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

Integer-spin components induced by the geometry

Three distinct holonomy ledgers › Integer-spin components induced by the geometry · lines 1525–1562

I-B-G@2#eq:helicity

For an integer spin-weight component, the flat Levi-Civita connection can induce a nontrivial equivariance in a developing parallel frame. To fix the convention without ambiguity, consider covariant transverse tensors and define their $+ s$ component by contraction with $(e_{X} + i e_{Y})^{\otimes s}$ . Since

e_{X} + i e_{Y} = e^{i α φ} (e_{r} + i e_{φ^}),

a single-valued polar component gives

A_{+ s} (φ + 2 π) = e^{2 π i s α} A_{+ s} (φ) = e^{- i s μ} A_{+ s} (φ) .

Thus

β_{+ s} \equiv - \frac{s μ}{2 π} (mod 1), β_{- s} \equiv + \frac{s μ}{2 π} (mod 1) .

Opposite component conventions interchange the signs. For the rough tensor equation

- \nabla^{a} \nabla_{a} A + m^{2} A = F,

each parallel spin-weight component obeys the scalar equation. General tensor bundles may contain several spin weights; $\pm s$ denote the specified pure helicity sectors, not an undeclared decomposition of every tensor into only two components.

Displayed formulas in this passage

I-B-G@2#eq:helicity · line 1546
$β_{+ s} \equiv - \frac{s μ}{2 π} (mod 1), β_{- s} \equiv + \frac{s μ}{2 π} (mod 1) .$
unlabeled · line 1535
$e_{X} + i e_{Y} = e^{i α φ} (e_{r} + i e_{φ^}),$
unlabeled · line 1540
$A_{+ s} (φ + 2 π) = e^{2 π i s α} A_{+ s} (φ) = e^{- i s μ} A_{+ s} (φ) .$
unlabeled · line 1554
$- \nabla^{a} \nabla_{a} A + m^{2} A = F,$

Integer-spin components induced by the geometry

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