Manuscript V@1 · V.tex · section · e7ddf37d

Raw current and complete gauge defect

Raw current and complete gauge defect · lines 630–668

V@1#eq:u, V@1#eq:rawadjoint

For this section and the classification, translate the initial surface to $t_{0} = 0$ . Fix a real free scalar

P ϕ = 0

with compact smooth Cauchy data, and a real

f \in C_{c}^{\infty} ((0, T) \times R^{3}) .

The scalar is fixed as a classical parameter.

Define

B_{raw} (h) = \int f S_{0} (h) ϕ d^{4} x .

Let

u (s) = \int_{s}^{T} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t) d t = - G_{A} f (s) .

It has zero terminal data and $P u = - f$ . It is smooth, including across a time later than the support of $f$ , where it vanishes.

Self-adjointness of the spatial sine operator gives

B_{raw} (h) = \int_{0}^{T} \int_{R^{3}} u L_{h} ϕ d^{3} x d t .

Put $V = Θ (t) u$ . Its support is contained in the compact set

K_{f} = J^{-} (supp f) \cap {0 \leq t \leq T} .

Displayed formulas in this passage

V@1#eq:u · line 649
$u (s) = \int_{s}^{T} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t) d t = - G_{A} f (s) .$
V@1#eq:rawadjoint · line 659
$B_{raw} (h) = \int_{0}^{T} \int_{R^{3}} u L_{h} ϕ d^{3} x d t .$
unlabeled · line 635
$P ϕ = 0$
unlabeled · line 639
$f \in C_{c}^{\infty} ((0, T) \times R^{3}) .$
unlabeled · line 645
$B_{raw} (h) = \int f S_{0} (h) ϕ d^{4} x .$
unlabeled · line 665
$K_{f} = J^{-} (supp f) \cap {0 \leq t \leq T} .$