Holomorphic quantization of linear field theory in the general boundary formulation
A General Boundary Formulation of quantum field theory is a generalization of the, mathematically better known, notion of a topological field theory. It associates a Hilbert space of states to a space manifold, and a linear map to a space-time cobordism between two space like manifolds. It may require extra structure, and the results may depend on more than the topology, but it does not require a background metric. The paper begins by formulating a list of axioms for such a formulation. It then describes how a classical action functional on fields gives rise to a \(1\)-form on the space of germs of classical solutions restricted to space-like slices. From here R. Oeckl, describes the natural symplectic form on the space of these germs, and explains why the restriction of the space of solutions to an initial of final surface is a Lagrangian subspace. He then describes holomorphic quantization in the context of geometric quantization. Basically, the local solutions inherits a sesquilinear form constructed out of the natural symplectic form and a compatible almost complex structure. Completing this space gives a first (pre-quantum) Hilbert space. The state space is the Hilbert space of holomorphic functions on the first Hilbert sapce. The next section of the paper describes a theory of Gaussian integration on the projective limit of subspaces of the pre-quantum Hilbert space endowed with compatible measures. He continues with the definition of holomorphic functions on the projective limit leading to the construction of the hilbert space of states. The next part of the paper describes how to pass from the data of a classical system to a quantum theory satisfying the GBF axioms. It is interesting to note that his formula for the amplitude is a Gaussian integral (with exponential of a negative definite form -- not an imaginary form) over the space of classical solutions. This is similar to, but distinct from the Feynman path integral that one might expect. There are certainly interesting questions to consider related to this theory, both physically and mathematically.
scientific article; zbMATH DE number 6184277