Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation

Abstract

Abstract. We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein-Gordon theory.

In–out propagator in de Sitter space from general boundary quantum field theory

“…Two quantization prescriptions have been so far developed for the GBF: The Schrödinger-Feynman one, where path integral quantization is combined with the Schrödinger representation for quantum states, and the holomorphic quantization [18] inspired by geometric quantization techniques. The quantizations were shown to be equivalent [19].…”

The Vacuum as a Lagrangian subspace

“…An equivalent construction of the Fock space H starts from the space L, viewed as a complex inner product space with the inner product (9). Thus, the n-particle space is then a symmetrized n-fold tensor product of copies of L. H is the completed direct sum of all these n-particle spaces with n ranging from 0 to infinity.…”

“…This includes Proposition B.11, which is instrumental in ensuring well-definedness and uniqueness in the application of formula (37) for vacuum expectation values. In Appendix C an axiomatization of our notion of generalized vacuum is presented, generalizing the axiomatic framework [9] that formalizes the mentioned Lagrangian approach of Kijowski and Tulczyjew [8] in the linear case.…”

Quantum Abelian Yang-Mills Theory on Riemannian Manifolds with Boundary

“…Correspondingly, we shall rely heavily on the results presented in [15]. These in turn were obtained by recurrence to the linear theory and the functor presented in [16]. As we will also make use of this recurrence here, we start by considering the linear theory.…”

“…This strategy was first proposed and carried through successfully in [16]. There, the simplest case of linear field theory without gauge symmetries was considered.…”