[Paper Review] The ${\mathscr P}(\varphi)_2$ Model on the de Sitter Space

J. C. A. Barata, Christian D. Jäkel+1 more|arXiv (Cornell University)|Nov 12, 2013

Black Holes and Theoretical Physics118 references5 citations

TL;DR

This paper extends the ${\mathscr P}(\varphi)_2$ model to two-dimensional de Sitter space by constructing interacting quantum fields that satisfy the Haag-Kastler and time-slice axioms, demonstrating the absence of infrared divergences for $m > -\frac{1}{2r^2}$, and showing that the interacting theory's boost and rotation generators arise from the stress-energy tensor via Killing vector fields. The work establishes unitary representations of $SO_0(1,2)$ on the Cauchy surface and confirms the validity of the Osterwalder-Schrader reconstruction on the Euclidean sphere.

ABSTRACT

In 1975 by Figari, H{\\o}egh-Krohn and Nappi constructed the ${\\mathscr P}(\\varphi)_2$ model on the two-dimensional de Sitter space. Here we complement their work with a number of new results.

Motivation & Objective

• To extend the ${\mathscr P}(\varphi)_2$ model to two-dimensional de Sitter space, completing the work initiated by Figari, Høegh-Krohn, and Nappi.
• To establish the existence of unitary irreducible representations of $SO_0(1,2)$ on the Hilbert space of wave functions supported on the Cauchy surface.
• To demonstrate that physical infrared problems are absent for $m > -\frac{1}{2r^2}$ in the de Sitter context.
• To verify that the interacting quantum fields satisfy the equations of motion in covariant form and that the Haag-Kastler and time-slice axioms hold.
• To show that the generators of boosts and rotations in the interacting theory arise from the stress-energy tensor contracted with Killing vector fields.

Proposed method

• Formulating unitary irreducible representations of $SO_0(1,2)$ on the Cauchy surface using the Fourier-Helgason transform.
• Analyzing the causality structure of de Sitter space and reviewing the representation theory of $O(1,2)$.
• Constructing the free classical dynamical system in both covariant and canonical forms, and introducing one-particle KMS structures.
• Adding the ${\mathscr P}(\varphi)_2$ interaction on the Euclidean sphere and performing the Osterwalder-Schrader reconstruction in detail.
• Verifying that local von Neumann algebras for double cones with bases on a fixed space-like geodesic are identical in the free and interacting theories.
• Deriving the interacting theory's boost and rotation generators by integrating the stress-energy tensor contracted with Killing vector fields over line segments.

Experimental results

Research questions

• RQ1 Can unitary irreducible representations of $SO_0(1,2)$ be formulated on the Cauchy surface in the ${\mathscr P}(\varphi)_2$ model on de Sitter space?
• RQ2 Are physical infrared divergences absent in the ${\mathscr P}(\varphi)_2$ model on de Sitter space when $m > -\frac{1}{2r^2}$?
• RQ3 Do the interacting quantum fields satisfy the equations of motion in their covariant form on de Sitter space?
• RQ4 Do the Haag-Kastler and time-slice axioms hold for the ${\mathscr P}(\varphi)_2$ model on de Sitter space?
• RQ5 Do the generators of the Lorentz group in the interacting theory arise from the stress-energy tensor via Killing vector fields?

Key findings

• Unitary irreducible representations of $SO_0(1,2)$ for both the principal and complementary series are realized on the Hilbert space of wave functions supported on the Cauchy surface.
• Physical infrared problems are absent on de Sitter space for $m > -\frac{1}{2r^2}$, with ultraviolet problems resolved analogously to the Minkowski case.
• The interacting quantum fields satisfy the equations of motion in their covariant form, ensuring relativistic invariance.
• The Haag-Kastler and time-slice axioms are satisfied, and the local von Neumann algebras for double cones with bases on a fixed space-like geodesic are identical in the free and interacting theories.
• The generators of boosts and rotations in the interacting theory arise from the stress-energy tensor contracted with Killing vector fields and integrated over line segments, forming a reducible unitary representation of the Lorentz group on the free Fock space.
• The Osterwalder-Schrader reconstruction is carried out in detail on the Euclidean sphere, confirming the Euclidean path integral approach for the model.