I suppose this question has been asked many times. I have been told that an elementary particle is a (moving) point, or (a section from) some field, or an excitation from some field. But now, I am told that a particle IS (from the verb "to be") an irreducible representation of the Poincaré or the Lorentz group. Surely this definition may be useful, and you can extract some information from that, namely mass and spin of something you may decide to call a particle, but curiously not the electric charge. Unfortunately the physics article and books I have at hand do not bother to make the connection between these points of view. You can't rely on the fact that something is a half-integer and has been called spin by Pauli or Uhlenbeck and that another half-integer encountered by Wigner in a completely different field of knowledge has also been called spin to force the student to believe that it's the same thing! So, can you explain this connection in a few sentences, or give me a good reference? PS: I am a mathematician.
3 Answers
But how else do you define a particle? Something that has a definite position or momentum, definite mass, and definite spin? Just like, erm, a representation of the Poincare group?
(By the way, a particle doesn't need to have a definite charge.)
With that in mind, you can then go on and identify what is particle and what is not in a given theory.
Since you are a mathematician, I imagine you are very familiar to this kind of thing already. You started out with some vague intuitive idea that you try to describe, and after a lot of careful thought, come up with a definition that is all but unrecognizable. Happens all the time in math, no?
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There's a lot to reply so I might as well do a proper edit.
The OP asked for a definition, not a vague intuitive idea. Note that most of the QFT books written by physicist and for physicist don't even try to define what a particle is. These books rely on your vague intuitive notion of what a particle should be.
This exactly mirrors the situation with real number in high school: you "kind of know" what it is, but none of what you know constitutes a true definition. And most of us live happily ever after and even sometimes do-- gasp-- contour integrations without being taught Dedekind cut. But if you asked for a constructive definition, what else can I tell you?
As for the charge thing, again to compare with high school math: yeah I know there are acute and obtuse triangles, but I am only talking about triangle now.
And finally, I want to comment on the "definition" that a particle is a quantized excitation of some quantum field. This one is missing a quantifier: "within a QFT", and consequently does not actually define what we intuitively perceive as a particle.
I can't come up with a metaphor with high school math, so let's try finance and economics instead: while everyone else is talking about money and currency in general, this one defines legal tender money in the US as the dollar issued by the Federal Reserve. It is a good definition provided that you already know what money is.
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I think when someone says “a particle IS a representation of some group” in the context of quantum field theory it would always be more precise to say, “at any point in space time, the state space of this quantum field is a representation of some group.” I think a further qualification is needed that the transformations are smooth in the manifold, that is, point wise transformation is not sufficient. Need to think about how to say this precisely and concisely.
In the context of non relativistic quantum mechanics where there are actually point particles and not just fields, we can say that “the state space of this particle is a representation of some field.”
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Saying that particles are irreducible representation of Poincare group is like saying that energy is a number, not a full story. Also note that a particle is a tensor field.
