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Dirac rest projections and two spin-component sectors

Example

Take the fiber matrices of Dirac Clifford matrices and energy projections at $k = 0$ with $μ > 0$ . Then $ω = μ$ and the positive and negative matrix eigenspaces each have dimension two. Define the dimensionless third spin-component matrix in this Pauli-block representation by $S_{3} : = \frac{1}{2} diag (σ_{3}, σ_{3})$ ; here the two spin-component sectors mean its $+ \frac{1}{2}$ and $- \frac{1}{2}$ eigenspaces within each energy eigenspace. The coordinate spinors below are vectors in the finite-dimensional fiber $C^{4}$ ; they are not point-supported, normalizable vectors in the continuum Hilbert space $H_{D}$ . For $μ = 0$ , the point $(ω, k) = (0, 0)$ is the omitted mass-shell apex, so this example makes no massless rest-state claim.

Facts & Assumptions

Given: $μ > 0$ , $k = 0$ , the fixed Pauli-block matrices and energy projections, and the ordinary Hermitian inner product on $C^{4}$ .

[F1]

$β = diag (I_{2}, - I_{2})$ , $h_{D} (k) = α \cdot k + β μ$ , $ω = \sqrt{∣ k ∣^{2} + μ^{2}}$ , and $P_{\pm} = (I_{4} \pm h_{D} / ω) / 2$ . Dirac Clifford matrices and energy projections

[F2]

$P_{\pm}$ are complementary orthogonal projections of rank two. Dirac matrix identities and orthogonal energy projections

Verification

Given: The data and matrix conventions in [F1]–[F2].

1.1F1givenalgebra

At $k = 0$ , the definition gives $h_{D} (0) = β μ$ and $ω (0) = μ$ . Since $μ > 0$ , $h_{D} (0) / ω (0) = β$ , so $P_{+} (0) = (I_{4} + β) / 2 = (\begin{array}{cc} I_{2} & 0 \\ 0 & 0 \end{array})$ and $P_{-} (0) = (I_{4} - β) / 2 = (\begin{array}{cc} 0 & 0 \\ 0 & I_{2} \end{array})$ .

2.1F1F2step 1.1algebra∎

Write $e_{1} = (1, 0, 0, 0)^{T}$ , $e_{2} = (0, 1, 0, 0)^{T}$ , $e_{3} = (0, 0, 1, 0)^{T}$ , and $e_{4} = (0, 0, 0, 1)^{T}$ . The first two standard coordinate vectors span $ran P_{+} (0)$ and the last two span $ran P_{-} (0)$ ; they are orthonormal in $C^{4}$ . The matrix $S_{3}$ has eigenvalue $+ \frac{1}{2}$ on $e_{1}, e_{3}$ and $- \frac{1}{2}$ on $e_{2}, e_{4}$ . Thus each of the two rank-two energy eigenspaces is the orthogonal sum of its two one-dimensional spin-component eigenspaces, agreeing with [F2].

Dirac rest projections and two spin-component sectors

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources