Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

Affine conventions and packets

Spinless packets and the matching contract › Affine conventions and packets · lines 357–405

VIII-B@3#eq:rotation, VIII-B@3#eq:packet, VIII-B@3#eq:conjugated-packet

Use coordinates $z = (T, X, Y)$ and

η = diag (- 1, 1, 1), t = (1, 0, 0)^{T} .

Affine maps and their composition are

(L, b) z = L z + b, (L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b) .

Define

R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ, J (μ) = (R (μ), 0) .

The reference worldline and rest vector are

W_{0} = R t, u_{0} = t .

The declared frame is restricted to

A = (C, a_{0}) \in ISO^{+} (2, 1), C \in SO^{+} (2, 1) .

It preserves the metric, orientation, and time orientation. For a fixed particle identifier $p$ , put

Π_{A} (μ) = (p, μ, A W_{0}, C t, A, H_{A} (μ)), H_{A} (μ) = A J (μ) A^{- 1} .

Explicitly,

H_{A} (μ) = (C R (μ) C^{- 1}, a_{0} - C R (μ) C^{- 1} a_{0}) .

Displayed formulas in this passage

VIII-B@3#eq:rotation · line 369
$R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ, J (μ) = (R (μ), 0) .$
VIII-B@3#eq:packet · line 391
$Π_{A} (μ) = (p, μ, A W_{0}, C t, A, H_{A} (μ)), H_{A} (μ) = A J (μ) A^{- 1} .$
VIII-B@3#eq:conjugated-packet · line 399
$H_{A} (μ) = (C R (μ) C^{- 1}, a_{0} - C R (μ) C^{- 1} a_{0}) .$
unlabeled · line 360
$η = diag (- 1, 1, 1), t = (1, 0, 0)^{T} .$
unlabeled · line 364
$(L, b) z = L z + b, (L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b) .$
unlabeled · line 381
$W_{0} = R t, u_{0} = t .$
unlabeled · line 386
$A = (C, a_{0}) \in ISO^{+} (2, 1), C \in SO^{+} (2, 1) .$