Understanding SU(2) Representations and Their Role in Particle Physics

In summary, the conversation discusses SU(2) as a group of 2x2 matrices with Pauli matrices as generators that act on a vector space. It is noted that SU(2) can be represented by any n x n matrix, but the author is confused about why it is still called SU(2). The concept of representation of the Hilbert space itself is introduced, and the question is posed on why it is necessary. The conversation also touches on the concept of abstract groups and their representations, as well as the use of (j, m) notation for representations of the Hilbert space. It is suggested to look at discussions on quantum-mechanical angular momentum for more information on representations of SU(2).
  • #1
Silviu
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Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a vector space. Also you can have any n x n representation of SU(2) (here I am a bit confused why is it called SU(2) if you can represent it as any n x n matrix).
So I assumed that this means they act (in the 2 x 2 case) on 2x1 column matrices. But, from what i read in the attached screenshots, I understand that the Hilbert space where these matrices act is the one that undergoes a representation, as they say "to completely reduce the Hilbert space of the world to block diagonal form". And also the (j, m) notation I thought it was for eigenvectors, but they say that they use that notation for the representations of the Hilbert space. From what I read (I just started all this stuff with representation theory so I really need help to understand this), I understood that the matrices that act on the Hilbert space can have different representations, but the Hilbert space itself (formed of vectors) stays the same.
So, to resume, my question is why you need to have a representation of the Hilbert space itself and not just of the operators (the Pauli matrices in this case), or in case I misunderstood what I read, what does the author means by the parts I attached? Thank you!
 

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  • #2
Maybe the question can be boiled down to "Why do we need abstract groups and can we describe everything about a group with a specific set of generator transformations?" Some points
  • There may be multiple choices of the field type (##\mathbb{R}## or ##\mathbb{C}##) and dimension of linear transformations which obey the abstract group algebra. As you mention ##SU(2)## can be represented by ##n\times n## matrices as long as ##n\geq2##.It may be useful to see matrix representation as a separate notion from the group itself.
  • Pauli matrices have an element of arbitrary choice. Even with field and dimension fixed there is an element of arbitrary parameter-ization which doesn't reflect any new group structure.
  • Not all Lie groups are simply connected and can be expressed as exponential expansion. The Lorentz group is a common example of a Lie group that is not simply connected.
As for the statement about ##(j,m)## - they are looking at representations that consist of operators on a subspace of the ##\mid j,m\rangle## states. They label the representations by ##j##, which is the highest ##J_3## value in that subspace. The dimension of a subspace of fixed ##j## is ##2j+1##.

So a ##j=\frac{1}{2}## representation would be ##2\frac{1}{2} + 1 = 2## dimensional and thus we would use ##2\times2## matrices.
 
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  • #3
A good place to look for stuff on representations of SU(2) is discussions of quantum-mechanical angular momentum. Each possible value of angular momentum j corresponds to a representation (rep) of SU(2), an irreducible representation (irrep) with dimension 2j+1. Each of the dimensions in it corresponds to a value of the projected angular value or magnetic quantum number m. The matrices of the rep are given by the Wigner D-matrices, matrices for doing 3D rotations in that rep. Addition of angular-momentum values corresponds to a product representation constructed from each value's rep. The resulting angular-momentum values correspond to irreps in that rep.
 

Related to Understanding SU(2) Representations and Their Role in Particle Physics

What is SU(2) representation?

SU(2) representation is a mathematical concept used in the study of group theory. It refers to the way in which a group element can be represented by a matrix. In SU(2) representation, the matrices are 2x2 unitary matrices with a determinant of 1.

What are the applications of SU(2) representations?

SU(2) representations have many applications in physics, particularly in the study of quantum mechanics and particle physics. They are also used in other areas of mathematics, such as differential geometry and topology.

What is the difference between SU(2) and SO(3)?

SU(2) and SO(3) are both groups with important applications in physics. The main difference between them is that SU(2) is a complex group, while SO(3) is a real group. This means that SU(2) has more degrees of freedom and is better suited for describing quantum systems.

How are SU(2) representations classified?

SU(2) representations are classified by their dimension, which is given by a positive integer. The dimension of a representation determines the number of basis vectors needed to describe it. This classification is important in understanding the properties of the representation.

What are the physical significance of SU(2) representations?

SU(2) representations have physical significance because they are used to describe the properties of particles in quantum mechanics. In particular, they are used to describe the spin of particles, which is a fundamental property that determines how they behave in the presence of external fields.

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