Supersymmetric distributions, Hilbert spaces of supersymmetric functions and quantum fields

Abstract

The recently investigated Hilbert-Krein and other positivity structures of the superspace are considered in the framework of superdistributions. These tools are applied to problems raised by the rigorous supersymmetric quantum field theory.

Supersymmetric Quantum Field Theory: Indefinite Metric

“…For precise definitions see Section 10 below. The quest of an indefinite metric inducing the physical Hilbert space in supersymmetry was asked and answered affirmatively in [1]. Recognizing the Hilbert space of supersymmetry as being generated by the indefinite metric may have applications to rigorous supersymmetric quantum field theory outside path integrals which includes supersymmetric canonical quantization [15].…”

“…Recognizing the Hilbert space of supersymmetry as being generated by the indefinite metric may have applications to rigorous supersymmetric quantum field theory outside path integrals which includes supersymmetric canonical quantization [15]. There are several proofs of (7.6),(7.7), some of which were sketched in [1]. In this paper we provide a simple proof which gives not too much insight into the matter and a second one, computationally more involved, worked out in every detail, which provides much more information then the first proof.…”

“…The subject was already touched in [15] but some terms in the representation were missed. Suppose that general principles of quantum field theory defined in Hilbert space [8] survive in the supersymmetric setting up [16,1,15]. Then the two point function W (z 1 , z 2 ) of a scalar neutral (or even complex) quantum field must satisfy the following requirements: i) it must be a superdistribution (i.e.…”

“…It gives rise by standard methods to an invariant Hilbert space realized on supersymmetric functions (supersymmetric Hilbert space). The pair of these two spaces was called in [1] the Krein-Hilbert (or Hilbert-Krein) structure of supersymmetry. In this paper we present several detailed simple proofs of this assertion together with the first steps towards the study of quantum supersymmetric fields outside path integral methods.…”

“…It was claimed [1] that the N = 1 superspace in four space-time dimensions hides an inherent indefinite metric which can be realized as a Krein space. It gives rise by standard methods to an invariant Hilbert space realized on supersymmetric functions (supersymmetric Hilbert space).…”