[Paper Review] Wigner's Last Papers on Spacetime Symmetries

Y. S. Kim|ArXiv.org|Dec 19, 1995

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper reviews the final collaborative works of Eugene Wigner on spacetime symmetries, focusing on the unification of internal and spacetime symmetries for both massive and massless particles. Using group-theoretical methods rooted in the inhomogeneous Lorentz group, the authors explore how Wigner's 1939 classification of relativistic particles can be extended to include internal symmetries, leading to a deeper understanding of particle classification and gauge invariance in quantum field theory.

ABSTRACT

Wigner's 1939 paper on representations of the inhomogeneous Lorentz group is one of the most fundamental papers in physics. Wigner maintained his passion for this subject throughout his life. In this report, I will review the papers which he published with me on this subject. These papers deal with the question of unifying the internal space-time symmetries of massive and massless particles.

Motivation & Objective

• To examine the final collaborative research of Eugene Wigner on spacetime symmetries, particularly the unification of internal and spacetime symmetries.
• To clarify the role of the inhomogeneous Lorentz group in classifying massive and massless particles.
• To extend Wigner's 1939 representation theory to include internal symmetries such as those associated with gauge fields.
• To explore the implications of these symmetries for quantum field theory and the structure of elementary particles.
• To preserve and disseminate Wigner's enduring contributions to the foundational understanding of relativistic quantum mechanics.

Proposed method

• Application of group representation theory to the inhomogeneous Lorentz group, particularly focusing on little groups for massive and massless particles.
• Analysis of the little group structure for massless particles, including the role of helicity and gauge invariance.
• Use of the Wigner's 1939 paper as a foundation to unify internal symmetries with spacetime symmetries.
• Investigation of the unitary representations of the Poincaré group and their physical interpretation in terms of particle states.
• Examination of how internal symmetries, such as those in gauge theories, can be embedded within the framework of spacetime symmetry groups.
• Use of the Wigner's little group for massless particles to derive the transformation properties of gauge bosons and their relation to helicity states.

Experimental results

Research questions

• RQ1 How can internal symmetries be consistently unified with spacetime symmetries in the framework of relativistic quantum mechanics?
• RQ2 What is the role of the little group in classifying massless particles and how does it relate to gauge invariance?
• RQ3 How does Wigner's 1939 classification of the Poincaré group representations extend to include internal degrees of freedom?
• RQ4 In what way do the unitary representations of the inhomogeneous Lorentz group accommodate both massive and massless particles?
• RQ5 What are the physical implications of extending spacetime symmetry to include internal symmetries in quantum field theory?

Key findings

• The paper confirms that the little group for massless particles, which is isomorphic to the Euclidean group in two dimensions, naturally leads to the concept of helicity and gauge invariance.
• The unification of internal and spacetime symmetries is achieved through the extension of Wigner's 1939 classification to include additional quantum numbers associated with internal symmetry groups.
• The structure of the little group for massless particles explains why gauge bosons have only two physical polarization states, consistent with helicity ±1 for photons.
• The framework provides a group-theoretical foundation for understanding the spin-statistics connection and the emergence of gauge invariance from spacetime symmetry.
• The results demonstrate that the internal symmetry structure of particles can be derived from the representation theory of the Poincaré group, particularly through the little group decomposition.
• The work preserves and clarifies Wigner's final insights, showing that the unification of internal and spacetime symmetries is deeply rooted in the representation theory of the inhomogeneous Lorentz group.