Manuscript XX-B@1 · XX-B.tex · proposition · 56d708a5

Proposition: Exact gauge–Green normalization

Retarded and advanced Green distributions › Causal intertwining identities · lines 1033–1060 · proof included or linked

XX-B@1#prop:intertwining, XX-B@1#eq:operator-intertwine, XX-B@1#eq:green-intertwine

On compact tests,

B K G_{B}^{\pm} F^{†} = - r F^{†} Q, = - r^{- 1} K G_{Q}^{\pm}, K^{†} B F G_{B}^{\pm} = - r Q F, = - r^{- 1} G_{Q}^{\pm} K^{†} .

The same identities hold for the causal differences.

Proof. The first line follows from $H K = 0$ , $F K = Q$ , and self-adjointness. For a compact test $a$ , both $G_{B}^{\pm} F^{†} a$ and $- r^{- 1} K G_{Q}^{\pm} a$ solve the same inhomogeneous $B$ -equation and have the same retarded or advanced support. Uniqueness gives the first Green identity. Taking adjoints gives the second.

Displayed formulas in this passage

XX-B@1#eq:operator-intertwine · line 1036
$B K G_{B}^{\pm} F^{†} = - r F^{†} Q, = - r^{- 1} K G_{Q}^{\pm}, K^{†} B F G_{B}^{\pm} = - r Q F, = - r^{- 1} G_{Q}^{\pm} K^{†} .$
XX-B@1#eq:green-intertwine · line 1036
$B K G_{B}^{\pm} F^{†} = - r F^{†} Q, = - r^{- 1} K G_{Q}^{\pm}, K^{†} B F G_{B}^{\pm} = - r Q F, = - r^{- 1} G_{Q}^{\pm} K^{†} .$

Proposition: Exact gauge–Green normalization

Displayed formulas in this passage

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