Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

The already completed scalar sector

Frozen data, completed tests, and the topology sheet › The already completed scalar sector · lines 426–520

X-D-ST@1#eq:green, X-D-ST@1#eq:Euler, X-D-ST@1#eq:bare, X-D-ST@1#eq:completed, X-D-ST@1#eq:moller

The Klein–Gordon operator and its positive Green operators are

P_{m} = □ + m^{2}, E_{R}, E_{A},

with

P_{m} E_{A} c = P_{m} E_{R} c = c, E_{A} P_{m} a = E_{R} P_{m} a = a, E_{R}^{T} = E_{A}

for compact smooth tests. Their supports are respectively advanced and retarded, and their test-to-smooth maps are continuous BGP.

The regulator-independent Euler coefficients are

L_{1} u L_{2} u = \partial_{μ} (g a^{μν} \partial_{ν} u) + g m^{2} b u = g [\partial_{μ} (a^{μν} \partial_{ν} u) + m^{2} b u] + (\partial_{μ} g) a^{μν} \partial_{ν} u, = \partial_{μ} (g A^{μν} \partial_{ν} u) + g m^{2} B u = g [\partial_{μ} (A^{μν} \partial_{ν} u) + m^{2} B u] + (\partial_{μ} g) A^{μν} \partial_{ν} u .

In particular, $L_{1}, L_{2}$ do not depend on $ε$ . Their action-Hessian signs are $- L_{1}, - L_{2}$ , as fixed in X-B-ST/X-C-ST.

Set

B_{ε}^{μ} = w_{ε}^{ν} \partial_{ν} w_{ε}^{μ} - v_{ε}^{μ},

and define the actual bare representatives

c_{1, ε} c_{2, ε} = \partial_{μ} (f w_{ε}^{μ}), = - \partial_{μ} (f B_{ε}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w_{ε}^{μ} w_{ε}^{ν}) .

They are supported in $K$ .

The completed tests inherited from X-C-ST are

d_{1, ε} d_{2, ε} = c_{1, ε} - L_{1} E_{A} f, = c_{2, ε} - L_{1} E_{A} c_{1, ε} - L_{2} E_{A} f + L_{1} E_{A} L_{1} E_{A} f .

Write

c_{ε} = f + κ c_{1, ε} + κ^{2} c_{2, ε}, e_{ε} = f + κ d_{1, ε} + κ^{2} d_{2, ε} .

Then

e_{ε} = R_{q} c_{ε}, R_{q} = (1 + L [q] E_{A})^{- 1}, L [q] = κ L_{1} + κ^{2} L_{2} (mod κ^{3}) .

This formula recalls the existing test recursion. It does not reconstruct its response maps. In particular, $- L_{2} E_{A} f + L_{1} E_{A} L_{1} E_{A} f$ is the existing second-order pair, counted once.

All intermediate insertions remain actual off-shell tests. Replacing a supplied insertion by its free class before applying a response rule is not allowed:

R_{1} (V_{1, g}; F_{P_{m} a}) = ϕ (- L_{1} a)

need not vanish XBST,XCST.

Displayed formulas in this passage

X-D-ST@1#eq:green · line 434
$P_{m} E_{A} c = P_{m} E_{R} c = c, E_{A} P_{m} a = E_{R} P_{m} a = a, E_{R}^{T} = E_{A}$
X-D-ST@1#eq:Euler · line 445
$L_{1} u L_{2} u = \partial_{μ} (g a^{μν} \partial_{ν} u) + g m^{2} b u = g [\partial_{μ} (a^{μν} \partial_{ν} u) + m^{2} b u] + (\partial_{μ} g) a^{μν} \partial_{ν} u, = \partial_{μ} (g A^{μν} \partial_{ν} u) + g m^{2} B u = g [\partial_{μ} (A^{μν} \partial_{ν} u) + m^{2} B u] + (\partial_{μ} g) A^{μν} \partial_{ν} u .$
X-D-ST@1#eq:bare · line 470
$c_{1, ε} c_{2, ε} = \partial_{μ} (f w_{ε}^{μ}), = - \partial_{μ} (f B_{ε}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w_{ε}^{μ} w_{ε}^{ν}) .$
X-D-ST@1#eq:completed · line 483
$d_{1, ε} d_{2, ε} = c_{1, ε} - L_{1} E_{A} f, = c_{2, ε} - L_{1} E_{A} c_{1, ε} - L_{2} E_{A} f + L_{1} E_{A} L_{1} E_{A} f .$
X-D-ST@1#eq:moller · line 500
$e_{ε} = R_{q} c_{ε}, R_{q} = (1 + L [q] E_{A})^{- 1}, L [q] = κ L_{1} + κ^{2} L_{2} (mod κ^{3}) .$
unlabeled · line 430
$P_{m} = □ + m^{2}, E_{R}, E_{A},$
unlabeled · line 464
$B_{ε}^{μ} = w_{ε}^{ν} \partial_{ν} w_{ε}^{μ} - v_{ε}^{μ},$
unlabeled · line 493
$c_{ε} = f + κ c_{1, ε} + κ^{2} c_{2, ε}, e_{ε} = f + κ d_{1, ε} + κ^{2} d_{2, ε} .$
unlabeled · line 516
$R_{1} (V_{1, g}; F_{P_{m} a}) = ϕ (- L_{1} a)$