Manuscript Em06-A@1 · Em06-A-v1.tex · lemma · ee9923c6

Lemma: Density and pure-gauge identities

Classical gauge legitimacy and support · lines 338–372 · proof included or linked

Em06-A@1#lem:gaugeidentities, Em06-A@1#eq:CK, Em06-A@1#eq:QK

For every smooth $ξ$ ,

C (K ξ) Q_{K ξ} = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .

Thus, on a free scalar solution,

Q_{K ξ} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .

Moreover, $Q_{h}$ is formally self-adjoint for real $h$ .

Proof. The trace of $K ξ$ is $2 \partial_{μ} ξ^{μ}$ . Differentiating (eq:C) gives (eq:CK).

Expanding (eq:Q) gives

Q_{h} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m} .

Also,

[P_{m}, X_{ξ}] = (□ ξ^{ν}) \partial_{ν} + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} .

Substitution proves (eq:QK). The on-shell identity follows from $P_{m} ϕ_{0} = 0$ . Formal self-adjointness follows by integration by parts in the divergence-form expression (eq:Q).

Displayed formulas in this passage

Em06-A@1#eq:CK · line 341
$C (K ξ) Q_{K ξ} = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .$
Em06-A@1#eq:QK · line 341
$C (K ξ) Q_{K ξ} = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .$
unlabeled · line 348
$Q_{K ξ} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .$
unlabeled · line 359
$Q_{h} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m} .$
unlabeled · line 364
$[P_{m}, X_{ξ}] = (□ ξ^{ν}) \partial_{ν} + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} .$

Lemma: Density and pure-gauge identities

Displayed formulas in this passage

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