Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Graviton tensor factor and endpoint maps

Four contraction kernels in the actual relative slots › Graviton tensor factor and endpoint maps · lines 1670–1742

Em01-F@3#eq:Nfactor, Em01-F@3#eq:Hdictionary

Define

Π_{α β ρ σ} = \frac{1}{2} (η_{α ρ} η_{β σ} + η_{α σ} η_{β ρ} - η_{α β} η_{ρ σ}) .

It is the inverse tensor pairing determined by $- □_{η} P$ in (eq:quadratic).

The endpoint maps raise the tensor at $y$ and extract the component at $a$ :

L^{μν, α β} = η^{μα} η^{ν β}, R_{I}^{ρ σ} = (e^{I})^{ρ σ} .

There are no endpoint derivatives or extra trace-reversal maps. Set

N^{μν}_{I} = η^{μα} η^{ν β} Π_{α β ρ σ} (e^{I})^{ρ σ}, ϑ_{I} = η_{ρ σ} (e^{I})^{ρ σ} .

In the relative variables

x_{1} = y - a, x_{2} = y - b,

the four lines are

H_{γ}^{μν}_{I} (x_{1}) H_{T, γ}^{μν}_{I} (x_{1}) H_{ϕ} (x_{2}) H_{T, ϕ} (x_{2}) = ℏ N^{μν}_{I} W_{0} (x_{1}), = ℏ N^{μν}_{I} D_{T, 0} (x_{1}), = ℏ W_{m} (x_{2}), = ℏ D_{T, m} (x_{2}) .

Each includes $ℏ$ , with no additional $i$ or $λ$ . Composition with the tensor Hessian gives $i ℏ$ times its identity kernel, and its antisymmetric part is $i ℏ$ times $- P (E_{0}^{R} - E_{0}^{A})$ .

The covariant graviton factor is indefinite on unrestricted metric components. It is a gauge-fixed quasifree contraction prescription, not a positive state on the whole metric-field algebra. No physical-state quotient is constructed.

Endpoint derivatives satisfy

\partial_{y} H (y - a) = \partial_{x_{1}} H (x_{1}), \partial_{a} H (y - a) = - \partial_{x_{1}} H (x_{1}),

and similarly for $x_{2} = y - b$ . Although $D_{T, m} (- x) = D_{T, m} (x)$ , the same equality is false for $W_{m}$ in general.

Displayed formulas in this passage

Em01-F@3#eq:Nfactor · line 1696
$N^{μν}_{I} = η^{μα} η^{ν β} Π_{α β ρ σ} (e^{I})^{ρ σ}, ϑ_{I} = η_{ρ σ} (e^{I})^{ρ σ} .$
Em01-F@3#eq:Hdictionary · line 1711
$H_{γ}^{μν}_{I} (x_{1}) H_{T, γ}^{μν}_{I} (x_{1}) H_{ϕ} (x_{2}) H_{T, ϕ} (x_{2}) = ℏ N^{μν}_{I} W_{0} (x_{1}), = ℏ N^{μν}_{I} D_{T, 0} (x_{1}), = ℏ W_{m} (x_{2}), = ℏ D_{T, m} (x_{2}) .$
unlabeled · line 1673
$Π_{α β ρ σ} = \frac{1}{2} (η_{α ρ} η_{β σ} + η_{α σ} η_{β ρ} - η_{α β} η_{ρ σ}) .$
unlabeled · line 1687
$L^{μν, α β} = η^{μα} η^{ν β}, R_{I}^{ρ σ} = (e^{I})^{ρ σ} .$
unlabeled · line 1707
$x_{1} = y - a, x_{2} = y - b,$
unlabeled · line 1735
$\partial_{y} H (y - a) = \partial_{x_{1}} H (x_{1}), \partial_{a} H (y - a) = - \partial_{x_{1}} H (x_{1}),$