Pauli Matrices as generators of SU(2)

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SUMMARY

The Pauli spin matrices serve as the generators of the SU(2) group, which is crucial in quantum mechanics for representing spin in three dimensions. The 2x2 representation is utilized due to its simplicity, and the connection between SU(2) and Spin(3) is established through their isomorphic relationship, where SU(2) acts as a double cover of the rotation group in three dimensions. The Lie algebra of SU(2) mirrors that of the rotation group, confirming the foundational role of the Pauli matrices in this symmetry. Understanding the derivation of these matrices requires advanced concepts related to Clifford algebra.

PREREQUISITES
  • Understanding of SU(2) group theory
  • Familiarity with Pauli spin matrices
  • Knowledge of Lie algebras
  • Basic concepts of Clifford algebra
NEXT STEPS
  • Study the derivation of Pauli matrices from Clifford algebra
  • Explore the relationship between SU(2) and Spin(3)
  • Learn about the representation theory of Lie groups
  • Investigate applications of SU(2) in quantum mechanics
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students studying quantum mechanics, particularly those interested in the mathematical foundations of spin and symmetry in quantum systems.

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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?
 
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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.
 

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