Spin 1 Particle Representations of SO(3) and SU(2)

In summary: So the answer is no, SU(3) is not a double cover for SO(3) but it is a cover. Given a representation of SO(3) on C^3 you can define a representation of SU(3) on the same C^3, which is irreducible because it is a covering space for SO(3).[...]In summary, the conversation discusses the groups related to the Dirac Equation for spin 1/2 particles and the reason behind the 2-dimensional Hilbert Space for these particles. It also raises a question about the spin vector for spin 1 particles and whether SU(3) maps to SO(3) 1-to-1. The conversation concludes by discussing the representation of SO(
  • #1
LarryS
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I am still learning about all the Groups related to the Dirac Equation for spin 1/2 particles. Apparently, the reason that the Hilbert Space for spin 1/2 particles is 2-dimensional is because when you try to map SU(2) to SO(3), the mapping is 2-to-1, i.e. SU(2) is a double cover for SO(3).

What about spin 1 particles. The spin vector space for those particles is 3-dimensional. Does that mean that SU(3) maps to SO(3) 1-to-1?

As usual, thanks in advance.
 
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  • #2
Having just posted the above question, I believe I see an error in my reasoning. I am rephrasing the question:

I am still learning about all the Groups related to the Dirac Equation for spin 1/2 particles. Apparently, the reason that the spin vector for a spin 1/2 particle goes to its negative after a 2-pi rotation is because when you try to map SU(2) to SO(3), the mapping is 2-to-1, i.e. SU(2) is a double cover for SO(3).

What about spin 1 particles? The spin vector for that particle goes to itself after a 2-pi rotation. Does that mean that SU(3) maps to SO(3) 1-to-1?

As usual, thanks in advance.
 
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  • #3
Spin 1 means the angular momentum can be -1, 0 or +1? I think that makes it a quantum trit with operations in SU(3)? I would expect that to increase the many-to-one-ness when mapping to SO(2) instead of decreasing it. More specifically, I'd expect SU(3) to relate to SO(4) the way SU(2) relates to SO(3).

Please correct me if the above is wrong.

Related question: is the "SU(2) to SO(3) mapping is 2-to-1" statement intentionally ignoring global phase factors except for -1? Because there are four directions of rotation in SU(2) instead of three like in SO(3). There's x-wise, y-wise, z-wise, and phase-wise. At least, that's the issue when I recently tried to use quaternion spherical interpolation to gradually change between two unitary matrices: until I tweaked it to deal with the phase, the intermediate matrices ended up non-unitary.
 
  • #4
referframe said:
[...] Apparently, the reason that the spin vector for a spin 1/2 particle goes to its negative after a 2-pi rotation is because when you try to map SU(2) to SO(3), the mapping is 2-to-1, i.e. SU(2) is a double cover for SO(3).

What about spin 1 particles? The spin vector for that particle goes to itself after a 2-pi rotation. Does that mean that SU(3) maps to SO(3) 1-to-1?
.

Not the spin vector, but rather the wavefunction itself. For spin 1 particles, the symmetry group is still SU(2), no SU(3), but the <spin space> is 3 dimensional, no longer 2 dimensional, as for spin 1/2. This is from a purely Galilean perspective. .
 
  • #5
The symmetry group for any spin s particle is always SO(3) or SU(2), as they correspond to physical rotations. However the corresponding operators are different because you are representing SO(3) (SU(2)) in different spaces. For spin 1 you just have to find a representation of SO(3) on C^3.
 

FAQ: Spin 1 Particle Representations of SO(3) and SU(2)

What is SU(3)?

SU(3) is a mathematical group known as the special unitary group of degree 3. It consists of 3x3 unitary matrices with determinant 1.

How is SU(3) related to spin 1 particles?

SU(3) is a symmetry group that describes the behavior of spin 1 particles, such as the proton and neutron. The particles can be represented by 3-dimensional vectors, which transform under SU(3) transformations.

What is the significance of SU(3) for spin 1 particles in particle physics?

SU(3) is an important mathematical tool in particle physics as it helps to explain the strong nuclear force, which holds together the particles in the nucleus of an atom. The theory of SU(3) symmetry for spin 1 particles is also an essential component of the Standard Model of particle physics.

How does SU(3) symmetry manifest in the properties of spin 1 particles?

SU(3) symmetry is reflected in the properties of spin 1 particles, such as their mass, charge, and spin. These properties are related to the symmetries of the underlying SU(3) group. For example, the proton and neutron, which are both spin 1 particles, have similar properties due to their underlying SU(3) symmetry.

Are there any experimental evidences for SU(3) symmetry in spin 1 particles?

Yes, there have been numerous experimental studies that have confirmed the SU(3) symmetry in spin 1 particles. One example is the discovery of the eightfold way, which showed that the properties of particles could be organized into groups of eight based on their SU(3) symmetries. Additionally, particle accelerators have also provided evidence for SU(3) symmetries in the behavior of spin 1 particles.

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