What are Elementary Particles?

Ontologically, an elementary particle is an irreducible representation of the Poincaré algebra (the Lie algebra of the Lorentz group). This means that it cannot be decomposed into simpler representations, reflecting the fact that elementary particles themselves are indivisible. In contrast, reducible representations correspond to composite particles, which are made up of multiple elementary constituents.

The dimension of the representation—for example, over \( \mathbb{C}^k \)—determines the number of internal states a particle can have, such as its possible spin orientations. This dimensionality is crucial in understanding the intrinsic properties of particles. In the Standard Model, a "type of particle" is defined as an irreducible representation of the the Poincaré Lie algebra \( \mathfrak{su}(3) + \mathfrak{su}(2) + \mathfrak{u}(1) \). These internal symmetries correspond to the strong, weak, and electromagnetic interactions, respectively. While mathematics permits a vast number of possible irreducible representations of these symmetry groups, only some of these representations correspond to the particles we observe in nature.

Viewing elementary particles primarily as mathematical structures, rather than concrete physical objects, offers conceptual clarity. When we regard particles as irreducible representations of symmetry groups, we naturally account for their quantum properties like superposition and entanglement, which defy classical intuitions about localized objects. This mathematical perspective also elegantly explains why particles of the same type are truly identical - they are multiple instances of the same abstract representation, rather than distinct physical entities that happen to share properties.

This view can initially seem unintuitive because our everyday experience leads us to imagine particles as localized physical objects, rather than abstract mathematical constructs. However, if we adopt an instrumentalist stance—that judges scientific theories by their effectiveness as tools for prediction rather than their literal truth—then this mathematical model becomes not just acceptable, but arguably the only viable approach. This selective correspondence suggests that there are additional principles beyond pure symmetry that dictate which mathematical possibilities are physically realized.