casimir element Definition and 1 Threads

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.
More generally, Casimir elements can be used to refer to any element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism.
The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.

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  1. redtree

    I The Pauli-Lubanski vector and angular momentum

    Given $$M_{\rho \sigma} = i (x^{\rho} \partial_{\sigma} - x^{\sigma} \partial_{\rho})$$ and $$W^{\mu} = \frac{1}{2} \epsilon^{\mu \nu \rho \sigma} P_{\nu} M_{\rho \sigma}$$ Why does ##W^{\mu}## pick up only the spin part of the total angular momentum?
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