Group Theory Notation: SO(n,1) & Poincare Groups Explained

[arroy_0205]

1. I understand the meaning of group SO(3) etc, but what is meant by say SO(n,1) or Poincare(n,1)group?

2. what is the importance of double cover of a group in physics?

 

[humanino]

I'll give my two cents, since this has not received any answer yet.

1. I understand the meaning of group SO(3) etc, but what is meant by say SO(n,1) or Poincare(n,1)group?

I never saw the notation Poincare(n,1), and I don't know if it is standard.
SO(n,1) correspond to the so-called indefinite orthogonal group in which some of the signs in the signature have been flipped.

2. what is the importance of double cover of a group in physics?

This is a very general question. Covering groups apprear everywhere really, from elementary spin in quantum mechanics to orbifold and Thurston's geometrization conjecture. A group and its cover have identical local properties but can have global (topological) different features. The basics in quantum mechanics for particle physics is that you are interested in irreducible representations of Lie groups. There is a unique simply connected group given the corresponding Lie algebra. As far as I understand, further topological properties will give you additional discrete (quantum numbers) symmetries, the breaking of which are usually studied separately from the representation theory.

This is not very clear to me, I hope I do not confuse you more than I help

 

[masudr]

SO(3) is the group of transformations on R^3 which preserves the bilinear form $x_1^2 + x_2^2 + x_3^2$ and does not perform an inversion on the space.

SO(3,1) is the group which preserves $x_0^2 - x_1^2 - x_2^2 - x_3^2$, i.e. the Lorentz invariant form.

SO(n,1) is the obvious generalization of that.

I've never seen the notation Poincaré(n,1) either. However, I know that the Poincaré group is SO(3,1) and the translation group in 4 dimensions (i.e. all the symmetries of special relativity), so I assume Poincaré(n,1) is SO(n,1) along with the translation group in n+1 dimensions.

 

[arroy_0205]

Thanks for the explanations.
In case you are interested, the terms "poincare(n,1)" or "poincare(2,1)" are used in the Wikipedia article on anyon. See:
http://en.wikipedia.org/wiki/Anyon#Topological_basis

 

[humanino]

From wikipedia, it appears that masudr is right : Poincare(n,1) is the isometric or affine of spacetime with signature (n,1). This notation of the author ot the anyon article directs to "Poincare group" where this notation is not used. The reason is was not positive that it is simply the Poincare group is that this notation is kind of odd. The raison d'etre of this notation is to allow for Poincare(p,q) with q time dimensions. I don't know that anybody really uses that, so does it really deserve a notation on wiipedia ?

Not even to mention the fact that everybody would use the (semi-direct) product of SO(p,q) with $$\mathbb{R}^{p,q}$$, so there is already such a notation in principle.