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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(e.g. Minkowski space), it must change asgroup

[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, which I thought would remain unchanged.algebra

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

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# Proving the Jacobi identity from invariance

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