Manuscript X-C-ST@1 · X-C-ST.tex · proposition · b679954e

Proposition: Pure-gauge two-jet closes the predecessor remainder

Diagnostics of the completed coefficient › Complete pure-gauge cancellation · lines 1858–1932 · proof included or linked

X-C-ST@1#prop:puregauge, X-C-ST@1#eq:pg-new, X-C-ST@1#eq:pg-cancel, X-C-ST@1#eq:pg-operators

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Let $ξ \in g_{p}$ , set $ζ = 0$ , and take

h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η .

Write

X = X_{ξ}, D = D_{ξ}, L = L_{1, g} [K ξ], M = L_{2, g} [K ξ, \frac{1}{2} L_{ξ}^{2} η] .

Then the new pair has the representative identity

L E_{A} L E_{A} f - M E_{A} f = \frac{1}{2} D^{2} f + P_{m} (\frac{1}{2} X^{2} E_{A} f - X E_{A} D f) .

Consequently, in the free algebra,

A_{2} [K ξ, \frac{1}{2} L_{ξ}^{2} η] = (\frac{1}{2} - 1 + \frac{1}{2}) ϕ (D^{2} f) = 0.

The bare second dressing is retained.

Proof. On the flat starting background, plateau Euler covariance gives, by expansion,

L = D P_{m} - P_{m} X, M = \frac{1}{2} D^{2} P_{m} + \frac{1}{2} P_{m} X^{2} - D P_{m} X .

These operator identities can also be obtained directly by differentiating the density naturality formula. They are identities on arbitrary smooth scalars.

Using $P_{m} E_{A} f = f$ , and $E_{A} P_{m} a = a$ for compact $a$ , one obtains

L E_{A} f L E_{A} L E_{A} f M E_{A} f = D f - P_{m} (X E_{A} f), = D^{2} f - P_{m} (X E_{A} D f) - D P_{m} (X E_{A} f) + P_{m} (X^{2} E_{A} f), = \frac{1}{2} D^{2} f + \frac{1}{2} P_{m} (X^{2} E_{A} f) - D P_{m} (X E_{A} f) .

The uses of $E_{A} P_{m} = 1$ are legitimate because $X E_{A} f$ and its further $X$ -derivatives have compact generator support. Subtraction proves (eq:pg-new).

The supplied coordinate laws give $w = ξ$ , $v = \frac{1}{2} X ξ$ , and hence

c_{1} = D f, c_{2} = \frac{1}{2} D^{2} f .

The mixed inherited term has representative

- L E_{A} D f = - D^{2} f + P_{m} (X E_{A} D f) .

Adding it and the bare second test to (eq:pg-new) gives the stronger test identity

e_{2} = P_{m} (\frac{1}{2} X^{2} E_{A} f) .

Taking its free class proves (eq:pg-cancel).

Displayed formulas in this passage

X-C-ST@1#eq:pg-new · line 1871
$L E_{A} L E_{A} f - M E_{A} f = \frac{1}{2} D^{2} f + P_{m} (\frac{1}{2} X^{2} E_{A} f - X E_{A} D f) .$
unlabeled · line 1902
$L E_{A} f L E_{A} L E_{A} f M E_{A} f = D f - P_{m} (X E_{A} f), = D^{2} f - P_{m} (X E_{A} D f) - D P_{m} (X E_{A} f) + P_{m} (X^{2} E_{A} f), = \frac{1}{2} D^{2} f + \frac{1}{2} P_{m} (X^{2} E_{A} f) - D P_{m} (X E_{A} f) .$
unlabeled · line 1861
$h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η .$
unlabeled · line 1865
$X = X_{ξ}, D = D_{ξ}, L = L_{1, g} [K ξ], M = L_{2, g} [K ξ, \frac{1}{2} L_{ξ}^{2} η] .$
X-C-ST@1#eq:pg-cancel · line 1879
$A_{2} [K ξ, \frac{1}{2} L_{ξ}^{2} η] = (\frac{1}{2} - 1 + \frac{1}{2}) ϕ (D^{2} f) = 0.$
X-C-ST@1#eq:pg-operators · line 1891
$L = D P_{m} - P_{m} X, M = \frac{1}{2} D^{2} P_{m} + \frac{1}{2} P_{m} X^{2} - D P_{m} X .$
unlabeled · line 1919
$c_{1} = D f, c_{2} = \frac{1}{2} D^{2} f .$
unlabeled · line 1923
$- L E_{A} D f = - D^{2} f + P_{m} (X E_{A} D f) .$
unlabeled · line 1928
$e_{2} = P_{m} (\frac{1}{2} X^{2} E_{A} f) .$

Nearby in this section

section: Complete pure-gauge cancellation