# Group Theoretic definition of a particle

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.?

• 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
• Closely related: physics.stackexchange.com/q/73593 – joshphysics Sep 30 '14 at 20:26
• A bit more info: en.wikipedia.org/wiki/… – Bubble Oct 1 '14 at 12:12
• 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 1 '14 at 13:28

## 1 Answer

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.

• 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 Mitra Oct 1 '14 at 12:04
• @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 Mitra Oct 1 '14 at 17:05