Manuscript XX-A@1 · XX-A.tex · section · 3a586110

Clock-slice inversion

An explicit on-shell background and the perturbative freeze › Clock-slice inversion · lines 381–403

XX-A@1#eq:clock-coeff, XX-A@1#eq:time-inverse

Write

T_{s} = t + sχ + \frac{s ^{2}}{2} ζ, θ_{s} (τ, x) = τ + s u_{1} + \frac{s ^{2}}{2} u_{2},

where $T_{s} (θ_{s} (τ, x), x) = τ$ . Taylor expansion gives

χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .

All coefficients on the right of the inverse formula are evaluated at $t = τ$ .

Displayed formulas in this passage

XX-A@1#eq:clock-coeff · line 391
$χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .$
XX-A@1#eq:time-inverse · line 391
$χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .$
unlabeled · line 384
$T_{s} = t + sχ + \frac{s ^{2}}{2} ζ, θ_{s} (τ, x) = τ + s u_{1} + \frac{s ^{2}}{2} u_{2},$