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

Proposition: The signs and normalization of improvement

Improvement and conformal Killing fields · lines 716–764 · proof included or linked

Oai282-B@1#prop:improvement, Oai282-B@1#eq:improvementdata, Oai282-B@1#eq:t

On $D_{E}$ ,

T^{μ}_{μ} = - K, □ Q = 2 K, V_{μ} = - \frac{1}{2} \partial_{μ} Q, L = - \frac{1}{2} Q .

Consequently

t_{μν} = T_{μν} + \frac{1}{3} (\partial_{μ} \partial_{ν} - η_{μν} □) L = T_{μν} + \frac{1}{6} (η_{μν} □ - \partial_{μ} \partial_{ν}) Q

is symmetric, conserved, and traceless.

Proof. Vacuum normal ordering commutes with constant-coefficient differentiation. Thus

□ Q = 2 : \partial_{ρ} ϕ \partial^{ρ} ϕ : + 2 : ϕ □ ϕ : = 2 K .

The trace of $T$ in four dimensions is $K - 2 K = - K$ . The remaining identities in (eq:improvementdata) follow.

The divergence of the improvement operator vanishes identically:

\partial^{μ} (η_{μν} □ - \partial_{μ} \partial_{ν}) Q = 0.

The canonical tensor is conserved by the free equation. The trace of the improvement is $\frac{1}{2} □ Q = K$ , which cancels the canonical trace. Symmetry is immediate.

Displayed formulas in this passage

Oai282-B@1#eq:improvementdata · line 719
$T^{μ}_{μ} = - K, □ Q = 2 K, V_{μ} = - \frac{1}{2} \partial_{μ} Q, L = - \frac{1}{2} Q .$
Oai282-B@1#eq:t · line 730
$t_{μν} = T_{μν} + \frac{1}{3} (\partial_{μ} \partial_{ν} - η_{μν} □) L = T_{μν} + \frac{1}{6} (η_{μν} □ - \partial_{μ} \partial_{ν}) Q$
unlabeled · line 746
$□ Q = 2 : \partial_{ρ} ϕ \partial^{ρ} ϕ : + 2 : ϕ □ ϕ : = 2 K .$
unlabeled · line 757
$\partial^{μ} (η_{μν} □ - \partial_{μ} \partial_{ν}) Q = 0.$

Proposition: The signs and normalization of improvement

Displayed formulas in this passage

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