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.
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2$\begingroup$ Of course we can't automatically conclude that the Pauli-Uhlenbeck spin is the same as Wigner's spin. We figured that out from consistency with experimental results, which is a core part of physics. $\endgroup$– knzhouCommented Aug 24 at 17:53
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$\begingroup$ You might find these two answers of mine useful, hopefully: physics.stackexchange.com/a/422088/84967, physics.stackexchange.com/a/455975/84967. $\endgroup$– AccidentalFourierTransformCommented Aug 24 at 18:45
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$\begingroup$ Answer to AccidentalFourierTransform. I shall study your answers. But let me ask a question. Does Deligne's theorem put a limit on the number of different elementary particle which may exist? $\endgroup$– André BellaïcheCommented Aug 24 at 21:41
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$\begingroup$ do not believe all the bull... that physicists tell you. We have a tendency to confuse a description with a metaphysical truth. $\endgroup$– Pato GalmariniCommented Aug 25 at 1:18
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2$\begingroup$ This question is similar to: What does it mean for particles to "be" the irreducible unitary representations of the Poincare group?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. See also the links therein $\endgroup$– Tobias FünkeCommented Aug 25 at 7:41
3 Answers
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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1$\begingroup$ @James If you have a question, ask it properly in this forum, not in a comment section of an answer unrelated to your questions. $\endgroup$ Commented Aug 25 at 5:57
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1$\begingroup$ @James feel free to ask a separate question and link to it here and I’ll have a look. Short story: in quantum field theory there are no particles, just fields. Hence the name. So we don’t talk about a particle being somewhere at a certain time, we talk about a field having amplitude in a region of space time. $\endgroup$ Commented Aug 25 at 10:48
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?
--edit--
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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$\begingroup$ Well, it happens all the time in math, but it shouldn't happen in physics. In fact, some of us say, “I'm the teacher, or the author, and I give the definitions I want.” The result is hard-to-follow courses for good students, and unreadable books for physicists, statisticians and engineers. But the majority seek to motivate, to justify their definitions. And some definitions are better than others. $\endgroup$ Commented Aug 24 at 21:12
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$\begingroup$ A university professor who refuses to use students' geometric knowledge acquired in high school, on the pretext that it's based on ungiven definitions (length of a curve) and muddy demonstrations, and who prefers to define sine by $$\sin x = x -\frac{x^3{3!} + \frac{x^5{5!} + \cdots $$ is a bad teacher. I know one. Personally, I prefer to use the differential equation $z'= iz$ which describes the motion of a point moving at speed 1 on the unit circle. $\endgroup$ Commented Aug 24 at 21:13
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$\begingroup$ "By the way, a particle doesn't need to have a definite charge." So, there is no difference between an electron and a positron ? $\endgroup$ Commented Aug 24 at 21:14
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$\begingroup$ I cannot read previous comments. But if I add a new comment, the 3 previous comments become readable. Strange behaviour. Surely, I miss something. $\endgroup$ Commented Aug 24 at 21:51
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$\begingroup$ @AndréBellaïche electrons an positrons are special cases because the positron was the first experimentally discovered antiparticle and there's no "better name". Just take the heavier electron, the muon. We can talk about the muon, or about the $\mu^+$ or $\mu^-$ respectively, but both of them are muons, just with a different charge. The same is true for example for delta baryons, where you have 4 different charges available but all of them are deltas. $\endgroup$ Commented Aug 25 at 7:09
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.
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$\begingroup$ or a scalar like the higgs, or a tensor like the graviton $\endgroup$ Commented Aug 25 at 1:49
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$\begingroup$ Tensor fields to be accurate. $\endgroup$ Commented Aug 25 at 2:44
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$\begingroup$ you were not that accurate in your answer, but you want me to be in a comment? $\endgroup$ Commented Aug 25 at 2:49