Supersymmetric Canonical Commutation Relations

Abstract

We discuss unitarily represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral, antichiral and vector fields. The canonical quantization shows some new facets which do not appear in the non supersymmetric case. Our tool is the supersymmetric positivity generating the Hilbert-Krein structure of the N = 1 superspace.

Supersymmetric Quantum Field Theory: Indefinite Metric

“…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]. There are several proofs of (7.6),(7.7), some of which were sketched in [1].…”

“…At the first glance canonical quantization in supersymmetry is hampered by the presence of so called auxiliary fields which seem to be non-quantizable because they are non-propagating fields. Based on the Krein-Hilbert structure it was possible to show that this is not the case at least at the level of canonical commutation relations [15].…”

“…It is related to the celebrated Källen-Lehmann representation. 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].…”

“…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.…”

“…The reader can solve it easily by going to the new variables θ = (θ 1 +θ 2 ),ζ =θ 1 −θ 2 as well as to x = x 1 − x 2 by translation invariance. The result is [15] …”