Manuscript XII-A@2 · XII-A-v2.tex · section · 955d4b22

Equation geometry versus action normalization

Frozen conventions and the operator split › Equation geometry versus action normalization · lines 323–377

XII-A@2#eq:Keq, XII-A@2#eq:shell, XII-A@2#eq:Kact, XII-A@2#eq:samekernel, XII-A@2#eq:reduction

The off-shell configuration space $C$ consists of smooth multiplets $Φ = (γ, ϕ, c, c ˉ, b)$ . Define the equation operator by

K_{eq} Φ = (□ γ, P_{m} ϕ, □ c, □ c ˉ, b - D γ) .

It is not asserted to be formally self-adjoint. Its kernel is the complete gauge-fixed free shell

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

The action Hessian is a different operator. In the reordered blocks $(γ, b) \oplus ϕ \oplus (c, c ˉ)$ ,

K_{act} = (A 2 D 2 D^{*} - 2 id) \oplus (- P_{m}) \oplus (0 i □ - i □ 0) .

The ghost block is interpreted with the graded variational convention. Ordinary transpose identities for bosonic matrices are not substituted for graded transpose identities.

Writing $ρ = b - D γ$ , the first two action equations become

- □ P γ + 2 D^{*} ρ = 0, - 2 ρ = 0.

The scalar sign and the invertible ghost mixing do not change their kernels. Thus

ker K_{eq} = ker K_{act} = C_{sol} .

For solution geometry only, use

Y = (γ, ϕ, c, c ˉ), J Y = (γ, ϕ, c, c ˉ, D γ), P = diag (□, P_{m}, □, □) .

Then $J : ker P \to C_{sol}$ is an isomorphism. The variable $b$ remains independent off shell; $J$ is not imposed on the off-shell carrier.

Displayed formulas in this passage

XII-A@2#eq:Keq · line 328
$K_{eq} Φ = (□ γ, P_{m} ϕ, □ c, □ c ˉ, b - D γ) .$
XII-A@2#eq:shell · line 335
$C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .$
XII-A@2#eq:Kact · line 344
$K_{act} = (A 2 D 2 D^{*} - 2 id) \oplus (- P_{m}) \oplus (0 i □ - i □ 0) .$
XII-A@2#eq:samekernel · line 362
$ker K_{eq} = ker K_{act} = C_{sol} .$
XII-A@2#eq:reduction · line 368
$Y = (γ, ϕ, c, c ˉ), J Y = (γ, ϕ, c, c ˉ, D γ), P = diag (□, P_{m}, □, □) .$
unlabeled · line 357
$- □ P γ + 2 D^{*} ρ = 0, - 2 ρ = 0.$