Pauli Matrices as generators of SU(2)

[OSUPhysics]

Why is it that the Pauli spin matrices ( the operators of quantum spin in x,y,z) are the generators of a representation of SU(2)? I understand that we use the 2X2 representation as it is the simplest, but why is it that spin obeys this SU(2) symmetry and how is it that we come up with the Pauli matrices for the spin operators?

 

[Ben Niehoff]

SU(2) is isomorphic to Spin(3), which is the double cover of the rotation group in three dimensions. That is, its Lie algebra is identical to the rotation group, but the map between the groups themselves is 2-to-1.

The real group of interest here is Spin(3), which is constructed via Clifford algebra. In this case, it happens to be equivalent to SU(2).

Deriving the Pauli matrices from scratch involves some slightly more advanced ideas that maybe someone else has more time to relate.

 

[Entropia]

The Pauli spin matrices, also known as the Pauli matrices, are a set of three 2x2 matrices that represent the operators of quantum spin in the x, y, and z directions. These matrices are important in quantum mechanics as they provide a mathematical framework for understanding the behavior of spin particles, such as electrons.

The reason why the Pauli spin matrices are the generators of a representation of SU(2) is due to the underlying symmetry of the quantum spin system. In physics, symmetry refers to the invariance of a physical system under certain transformations. In the case of spin particles, the system is invariant under rotations in three-dimensional space. This means that the physical properties of the system remain the same regardless of how it is oriented in space.

The group of rotations in three-dimensional space is known as the special unitary group SU(2). This group has important mathematical properties that make it useful in understanding the behavior of quantum spin particles. In particular, SU(2) is a Lie group, which means that it can be represented by a set of generators. The Pauli spin matrices are these generators for the SU(2) group.

To understand how the Pauli matrices arise as the generators of SU(2), we need to look at the mathematical properties of the group. The SU(2) group is a special case of the more general SU(n) group, which represents rotations in n-dimensional space. The generators of the SU(n) group are a set of n x n matrices that satisfy certain algebraic conditions. For the case of SU(2), the generators are 2x2 matrices, and the Pauli matrices are a specific set of these generators that satisfy the necessary algebraic conditions.

In summary, the Pauli spin matrices are the generators of SU(2) because they arise from the underlying symmetry of the quantum spin system, and they satisfy the necessary mathematical properties of the SU(2) group. This connection between the Pauli matrices and SU(2) is essential in understanding the behavior of spin particles and has implications in various areas of physics, such as quantum computing and particle physics.