Is a particle an irreducible representation of the Poincaré group? (2410.02354v1)

Abstract: The claim that a particle is an irreducible representation of the Poincar\'e group -- what I call \emph{Wigner's identification} -- is now, decades on from Wigner's (1939) original paper, so much a part of particle physics folklore that it is often taken as, or claimed to be, a definition. My aims in this paper are to: (i) clarify, and partially defend, the guiding ideas behind this identification; (ii) raise objections to its being an adequate definition; and (iii) offer a rival characterisation of particles. My main objections to Wigner's identification appeal to the problem of interacting particles, and to alternative spacetimes. I argue that the link implied in Wigner's identification, between a spacetime's symmetries and the generator of a particle's space of states, is at best misleading, and that there is no good reason to link the generator of a particle's space of states to symmetries of any kind. I propose an alternative characterisation of particles, which captures both the relativistic and non-relativistic setting. I further defend this proposal by appeal to a theorem which links the decomposition of Poincar\'e generators into purely orbital and spin components with canonical algebraic relations between position, momentum and spin.

• The paper critically examines Wigner's identification of particles as irreducible representations of the Poincaré group, arguing this definition is inadequate for interacting particles and particles in alternative spacetimes.
• The author proposes the QPS (Position, Momentum, Spin) algebra as an alternative, dynamically and geometrically agnostic framework for particle characterization that is more universal.
• The QPS algebra challenges the traditional particle-spacetime symmetry link and offers a basis for QFTs describing interactions in diverse spacetimes.

Analysis of "Is a particle an irreducible representation of the Poincaré group?"

The paper critically examines the prevalent conceptualization in particle physics, referred to as Wigner’s identification, which posits that particles are irreducible representations of the Poincaré group. This idea, rooted in Wigner's 1939 work, is deeply entrenched in particle physics but is seldom interrogated as a definitive characterization. The author, Adam Caulton, explores the theoretical underpinnings of this identification, raises objections against its adequacy, and proposes an alternative characterization that aims to transcend the limitations identified.

Critique of Wigner’s Identification

The central critique of Wigner’s identification is its failure to account for interacting particles and particles in alternative spacetimes. The traditional view ties particle states strictly to the spacetime symmetries; however, Caulton argues that this association is both misleading and insufficient for characterizing particles across different physical scenarios. Wigner’s theorem, which underpins the identification, connects symmetry operations that preserve transition probabilities to unitary or anti-unitary operators. Caulton suggests that the categorical treatment of particles as irreducible Poincaré group representations is not extensionally adequate, as it cannot accommodate the diversity of particle behaviors, particularly in non-Minkowski spacetimes or when particles interact.

Proposed Alternative: The QPS Algebra

Caulton introduces the QPS (Position, Momentum, Spin) algebra as an alternative framework for characterizing particles. Unlike the traditional view restricted to specific spacetime symmetries, the QPS approach remains agnostic to dynamical and geometrical constraints, thus enabling a more universal characterization of particles. This method maintains compatibility with various Hamiltonians and geometric configurations, making it both dynamically and geometrically innocent. The crux of this approach is its underlying algebraic structure that supports the representation of quantum systems beyond the confines of free particles in Minkowski space.

Theoretical and Practical Implications

The implications of adopting the QPS algebra are significant on both theoretical and practical fronts. Theoretically, it challenges the deeply rooted assumption that particle properties must fundamentally reflect the symmetries of the encompassing spacetime, promoting a more flexible view that aligns with quantum mechanics' capabilities of handling diverse quantum systems. Practically, it opens the avenue for more applicable quantum field theories that adequately describe interacting particles and systems in various gravitational contexts. This approach can potentially yield new insights into quantum field localization, addressing long-standing issues like superluminal propagation peculiar to traditional quantum field theory.

Future Developments and Speculation

The proposition of the QPS algebra lays a foundation for further exploration into the ontology of particles within the quantum field, particularly in regimes previously deemed incompatible with Wigner’s framework. Future research may explore formulating quantum fields that inherently support particles in generalized spacetimes or probing the quantum gravity intersection, where the symmetry-language connection becomes increasingly complex. Additionally, empirical studies examining the QPS approach’s predictive power concerning fine-grained phenomena could substantiate its validity and inspire adaptations in standard model theories sited in nonconventional spacetime structures.

In conclusion, this paper delivers a cogent critique of Wigner’s identification, suggesting a holistic framework that pivots from symmetry-bound representations toward more universally adaptable particle characterizations. By embracing a broader conceptualization of particles through the QPS algebra, the paper not only challenges conventional wisdom but also encourages a re-examination of the foundational postulates in theoretical physics.