Massless Representations in Conformal Space and Their de Sitter (dS) Restrictions

TL;DR

This work develops a mathematically rigorous, Clifford-algebra–based framework for massless conformal representations of the unitary group U(2,2) and their restriction to de Sitter symmetry Sp(2,2). Central to the approach is a split-octonion realization of an 8-component Majorana spinor representation of the conformal Clifford algebra cl(4,2), from which conformal and de Sitter structures are unified inside a single algebraic setting. The authors construct positive-energy ladder representations of u(2,2), classify them by helicity λ, compute invariant bilinear forms and Casimir operators, and demonstrate irreducibility upon restriction to the de Sitter subalgebra so(4,1). They provide explicit vertex-operator constructions and two-point functions for low-helicity fields and develop a 6D conformal–4D de Sitter analytic framework, culminating in a conformal vertex algebra perspective for massless fields in both flat and curved spacetimes. The treatment links foundational representation theory, explicit operator realizations, and cosmological applications, offering a transparent, computationally explicit toolkit with potential implications for higher-spin QFT, holography, and early-universe cosmology.

Abstract

The monograph develops a self-contained, mathematically rigorous framework for massless representations of the conformal group U(2,2) and their restriction to the de Sitter group Sp(2,2). It systematically constructs ladder (massless) representations, computes invariant bilinear forms and Casimir operators, and provides explicit constructions of vertex operators and two-point functions for low-helicity fields. The monograph employs a novel Clifford-split-octonion approach, embedding 8-component Majorana spinors within an alternative composition algebra, which allows for fully explicit algebraic, spinorial, and geometric constructions. The text bridges foundational representation theory, explicit computational methods, and physical applications in quantum field theory and cosmology. While targeted at researchers in mathematical physics, quantum cosmology, and quantum field theory, the exposition is pedagogically structured to guide advanced graduate students through subtle algebraic and representation-theoretic ideas, making complex constructions accessible without sacrificing rigor.

Figures (4)

• Figure 1: Professor Ivan Todorov (1933-2025), whose pioneering contributions to CFT, ladder representations, and massless quantum systems profoundly shaped mathematical physics.
• Figure 2: Schematic illustration of the relationships among the Poincaré, de Sitter (dS), and anti-de Sitter (anti-dS) groups as subgroups of the conformal group. Both dS and anti-dS groups contract to the Poincaré group in the flat (zero-curvature) limit $\Lambda\to 0$. Conversely, for nonzero curvature, the dS and AdS groups can be regarded as deformations of the Poincaré group. The conformal group provides a unifying framework encompassing all three.
• Figure 3: Nested hierarchy of algebraic structures illustrating the relationships among Clifford and Lie algebras relevant to the conformal and spin groups in signature $(4,2)$. The full Clifford algebra $\mathfrak{cl}(4,2)$ contains its even subalgebra $\mathfrak{cl}^{\mathrm{even}}(4,2) \cong \mathfrak{cl}(4,1)$, which in turn includes the Lie algebra $\mathfrak{u}(2,2) \cong \mathfrak{su}(2,2) \oplus \mathfrak{u}(1)$. The special unitary algebra $\mathfrak{su}(2,2)$ is isomorphic to the conformal algebra $\mathfrak{so}(4,2)$ and contains the dS subalgebra $\mathfrak{sp}(2,2) \cong \mathfrak{so}(4,1)$.
• Figure :

Theorems & Definitions (75)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 2.1
• Remark 2.2
• Remark 2.3
• proposition thmcounterproposition