Manuscript Oai282-B@1 · Oai282-B.tex · section · f2956de5

Lie brackets and the sign convention

Lie brackets and the sign convention · lines 1363–1393

Oai282-B@1#eq:deltasign

The vector-field bracket is

[X, Y]^{μ} = X^{ν} \partial_{ν} Y^{μ} - Y^{ν} \partial_{ν} X^{μ} .

Its conformal factor obeys

σ_{[X, Y]} = X^{μ} \partial_{μ} σ_{Y} - Y^{μ} \partial_{μ} σ_{X} .

The current commutator in (eq:wickaction) gives $v_{X}$ , with positive point-field action $d_{X} ϕ$ . As an evolutionary action, it keeps the numerical coefficients fixed. Its bracket is therefore an anti-representation. To state a representation with the vector-field bracket, define

δ_{X} B = - v_{X} B = - \frac{i}{ℏ} [q_{X} [a, χ], B]

on a localized word. The result is independent of any choice satisfying the collar contract.

Displayed formulas in this passage

Oai282-B@1#eq:deltasign · line 1383
$δ_{X} B = - v_{X} B = - \frac{i}{ℏ} [q_{X} [a, χ], B]$
unlabeled · line 1367
$[X, Y]^{μ} = X^{ν} \partial_{ν} Y^{μ} - Y^{ν} \partial_{ν} X^{μ} .$
unlabeled · line 1371
$σ_{[X, Y]} = X^{μ} \partial_{μ} σ_{Y} - Y^{μ} \partial_{μ} σ_{X} .$

Lie brackets and the sign convention

Displayed formulas in this passage

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