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Creation Operators application

  1. Jun 6, 2013 #1
    Can creation operators be used to find a matrix representation in a larger dimension? Is that maybe how I could find the 3D representation for SU(2) ?
     
  2. jcsd
  3. Jun 6, 2013 #2

    tom.stoer

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    Suppose you have generators, i.e. su(n) matrices, Taik in the fundamental representation with the commutation relations

    ##[T^a,T^b] = if^{abc}T^c##
    ##a = 1 \ldots n^2 - 1##

    Then one can define the operators

    ##Q^a = \sum_{ik}a^\dagger_i\,T^a_{ik}\,a_k##

    using creation and annihilation operators of the n-dim. harmonic oscillator.

    These operators form a representation acting on the Hilbert space; the commutation relations are

    ##[Q^a,Q^b] = if^{abc}Q^c##

    Therefore all algebraic relations for the T's can be translated for the Q's; the eigenstates can be classified according to the representations of the su(n) algebra.

    A nice side-effect is that the n-dim. harmonic oscillator has a much larger symmetry group, namely not SO(n) but SU(n). One can show that all Q's a conserved, i.e. that

    ##[Q^a,H] = 0##

    holds
     
  4. Jun 7, 2013 #3
    Thanks, Tom... I still have a lot to learn!!! Thanks for mentioning Hilbert Space... When does Minkowski space enter the picture? Does QFT use both Hilbert and Minkowski space, depending on the speed of the particles?
     
  5. Jun 7, 2013 #4

    tom.stoer

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    Hilbert space and Minkowskli space have nothing to do with each other.

    Hilbert space is the ∞-dim space of quantum states, e.g. harmonic oscillator states |n> or wave functions ψ.
    Minkowski space is the 4-dim. space of spacetime points x.
     
  6. Jun 7, 2013 #5
    :blushing:Thanks... by the way, I appreciate your previous post... it was just a little too advanced, and of course, that is my problem to fix!
     
  7. Jun 8, 2013 #6

    tom.stoer

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    do you mean #2 or #4? what's the problem? don't hesitate to ask
     
  8. Jun 8, 2013 #7
    Thanks for the nudge!!! #2!! I have read that the azimuthal quantum number is equal to 1 and therefore the magnetic quantum number (does the magnetic number always define the dimension of the space?) is equal to -1,0,1 for the 3D rep of Pauli matrices, but I don't know exactly how the value of l was found.... and I also don't know why we would construct the z component first, then use the ladder operators to find the other two matrices?? I have not taken QM (probably should've mentioned that!) :smile:
     
  9. Jun 9, 2013 #8

    tom.stoer

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    Now it seems that you are asking for the 3-dem. rep. of the su(2) matrices; the 2-dim. rep. are just the Pauli matrices (times 1/2) which give you s(s+1) = 3/4 and sz = 1/2. For all other n-dim. reps. procedure is always the same: find the matrices satisfying the commutation relations for su(2), calculate the eigenvalues for sz and calculate the the total spin s.

    All this has nothing to do with the harmonic oscillator but is valid on one the algebraic level, i.e. for matrices.

    In #2 I did two things: I generalized from su(2) to su(n), and I constructed the operator Q from the matrices T. This works only in the n-dim. rep. of su(n).

    Please note that the n-dim. rep. of su(2) and su(n) are two different things. You where asking in #1 for the 3D rep. of su(2) using the 3-dim. harmonic oscillator. My answer was that you don't get the 3-dim. rep. of su(2) but the 3-dim. rep. of su(3) which is not the same.
     
  10. Jun 10, 2013 #9
    Thanks for pointing out there is a difference between the two representations...
     
  11. Jun 10, 2013 #10

    tom.stoer

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    There is not only a difference of the reps, we are talking about different groups
     
  12. Jun 10, 2013 #11
    Yes, I see... Do the 3 Pauli matrices have any relation to the three force carriers W+ W- and Z0?
     
  13. Jun 10, 2013 #12

    tom.stoer

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    Yes.

    In an SU(n) gauge theory you have n2-1 gauge bosons; and there are n2-1 matrices Ta with a=1,2,...,n2-1. For SU(2) you have 22-1 Pauli matrices, for SU(3) you have 32-1 Gell-Mann matrices.

    The gauge fields are described by n2-1 vector fields Aaμ where again a=1,2,...,n2-1 is an su(n) index, and where μ=0,1,2,3 is a spacetime index. One can introduce matrix-values fields

    ##A_\mu = \sum_a A_\mu^a\,T^a##

    to describe the gauge fields which are then "rotated" by SU(n) matrices constructed from angles θa as

    ##U(\theta) = \exp[\sum_a \theta^a\,T^a]##

    So this how to describe the 3 gauge fields for W and Z in weak interactions, and the 8 gluon fields n QCD.
     
    Last edited: Jun 11, 2013
  14. Jun 11, 2013 #13
    Great! I was wondering where spacetime fit in... and I appreciate the clear distinction that there are the same number of generators as gauge bosons...
     
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