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We intuitively have a sense of what a particle means in the conventional sense. But is it possible to have a group theoretical definition of a particle, I mean in terms of irreducible representations etc.?

User Anonymous
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    Have you read about Wigner's classification? – ACuriousMind Sep 30 '14 at 16:23
  • @ACuriousMind - Actually, I haven't. May be I will follow your link first, and then get back to the question. Thanks :) – User Anonymous Sep 30 '14 at 16:24
  • Have you done experiments on particles? If not, I doubt that you can develop an "intuitive" understanding for what a particle is. It can certainly not follow from a theoretical definition, which leaves you with months worth of calculations between the theory and the phenomenology. – CuriousOne Sep 30 '14 at 18:30
  • @CuriousOne - I meant the casual one - the textbook stuff + the fact that resonances in particle experiments can be deemed as particles, if I'm right. What exactly is the point that I'm missing? – User Anonymous Sep 30 '14 at 19:08
  • I think you are missing the phenomenology of particle physics, which is very rich. Is a muon a long lived resonance? Is a short lived resonance that never reaches a particle detector a particle? Is the core of a neutron star a particle? Where do you draw the line? – CuriousOne Sep 30 '14 at 19:22
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    Closely related: http://physics.stackexchange.com/q/73593/ – joshphysics Sep 30 '14 at 20:26
  • @UserAnonymous: like I said, you are missing the phenomenology of the vacuum, even at the textbook level. It's not in the math, but in the diversity of phenomena described by the math. Group theory is not going to help with that in the slightest, since it has solutions that don't exist in nature and it can't describe solutions that do (like neutron stars). – CuriousOne Sep 30 '14 at 23:14
  • @CuriousOne - Yes, but we could still know about it. May be not very practical, but still may be useful somewhere or the other. Thanks for the information nevertheless, that was very helpful :) – User Anonymous Oct 01 '14 at 05:00
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    A bit more info: http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory – Bubble Oct 01 '14 at 12:12
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    I agree that particle phenomenology is extremely rich and there are nonperturbative states that cannot be usefully classified as particles (there is more to physics than group theory!)... But I think it's too strong to say Wigner's classification is useless. It's a deep statement about what we mean by particles quantum mechanically. I'll agree it's also not the best starting point for building intuition about particle physics though. – Andrew Oct 01 '14 at 13:28

1 Answers1

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An elementary particle is defined as an irreducible representation of the Poincar\'e group. These were classified by Wigner in 1939. This was done via the little group construction. The important representations are (metric signature $(-,+,+,+)$

  1. $p^2 = 0$, $p^0 < 0$ - The little group is ISO(2). All finite dimensional representations of this group are one-dimensional and labelled by a single number $h$ (called helicity). Topological considerations require that $h$ be a half-integer. Under parity, the representation $h$ is rotated to $-h$. Thus, a massless particle that has parity and proper Lorentz invariance, has two degrees of freedom and is labelled by its helicity $|h|$.

  2. $p^2 = - m^2 < 0$, $p^0 < 0$ - The little group is $SO(3)$. All representations of this are finite dimensional and are labelled by a single number $j$ with dimension $2j+1$ ($j$ is called spin). Thus, a massive particle of spin $j$ has $2j+1$ degrees of freedom.

Prahar
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  • Thanks very much. This is a lot deeper than I'd imagined. I'm not sure if I understand it fully, so apart from ACuriousMind's wiki link, I would be very happy if you could suggest a pedagogic reference (like some book) to these issues. Thanks in advance. – User Anonymous Sep 30 '14 at 19:03
  • Secondly, I've upvoted, but if it doesn't bother you, can I please defer "accepting" till the time I make good sense of it? – User Anonymous Sep 30 '14 at 19:04
  • It may be good to append the qualifier elementary to the definition given that you've restricted to irreps. – joshphysics Sep 30 '14 at 20:25
  • @joshphysics - of course. Done! – Prahar Oct 01 '14 at 12:04
  • @UserAnonymous - No problem. You can ask questions if you have doubts in my answer. – Prahar Oct 01 '14 at 12:05
  • @Prahar - Is there any introductory/pedagogic text dealing with these issues, and discussing this (in particular) in detail? – User Anonymous Oct 01 '14 at 16:42
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    @UserAnonymous The construction of representations of Poincare group is done very well in Section 2.5 of Wieinberg. He discusses representations of ISO(2). Representations of SO(3) can be found in any standard textbook on quantum mechanics (for instance, Section 4.3 of Griffiths). – Prahar Oct 01 '14 at 17:05
  • @Prahar - Thank you so much for the answer and the references. – User Anonymous Oct 02 '14 at 02:26