The quadratic Fock functor

We construct the quadratic analog of the boson Fock functor. While in the first order (linear) 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. The encouraging fact is that it contains, as proper subgroups (i.e., the contractions), all the gauge transformations of second kind and all the a. e. invertible maps of R(d) into itself leaving the Lebesgue measure quasi-invariant (in particular, all diffeomorphism of R(d)). This allows quadratic two-dimensional quantization of gauge theories, of representations of the Witt group (in fact it continuous analog), of the Zamolodchikov hierarchy, and much more. Within this semigroup we characterize the unitary and the isometric elements and we single out a class of natural contractions. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3294771]