# Wigner-Eckart Theorem: Rigorous Math Treatment

• WiFO215
In summary, the conversation is about the search for material that covers the Wigner Eckart theorem and mathematics of Angular Momentum in a more rigorous manner. The person is open to books, papers, and articles that approach the topic from a mathematical perspective, particularly using Lie algebras and groups. A recommendation for Fuchs & Schweigert's "Symmetries, Lie Algebras and Representations" is provided.
WiFO215
I originally posted this in the Science Book and discussion forum but received no help. Am posting it here, hoping that I will.

I was looking for material that would go over the Wigner Eckart theorem and mathematics of Angular Momentum in more rigor than the traditional texts do (in specific Ballentine). I am not only looking for books, but any papers, articles which treat the following in a more rigorous mathematical footing would be appreciated.

What kind of rigor are you looking for?

From the point of view of Lie algebras and groups. Hopefully, one that develops the material from scratch.

For a mathematical treatment of this kind of thing, I have found Fuchs & Schweigert "Symmetries, Lie Algebras and Representations" good.

Thanks henry. I'll look into it.

## 1. What is the Wigner-Eckart Theorem?

The Wigner-Eckart Theorem is a mathematical expression that relates the matrix elements of a unitary irreducible representation of a group to the matrix elements of a reference representation of that group. It is commonly used in physics to simplify calculations related to angular momentum and symmetry.

## 2. How is the Wigner-Eckart Theorem used in physics?

The Wigner-Eckart Theorem is used to simplify calculations related to angular momentum and symmetry in physics. It allows for the determination of matrix elements in terms of simpler reference matrix elements, making complex calculations more manageable.

## 3. Can you explain the rigorous mathematical treatment of the Wigner-Eckart Theorem?

The rigorous mathematical treatment of the Wigner-Eckart Theorem involves using group theory and representation theory to derive the general form of the theorem. This includes defining the irreducible representations of a group, determining the transformation properties of these representations, and using these properties to derive the Wigner-Eckart Theorem.

## 4. What is the significance of the Wigner-Eckart Theorem in quantum mechanics?

The Wigner-Eckart Theorem is significant in quantum mechanics because it allows for the simplification of complicated calculations involving angular momentum and symmetry. It also provides a way to connect different representations of a group, making it a powerful tool in understanding the underlying symmetries of physical systems.

## 5. Are there any limitations to the Wigner-Eckart Theorem?

The Wigner-Eckart Theorem has some limitations, particularly in cases where the system under study does not possess the full symmetry of the group being considered. In these cases, the theorem may not be applicable and alternative methods must be used. Additionally, the theorem is limited to unitary irreducible representations, so it cannot be applied to all groups and physical systems.

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