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Construction of Hamiltonian from Casimir operators

  1. Jun 22, 2011 #1
    In Greiner & Muller's 'Quantum Mechanics: Symmetries' (section 3.5) they explain that where a system possesses a symmetry, the corresponding Hamiltonian must be 'built up' from the Casimir operators of the corresponding symmetry group.

    Does anyone know of a reference where this is gone into in any detail?

    Does anyone know what happens when the system possesses a product of symmetries (such as, say, Poincare x SU(2))?

    Any help / references appreciated!
     
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  3. Jun 22, 2011 #2

    Bill_K

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    That statement seems a bit of a stretch. A weaker form of it might be, "the group operators can only appear in the Hamiltonian in combinations corresponding to the Casimir operators." But even this seems too strong. I don't see any Casimir operators in a Hamiltonian with a gauge symmetry.
     
  4. Jun 23, 2011 #3

    strangerep

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    Are you sure you quoted the correct book and section?

    In my 1989 (1st edition) copy of G+M sect 3.5 is indeed about Casimir operators,
    but I don't see anything in that (small) section about constructing Hamiltonians.
    I also checked the 2004 edition on Amazon, but it's much the same. Maybe you
    were thinking of a different book or section?

    For a completely integrable system, one can express the Hamiltonian in terms
    of conserved quantities, but they're different from Casimirs in general.

    I guess that you're talking about the meaning of "symmetry" as a conserved
    (time-invariant) symmetry. In that case the Hamiltonian commutes with all
    the generators of the symmetry and therefore must be a function of the
    Casimirs of the symmetry group (else the Hamiltonian would simply be
    another Casimir). G+M go into more detail about why this is so in their
    later section 3.8.

    Oh, here's the passage you probably meant. In my edition it's at
    the end of section 3.11 (Completeness Relation for Casimir Operators).
    They're talking in the context of multiplets (i.e., a subspace of the
    Hilbert space which transforms into itself under the action of the
    symmetry operators).

    I'm not sure what precisely you find unclear about their presentation
    so if you're still not happy you'll need to ask a more specific question...
     
  5. Jun 23, 2011 #4

    Bill_K

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    strangerep, Since you seem to agree with the original poster, maybe you can clear up my doubts on this. For every example I've been able to think of, the statement is not true.

    1) Schrodinger particle in a spherically symmetric potential. Hψ = h2/2m [- (1/r2)∂/∂r(r2∂ψ/∂r) + L2/r2] ψ + V(r) ψ. I see the Casimir operator L2 in there, but H is certainly not a function of L2.

    2) Spin-orbit coupling, in which a L·S term appears in the Hamiltonian. This is not a function of the Casimir operator J2 for the rotation group. While you can write J = L + S and therefore L·S = (J2 - L2 - L2)/2, you now have terms L2 and S2 which are not separately conserved and moreover not the Casimir operators of anything.

    3) The Dirac Equation, Hψ = β mc2 ψ - ihc α·∇ ψ. This is invariant under the Poincare group, whose Casimir operators P2 and W2 do not appear.

    4) Electromagnetism, L = - Aμ,ν Aμ,ν, invariant under gauge transformations Aμ → Aμ + λ. How does the statement about Casimir operators apply to this case?

    Quoting what you said,
    I don't see that this statement is true, and I think the examples above illustrate what's wrong with it. In each case, H commutes with all the generators of the symmetry group, but is not a function of the Casimir operators.
     
    Last edited: Jun 23, 2011
  6. Jun 23, 2011 #5
    That the Hamiltonian commutes with all the generators of the symmetry group doesn't mean that it is actually part of the group and therefore somehow related to the Casimir operators.
     
  7. Jun 23, 2011 #6

    strangerep

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    Actually, I was trying to encourage the OP to re-study the context in G&M in
    which their statement was made.

    G&M are talking strictly in the context of multiplets on which the Hamiltonian
    is degenerate.

    But more general cases are more complicated. E.g., for the spherically-symmetric
    Coulomb potential (non-rel hydrogen atom, let's say) the full dynamical group is SO(4,2)
    not merely SO(3), though this is a subgroup of SO(4,2). So in that case, one must
    work with the 3 Casimirs of SO(4,2) to get the full picture.

    In other words, the statement that "H itself has to be built up from invariant
    operators of the symmetry group" should be understood to mean the largest
    dynamical group applicable for that Hamiltonian.

    In a specific representation, the Casimirs take on numerical values appropriate to
    that representation. In the Dirac case, these are mass=m and spin=1/2, and
    there's obviously an "m" in the Dirac Hamiltonian.

    To say anything on the spin-orbit case, you'd need to specify the particular
    Hamiltonian. Generically, the dynamical group gets bigger, hence (probably)
    more Casimirs.

    [Gotta run, sorry. I hope Arnold Neumaier stops by. He could answer this
    much better than me.]
     
    Last edited: Jun 23, 2011
  8. Jun 24, 2011 #7

    A. Neumaier

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    Probably you meant ''may be''. The technique assumes that the Hamiltonian can be written in terms of generators from a representation of a finite-dimensional so-called ''dynamical group'', of which the symmetry group is a subgroup. In the simplest case of a chain of only two Lie algebras,, one writes H=H_0+V, where H_0 is invariant under the whole group, and the perturbation V only under the subgroup.

    The values of the possible Casimirs define the irreducible representations involved, and the generators of a Cartan subalgebra together with Casimirs of the chain of subalgebras define a commutative algebra in a concrete diagonal representation in which H_0 is diagonal. The problem then becomes a small matrix diagonalization problem.

    The details are outlined in Section 23.6 of my book

    Arnold Neumaier and Dennis Westra,
    Classical and Quantum Mechanics via Lie algebras,
    2008, 2011. http://lanl.arxiv.org/abs/0810.1019

    Much more detailed treatments, complete with examples, can be found for example in papers and books by Jachello.
     
  9. Jun 24, 2011 #8

    Bill_K

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    Ok, this has gradually evolved to a statement that I can agree with. IF the Hamiltonian can be written in terms of the generators of a finite-dimensional symmetry group, then those terms can be manifestly expressed as a combination of the Casimir operators.

    But I do not think it's tenable to claim that the Dirac equation is "built from" the values m and 1/2.
     
  10. Jun 25, 2011 #9

    samalkhaiat

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    Last edited: Jun 25, 2011
  11. Jun 25, 2011 #10

    samalkhaiat

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