On the notion of elementary particles on curved space-times

Wigner’s program that identified elementary particles with unitary irreducible representations of Poincaré group, is extended, inspired by the modern categorical/groupoidal picture of Quantum Mechanics, to general Lorentzian space-times.

💡 Research Summary

The paper tackles the long‑standing problem of defining elementary particles on generic curved Lorentzian space‑times, where the usual reliance on the global Poincaré symmetry breaks down. In flat Minkowski space, Wigner’s program identifies particles with irreducible projective unitary representations of the restricted Poincaré group, and quantum field theory (QFT) builds on a globally defined vacuum and creation/annihilation operators. On a generic space‑time, however, the isometry group may be trivial, bundles may be non‑trivial, and a global vacuum state cannot be defined, rendering both traditional definitions inapplicable.

To overcome this, the authors introduce the Wigner groupoid, a Lie groupoid whose objects are the points of the space‑time and whose morphisms are the local isometries that exist between pairs of points. This structure encodes the “symmetry” of a space‑time even when no global symmetry group exists, allowing one to glue together point‑wise symmetry data into a coherent geometric object.

The second major contribution is a generalisation of Mackey’s theory of induced representations to the setting of Lie groupoids and projective representations. Starting from an irreducible projective representation of an isotropy (or “little”) group at a point, the authors construct an induced representation of the whole groupoid. They analyse topological obstructions to this construction, showing that they are captured by a Dixmier‑Douady class in (H^{3}(M)). For the Wigner groupoid this class always vanishes, guaranteeing that every little‑group representation lifts to a full groupoid representation.

Applying this machinery, the paper reproduces the standard Wigner classification on Minkowski space: massive particles correspond to orbits with (p^{2}=m^{2}>0) and little group (SU(2)) (spin (s\in\frac{1}{2}\mathbb{N})); massless particles correspond to light‑like orbits with little group the double cover of the Euclidean group (E(2)), giving helicity (\lambda\in\mathbb{Z}/2). In addition, a new family of mass‑less representations emerges, associated with a non‑trivial central extension of (E(2)). These “magnetic” representations carry an intrinsic parameter (\mu\neq0) that behaves like a background magnetic‑like field. The authors discuss the physical plausibility of these new representations, noting a similarity to the historically controversial continuous‑spin representations, and leave their experimental relevance to future work.

Overall, the paper demonstrates that by replacing the global isometry group with the Wigner groupoid and by extending Mackey’s induction to groupoids, one can formulate a robust definition of elementary particles on any strongly causal, oriented Lorentzian manifold. This framework preserves the familiar mass‑spin‑helicity labels in the flat limit while predicting novel particle‑like excitations tied to the topology and geometry of curved space‑times. The work thus bridges a conceptual gap between general relativity and quantum field theory and opens new avenues for exploring particle physics in non‑trivial gravitational backgrounds.