Manuscript X-F@1 · X-F.tex · definition · f35cc129

Definition: Field content and the second-order quantum metric

Frozen sheet › Field content, kinetic operator, switched source, response · lines 532–568

X-F@1#def:content, X-F@1#eq:metric, X-F@1#eq:Q-def, X-F@1#eq:source, X-F@1#eq:khat

q = η + κ (h_{cl} + h) + κ^{2} (k_{cl} + k), q ˉ = η + κ h_{cl}, u := h + κ k,

with $ϕ$ the free Klein–Gordon field, $h$ the free flat gauge-fixed graviton of xe:algebra, $h_{cl}, k_{cl}$ smooth classical off-shell backgrounds, and $k$ defined as follows.

The kinetic operator of the fluctuation at background $q ˉ$ is the Euler derivative of the part of $S_{gr} [q ˉ + κ u] + S_{gf} [u; q ˉ]$ quadratic in $u$ ,

Q [q ˉ] u := \frac{1}{2} \frac{d}{d s} E [q ˉ + s u]_{s = 0} + F [u; q ˉ], Q [η + κ h_{cl}] = P_{gr} + κ Q^{(1)} [h_{cl}] + O (κ^{2}),

where the expansion and the identification $Q^{(1)} [h_{cl}] u = G_{2} (h_{cl}, u) + F^{(1)} [h_{cl}] u$ are Lemma lem:fluct.

The switched source is

S := \frac{1}{2} g Θ - σ_{ε} Q^{(1)} [h_{cl}] h - \frac{1}{2} σ_{ε} : G_{2} [h, h] :, Θ^{μν} = : T^{μν} [ϕ] :,

and the second-order quantum metric is its retarded response,

k := Δ_{R}^{gr} S = - 4 P E_{0}^{R} S .

Displayed formulas in this passage

X-F@1#eq:metric · line 534
$q = η + κ (h_{cl} + h) + κ^{2} (k_{cl} + k), q ˉ = η + κ h_{cl}, u := h + κ k,$
X-F@1#eq:Q-def · line 546
$Q [q ˉ] u := \frac{1}{2} \frac{d}{d s} E [q ˉ + s u]_{s = 0} + F [u; q ˉ], Q [η + κ h_{cl}] = P_{gr} + κ Q^{(1)} [h_{cl}] + O (κ^{2}),$
X-F@1#eq:source · line 557
$S := \frac{1}{2} g Θ - σ_{ε} Q^{(1)} [h_{cl}] h - \frac{1}{2} σ_{ε} : G_{2} [h, h] :, Θ^{μν} = : T^{μν} [ϕ] :,$
X-F@1#eq:khat · line 564
$k := Δ_{R}^{gr} S = - 4 P E_{0}^{R} S .$

Nearby in this section

section: Field content, kinetic operator, switched source, response