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Proving the Jacobi identity from invariance

  1. Apr 17, 2012 #1
    "Proving" the Jacobi identity from invariance

    Hi all,

    In an informal and heuristic manner, I have heard that the "change" in something is the commutator with it, i.e. [itex]\delta A =[J,A][/itex] for an operator [itex]A[/itex] where the change is due to the Lorentz transformation [itex]U = \exp{\epsilon J} = 1 + \epsilon J + \ldots[/itex] where [itex]J[/itex] is one of the six generators of the Lorentz group (rotation or boost). That is, if we have an operator [itex]\phi\ :\ G\to G[/itex], where [itex]G[/itex] is the vector space spanned by the size generators [itex]J_i,K_i[/itex] of the Lorentz group, (i.e. [itex]G[/itex] is the vector representation of the Lorentz algebra) then
    [tex]\delta (\phi(T)) = \delta\phi (T) + \phi (\delta T)[/tex]
    so, using the above definition of "change"
    [tex][J,\phi(T)] = \delta\phi (T) + \phi ([J,T])[/tex].

    We can then define [itex]\phi[/itex] to be invariant by saying that [itex]\delta\phi = 0[/itex], and hence

    [tex][J,\phi(T)] = \phi([J,T])[/tex].

    If one does the same for a Lie product [itex]\mu(X,Y) = [X,Y][/itex] then

    [tex]\delta\mu(Y,Z) =\delta\mu (Y,Z) + \mu(\delta Y, Z) + \mu(Y,\delta Z)[/tex]

    We say that [itex]\mu[/itex] is invariant and set [itex]\delta\mu = 0[/itex] and hence

    [tex][J,\mu(Y,Z)] =\mu([J,Y], Z) + \mu(Y,[J, Z])[/tex]
    or

    [tex][J,[Y,Z]] =[[J,Y], Z] + [Y,[J, Z]][/tex]

    which is the Jacobi identity. This seems great, but I don't understand a few points.

    1. I believe the Lie product commutator enters as if we have an operator [itex]A[/itex] on the vectors in the Lorentz group (e.g. Minkowski space), it must change as
    [tex]A\to A' = UAU^{-1} = A + \epsilon [J,A] + \ldots[/tex]
    correct? But in the above description with [itex]\phi[/itex] and [itex]\mu[/itex], these are operators on the Lorentz algebra, which I thought would remain unchanged.

    2. Is the expression
    [tex]\delta\mu(Y,Z) =\delta\mu (Y,Z) + \mu(\delta Y, Z) + \mu(Y,\delta Z)[/tex]
    rigourous? What about terms like [itex]\mu(\delta Y, \delta Z)[/itex]? Or are those second order?

    Any help would be great,

    Ianhoolihan
     
  2. jcsd
  3. Apr 22, 2012 #2

    samalkhaiat

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    Science Advisor

    Re: "Proving" the Jacobi identity from invariance

    Can you please clarify the following?
    What is [itex]T[/itex]? Can you give me a specific example of [itex]\phi (T)[/itex]?
    Why is it that [itex]\delta[/itex] acts on both [itex]\phi[/itex] and its “argument” [itex]T[/itex]? It looks like that you defined [itex]\phi[/itex] to be a Lie algebra representation or a Lie algebra-valued operator! So, if [itex]\exp (\epsilon \phi)[/itex] is not the identity, what does it mean to set [itex]\delta \phi = 0[/itex]? The same goes for [itex]\mu[/itex]; I can take it to be the linear map [itex](\mu(X))(Y) = [X,Y][/itex], defined on the lie algebra such that (representing the lie algebra in itself)
    [tex]\mu([X,Y]) = [\mu(X),\mu(Y)].[/tex]
    You can check that such map guarantees the Jacobi identity; expand both sides of
    [tex]\mu([X,Y])(Z) = [\mu(X),\mu(Y)](Z).[/tex]
    So, what does it mean to set [itex]\delta \mu = 0[/itex]?
    In the same way, we can define the action of [itex]\delta_{J}[/itex] on lie algebra element [itex]X[/itex](or, on a lie algebra-valued function [itex]f(x;X)[/itex]) by
    [tex]\delta_{J}X = [J,X].[/tex]
    This means that [itex]\delta_{J}[/itex] acts as derivation, i.e., it guarantees Jacobi identity. This is because Lie brackets are derivations;
    [tex]\delta_{J}[X,Y] = \delta_{J}(XY) - \delta_{J}(YX),[/tex]
    This gives the Jacobi identity:
    [tex]\delta_{J}[X,Y] = [J,XY] - [J,YX] = [X,\delta_{J}Y] + [\delta_{J}X, Y].[/tex]

    Sam
     
  4. Apr 22, 2012 #3
    Re: "Proving" the Jacobi identity from invariance

    Thanks Sam.

    [itex]T[/itex] is a vector in the vector space of generators for the algebra. An example would be [itex]T=J_i, K_i[/itex] in the Galilean algebra.

    Say, the trivial one: [itex]\phi(T)=T[/itex] or [itex]\phi(T) = c T[/itex] for some constant [itex]c[/itex].

    That is my question 2. I can only reason along the lines of the chain/product rule. A variation in an evaluated function must depend on the variation in the function, and the variation in the thing it is acting on. Formalising this is what I'm looking for.

    [itex]\exp (\epsilon \phi)[/itex] is not the identity. I believe the point is what I made above --- under a given transformation of the vector space, both the vectors [itex]\phi[/itex] is operating on, and the 1--cochain [itex]\phi[/itex] may depend on the transformation. In differential geometric language, this would be like saying that both the 1--form and vector bases transform, and so may the "coefficients". Setting [itex]\delta \phi = 0[/itex] means the coefficients do not change. I'm not even 50% sure myself, however.

    I do not know what you mean by representing a lie algebra in itself, but, as far as I am concerned, the lie product [itex]\mu[/itex] is a bilinear antisymmetric 2--cochain. Yours is a 1--cochain it seems...?

    Anyway, it turns out that I wasn't as clued up about Lie algebras etc as I thought. I still am not, but will look again at Kirillov soon. If you've got any comments on this question, however, they'd still be much appreciated.

    Ianhoolihan
     
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