Manuscript Oai282-B@1 · Oai282-B.tex · lemma · f2956de5

Lemma: Infinitesimal covariance of the contraction

Compact-current implementation on the complete field algebra › Normal-ordering compatibility · lines 1215–1244 · proof included or linked

Oai282-B@1#lem:Wcov, Oai282-B@1#eq:Wcov

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For every conformal Killing field,

(d_{X}^{x} + d_{X}^{y}) W (x - y) = 0

as a tempered distribution.

Proof. For the fifteen displayed fields, direct polynomial calculation gives

(X (x) \cdot \partial_{x} + X (y) \cdot \partial_{y}) (x - y)^{2} = (σ_{X} (x) + σ_{X} (y)) (x - y)^{2} .

Apply this identity to the rational inverse of the squared separation. The weight terms in $d_{X}^{x} + d_{X}^{y}$ cancel its derivative.

To pass the identity to (eq:W), take independent complex point variables in a positive-energy tube. The same polynomial identity holds there, before taking a boundary value. The rational two-point function has the tempered boundary value (eq:W); differentiation and polynomial multiplication commute with this boundary limit. This proves (eq:Wcov), including the light-cone singularity.

Displayed formulas in this passage

Oai282-B@1#eq:Wcov · line 1218
$(d_{X}^{x} + d_{X}^{y}) W (x - y) = 0$
unlabeled · line 1228
$(X (x) \cdot \partial_{x} + X (y) \cdot \partial_{y}) (x - y)^{2} = (σ_{X} (x) + σ_{X} (y)) (x - y)^{2} .$

Lemma: Infinitesimal covariance of the contraction

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