Fock Space: A Model of Linear Exponential Types

R.F. Blute, Prakash Panangaden, R.A.G. Seely

It has been observed by several people that, in certain contexts, the free symmetric algebra construction can provide a model of the linear modality !. This construction arose independently in quantum physics, where it is considered as a canonical model of quantum field theory. In this context, the construction is known as (bosonic) Fock space. Fock space is used to analyze such quantum phenomena as the annihilation and creation of particles. There is a strong intuitive connection to the principle of renewable resources, which is the philosophical interpretation of the linear modalities.
In this paper, we examine Fock space in several categories of vector spaces. We first consider vector spaces, where the Fock construction induces a model of the $ \otimes, &, ! $ fragment in the category of symmetric algebras. When considering Banach or Hilbert spaces, the Fock construction provides a model of weakening cotriple in the sense of Jacobs. While the models so obtained have weaker properties, it is closer in practice to the structures considered by physicists. In this case, Fock space has a natural interpretation as a space of holomorphic functions. This suggests that the ``nonlinear'' functions we arrive at via Fock space are not merely continuous but analytic.
Finally, we also consider fermionic Fock space, which corresponds algebraically to skew symmetric algebras. By considering fermionic Fock space on the category of finite-dimensional vector spaces, we obtain a model of full propositional linear logic, although the model is somewhat degenerate in that certain connectives are equated.