A superspace embedding of the Wess–Zumino model

Abstract

We embed the Wess-Zumino (WZ) model in a wider superspace than the one described by chiral and anti-chiral superfields.1. There is a systematic and interesting formalism for embedding, developed by Batalin, Fradkin, Fradkina, and Tyutin (BFFT) [1], where theories with second-class constraints [2] are transformed into more general (gauge) theories where all constraints become first-class. The transformation of constraints from second to first-class is achieved after extending the phase space by means of auxiliary variables under the general rule that there is one pair of canonical variables for each second class constraint. The method is iterative and can stop in the first step [3] or can go on indefinitely [4,5]. In any case, after all constraints have been transformed into first-class, it is necessary to look for the Hamiltonian corresponding to this new theory. The method also permit us to obtain any involutive quantity that has zero Poisson brackets with all the constraints. The embedding Hamiltonian can be obtained in this way, starting from the initial canonical Hamiltonian and iteratively calculating the corresponding corrections.There is another manner to obtain an embedding Hamiltonian, which consists in using the BFFT method to obtain involutive coordinates [4]. The canonical Hamiltonian is then rewritten in terms of these new coordinates that automatically give it the involutive condition. Of course, the embedding Hamiltonians obtained from these two different ways are not necessarily equal. This means that for some specific theory there may exist more than one possible embeddings. It is also opportune to say, on the other hand, that there are theories which cannot be embedded [6].One of the interesting problem that the BFFT method could be addressed is the covariant quantization of superparticles and superstrings, that remained opened for a long time. This problem has been solved in a embedding procedure but differently of the BFFT method [7]. In fact, the meaning of embedding in field theory can be taken as much wider than the cases described by the BFFT method. The important point is that the embedding theory contains all the physics of the embedded one.We mention, for example, even the general procedure of supersymmetrization is an example of embedding.We would like to address the present paper to this point of view of considering the embedding procedure in a wider way. We concentrate on the WZ model [8] in superfield language [9,10]. Conventionally, the WZ model is always developed in terms of chiral and antichiral superfields, that are examples of irreducible superfields. We shall consider here a kind of embedding where we describe the WZ model by using a more general superfield representation. We shall see that, contrarily to the bosonic nature of the chiral and antichiral superfields, the general superfield we have to use is fermionic. We shall also see that there are two possible terms that can figure in the Lagrangian and having a relative parameter between them. The consistency of th...