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Meaning of this operator

by Karlisbad
Tags: meaning, operator
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Oct5-06, 02:22 AM
P: 134
From the "Lie Group" theory point of view we know that:

[tex] p [/tex] := is the generator for traslation (if the Lagrangian is invariant under traslation then p is conserved)

[tex] L [/tex]:= s the generator for rotation (if the Lagrangian is invariant under traslation then L is conserved)

(I'm referring to momentum p and Angular momentum L, although the notation is obvious )

My question is if we take the "Lie derivative" and "covariant derivative" as a generalization of derivative for curved spaces.. if we suppose they're Lie operators..what's their meaning?..if the momentum operator acts like this:

[tex] pf(x)\rightarrow \frac{df}{dx} [/tex] derivative of the function..could the same holds for Lie and covariant derivative (covariant derivative is just a generalization to gradient, and i think that Lie derivatives can be expressed in some cases as Covariant derivatives, in QM the momentum vector applied over the wave function is just the gradient of the [tex] \psi [/tex]
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