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

Remark: Do not replace the kernel by its real part

Noether normalization and renormalized current › Point splitting and the complex first jet · lines 540–560

I-B-VP-0@1#rem:imaginary

The stress tensor in I-B-VP uses $2 Re H_{β}$ . The current does not. Its information is in the imaginary, endpoint-antisymmetric first jet of the complex kernel. Hermiticity gives

[(\partial_{μ} - \partial_{μ^{'}}) H_{β}] = 2 i Im [\partial_{μ} H_{β}],

and consequently

J_{μ} = - 2 Im [\partial_{μ} H_{β}] .

Replacing $H_{β}$ by its real part would incorrectly set this current to zero. Symmetrizing operator order is not averaging the two charge-conjugate bundle sectors.

Displayed formulas in this passage

unlabeled · line 546
$[(\partial_{μ} - \partial_{μ^{'}}) H_{β}] = 2 i Im [\partial_{μ} H_{β}],$
unlabeled · line 553
$J_{μ} = - 2 Im [\partial_{μ} H_{β}] .$

Nearby in this section

definition: Definition: Renormalized one-point current
section: Point splitting and the complex first jet
section: Point splitting and the complex first jet
assumption: Assumption: Renormalization class
proposition: Proposition: Well-definedness, gauge invariance, and conservation
remark: Remark: A current-specific ambiguity statement