Quantum fields on projective geometries

TL;DR

This work develops an axiomatic framework for projective quantum fields on homogeneous four-dimensional space-time geometries realized as subgeometries of the ambient real projective space $(\mathbb{RP}^4,\mathrm{PGL}_5\mathbb{R})$. It shows that projective correlators and their expectation values remain well-defined under geometry deformations and limits, with possible dimensional reduction to lower-dimensional boundaries, and that ultraviolet and infrared limits can be analyzed within a single ambient setting. By classifying superselection sectors via multipliers of the ambient projective representation, it identifies Dirac fermions as the spin-statistics-consistent irreducible case and constructs composite and Schur-irreducible projective fields, providing a projective-extension of ambient-space techniques familiar from AdS/CFT. The framework unifies geometric deformations, limit procedures, and representation-theoretic constructions, offering a principled route to explore holographic correspondences, renormalization behavior, and the role of ambient geometry in QFT beyond traditional fixed backgrounds.

Abstract

Considering homogeneous four-dimensional space-time geometries within real projective geometry provides a mathematically well-defined framework to discuss their deformations and limits without the appearance of coordinate singularities. On Lie algebra level the related conjugacy limits act isomorphically to concatenations of contractions. We axiomatically introduce projective quantum fields on homogeneous space-time geometries, based on correspondingly generalized unitary transformation behavior and projectivization of the field operators. Projective correlators and their expectation values remain well-defined in all geometry limits, which includes their ultraviolet and infrared limits. They can degenerate with support on space-time boundaries and other lower-dimensional space-time subspaces. We explore fermionic and bosonic superselection sectors as well as the irreducibility of projective quantum fields. Dirac fermions appear, which obey spin-statistics as composite quantum fields. The framework systematically formalizes and generalizes the ambient space techniques regularly employed in conformal field theory.

Figures (1)

• Figure 1: Limiting projective correlator algebras for a selection of geometry limits of projective de Sitter geometry $\mathsf{G}(4,1)$ and projective anti-de Sitter geometry $\mathsf{G}(3,2)$. Convergence of algebras is defined as element-wise convergence. $\mathfrak{A}_\textrm{deg}$ denotes an algebra of projective correlators, which are degenerate with respect to their field operator components. The rows starting with representations $\rho=\ldots$ denote the non-trivial dependence on field operator components of the degenerate projective correlator algebra. $R(\mathbb{CP}^4_{\mathrm{PGL}_5\mathbb{R}})$ denotes the action of $\overline{\mathrm{PGL}_5\mathbb{R}}$ on $\mathbb{CP}^4$ by applying the cover projection $\overline{\mathrm{PGL}_5\mathbb{R}}\to\mathrm{PGL}_5\mathbb{R}$ and inverse right multiplication. The space $i^0_p:=\{[0,0,x_2,x_3,x_4]\in \mathbb{RP}^4\}\cong \mathbb{RP}^2$ is projective spatial infinity, $i^+_p:=\{[x_0,x_1,0,0,0]\in \mathbb{RP}^4\,|\, x_0\neq 0\}= \mathbb{R}$ is projective time-like infinity and $i_p^0 \vee i^+_p\setminus [0]$ indicates their wedge product at $[0]$ with $[0]$ removed.

Theorems & Definitions (75)

• Definition : Shortened \ref{['DefFieldOperator']}
• Theorem : Shortened \ref{['ThmCorrelatorDegenerationInLimits']}
• Theorem : \ref{['ThmSpinStatIrredPoincareIrred']}
• Example 2.1
• Remark
• Example 2.2
• Example 2.3
• Lemma 2.4
• Theorem 2.5
• Remark