# Completeness and orthogonality of unitary irreducible representations

• hideelo
In summary, Georgi proves in section 1.12 of "Lie Algebras and Particle Physics" that the matrix elements of the unitary irreducible representations form a vector space of dimension N, where N is the order of the group. This is shown by the fact that each matrix element can be described as an N-tuple of complex numbers, and that these matrix elements form an orthonormal set of vectors. The regular representation of the group, which is the space of all functions on the group elements, is also of dimension N, and is spanned by the orthonormal set of matrix elements. This is a result of the completeness and orthogonality of the matrix elements.
hideelo
I'm reading Lie Algebras and Particle Physics by Howard Georgi. He is trying to prove (section 1.12) that the matrix elements of the unitary irreducible representations (irreps) form a vector space of dimension N where N is the order of the group. For example for the matrix of the kth unitary irrep, consider element ij, it can be described as an N tuple of complex numbers and is therefore a vector in an N dim space. He shows that the collection of all such matrix elements form an orthonormal set of vectors.

Now he has to show that these vectors span an N dimensional space. He does this by first raising the point that the space of functions f:G--->C is N dimensional since in the regular representation we can write every function on the group elements as a functional, or a covector. I don't see however how this space is spanned by the vectors from the orthonormal proof.

Can anyone please help me out, and also I need some confirmation that what I think I understand, I actually got right.

Thanks

Hello there,

Thank you for sharing your thoughts on this topic. As a scientist who is familiar with Lie algebras and particle physics, I would like to provide some clarification and confirmation on your understanding.

Firstly, I agree with you that the vector space formed by the matrix elements of the unitary irreducible representations is indeed of dimension N, where N is the order of the group. This is because each matrix element can be described as an N-tuple of complex numbers, which can be seen as a vector in an N-dimensional space. And as you mentioned, Georgi shows that these matrix elements form an orthonormal set of vectors.

To address your question about how these vectors span an N-dimensional space, let me explain it in more detail. The regular representation of a group G is the set of all functions f: G -> C, where C is the set of complex numbers. In other words, it is the space of all possible functions on the group elements. Now, for each function f in this space, we can define a corresponding functional, or covector, by taking the inner product with each of the matrix elements from the unitary irreducible representations. This results in an N-dimensional space, as there are N matrix elements in each unitary irrep.

So, in essence, the orthonormal set of matrix elements spans this N-dimensional space of functions on the group elements, which is the regular representation. This can be seen as a consequence of the fact that the matrix elements form a complete and orthonormal set of vectors.

I hope this helps clarify your understanding. If you have any further questions, please feel free to ask. Keep up the good work in your studies!

## 1. What is the concept of completeness in unitary irreducible representations?

The concept of completeness in unitary irreducible representations refers to the idea that all possible states of a physical system can be represented by a linear combination of the basis states of a particular representation. In other words, the basis states form a complete set of states that can fully describe the system.

## 2. How are unitary irreducible representations related to the concept of orthogonality?

Unitary irreducible representations are related to orthogonality in that the basis states of a representation are mutually orthogonal. This means that the inner product of any two basis states is zero, indicating that they are perpendicular to each other. This property is important in quantum mechanics as it allows for the determination of probabilities for different states.

## 3. What is the significance of unitarity in unitary irreducible representations?

The term "unitary" in unitary irreducible representations refers to the fact that the transformations between different basis states in a representation preserve the inner product. This means that the transformations are reversible and do not change the overall structure of the representation. In quantum mechanics, unitarity is important as it ensures the conservation of probability.

## 4. How do unitary irreducible representations relate to the symmetry of a physical system?

Unitary irreducible representations are closely related to the symmetry of a physical system. In particular, they can be used to classify different symmetries and to study the effects of symmetry operations on a system. The basis states of a representation will transform in a predictable way under these operations, allowing for a deeper understanding of the system's symmetries.

## 5. What is the role of unitary irreducible representations in quantum field theory?

Unitary irreducible representations play a crucial role in quantum field theory. They are used to represent the states of particles and fields, and to study the interactions between them. In particular, they are used to construct the quantum states of particles and to determine the probabilities of different particle interactions. Without the use of unitary irreducible representations, the mathematical framework of quantum field theory would not be possible.

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