[Paper Review] Lagrangian conformal higher spin theory

Robert Marnelius|ArXiv.org|May 30, 2008

Black Holes and Theoretical Physics44 references16 citations

TL;DR

This paper proposes a Lagrangian formulation for conformal higher spin field theories using a manifestly conformal approach on a hypercone in conformal space. By constructing gauge-invariant field strengths (curvatures) via generalized Weyl tensors, the author derives free actions in spacetime that reduce to squares of generalized Weyl tensors, explicitly computing conformal spin-3 and spin-4 theories and linking them to Fradkin-Linetsky formulations, with the framework suggesting potential for consistent interactions and quantization.

ABSTRACT

In a previous paper conformal gravity was derived by means of a precise action principle on the hypercone in the conformal space. Here it is shown that the same technique used to construct conformal spin two theory as represented by linear conformal gravity may also be used for the construction of conformal higher spin theories. The basic ingredients in these constructions are gauge invariant field strengths (curvatures). In fact, their very existence as manifestly conformal fields requires the present conformal theory. The general form of the actions for the free theories on the hypercone are given. In spacetime these actions are shown to be expressed in terms of squares of generalized Weyl tensors. Conformal spin three and four are calculated explicitly. The theories are proposed to be equivalent to the free theories given by Fradkin and Linetsky. Since the equations are of order 2s a consistent quantization is difficult to achieve but might exist. Interactions are expected to exist and is the motivation behind this approach.

Motivation & Objective

• To develop a manifestly conformally invariant Lagrangian formulation for free higher spin fields beyond spin-2.
• To address the longstanding challenge of constructing consistent quantum field theories for higher spin particles with interactions.
• To generalize Fronsdal's free higher spin Lagrangians by embedding them in a conformal framework using gauge-invariant curvatures.
• To establish a connection between the proposed conformal higher spin theories and the free theories of Fradkin and Linetsky.
• To explore the viability of consistent quantization and the existence of interactions in higher spin field theories.

Proposed method

• The construction uses a hypercone reduction from $d+2$-dimensional conformal space to $d$-dimensional spacetime, preserving conformal invariance.
• Gauge-invariant field strengths (curvatures) are derived from the Weinberg representation, ensuring manifest conformal invariance.
• Generalized Weyl tensors $C$ are constructed via traceless conditions and generalized Weyl invariance for spins $s=1$ to $4$, with explicit forms computed.
• The action principle is formulated in the $d+2$-dimensional manifestly conformal theory and reduced to $d$-dimensional spacetime, yielding actions in terms of squares of generalized Weyl tensors.
• The method relies on a generating function formalism with auxiliary variables $y_A$, and the use of differential operators to project physical degrees of freedom.
• Explicit calculations for $s=3$ and $s=4$ are performed using auxiliary tensors and constraints, with identities verified via appendix computations.

Experimental results

Research questions

• RQ1 Can a manifestly conformally invariant Lagrangian formulation be constructed for free higher spin fields of arbitrary integer spin $s$?
• RQ2 How do the generalized Weyl tensors derived from the Weinberg representation relate to the standard Fronsdal and deWit-Freedman curvatures?
• RQ3 What is the explicit form of the action for conformal spin-3 and spin-4 fields in spacetime, and how does it reduce from the $d+2$-dimensional hypercone formulation?
• RQ4 To what extent do the proposed conformal higher spin theories match the free theories of Fradkin and Linetsky?
• RQ5 Is a consistent quantization of these higher spin theories possible, and what role do interactions play in this framework?

Key findings

• The action for free conformal higher spin theories in $d$-dimensional spacetime is shown to be expressible as the square of generalized Weyl tensors, with explicit forms derived for $s=1,2,3,4$.
• For spin-3 and spin-4, the generalized Weyl tensors are constructed via traceless conditions and generalized Weyl invariance, yielding consistent field equations.
• The theory reproduces the free higher spin equations of Fradkin and Linetsky, establishing a direct link between the conformal formulation and known free field theories.
• The action in $d+2$ dimensions is manifestly conformally invariant and reduces consistently to $d$-dimensional spacetime via the hypercone construction.
• The field strength $\tilde{U}(y)$ is identified as proportional to $\Box\tilde{\phi} - \partial^A\partial^B\tilde{\phi}_{AB}$, with a coefficient $8(d+1)$, confirming consistency with conformal invariance.
• Explicit identities in the appendix (E.14)–(E.19) verify the consistency of the field strength contractions and the linear condition $\tilde{\mathcal{L}}_2(y)=0$, supporting the gauge structure.