TITLE: What is a particle mathematically?
QUESTION [10 upvotes]: In quantum field theory, what is a particle mathematically? 
How would you explain to someone who kows alot of math but no physics what a particle is?
A simple example model would suffice.

REPLY [3 votes]: A possible answer is 'an irreducible unitary representation of the Poincar\'e group for which the eigenspaces of the generators of translation are finite-dimensional'. (At least, this is the space of states of a single particle of a single type).
This isn't a perfect answer, because not every QFT is relativistic (nor indeed does every QFT have particles), but hopefully it's useful nonetheless.

A bit more reasoning why:
Translations are mutually commuting, so pick a basis where they're all diagonal. This labels states by their momentum and energy. You can do this for any state, but  a single particle state is distinguished by having only finitely many remaining degrees of freedom. This is why the 'finite-dimensional' is in there. The 'irreducible' is there because if I can't rotate/boost from one set of states into another, there is no clear sense in which they are the same: they're different sorts of particle.
[Note that this includes bound states (like a hydrogen atom ground state) on the same footing, as single particles in their own right. This is sensible because there need not be any clear definition of what an `elementary particle' is: this concept makes reference to some free limit, which needn't exist].
Why symmetries require unitary representations:

States of a quantum system live in projective Hilbert space
A symmetry is an invertible map between states that preserves the modulus of inner products, which are important because of their relation to measurable quantities
Wigner's theorem: any symmetry can be lifted to a unitary or antiunitary operator on Hilbert space
As a consequence, a group of symmetries will have a (projective) representation on Hilbert space. For the connected part of a Lie group, it'll be unitary (the identity is unitary!)