[Paper Review] Mathematical definition of quantum field theory on a manifold

A. V. Stoyanovsky|ArXiv.org|Oct 13, 2009

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper presents a rigorous mathematical definition of quantum field theory (QFT) on a pseudo-Riemannian manifold, grounded in canonical commutation relations and covariant Hamiltonian dynamics. It establishes a framework where Hilbert spaces are assigned to space-like hypersurfaces, transition amplitudes are defined via manifolds with boundary, and quantization is realized through a flat connection that reproduces classical Poisson brackets in the h→0 limit, ensuring consistency with standard QFT in flat space.

ABSTRACT

We give a mathematical definition of quantum field theory on a manifold, and definition of quantization of a classical field theory given by a variational principle.

Motivation & Objective

• To provide a mathematically rigorous definition of quantum field theory on a general pseudo-Riemannian manifold, extending standard QFT in Minkowski space.
• To formalize the quantization of classical field theories defined by a variational principle, ensuring compatibility with canonical commutation relations.
• To ensure consistency with the dynamical evolution of quantum states across space-like surfaces via a flat, integrable connection on the Hilbert bundle.
• To incorporate sources and Green's functions through a modified flat connection, enabling perturbative calculations.

Proposed method

• Assigns a Hilbert space H_C to each co-oriented, closed, connected space-like hypersurface C in the manifold M.
• Constructs tensor products of Hilbert spaces for disconnected surfaces, using bounded tensor products and identifying H̄_C ⊗ H_C' with bounded operators between spaces.
• Defines transition amplitudes Ψ_N ∈ H_C for manifolds N with boundary ∂N = C, ensuring consistency under gluing of manifolds.
• Imposes that the Hilbert spaces H_C are irreducible unitary representations of canonical commutation relations, ensuring quantum structure.
• Derives a flat connection ∇₁ on End(H_C) that, in the classical limit h→0, reproduces the covariant Hamiltonian evolution via Poisson brackets.
• Introduces a source-dependent flat connection ∇_j = ∇ + (1/ih)∫_C j(x(s)) φ̂(s) ds, preserving flatness for Green's function construction.

Experimental results

Research questions

• RQ1 How can quantum field theory be rigorously defined on a general pseudo-Riemannian manifold, independent of a fixed background spacetime?
• RQ2 What conditions ensure that the Hilbert space structure on space-like surfaces is consistent with canonical quantization of classical field theories?
• RQ3 How can the dynamical evolution of quantum states across different spatial slices be encoded in a geometric and covariant way?
• RQ4 In what sense does the proposed connection reproduce the classical Poisson bracket structure in the h→0 limit?
• RQ5 How can sources be consistently coupled to the theory while preserving the flatness of the connection necessary for Green's functions?

Key findings

• The theory assigns unitary Hilbert spaces to space-like hypersurfaces, with tensor product structure for disconnected components, ensuring consistency under topological gluing.
• The transition amplitude Ψ_N associated with a manifold N with boundary C is an element of H_C, and its composition under gluing reproduces the expected unitary evolution.
• The flat connection ∇₁ on End(H_C) reproduces the classical Hamiltonian evolution up to O(h), with the quantum correction term involving the commutator with the Hamiltonian density.
• The connection ∇₁ reduces to the classical Poisson bracket structure in the h→0 limit, confirming the correspondence principle.
• The inclusion of a source term via ∇_j preserves flatness, enabling the construction of time-ordered correlation functions as in standard perturbative QFT.
• The framework ensures G-invariance of all structures, making it suitable for general relativity-compatible QFT formulations.