Democritus famously proposed that all matter consists of microscopic particles, called 'atoms', which are not themselves composed of other particles. By historical accident, it is the chemical elements which are referred to as atoms, despite the fact that they are composed of smaller particles. Nevertheless, it is currently believed that there are particles which are not composed of other particles, and these entities, which include electrons, photons and quarks, are dubbed 'elementary particles'. The Democritean vision of elementary particles as miniature snooker balls, however, has been somewhat vitiated by quantum theory, and it is not merely the classical notion of a particle as a localisable entity which has been undermined, but the mereological notion that a composite system has a unique decomposition into elementary entities.
According to modern theoretical physics, the fundamental types of things which exist are quantum fields, and particles are merely excited states of the underlying quantum field. Given that these modes of excitation satisfy the principles of quantum theory, they are often dubbed 'excitation quanta'. Even when there are no particles present, the quantum field is simply in its lowest-energy state, and this non-zero energy of the so-called 'vacuum state' duly has a detectable effect.
Because particles are excitation quanta of an underlying field, their identity conditions are more akin to those of waves or vibrations in a continuous medium than miniature snooker balls. For example, if one begins with a number of separate travelling waves on the surface of a body of water, and they merge together to form a standing wave, then the individual identities of the original constituent waves would be lost. This has some similarity with quantum phenomena: For example, there are conditions under which one can say that there is an N-particle state of a quantum field, but in which it appears to be impossible to individuate N distinguishable particles; there are states of a quantum field in which there are simply an indefinite number of particles present; and given a quantum field state in which no particles are present in one reference frame, this is the same state for which there will be many particles present in an accelerated reference frame.
The analogy with vibrations or oscillations in a continuous medium also leads to a better understanding of what an elementary particle might be. On a classical level, Fourier analysis treats each wave as an element in a vector space. By selecting a basis for a vector space, one can decompose any element as a linear combination of basis vectors; select a different basis, and one can decompose the same element as a different linear combination. In Fourier analysis, plane waves are selected as the preferential basis vectors, and in a sense, these are the elementary waves. As Roberto Torretti points out, telecommunication companies use Fourier analysis to "literally superpose the electromagnetic renderings of many simultaneous long-distance messages in a single wave train that is echoed by satellite and then automatically analyzed at the destination exchange into its several components, each one of which is transmitted over a separate private telephone line...the signal could also be split into other, meaningless components if the analysis were not guided by human interests and aims." (The Philosophy of Physics, p393).
This clearly undermines the Democritean mereological concept of elementarity, in which a composite entity has a unique decomposition into a set of indivisible entities.
It is also well-known amongst physicists that the classical description of the oscillations in a continuous medium can be quantized. When the vibrations in a crystalline solid are so quantized, one obtains elementary modes of excitation called 'phonons'. (These are also considered to be the elementary modes of the sound waves in a solid). Similarly, when the oscillations of the electrons in a plasma are subjected to a quantum description, the elementary modes of excitation are called 'plasmons'. In this context, elementarity seems to correspond to nothing more than the choice of a particularly convenient decomposition of oscillatory behaviour.
Richard Feynman once described the function of high-energy particle colliders as akin to smashing watches together, and then looking at the gears, cogs and springs which fly out, in order to better understand how the watches are put together. Given the considerations above, a better analogy might be to imagine high-energy incoming waves, which collide and merge, and then split apart into different, smaller, outgoing waves. Certainly, when elementary particles and anti-particles collide, and transform into different types of outgoing particles, the use of an analogy which employs composite systems, such as watches, breaks down somewhat.
So these are the mereological questions which beset elementary particles, but even if we successfully elucidate the quantum concepts of parts and wholes, we are still left with the question, 'What is an elementary particle?'. Attempts to answer this question have employed the notions of intrinsic properties and extrinsic properties.
An intrinsic property of an object can be defined to be a property which the object possesses independently of its relationships to other objects. In contrast, an extrinsic property can be defined to be a property which an object possesses depending upon its relationships with other objects. Thus, one might deem that a particle's mass and charge are intrinsic properties, whilst its velocity is an extrinsic property, depending as it does upon the reference frame chosen.
