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I know spin-1/2 Pauli matrices satisfy the anticommutation relationship $\{\sigma_i, \sigma_j\}=2\delta_{ij} \mathbb{I}$. I wonder how this can be proved without writing down the matrix representation of these matrices and performing matrix multiplication. As the matrix representation of angular momentum operator (and hence Pauli matrix) can be written down just using the commutation relationship $[\hat{J_x},\hat{J_y}]=i\hbar\hat{J_z}$ and its cyclic substitutions, I think there should be a way to prove this anticommutation relationship just using the commutation relationship and without using any specific matrix representation of Pauli matrices.

I tried to follow a similar fashion as in determining matrix representation of $\hat{J_x}$ and $\hat{J_y}$, but the use of the lowering and raising operators $\hat{J_-}$ and $\hat{J_+}$ (which I believe may be useful in the proof) only occurs when evaluating the matrix entry $\langle s,m'|J_\pm|s,m\rangle$, which is something I want to avoid. As a result, I failed to finish the proof.

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2 Answers 2

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It will not be possible to derive the anti-commutators from the commutation relations alone, because not every representation of the commutation relations (i.e., of the algebra $\mathfrak{su}(2)$) satisfies $\{L_i, L_j\} = 2\delta_{ij}$. For example, $\sigma_x$ and $\sigma_y$ anti-commute, but the spin-1 matrices $$ L_x = \frac{\hbar}{\sqrt 2} \pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 1 \cr 0 & 1 & 0 } \qquad \text{and}\qquad L_y = \frac{\hbar}{\sqrt 2} \pmatrix{ 0 & -i & 0 \cr i & 0 & -i \cr 0 & i & 0 } $$ do not. It seems to me that the anti-commutation relations of the Pauli matrices are a coincidence.

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The pauli matrices are simply a matrix basis which (up to a factor of i) represents a basis for Cartesian bivectors. In this sense they are isomorphic to the quaternions, meaning they form a representation of the Clifford algebra of $\mathbb{R}^{3}$ .

As another answerer has pointed out, the dimension of one's representation of this algebra determines whether or not the the generators will anticommute. If one would like to study non-relativistic spin-1/2 fermions, one must use a representation which reflects the change of sign of the wave functions of spin-1/2 under a rotation of $2\pi$, and it turns out one can use a 2×2 complex hermitian representation to get this job done. In this sense one does not derive the anticommutation relations (which tell you about the representation being used) from the commutation relations (which are determined by the space being described) because one chooses the anticommuting representation to describe what one observes.

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