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

Proposition: Exact elementary conformal contact

The explicit current contact calculation › Retarded elementary insertion · lines 933–983 · proof included or linked

Oai282-B@1#prop:elementarycontact, Oai282-B@1#eq:contact

The insertion (eq:Cexplicit) has retarded support in $x - y$ and satisfies

\partial_{μ}^{x} C_{X}^{μ} (x, y) = δ^{(4)} (x - y) v_{X} ϕ (y) .

No conservation-to-implementation inference is used.

Proof. Temporarily write $u = ϕ (x)$ and $r = R (x - y)$ . The polarized improved tensor in the square brackets is

s^{μν} (u, r) = \partial^{μ} u \partial^{ν} r + \partial^{μ} r \partial^{ν} u - η^{μν} \partial_{ρ} u \partial^{ρ} r + \frac{1}{3} (η^{μν} □ - \partial^{μ} \partial^{ν}) (u r) .

Commuting derivatives shows

\partial_{μ} s^{μν} = (□ u) \partial^{ν} r + (□ r) \partial^{ν} u .

Its trace is

s^{μ}_{μ} = u □ r + r □ u .

Using $□ u = 0$ , $□ r = δ^{(4)} (x - y)$ , and (eq:CK), we obtain

\partial_{μ} (X_{ν} s^{μν}) = X_{ν} δ^{(4)} (x - y) \partial^{ν} ϕ (x) + σ_{X} δ^{(4)} (x - y) ϕ (x) = δ^{(4)} (x - y) (X^{ν} \partial_{ν} + σ_{X}) ϕ (y) .

All products are the already defined tensor products or delta insertion maps. Retarded support follows from the support of $R$ and its derivatives.

Displayed formulas in this passage

Oai282-B@1#eq:contact · line 937
$\partial_{μ}^{x} C_{X}^{μ} (x, y) = δ^{(4)} (x - y) v_{X} ϕ (y) .$
unlabeled · line 949
$s^{μν} (u, r) = \partial^{μ} u \partial^{ν} r + \partial^{μ} r \partial^{ν} u - η^{μν} \partial_{ρ} u \partial^{ρ} r + \frac{1}{3} (η^{μν} □ - \partial^{μ} \partial^{ν}) (u r) .$
unlabeled · line 958
$\partial_{μ} s^{μν} = (□ u) \partial^{ν} r + (□ r) \partial^{ν} u .$
unlabeled · line 964
$s^{μ}_{μ} = u □ r + r □ u .$
unlabeled · line 969
$\partial_{μ} (X_{ν} s^{μν}) = X_{ν} δ^{(4)} (x - y) \partial^{ν} ϕ (x) + σ_{X} δ^{(4)} (x - y) ϕ (x) = δ^{(4)} (x - y) (X^{ν} \partial_{ν} + σ_{X}) ϕ (y) .$