Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Trace reversal

Setting, source input, and signs › Trace reversal · lines 335–367

XVI-D@1#eq:P, XVI-D@1#eq:P2, XVI-D@1#eq:inverse

For a symmetric tensor $r$ , define

(P r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, tr_{η} r = η^{μν} r_{μν} .

The analogous raised-index formula is used for contravariant tensors. In four dimensions,

tr_{η} (P r) = - tr_{η} r, P^{2} = id .

These follow directly from $η^{μν} η_{μν} = 4$ . Trace reversal has constant coefficients and commutes with $□$ .

The selected free operator and compact-test inverse are

L = - □ P, G_{gr}^{R} = - P E_{0}^{R} .

The coefficient is $- 1$ , not $- 4$ XIB. The placement of $P$ in this notation does not enlarge the domain of the scalar operator.

Displayed formulas in this passage

XVI-D@1#eq:P · line 338
$(P r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, tr_{η} r = η^{μν} r_{μν} .$
XVI-D@1#eq:P2 · line 347
$tr_{η} (P r) = - tr_{η} r, P^{2} = id .$
XVI-D@1#eq:inverse · line 357
$L = - □ P, G_{gr}^{R} = - P E_{0}^{R} .$