Manuscript X-L-ST@1 · X-L-ST.tex · section · 20a32a22

Fields, pairings, and operators

The normalized free auxiliary–ghost package › Fields, pairings, and operators · lines 326–371

X-L-ST@1#eq:operatoridentities, X-L-ST@1#eq:freeoperators

Use the field order

Φ = (r_{μν}, b_{μ}, ϕ, c^{μ}, c ˉ_{μ}) .

The fields $r, b, ϕ$ are even and have ghost number zero. The ghosts $c, c ˉ$ are odd, with

gh (c) = 1, gh (c ˉ) = - 1.

All pairings use $η$ , the coordinate density, and the bilinear formal transpose, without complex conjugation.

For a symmetric tensor $r$ , define

(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} r^{ρ}_{ρ},

(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, (C r)_{ν} = \partial^{μ} r_{μν} - \frac{1}{2} \partial_{ν} r^{μ}_{μ} .

In four dimensions,

P_{tr}^{2} = I, C K = □, C^{*} = - \frac{1}{2} P_{tr} K .

Set

P_{gr} = - \frac{1}{4} P_{tr} □, A = P_{gr} - \frac{1}{2} C^{*} C, P_{m} = □ + m^{2}, m \geq 0.

Then

A K = 0, P_{gr} = A + \frac{1}{2} C^{*} C .

Displayed formulas in this passage

X-L-ST@1#eq:operatoridentities · line 353
$P_{tr}^{2} = I, C K = □, C^{*} = - \frac{1}{2} P_{tr} K .$
X-L-ST@1#eq:freeoperators · line 360
$P_{gr} = - \frac{1}{4} P_{tr} □, A = P_{gr} - \frac{1}{2} C^{*} C, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 329
$Φ = (r_{μν}, b_{μ}, ϕ, c^{μ}, c ˉ_{μ}) .$
unlabeled · line 334
$gh (c) = 1, gh (c ˉ) = - 1.$
unlabeled · line 341
$(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} r^{ρ}_{ρ},$
unlabeled · line 345
$(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, (C r)_{ν} = \partial^{μ} r_{μν} - \frac{1}{2} \partial_{ν} r^{μ}_{μ} .$
unlabeled · line 367
$A K = 0, P_{gr} = A + \frac{1}{2} C^{*} C .$