Manuscript VI@1 · VI.tex · section · b5b60afb

A free gauge sector does not complete the interacting one

Second-order retarded bookkeeping and the missing quantum step › A free gauge sector does not complete the interacting one · lines 1921–1952

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For orientation, a free BRST complex can be written with ghost $c$ , antighost $c ˉ$ , and auxiliary field $b$ :

s_{0} h = K c, s_{0} φ = 0, s_{0} c = 0, s_{0} c ˉ = b, s_{0} b = 0.

A harmonic gauge fermion is

Ψ_{α} = \int c ˉ_{ν} (D (h)^{ν} + \frac{α}{2} b^{ν}) d^{4} x, α \neq = 0.

Its variation gives

s_{0} Ψ_{α} = \int (b_{ν} D (h)^{ν} + \frac{α}{2} b_{ν} b^{ν} - c ˉ_{ν} □_{η} c^{ν}) d^{4} x .

Eliminating $b$ gives $- \frac{1}{2 α} \int D (h)_{ν} D (h)^{ν}$ .

These formulas identify a free gauge sector. They do not supply the interacting BRST differential, its switched identity, or the renormalized products of the relational functional. No quantum theorem below relies on pretending otherwise.

Displayed formulas in this passage

unlabeled · line 1925
$s_{0} h = K c, s_{0} φ = 0, s_{0} c = 0, s_{0} c ˉ = b, s_{0} b = 0.$
unlabeled · line 1930
$Ψ_{α} = \int c ˉ_{ν} (D (h)^{ν} + \frac{α}{2} b^{ν}) d^{4} x, α \neq = 0.$
unlabeled · line 1937
$s_{0} Ψ_{α} = \int (b_{ν} D (h)^{ν} + \frac{α}{2} b_{ν} b^{ν} - c ˉ_{ν} □_{η} c^{ν}) d^{4} x .$