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

In summary, the standard model incorporates U(1), SU(2), and SU(3) as internal symmetry groups in its Lagrangians, while SO(3) is used for space-time symmetry in the Poincare group. The SU(3)xSU(2)xU(1) describes the gauge group of the standard model, which operates in abstract spaces to describe charge-like quantities. These groups do not specify linear translation in space, but can be used to construct rules that comply with groups such as the Galilean, Lorentz, and Poincare groups.
  • #1
lkwarren01
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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
 
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  • #2
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".
 
  • #3
thanks very much for you help
 
  • #4
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.
 
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FAQ: Can SO(3) be used for Poincare spacetime symmetry in the standard model?

1. What is SO(3) in the standard model?

SO(3) refers to the special orthogonal group in three dimensions, which is a mathematical concept used in the standard model of particle physics. It is a group of rotations in three-dimensional space, and it plays a crucial role in describing the symmetries of the fundamental particles and their interactions in the standard model.

2. How does SO(3) relate to the gauge group in the standard model?

SO(3) is a subgroup of the gauge group in the standard model, which is known as SU(2). The gauge group is responsible for the weak nuclear force, and SO(3) is specifically related to the weak isospin component of this force.

3. What does the number 3 represent in SO(3)?

The number 3 in SO(3) refers to the three spatial dimensions in which rotations can occur. These dimensions are often represented by the x, y, and z axes in a Cartesian coordinate system.

4. How does SO(3) relate to the Higgs mechanism in the standard model?

SO(3) plays a crucial role in the Higgs mechanism, which is responsible for giving particles their mass in the standard model. The Higgs field is a complex scalar field that transforms under the gauge symmetry of SO(3) and is necessary for the spontaneous breaking of this symmetry, leading to the generation of mass for particles.

5. How is SO(3) experimentally verified in the standard model?

SO(3) symmetries can be experimentally verified through the study of particle interactions and decays. The predictions made by the standard model, which includes the use of SO(3) symmetries, have been extensively tested and confirmed through experiments at particle accelerators such as the Large Hadron Collider (LHC).

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