Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Bundle covariance and local gauges

The Sommerfeld–Carslaw–Dowker kernel › Bundle covariance and local gauges · lines 1320–1360

I-B-G@2#eq:lift-factor

The kernel is a section of $Hom (L_{β, x^{'}}, L_{β, x})$ . For example,

G_{+} (t; r, φ + 2 π, z; x^{'}) = e^{2 π i β} G_{+} (t; r, φ, z; x^{'}) .

ϑ = ψ + 2 π α ℓ, ℓ \in Z,

then every formula reduced to $ψ$ acquires the overall factor

G_{+} (t; ϑ) = e^{2 π i β ℓ} G_{+} (t; ψ) .

The same identity holds for $Φ_{β}$ . Omitting this factor would destroy bundle covariance.

On a simply connected chart, the lifted representation is a parallel trivialisation; the differential operator is the ordinary scalar Klein–Gordon operator. By contrast, the periodic gauge

v = e^{- i β φ} u

has covariant derivative $\partial_{φ} + i β$ , and its kernel is

K_{per} (x, x^{'}) = e^{- i β φ} K_{par} (x, x^{'}) e^{i β φ^{'}} .

In that gauge the direct term has the corresponding endpoint phase. It is not the ordinary flat kernel with unit direct weight. All flat comparisons below use a local parallel trivialisation.

Displayed formulas in this passage

I-B-G@2#eq:lift-factor · line 1335
$G_{+} (t; ϑ) = e^{2 π i β ℓ} G_{+} (t; ψ) .$
unlabeled · line 1325
$G_{+} (t; r, φ + 2 π, z; x^{'}) = e^{2 π i β} G_{+} (t; r, φ, z; x^{'}) .$
unlabeled · line 1330
$ϑ = ψ + 2 π α ℓ, ℓ \in Z,$
unlabeled · line 1346
$v = e^{- i β φ} u$
unlabeled · line 1351
$K_{per} (x, x^{'}) = e^{- i β φ} K_{par} (x, x^{'}) e^{i β φ^{'}} .$