Understanding Casimir Operators: Explained for Non-Mathematicians

[Wiemster]

Can someone explain to me the concept of Casimir operators for someone who's not too familiar in abstract mathematics. E.g. What is the quadratic Casimir operator and why is it part of a maximally commuting set of operators?

 

[matt grime]

The casimir has the property that it commutes with all elements of the algebra essentially by construction. We construct such an object by brute force. Once we've done this they have the useful property that their generalizerd eigenspaces are eigenspaces for each element (this is why we make it commute), and by schur's lemma they act by scalar multiplication on their generalized eigenspaces. Or something like that. I'm a pure mathematician so not the best one to answer this question for you.

 

[Wiemster]

It's ok, I got the first part at least. So it just commutes with all the other operators by definition (is that what's called 'maximally commuting'?), do they always exist? And is there a simple way to find them (is it generally true that you can add the squares of all the other operators)?

 

[matt grime]

The 'maximally commuting subalgebra' is more properly called the centre (or improperly, the center). It is the set of all objects that commute with everything, hence the word maximally. The casimir is an element of the centre.

There are formulae for the casimir operators, and here it is:

http://en.wikipedia.org/wiki/Casimir_operator

or here

http://planetmath.org/encyclopedia/CasimirOperator.html

(just put the words you want to define into google).

I'm unsure about precisely what the 'quadratic' refers to.The casimir on sl_2 is ef+fe+2h^2, where e,f,h are the standard basis.

here is a useful link:

http://www.lepp.cornell.edu/spr/2002-07/msg0042697.html

 

[Wiemster]

Thx for the info! I liked the part: A physicist's "Casimir" is a mathematician's "element of the center of the universal enveloping algebra"...:tongue:

 

[zetafunction]

For a physicist a Casimir Operator is the 'one' that commutes with ALL the element of Algebra

several example

(classical Mechanics) $$H= a(p^{2} _x+p^{2} _y)$$

in fact the 'dispersion' relation $$E^{2}-p^{2}=m_{0}^{2}$$

in Special Relativity is just the Casimir operator of Lorentz Algebra

but..is not there any method or mathematical method to 'construct' a Casimir Operator ??