Understanding Casimir Operators: Explained for Non-Mathematicians

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In summary: The 'maximally commuting subalgebra' is more properly called the center (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 center.
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Wiemster
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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?
 
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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.
 
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  • #3
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)?
 
  • #4
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
 
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  • #5
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"...:-p
 
  • #6
For a physicist a Casimir Operator is the 'one' that commutes with ALL the element of Algebra

several example

(classical Mechanics) [tex] H= a(p^{2} _x+p^{2} _y) [/tex]

in fact the 'dispersion' relation [tex] E^{2}-p^{2}=m_{0}^{2} [/tex]

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 ??
 

FAQ: Understanding Casimir Operators: Explained for Non-Mathematicians

What are Casimir operators?

Casimir operators are mathematical operators that arise in the study of symmetries in physics. They play a crucial role in understanding the behavior of physical systems and can be used to find conserved quantities such as energy and momentum.

Why is it important to understand Casimir operators?

Understanding Casimir operators is important because they provide a powerful tool for analyzing physical systems and predicting their behavior. They are also essential in the development of quantum mechanics and have applications in many areas of physics, including particle physics and quantum field theory.

Do I need to be a mathematician to understand Casimir operators?

No, you do not need to be a mathematician to understand Casimir operators. While a basic understanding of mathematics is helpful, the concepts can be explained in a way that is accessible to non-mathematicians.

What are some real-world examples of Casimir operators?

Casimir operators have been used to study the behavior of atoms, molecules, and even particles in the Large Hadron Collider. They are also important in the development of new technologies, such as quantum computing.

Are there any practical applications of understanding Casimir operators?

Yes, understanding Casimir operators has many practical applications in physics, engineering, and technology. They can be used to predict the behavior of physical systems, design new materials, and develop new technologies. Additionally, they have applications in fields such as chemistry, biology, and computer science.

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