Manuscript Oai282-A@1 · Oai282-A.tex · theorem · 5b6134ec

Theorem: Flat-space improvement identities

Stage 2: conditional improvement and its affine charges › The independent field contract · lines 594–621 · proof included or linked

Oai282-A@1#thm:improvement, Oai282-A@1#eq:improvement-identities

Back to manuscript

Under Assumption ass:improvement, $t$ is Hermitian, symmetric, conserved, and traceless as a field distribution on $E$ . More generally, without assuming $T^{μ}_{μ} = □ L$ ,

\partial^{μ} t_{μν} = \partial^{μ} T_{μν}, t^{μ}_{μ} = T^{μ}_{μ} - □ L .

Proof. Distributional derivatives commute, so

\partial^{μ} I_{μν} [L] = \frac{1}{3} (□ \partial_{ν} - \partial_{ν} □) L = 0.

In four dimensions,

η^{μν} I_{μν} [L] = \frac{1}{3} (1 - 4) □ L = - □ L .

The remaining assertions follow directly from the declared Hermitian and symmetry properties.

Displayed formulas in this passage

Oai282-A@1#eq:improvement-identities · line 600
$\partial^{μ} t_{μν} = \partial^{μ} T_{μν}, t^{μ}_{μ} = T^{μ}_{μ} - □ L .$
unlabeled · line 610
$\partial^{μ} I_{μν} [L] = \frac{1}{3} (□ \partial_{ν} - \partial_{ν} □) L = 0.$
unlabeled · line 615
$η^{μν} I_{μν} [L] = \frac{1}{3} (1 - 4) □ L = - □ L .$

Theorem: Flat-space improvement identities

Displayed formulas in this passage

Nearby in this section