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Spin and symmetry

  1. Jul 24, 2012 #1
    I'm wondering how the spin of a particle, whether a particle is a fermion or a boson... how does this relate to the symmetry of a particle, U(1) or SU(2) or SU(3)? I'm trying to understand SUSY in relation to the other internal symmetries? Is there spin 1/2 and spin 1 particles associated with each of the symmetries U(1) or SU(2) or SU(3)? Or does each symmetry only have one possible spin?
  2. jcsd
  3. Jul 25, 2012 #2
    seems so,yes.....

    This comes from THE Lisa Randall [Harvard] book, WARPED PASSAGES, [2005], Chapter 19, Supersymmetry

    also, more generally:

    I'm no QM particle physics person, but if you are, the Randall book has gobs and gobs and gobs of detailed particle physics stuff neatly explained, no math. Used copies usually available on Amazon or your favorite source.
  4. Jul 25, 2012 #3
    I believe that U(1) symmetry leads to the electromagnetic field. All charged particles have this type of symmetry. SO(3) symmetry is normal rotational symmetry, a property shared by all bosons (having integer spin). SU(2) is the symmetry of fermions (spin 1/2). Finally, SU(3) is the symmetry of particles having "color"; i.e., quarks. Some particles have more than one type of symmetry. Quarks have U(1), SU(2) and SU(3) symmetries.
  5. Jul 25, 2012 #4
    Generally when you talk about "spin" you referre to SU(2) simmetry of a particle. When you say that a particle have 1/2 spin you mean that this particular particle transform under the fundamental representation of SU(2). This is a two dimensional representation and, in fact, a spin 1/2 particle can be represented by a vector:

    $$ \chi=\alpha\binom{1}{0}+\beta\binom{0}{1} $$

    and it can be transformed under SU(2) with the two dimensional fundamental representation.
    When you talk about a spin 1 particle, instead, you referre to a particle that transform under the adjoint representation of SU(2) which is a three dimensional one. And so on: spin 3/2 particle transform under a four dimensional representation of SU(2), ecc.

    So when you talk about spin 1/2, 1, ... you referre only to SU(2) symmetry.

    U(1) and SU(3) symmetries are referred to other degrees of freedom. As cygnet1 said, U(1) is the symmetry of em interaction (i.e. the quantum number associated is the electric charge), while SU(3) is the symmetry of color.
  6. Jul 25, 2012 #5
    Thank you. That's very clear. OK, so... there are many "representations" of the SU(2) symmetry. Is there one particular to the SM? And if so, does that limit spins to fermions or boson? Thank you.
  7. Jul 25, 2012 #6
    Yes, it is just like that. There exist a particle type for each representation of the symmetry group of the universe. They are different representations (homomorphism images) of the same group, though.

    The U(1) representation represents scalar (spin-0) particles. The SU(2) represents both vector (spin-1) and scalar (spin-0) particles. Higher groups describe particles of all lower spins. To exclude the lower spin particles, you must impose additional conditions.

    Bosons are associated with standard (vector-like) representations of the symmetry group. Fermions are associated with so called spin representations.
  8. Jul 26, 2012 #7
    Actually when you talk about SU(2) in the SM you are not referring to the symmetry that leads to spin. The SU(2) in SM is called "weak isospin". The mathematical threatment is exactly the same of the spin (it's the same group) but the physical meaning is different.

    In SM there are two different kind of fundamental particle. The "left" particles (i.e. with left handed chirality) and the "right" particles. Left particles form isospin doublets (they have 1/2 weak isospin just like electrons have 1/2 spin) and so transform with the fundamental representation of SU(2)[itex]_{weak}[/itex]. On the other hand right particle form isospin singlet and so they tranform under the identity representation.

    I'm not really sure but I guess that composed particle could form multiplets that transform under higher representation.
  9. Jul 27, 2012 #8
    According to the chart on the link below, all symmetries are involved with spin:


    Click on the "Field Content" in the table of contents.

    Can I conclude from this that spin is independent of whether the particle belongs to the symmetry of U(1), or SU(2), or SU(3)?
  10. Jul 27, 2012 #9


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    The SM has space-time symmetry as well as internal symmetry. Poincare group, [itex]SL(2,C)\odot T(4)[/itex], which is a spacetime symmetry, assigns SPIN and MASS to the particles. [itex]U(1)\times SU(2) \times SU(3)[/itex], which is the internal symmetry of the SM, gives rise to different CHARGES.

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