# 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? Sep 30, 2014 at 16:23
• @ACuriousMind - Actually, I haven't. May be I will follow your link first, and then get back to the question. Thanks :) Sep 30, 2014 at 16:24
• Closely related: physics.stackexchange.com/q/73593 Sep 30, 2014 at 20:26
• A bit more info: en.wikipedia.org/wiki/… Oct 1, 2014 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. Oct 1, 2014 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. Sep 30, 2014 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? Sep 30, 2014 at 19:04
• It may be good to append the qualifier elementary to the definition given that you've restricted to irreps. Sep 30, 2014 at 20:25
• @joshphysics - of course. Done! Oct 1, 2014 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). Oct 1, 2014 at 17:05