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What is internal space/internal symmetry?

  1. Feb 27, 2012 #1

    I have already been familiar with that exterior symmetry is the spacetime symmetry. Such kind of symmetry has been depicted by Poincare' group.

    Then I am still find the concept internal space/internal symmetry ambiguous.

    And I cannot understand why put Spin in exterior symmetry not internal symmetry, since there is no space-time variation for Spin.

    And I have heard someone told me that exterior symmetry has something to do with U(1) group. I cannot understand it. I think exterior symmetry is associated with Poincare' group.

    The last question, are the two concepts internal symmetry and intrinsic symmetry of the same meaning?

    Thanks very much.
    Last edited: Feb 27, 2012
  2. jcsd
  3. Feb 28, 2012 #2
    One of my questions has been resolved.

    Spin is considered in angular momentum term, so it should have spacetime variation. Thus, it is in exterior symmetry.
  4. Feb 29, 2012 #3


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    U(1), and gauge symmetries in general, are considered internal symmetries because the transformations are not occurring in spacetime. The symmetries correspond to internal "charges" (the charges are those quantities in the theory that are conserved under the symmetry transformations.) For example, the action of U(1) is to impart a phase shift to the wavefunction (or particle). Saying that two particles that differ by a phase shift are actually the same results in the existence of the electric charge. Mathematically, the theory describing the electromagnetic interaction is invariant under U(1) operations. This is an internal symmetry because there are no spacetime operations (like rotations or translations) that can change the phase of the wavefunction.
  5. Feb 29, 2012 #4

    How about SU(3), SU(5)? Are they all internal symmetry?
  6. Feb 29, 2012 #5


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    Yup. SU(3) operates on the "color" degrees of freedom which are internal to hadrons. Basically, SU(3) performs a rotation in color-charge space, allowing hadrons to swap colors. Invariance under this operation implies the existence of the strong force, which facilitates the color change.
  7. Apr 20, 2012 #6
    I was wondering about this, too. What kind of symmetry is the internal spin symmetry?

    It is not a spacetime symmetry, but a symmetry taking place at a point in spacetime. It rotates the spin of point particles. But then also it is in a finite dimensional representation of the Lorentz symmetry.

    So can we picture internal spin as acting on little tangent spaces attached at every point in spacetime, just as the U(1), SU(2) and SU(3) internal symmetries?

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