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

Proposition: Free-scalar improvement

Stage 5: the free massless scalar and the field-list issue › The normalization calculation · lines 1475–1512 · proof included or linked

Oai282-A@1#prop:free, Oai282-A@1#eq:free-improvement

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On the supplied Wick domain,

T^{μ}_{μ} = - K, □ Q = 2 K .

Thus

V_{μ} = - \frac{1}{2} \partial_{μ} Q, L = - \frac{1}{2} Q

satisfy $T^{μ}_{μ} = \partial^{μ} V_{μ} = □ L$ , and

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

is symmetric, conserved, and traceless.

Proof. Contraction in four dimensions gives $T^{μ}_{μ} = (1 - 2) K = - K$ . The Wick derivative rule and the free equation give

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

The definitions of $V, L$ therefore give the trace relation. Substitution into (eq:improvement) gives (eq:free-improvement). Conservation of $T$ follows from the free equation, and Theorem thm:improvement completes the calculation.

Displayed formulas in this passage

Oai282-A@1#eq:free-improvement · line 1488
$t_{μν} = T_{μν} + \frac{1}{6} (η_{μν} □ - \partial_{μ} \partial_{ν}) Q$
unlabeled · line 1478
$T^{μ}_{μ} = - K, □ Q = 2 K .$
unlabeled · line 1483
$V_{μ} = - \frac{1}{2} \partial_{μ} Q, L = - \frac{1}{2} Q$
unlabeled · line 1501
$□ : ϕ^{2} := 2 : \partial_{ρ} ϕ \partial^{ρ} ϕ : + 2 : ϕ □ ϕ := 2 K .$

Proposition: Free-scalar improvement

Displayed formulas in this passage

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