Extended supersymmetries in the causal approach

The scattering matrix is a formal series of operator valued distributions T(W1(x1), ... , Wn(xn)) which act in the Fock space of some collection of free fields; they are called chronological products and verify Bogoliubov axioms expressing the following properties: the initial condition, skew-symmetry in all arguments, Poincare invariance, causality and unitarity. The expressions Wj are arbitrary Wick monomials. The existence of solutions is rigorously established using a recursive procedure based on the causality axiom. Sometimes it is possible to supplement these axioms by other invariance properties. In this way supersymmetric invariance can be treated also in this formalism. A quantum extended supersymmetric theory is a collection of quantum relativistic free fields bj (fA, respectively) which are bosonic (fermionic, respectively) acting in the Hilbert space H together with the operators Qaj, Zjk (here a = 1, 2, j = 1, ... , N). We have denoted generically by b (f, respectively) arbitrary linear combinations of the Bose (Fermi, respectively) fields and their partial derivatives. These relations express the tensor properties of the fields with respect to (infinitesimal) supersymmetry transformations and the definition of a supersymmetric algebra with central charges Zjk. By definition, they commute with all other SUSY generators and U → ρ (U) is a N-dimensional representation of the group G. We will consider only the case ρ = Id. If these conditions are true we say that Qaj are super-charges and bj, fA are forming a quantum (extended) supersymmetric multiplet. Then supersymmetric invariance of the model can be expressed in an infinitesimal form analogue with gauge invariance for some chronological products T(X), Ta;lμ(X). One can establish by a direct analysis that the chronological products can be normalized such that these identities are true in all orders for toy models as it is Wess-Zumino model. One can introduce superfields for extended supersymmetric theories as in the more simple case of N = 1 (without central charges) and one can generalize the Bogoliubov axioms in the supersymmetric context. From the analysis of the irreducible representations of the supersymmetric algebra it is known that there are only five multiplets describing irreducible representations with particles of spin s≤1 namely: for N = 1 the chiral multiplet and the vector multiplet; for N = 2 the hyper-multiplet and a vector multiplet; for N = 4 a vector multiplet. The starting point of the analysis is not trivial. The construction of the corresponding quantum multiplet with extended supersymmetry is a non-trivial task because of the condition of positivity of the scalar product. This condition is not imposed in the standard literature where it is assumed that it follows automatically from the path-integral quantization procedure. (author)