Can SO(3) be used for Poincare spacetime symmetry in the standard model?

[lkwarren01]

I'm a layman trying to understand the symmetries used in the std model. I understand that
U(1), SU(2), & SU(3) are incorporated in the Lagrangians for internal symmetries. I've read that SO(3) is also used in the std model for Poincare spacetime symmetry. Is that true and if so, how is it applied...is it somehow in the Lagrangian too?
Thanks very much

 

[vanhees71]

The SO(3) is the group of spatial rotations and as such part of the space-time symmetry, which is the Poincare group, consisting of the Lorentz transformations [which contain the rotations and "boosts" (i.e., switching from one inertial frame to another one, which moves with constant velocity with respect to the former)] and space-time translations.

The SU(3)xSU(2)xU(1) describes the gauge group of the standard model of elementary particles. These are transformations not in space-time coordinates but in abstract spaces, describing charge-like quantities. E.g., the SU(3) consists of all complex 3 x 3-matrices which operate in "color-charge space" of the strong interactions. Each quarks comes in three copies (labeled as "red", "green", "blue") and each antiquark in three "anti-copies" ("anti-red", "anti-green", "anti-blue"). These matrices have determinant 1 and leave the scalar product in the three-dimensional color-vector space invariant. It is generated by 8 independent infinitesimal generators, and accordingly the gauge potential consists of an octet representation of this color SU(3) also known as "the adjoint representation".

 

[lkwarren01]

thanks very much for you help

 

[PhilDSP]

Hmmm, I'm thinking it's really necessary to get some clarity on this. Would you agree that the groups, SO(3), SU(2), U(1) for example, specify the representational logic of spin but not linear translation in space?

The term space above is not necessarily mundane physical "space" but may be Hilbert space for instance or potentially any mathematical space which allows the same rules as rotation groups. The rotation groups implicitly assume that the object being represented in terms of spatial extension is centered at the origin in the group's coordinate system. Or rather that the axial point of spin is the origin.

The groups above do not specify compliance with the Galilean, Lorentz or Poincare groups however rules or operations on the rotation groups can easily be constructed that are in compliance with groups such as the Galilean, Lorentz or Poincare groups. Those groups add the dimension of time as well as linear translation rules. Doing so gives at least a partial definition of how "space" behaves apart from rotation characteristics and leads to specific applications such as the behaviors within particular charge models.