On operator commutation

  1. Hi, I have a question,
    As it is said in QM, if two operators commute, they have so many common eigenstates that they form a basis. And the inverse is right.
    Now there is the question,
    if A,B,C are operators, [A,B]=0, [A,C]=0,
    then is "[B,C]=0" also right?

    If we simply say A and B, A and C both have common eigenstates, so B and C have common eigenstates, so [B,C]=0, it seems to be right.

    But in QFT, if x,y spacelike, then [\phi(x),\phi(y)]=0,
    if the above is right, then we can find a point z which is spacelike according to two non-spacelike point x,y to make any non-spacelike [\phi(x),\phi(y)]=0. It looks like a paradox.

    thank you!
     
  2. jcsd
  3. CarlB

    CarlB 1,204
    Science Advisor
    Homework Helper

    > If A,B,C are operators, [A,B]=0, [A,C]=0,
    > then is "[B,C]=0" also right?

    No, it's not right. For a counterexample in the usual QM variables, let [tex]A=x, B=y, C=p_y[/tex].

    For a counterexample in the Dirac gamma matrices,
    let [tex]A=\gamma^0, B=\gamma^1\gamma^2, C=\gamma^1\gamma^3[/tex].

    For a counterexample in QFT, replace the gamma matrices with your favorite four anticommuting field variables.

    In each of these counterexamples, A commutes with B and A commutes with C, but B and C do not commute.

    Carl
     
  4. No. The angular momentum operators give a counterexample: A = L^2, B = L_x, and C = L_y. Then [A,B] = [A,C] = 0. But [B,C]= [L_x, L_y] = ih L_z.

    It is true, however, that [B, C] commutes with A. This can be seen from the jacobi identity

    [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.
     
  5. CarlB

    CarlB 1,204
    Science Advisor
    Homework Helper

    > If we simply say A and B, A and C both have
    > common eigenstates, so B and C have common
    > eigenstates, so [B,C]=0, it seems to be right.

    If A has no degeneracy in its eigenvalues, then your logic works. In the presence of degeneracy, A can arrange to share a different set of eigenstates with B than it shares with C.

    Carl
     
  6. Galileo

    Galileo 2,001
    Science Advisor
    Homework Helper

    [A,B]=0 means you can find a set of eigenstates common to A and B.
    [A,C]=0 means you can find a set of eigenstates common to A and C.

    That doesn't imply these two sets are the same, so it will in general not give a set of eigenstates common to B and C.
     
  7. Thank you all, i see :)
     
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