Affine holomorphic quantization

Abstract

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin-Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S-matrix.

42 pages, LaTeX + AMS; v2: expanded to improve readability, new sections 3.1 (geometric data) and 3.3 (core axioms), minor corrections, update of references; v3: further update of references

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Cited by in corpus (22)

• The Spin Foam Approach to Quantum Gravity
• Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation
• The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory
• General boundary quantum field theory in de Sitter spacetime
• Free Fermi and Bose Fields in TQFT and GBF
• The new spin foam models and quantum gravity
• The Unruh Effect in General Boundary Quantum Field Theory
• Locality and General Vacua in Quantum Field Theory
• The vacuum as a Lagrangian subspace
• Quantum field theory on timelike hypersurfaces in Rindler space
• Classical Klein-Gordon solutions, symplectic structures and isometry actions on AdS spacetimes
• Multisymplectic effective General Boundary Field Theory
• Complex structures for an S-matrix of Klein-Gordon theory on AdS spacetimes
• Quantum Abelian Yang-Mills Theory on Riemannian Manifolds with Boundary
• Towards state locality in quantum field theory: free fermions
• General Boundary Formulation for $n$-Dimensional Classical Abelian Theory with Corners
• Dirichlet to Neumann operator for abelian Yang-Mills gauge fields
• Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution
• The Unruh-deWitt Detector and the Vacuum in the General Boundary formalism
• Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part I: Gaussian analysis and other Mathematical aspects
• Complex structures and quantum representations for scalar QFT in curved spacetimes
• General Boundary Quantum Field Theory in Anti de Sitter Spacetimes