# How can one (formally) determine the particle content of a free field theory?

Here's my question: Suppose I'm given a free field theory, where my fields are functions $\phi:\mathbb{R}^4 \rightarrow V$, and the equations of motion are a system of linear Lorentz-invariant differential equations (Given an action of Lorentz algebra on $V$). Is there a way of saying formally or abstractly what the 'particle content' of this field theory is? Ie, how many particles are there and what are there spins/masses?

I should mention that I understand how to determine this empirically- One can often just 'look at the Lagrangian' and figure out what the kinds of particles there are and what their masses are. I'm looking for a more formal way of saying this.

Yes, the solutions to your linear equations of motion will furnish linear representations of the Poincare group. Particles will correspond to the irreducible representations present. The unitary representations of the Poincare group were classified by Wigner using the method of induced representations [see Weinberg Ch. 2]. They are labeled by a mass and a representation of the little group (SU(2) for massive particles, ISO(2) or SO(2) for massless particles). These quantum numbers arise as casimirs of the algebra (mass is $P^{\mu}P_{\mu}$ and the Pauli Lubanski vector $W_{\mu}\approx\epsilon_{\mu \nu \rho \sigma}J^{\nu \rho}P^{\sigma}$ and its square tells you the little group and corresponding casimir.) The Casimirs are constant on representations of ISO(3,1) so each irreducible representation gives you a particle, and the casimirs tell you the mass and spin. You can examine your equations of motion more closely. If there are additional symmetries you may further decompose the particles according to the irreducible representations of the internal symmetries.