Manuscript XX-E@1 · XX-E.tex · section · cf7a8f40

Residual gauge invariance can be vacuous

Why the free input does not supply a quantum Ward operation › Residual gauge invariance can be vacuous · lines 694–721

XX-E@1#eq:free

Use the fixed free operators

H z F_{ν} z r A = (- \frac{1}{2} δ E_{μν} [z], δ E_{ϕ} [z]), = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h - M^{- 2} \partial_{ν} ϕ ˉ φ, = M^{2} /2, B = H - r F^{†} F, Q = F K, = diag (\frac{M ^{2}}{4} T, id), L = A^{- 1} B .

Here $T$ is four-dimensional trace reversal. The differential principal part of $L$ is $□_{g ˉ} id$ , and $Q$ is normally hyperbolic. These are the reproduced XX-B operators.

Displayed formulas in this passage

XX-E@1#eq:free · line 697
$H z F_{ν} z r A = (- \frac{1}{2} δ E_{μν} [z], δ E_{ϕ} [z]), = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h - M^{- 2} \partial_{ν} ϕ ˉ φ, = M^{2} /2, B = H - r F^{†} F, Q = F K, = diag (\frac{M ^{2}}{4} T, id), L = A^{- 1} B .$

Residual gauge invariance can be vacuous

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