What are Elementary Particles?

In primary school we're taught the naive intuition of particles as tiny, solid spheres propagating along well-defined trajectories collapses under the weight of quantum phenomena. Then in high-school we're taught that wave–particle duality shows that entities such as electrons and photons exhibit delocalized wave-like behavior described by a wavefunction $\psi(x)$ until a measurement forces a localized, particle-like manifestation. The indistinguishability of identical quanta—formalized by the need to symmetrize or antisymmetrize the multi-particle wavefunction in Hilbert space—renders them fundamentally indistinct, contrary to classical notions of separable objects. Then in university we're taught that Heisenberg’s uncertainty principle, precludes simultaneous precise knowledge of position and momentum, negating the picture of microscopic billiard balls. Then in graduate school we're taught that in quantum field theory particles are not primitive constituents but excitations of underlying fields. But there's another level of questioning beyond that.

In the QFT ontology, fields defined at each spacetime point are the primary entities, each carrying operator-valued degrees of freedom. What we term a "particle" is a quantized excitation of its associated field, created or annihilated by ladder operators acting on the Fock space. The dynamics are encoded in a Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi,\psi,\bar\psi,\dots)$, and physical processes such as scattering or decay correspond to perturbative expansions in Feynman diagrams. Creation and annihilation operators $a^\dagger_p$ and $a_p$ succinctly describe transitions between vacuum and excited states, replacing classical trajectories with changes in field occupancy.

Within the Standard Model, each particle species is identified by its transformation properties under the Poincaré group and the internal gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$. Wigner's classification associates elementary particles with irreducible representations of the Poincaré algebra, fixed by the Casimir invariants

$$
C_1 = P_\mu P^\mu = m^2,
\qquad
C_2 = W_\mu W^\mu = -m^2 s(s+1),
$$

which uniquely determine mass $m$ and spin $s$. Gauge charges such as color, weak isospin, and hypercharge arise from irreducible representations of the internal symmetry group. Thus an electron field transforms as a spin-$\tfrac12$ Weyl representation of the Lorentz group, carries hypercharge $-\tfrac12$ under $U(1)_Y$, and is a singlet under $SU(3)_c$.

Philosophically, structural realism offers a compelling ontology for this representation-theoretic view. Then there are two interpretations of structural realism: epistemic structural realism holds that scientific theories capture only relational structure, not the intrinsic nature of unobservables, while ontic structural realism asserts that structure is all there is: relations and symmetries compose reality, and objects are mere nodes in a relational web. Under this perspective, a particle is not a substratum but the instantiation of a mathematical structure defined by group representations and field equations.

The gauge symmetries of $SU(3)_c\times SU(2)_L\times U(1)_Y$ further illustrate how abstract structure dictates physical phenomena. Color charge and confinement in quantum chromodynamics arise from the nonabelian structure of $SU(3)_c$, while spontaneous symmetry breaking in the electroweak sector, achieved through the Higgs field acquiring a vacuum expectation value, endows particles with mass. These mechanisms underscore that what we call "particles" are emergent nodes in a tapestry woven from symmetry principles rather than independent substances.

We engage with these structures through the formal tools of Lagrangian mechanics, field operators, and perturbation theory, where we translate abstract algebraic relations into precise numerical predictions. The extraordinary empirical success of quantum electrodynamics, quantum chromodynamics, and the electroweak theory—verified to unprecedented accuracy—attests to the ontological priority of mathematical structure over picturable matter.

Particles are best understood as mathematical structures: irreducible representations of spacetime and internal symmetry groups that manifest as quantized field excitations. This structural ontology dissolves classical paradoxes of identity, localization, and determinacy, revealing a reality whose fundamental fabric is woven from abstract symmetries and relations. This view avoids speculating about an underlying "stuff" or "substratum" that has these properties. The properties, defined mathematically, constitute the particle ontologically.

Of course this begs the perennial question: Does this mean reality is mathematical? Or is mathematics just our best tool for describing a reality whose ultimate nature remains elusive? This ties back to the broader realism/anti-realism debates. But ultimatly these are not physics questions anymore, so that's best left to another department.