[Paper Review] S-matrix in de Sitter spacetime from general boundary quantum field theory

Daniele Colosi|ArXiv.org|Oct 15, 2009

Black Holes and Theoretical Physics11 citations

TL;DR

This paper proposes a new quantization scheme for a massive scalar field in de Sitter spacetime using the general boundary formulation (GBF) of quantum field theory. It derives an S-matrix from spatial asymptotic states on a hypercylinder and proves its equivalence to the standard S-matrix via an isomorphism between asymptotic state spaces, validating the GBF approach in curved spacetime and offering a new framework for de Sitter scattering processes.

ABSTRACT

A new quantization scheme for a massive scalar field in de Sitter spacetime is proposed, based on the general boundary formulation of quantum field theory. We show that the general interacting theory can be consistently described in terms of the S-matrix for spatial asymptotic states. The new S-matrix results to be equivalent to the standard one in situations where both apply. This is due to the existence of an isomorphism between the corresponding asymptotic state spaces.

Motivation & Objective

• To develop a consistent quantization scheme for a massive scalar field in de Sitter spacetime using the general boundary formulation (GBF).
• To define and compute the S-matrix for interacting quantum fields in de Sitter spacetime using asymptotic coherent states on a spacelike hypercylinder.
• To establish equivalence between the GBF-derived S-matrix and the standard S-matrix in regimes where both apply.
• To extend the applicability of GBF beyond flat spacetime to curved spacetimes, particularly de Sitter space.
• To provide a new framework for analyzing scattering processes in inflationary cosmology and dS/CFT correspondence.

Proposed method

• Formalism uses GBF to associate Hilbert spaces to arbitrary spacelike hypersurfaces and amplitudes to spacetime regions bounded by these surfaces.
• Amplitudes are defined via a path integral over field configurations matching boundary data, combining Feynman's path integral with the Schrödinger representation.
• Coherent states are constructed on equal conformal time hypersurfaces using complex functions in momentum space, with time-dependent normalization and Hankel functions in the wave functional.
• The asymptotic S-matrix is derived by taking the limit of large spatial radius, ensuring independence from the cutoff radius R.
• An isomorphism is established between the state space of the standard S-matrix (on two time slices) and the GBF state space (on a hypercylinder), mapping coherent states via solutions to the Klein-Gordon equation.
• The equivalence of free and interacting amplitudes is shown by matching the Feynman propagator and boundary conditions in both formulations, with no residual dependence on the radius R.

Experimental results

Research questions

• RQ1 Can the general boundary formulation (GBF) consistently describe interacting quantum fields in de Sitter spacetime?
• RQ2 How does the S-matrix derived from GBF on a hypercylinder relate to the standard S-matrix defined on initial and final time slices?
• RQ3 What is the role of the isomorphism between asymptotic state spaces in ensuring equivalence between the two S-matrix formulations?
• RQ4 Does the GBF approach allow for a cutoff-independent derivation of the S-matrix in de Sitter spacetime?
• RQ5 Can the GBF framework be used to study scattering processes in curved spacetime, particularly in inflationary models?

Key findings

• The GBF-derived S-matrix for a massive scalar field in de Sitter spacetime is equivalent to the standard S-matrix when both are applicable, due to an isomorphism between their respective asymptotic state spaces.
• The free amplitude in the GBF formulation matches the standard free amplitude under the isomorphism, confirming consistency in the non-interacting limit.
• The interacting amplitude in the GBF framework is expressed via a functional integral over the source field, with no dependence on the spatial cutoff radius R, allowing the limit R→∞ to be taken trivially.
• The same Feynman propagator appears in both formulations, ensuring matching dynamics for the interaction term.
• The isomorphism between state spaces is defined by the equality of complex solutions to the Klein-Gordon equation, namely ˆξ = ˆη, which maps coherent states in the two formulations.
• The physical probabilities extracted from the S-matrices in both formulations are identical, confirming the physical equivalence of the two descriptions.