Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

The classical normalization

Noether normalization and renormalized current › The classical normalization · lines 476–507

I-B-VP-0@1#eq:classicalcurrent

Use the local charged action

S [Φ] = - \int_{M} ((D_{μ} Φ)^{†} D^{μ} Φ + ξ R Φ^{†} Φ) d vol_{g} .

The global phase transformation $δ Φ = i ϵ Φ$ has Noether current

J_{μ} = i (Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ) .

The normalization is unit $U (1)$ charge. An electric current with charge $e$ is $e J_{μ}$ when the same connection and orientation conventions are retained.

Locally,

Φ = \frac{ϕ _{1} + i ϕ _{2}}{2}, J_{μ} = ϕ_{2} \partial_{μ} ϕ_{1} - ϕ_{1} \partial_{μ} ϕ_{2}

in a parallel frame. This is compatible with $T^{complex} = T [ϕ_{1}] + T [ϕ_{2}]$ . There is no additional factor of two to append to (eq:classicalcurrent).

Displayed formulas in this passage

I-B-VP-0@1#eq:classicalcurrent · line 486
$J_{μ} = i (Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ) .$
unlabeled · line 479
$S [Φ] = - \int_{M} ((D_{μ} Φ)^{†} D^{μ} Φ + ξ R Φ^{†} Φ) d vol_{g} .$
unlabeled · line 498
$Φ = \frac{ϕ _{1} + i ϕ _{2}}{2}, J_{μ} = ϕ_{2} \partial_{μ} ϕ_{1} - ϕ_{1} \partial_{μ} ϕ_{2}$