The quadratic Fock functor

TL;DR

This paper introduces a quadratic analogue of the boson Fock functor, exploring the limitations of second quantization in the quadratic case and characterizing its unitary and isometric elements.

Contribution

It constructs the quadratic Fock functor and characterizes the semigroup of contractions that admit quadratic second quantization, highlighting differences from the linear case.

Findings

The quadratic Fock functor is successfully constructed.

The semigroup of contractions allowing quadratic second quantization is characterized.

Unitary and isometric elements within this semigroup are identified.

Abstract

We construct the quadratic analogue of the boson Fock functor. While in the first order case all contractions on the 1--particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. Within this semigroup we characterize the unitary and the isometric elements.