[Paper Review] Particle Physics as Representations of the Poincare Algebra

Lars Brink|Chalmers Publication Library (Chalmers University of Technology)|Mar 4, 2005

Black Holes and Theoretical Physics22 references7 citations

TL;DR

This paper proposes that elementary particles and quantum field theories can be systematically derived as irreducible representations of the Poincaré algebra, extended to include coupling constants via light-cone quantization. By constructing interacting generators that close under the algebra, the method uniquely identifies N = 4 Yang-Mills and N = 8 Supergravity as the only consistent massless theories in four dimensions, with higher-dimensional analogues in d=10 and d=11, and shows that string theories emerge naturally as Poincaré-invariant functionals.

ABSTRACT

Eugene Wigner showed already in 1939 that the elementary particles are related to the irreducible representations of the Poincare algebra. In the light-cone frame formulation of quantum field theory one can extend these representations to depend also on a coupling constant. The representations then become non-linear and contain the interaction terms which are shown to have strong uniqueness. Extending the algebra to supersymmetry it is shown that two field theories stick out, N=4 Yang-Mills and N=8 Supergravity and their higher dimensional analogues. I also discuss string theory from this starting point.

Motivation & Objective

• To establish a systematic framework for deriving quantum field theories from the irreducible representations of the Poincaré algebra.
• To explore how coupling constants can be incorporated into these representations, leading to non-linear, interacting theories.
• To identify which quantum field theories are uniquely selected by the requirement of Poincaré invariance and closure of the algebra with interaction terms.
• To extend the formalism to supersymmetric theories and determine which theories—such as N = 4 Yang-Mills and N = 8 Supergravity—emerge as special cases.
• To investigate the role of continuous spin representations and their implications in higher dimensions, particularly in d=11 for M-theory connections.

Proposed method

• Use light-cone quantization to eliminate unphysical degrees of freedom, reducing the symmetry to the Poincaré group alone.
• Construct interacting generators of the Poincaré algebra by extending free generators with coupling-dependent terms that preserve closure of the algebra.
• Apply the method to massless theories by solving p² = 0 in light-cone coordinates, leading to p⁻ = p̄p/p⁺.
• Introduce additional generators Tⁱ (light-cone translations) to realize continuous spin representations (CSRs) when Tⁱ ≠ 0.
• Extend the formalism to supersymmetry by decomposing supercharges in d=11 into SO(8) spinors, leading to supermultiplets including IIA, IIB, and SYM theories.
• Treat string theories as Poincaré-invariant functionals of fields, showing that only a few such theories exist due to symmetry constraints.

Experimental results

Research questions

• RQ1 Which quantum field theories can be uniquely derived from the irreducible representations of the Poincaré algebra when interactions are included via coupling constants?
• RQ2 How do continuous spin representations (CSRs) arise in this framework, and what is their role in higher-dimensional field theories?
• RQ3 Why do N = 4 Yang-Mills and N = 8 Supergravity emerge as the only consistent massless theories in four dimensions under this formalism?
• RQ4 What is the connection between this Poincaré-based construction and higher-dimensional theories such as those in d=10 and d=11?
• RQ5 Can string theories be consistently derived as Poincaré-invariant functionals within this representation-theoretic framework?

Key findings

• The method uniquely identifies N = 4 Yang-Mills and N = 8 Supergravity as the only consistent massless quantum field theories in four dimensions that close under the Poincaré algebra with coupling constants.
• In d=11, the short little group is SO(8), and the supercharges decompose into two 8-component spinors, leading to supermultiplets that include the N=1, N=2, and N=8 theories.
• Continuous spin representations (CSRs) appear when Tⁱ ≠ 0, corresponding to massless states with an infinite number of helicity states labeled by a space-like vector ξⁱ.
• In higher dimensions, such as d=11, there are infinitely many CSRs, each labeled by the length of the Tⁱ vector and the Dynkin labels of the SO(d−3) short little group.
• The formalism shows that only a few string theories can be Poincaré-invariant functionals, indicating strong constraints on consistent string theories.
• The approach provides a perturbative framework that is useful for studying finiteness in quantum field theories but is not suitable for non-perturbative phenomena like solitons or branes.