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

The independent auxiliary field

Free normalization and the exact free field limit › The independent auxiliary field · lines 482–508

X-I-ST@1#eq:s0, X-I-ST@1#eq:Psi0, X-I-ST@1#eq:Ifree

Define the left odd derivation

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

It commutes with total spacetime derivatives.

The gauge fermion and free action are

Ψ_{0} I_{free} = \int c ˉ_{μ} ((C r)^{μ} - b^{μ}) d^{4} x, = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + s_{0} Ψ_{0} = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + \int (b \cdot C r - b^{2} - c ˉ \cdot □ c) d^{4} x .

These are variational actions with compact variations, not an assumption that their values are finite on every ambient smooth configuration.

Displayed formulas in this passage

X-I-ST@1#eq:s0 · line 485
$s_{0} r = K c, s_{0} ϕ = 0, s_{0} c = 0, s_{0} c ˉ = b, s_{0} b = 0.$
X-I-ST@1#eq:Psi0 · line 493
$Ψ_{0} I_{free} = \int c ˉ_{μ} ((C r)^{μ} - b^{μ}) d^{4} x, = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + s_{0} Ψ_{0} = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + \int (b \cdot C r - b^{2} - c ˉ \cdot □ c) d^{4} x .$
X-I-ST@1#eq:Ifree · line 493
$Ψ_{0} I_{free} = \int c ˉ_{μ} ((C r)^{μ} - b^{μ}) d^{4} x, = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + s_{0} Ψ_{0} = \frac{1}{2} ⟨ r, A r ⟩ - \frac{1}{2} ⟨ ϕ, P_{m} ϕ ⟩ + \int (b \cdot C r - b^{2} - c ˉ \cdot □ c) d^{4} x .$