Is the given Pauli matrix in SU(2)?

In summary, the Pauli matrix provided in the conversation is not in SU(2) form. However, Pauli matrices themselves are not elements of SU(2), but rather generators for infinitesimal SU(2) transformations. This is based on the exponentiation of a linear combination of Pauli matrices.
  • #1
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Can I check with someone - is the following pauli matrix in SU(2):

0 -i
i 0

Matrices in SU(2) take this form, I think:

a b
-b* a*

(where * represents complex conjugation)

It seems to me that the matrix at the top isn't in SU(2) - if b=-i, (-b*) should be -i...

However, my notes say otherwise (that all pauli matrices are in SU(2)).
 
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  • #2
Pauli matrices are not actually unitary matrices and thus are not actually themselves elements of SU(2). They are the traceless and Hermitian 'generators' of infinitesimal SU(2) transformations. I.e., an arbitrary SU(2) matrix is given by exponentiation of a linear combination of Pauli matrices.

This is if my memory serves me correctly. I'm sure someone will correct me if not.
 

Related to Is the given Pauli matrix in SU(2)?

1. What are Pauli matrices and what is their significance in SU(2)?

Pauli matrices are a set of three 2x2 complex matrices (σx, σy, σz) named after physicist Wolfgang Pauli. They are significant in SU(2), a special unitary group in mathematics, because they form a basis for the Lie algebra of SU(2) and play a crucial role in quantum mechanics and particle physics.

2. How are Pauli matrices related to spin and angular momentum?

Pauli matrices are closely related to spin and angular momentum in quantum mechanics. In fact, they represent the spin operators for spin-1/2 particles and can be used to calculate the components of angular momentum in a given direction.

3. What is the physical interpretation of the Pauli matrices?

The physical interpretation of the Pauli matrices is that they represent the spin states of a quantum system. This means that they can be used to describe the orientation of the spin of a particle in space.

4. What is the mathematical relationship between Pauli matrices and the special unitary group SU(2)?

The connection between Pauli matrices and SU(2) lies in their algebraic properties. The Pauli matrices, along with the identity matrix, form a basis for the Lie algebra of SU(2). This means that they can be used to generate all other matrices in SU(2) through linear combinations.

5. How do Pauli matrices relate to the concept of quantum entanglement?

Pauli matrices play a crucial role in the study of quantum entanglement, which is the phenomenon where two or more particles become connected in such a way that the state of one particle is dependent on the state of the others. Pauli matrices are used to describe the entanglement of spin states in quantum systems.

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