Manuscript X-I-ST@1 · X-I-ST.tex · section · 9745fd40

Fields, parities, and operators

Free normalization and the exact free field limit › Fields, parities, and operators · lines 416–481

X-I-ST@1#eq:freeidentities, X-I-ST@1#eq:A

The independent fields are

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

Their parities and ghost numbers are

Φ ∣Φ∣ gh (Φ) r 00 ϕ 00 c 11 c ˉ 1 - 1 b 0 0.

All are smooth off shell. Odd fields are interpreted in a supercommutative coefficient algebra; finite exterior algebras will be used for numerical coefficient extraction.

Set

□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.

For a covariant symmetric tensor $r$ , let

(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, (C r)^{μ} = \partial_{ν} (P_{tr} r)^{ν μ},

(K c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} .

Compact integration by parts gives

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

Use the parent normalization

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

Here the linearized Einstein tensor is defined by

(G_{lin} r)_{μν} = \frac{1}{2} (\partial_{ρ} \partial_{μ} r^{ρ}_{ν} + \partial_{ρ} \partial_{ν} r^{ρ}_{μ} - □ r_{μν} - \partial_{μ} \partial_{ν} tr r - η_{μν} (\partial_{ρ} \partial_{σ} r^{ρ σ} - □ tr r)) .

Equations (eq:A) follow by substitution. This is the normalization of X-G/X-G-ST XG,XGST.

Displayed formulas in this passage

X-I-ST@1#eq:freeidentities · line 450
$C^{*} = - \frac{1}{2} P_{tr} K, C K = □, P_{tr}^{2} = I .$
X-I-ST@1#eq:A · line 457
$A = \frac{1}{2} G_{lin}, A K = 0, A + \frac{1}{2} C^{*} C = P_{gr} = - \frac{1}{4} P_{tr} □ .$
unlabeled · line 419
$Φ = (r_{μν}, ϕ, c^{μ}, c ˉ_{μ}, b_{μ}) .$
unlabeled · line 423
$Φ ∣Φ∣ gh (Φ) r 00 ϕ 00 c 11 c ˉ 1 - 1 b 0 0.$
unlabeled · line 436
$□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 440
$(P_{tr} r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, (C r)^{μ} = \partial_{ν} (P_{tr} r)^{ν μ},$
unlabeled · line 446
$(K c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} .$
unlabeled · line 465
$(G_{lin} r)_{μν} = \frac{1}{2} (\partial_{ρ} \partial_{μ} r^{ρ}_{ν} + \partial_{ρ} \partial_{ν} r^{ρ}_{μ} - □ r_{μν} - \partial_{μ} \partial_{ν} tr r - η_{μν} (\partial_{ρ} \partial_{σ} r^{ρ σ} - □ tr r)) .$