described by a kind of non-linear Born-Infeld action [9]. Several aspects of the duality symmetry in the Born-Infeld theory have been developed recently [10], [11]. In particular, Perry and Schwarz [12] proposed a non-manifestly Lorentz invariant Born-Infeld action for a self-interacting self-dual antisymmetric tensor field in D = 6. Afterward, Pasti, Sorokin and Tonin presented a manifestly covariant formulation of this action [13], from which Berman [14], after dimensional reduction to four dimensions and