8B (Vectors and Matrices)

All three Pauli matrices are Hermitian. Direct multiplication gives

σ_{a}^{2} = I

and, for

a \neq = b

σ_{a} σ_{b} = i ε_{ab c} σ_{c}

, which combines as the displayed identity.

b

Because

(n \cdot σ)^{2} = I

, even and odd powers sum separately:

R = I cos θ + i (n \cdot σ) sin θ

. Then

R R^{†} = I

A = (x_{3} x_{1} + i x_{2} x_{1} - i x_{2} - x_{3})

, so

tr A = 0

and

det A = - ∣ x ∣^{2}

. Unitary conjugation preserves Hermiticity, trace, and determinant; every traceless Hermitian

2 \times 2

x' \cdot σ

with real

x'

, and

∣ x'∣ = ∣ x ∣

Using

(a \cdot σ) (b \cdot σ) = (a \cdot b) I + i (a \times b) \cdot σ

gives

x' = x cos (2 θ) - (n \times x) sin (2 θ) + n (n \cdot x) (1 - cos (2 θ)) .

The sign follows from the convention

R = e^{i θ n \cdot σ}

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