Manuscript XIII-A@1 · XIII-A.tex · section · f56e91c7

Geometry, pairings, and equations

A0: prerequisite sheet and inherited conventions › Geometry, pairings, and equations · lines 352–406

XIII-A@1#eq:operators, XIII-A@1#eq:shell

Use the supplied Minkowski conventions

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T, m \geq 0.

The ramp $σ_{ε}$ is smooth, zero for $t \leq 0$ , and one for $t \geq ε$ . The switch is $g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ , with $P_{g} = int {g = 1}$ . All fields are ambient smooth configurations. No spatial decay is imposed. Tensor indices and dual tests are identified using $η$ .

For a symmetric covariant tensor $r$ and a vector $v$ ,

P r (K v)_{μν} (D^{*} v)_{μν} = r - \frac{1}{2} η tr_{η} r, = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} . D^{ν} r R = \partial_{μ} (P r)^{μν}, = \frac{1}{2} K,

Parentheses include $\frac{1}{2}$ . The frozen identities are

P^{2} = id, D K = □, A + 2 D^{*} D = - □ P, P D^{*} = - R, A R = 0.

The inverse remains $G_{gr}^{R} = - P E_{0}^{R}$ , with coefficient $- 1$ XIB,XIIA.

The independent multiplet is $(γ, ϕ, c, c ˉ, b)$ . Only $c, c ˉ$ are odd; their ghost numbers are respectively $1, - 1$ . The other fields have ghost number zero. The shell is

C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .

Odd identities mean coefficientwise identities in every finite Grassmann algebra, as in XII-A. In particular, $b$ is an independent off-shell coordinate. Equation (eq:shell) is not an off-shell substitution rule.

Displayed formulas in this passage

XIII-A@1#eq:operators · line 372
$P r (K v)_{μν} (D^{*} v)_{μν} = r - \frac{1}{2} η tr_{η} r, = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} . D^{ν} r R = \partial_{μ} (P r)^{μν}, = \frac{1}{2} K,$
XIII-A@1#eq:shell · line 396
$C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .$
unlabeled · line 355
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},$
unlabeled · line 361
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T, m \geq 0.$
unlabeled · line 385
$P^{2} = id, D K = □, A + 2 D^{*} D = - □ P, P D^{*} = - R, A R = 0.$