Now, in terms of these concepts, classical physics offers a nice clear definition of an elementary particle: it is a system which has a unique intrinsic state. Souriau and Cushman-de Vries define an elementary system to be one in which the restricted Poincare group acts transitively upon its 'space of motions', (Structure of Dynamical Systems: A Symplectic View of Physics, p173). The space of motions here is the set of all possible histories of a system. Within special relativity, the Poincare group provides the group of all possible transformations between those reference frames which are unaccelerated, and therefore free from the influence of forces. (One also says that the Poincare group is the space-time symmetry group of special relativity). Those properties which change under the action of the Poincare group, must be extrinsic properties, and histories which are related by a Poincare transformation are the same intrinsic history. Now, if the (restricted) Poincare group acts 'transitively' upon the space of histories of a system, it entails that any two histories, v and w, are related by a Poincare transformation g, v=gw, and therefore there is only one intrinsic history. In classical physics, an elementary particle has a unique intrinsic state, and a unique intrinsic history.
However, the situation changes in a subtle fashion in quantum theory. Here, Wigner established that each type of elementary particle corresponds to an irreducible Hilbert space representation of the restricted Poincare group. (An irreducible representation is one in which there are no subspaces invariant under the group action, apart from the null vector and the entire Hilbert space). However, the irreducibility of a group representation does not entail the transitivity of the group action, and neglect of this fact has a tendency to lead some authors astray.
For example, the mathematician J.M.G. Fell adopted Wigner's notion that the irreducibility of a representation is the defining characteristic of an elementary particle representation, and argued that the ensuing group action is "essentially" transitive upon the state space of such a representation. He argued from this that an elementary particle has only one intrinsic state:
"It can never undergo any intrinsic change. Any change which it appears to undergo (change in position, velocity, etc.) can be 'cancelled out' by an appropriate change in the frame of reference of the observer. Such a material system is called an elementary system or an elementary particle. The word 'elementary' reflects our preconception that, if a physical system undergoes an intrinsic change, it must be that the system is 'composite', and that the change consists in some rearrangement of the 'elementary parts'," (Fell, J.M.G., and Doran, R.S. (1988). Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, p31).
Unfortunately, Wigner's irreducible representations are representations upon infinite-dimensional Hilbert spaces, whilst the Poincare group has only 10 dimensions. A group can only act transitively upon a space with the same dimension as the group itself, hence in quantum theory, the restricted Poincare group has many different 'orbits' upon the state spaces of elementary particles. Each orbit corresponds to a different intrinsic state of the elementary particle. Essentially, this is because a particle is represented in quantum theory by a wave-function, a field-like object, and the multitude of possible, locally-varying, intrinsic changes in such an object cannot be cancelled out by the rigid transformations of the Poincare group.
One needs to carefully distinguish the false notion that the space-time symmetry group acts transitively upon the quantum state space of an elementary system, from the correct notion that any vector in such a state space is 'cyclic' with respect to the action of the space-time symmetry group. If one takes the orbit of the action of the space-time symmetry group upon an arbitrary vector, and if one then takes the set of all superpositions of the elements in that orbit (technically, if one takes the topological closure of the complex linear span of all the elements in the orbit), then one obtains the entire state space. The vector chosen is said to be a cyclic vector, and the representation is said to be cyclic. In the case of an irreducible representation of the space-time symmetry group, the orbit of a single state takes one through a sufficient number of orthogonal states to span the entire infinite-dimensional state space. However, this doesn't entail that there is only one intrinsic state! The mathematical operation of taking a linear combination of a set of states does not correspond to a change of physical reference frame.
In the Stanford Encyclopedia of Philosophy, Meinard Kuhlmann asserts that:
The physical justification for linking up irreducible representations with elementary systems is the requirement that “there must be no relativistically invariant distinction between the various states of the system” (Newton & Wigner 1949). In other words the state space of an elementary system shall have no internal structure with respect to relativistic transformations. Put more technically, the state space of an elementary system must not contain any relativistically invariant subspaces, i.e., it must be the state space of an irreducible representation of the relevant invariance group. If the state space of an elementary system had relativistically invariant subspaces then it would be appropriate to associate these subspaces with elementary systems. The requirement that a state space has to be relativistically invariant means that starting from any of its states it must be possible to get to all the other states by superposition of those states which result from relativistic transformations of the state one started with. (2006, Section 5.1.1).
It is indeed true, by definition, that under an irreducible representation of the space-time symmetry group, there can be no non-trivial subspace which is invariant under the action of the symmetry group. However, for the reasons explained above, this does not entail that there can be no "relativistically invariant distinction between the various states of the system". There can indeed be such a distinction, defined by the different orbits of the symmetry group. Note also that Kuhlmann conflates an irreducible group representation with a cyclic representation; irreducibility is not the same thing as cyclicity.
In modern physics, then, elementarity is far from elementary.