Manuscript VII-B@1 · VII-B.tex · section · be632835

Gauge-fixed free multiplet

Quantum model and declared insertion domain › Gauge-fixed free multiplet · lines 474–561

VII-B@1#eq:trace-reversal, VII-B@1#eq:free-action, VII-B@1#eq:wightman, VII-B@1#eq:H-even, VII-B@1#eq:H-odd

The quantum field multiplet is

Ψ = (h_{μν}, ϕ, c^{μ}, c ˉ_{μ}, β_{μ}) .

The fields $c, c ˉ$ are odd; the remaining fields are even. The scalar has mass $m > 0$ .

Let the trace-reversal operator be

(T h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} h, T^{2} = id .

The free action, modulo an unswitched total divergence, is

S_{0} = \int [\frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} - \frac{1}{4} \partial_{λ} h \partial^{λ} h + \frac{1}{2} \partial_{λ} ϕ \partial^{λ} ϕ - \frac{1}{2} m^{2} ϕ^{2} - c ˉ_{μ} □ c^{μ} - \frac{1}{4} β_{μ} β^{μ}] d^{4} x .

This is the linear de Donder gauge with a decoupled, shifted Nakanishi–Lautrup field. An explicit nonlinear action representative and its switched vertices are specified in Appendix app:vertices.

With the metric used to identify dual tensor bundles, the free Hessian has diagonal even blocks

P_{hh} = - T □, P_{ϕϕ} = - (□ + m^{2}), P_{β β} = - \frac{1}{2} η,

and odd block, in the order $(c, c ˉ)$ ,

P_{odd} = (0 - □ □ 0) .

The displayed odd signs correspond to writing the quadratic action as $- c ˉ_{μ} □ c^{μ}$ .

For $m \geq 0$ , write

Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{m} ( k )} e^{- i ω_{m} (k) (x^{0} - y^{0}) + i k \cdot (x - y)}, ω_{m} (k) = ∣ k ∣^{2} + m^{2} .

Fix the graded two-point matrix by

H_{hh μν ρ σ} H^{c^{μ} c ˉ_{ν}} = T_{μν ρ σ} Δ_{0}^{+}, = δ^{μ}_{ν} Δ_{0}^{+}, H_{ϕϕ} H^{c ˉ_{ν} c^{μ}} = Δ_{m}^{+}, = - δ^{μ}_{ν} Δ_{0}^{+} .

All remaining blocks vanish. In particular, the algebraic auxiliary field has zero two-point function.

No positivity assertion is made for this gauge-fixed matrix. Moreover, gauge fixing does not impose $C [h] = 0$ as an identity on the off-shell configuration space.

Displayed formulas in this passage

VII-B@1#eq:trace-reversal · line 484
$(T h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} h, T^{2} = id .$
VII-B@1#eq:free-action · line 492
$S_{0} = \int [\frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} - \frac{1}{4} \partial_{λ} h \partial^{λ} h + \frac{1}{2} \partial_{λ} ϕ \partial^{λ} ϕ - \frac{1}{2} m^{2} ϕ^{2} - c ˉ_{μ} □ c^{μ} - \frac{1}{4} β_{μ} β^{μ}] d^{4} x .$
VII-B@1#eq:wightman · line 531
$Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{m} ( k )} e^{- i ω_{m} (k) (x^{0} - y^{0}) + i k \cdot (x - y)}, ω_{m} (k) = ∣ k ∣^{2} + m^{2} .$
VII-B@1#eq:H-even · line 542
$H_{hh μν ρ σ} H^{c^{μ} c ˉ_{ν}} = T_{μν ρ σ} Δ_{0}^{+}, = δ^{μ}_{ν} Δ_{0}^{+}, H_{ϕϕ} H^{c ˉ_{ν} c^{μ}} = Δ_{m}^{+}, = - δ^{μ}_{ν} Δ_{0}^{+} .$
VII-B@1#eq:H-odd · line 542
$H_{hh μν ρ σ} H^{c^{μ} c ˉ_{ν}} = T_{μν ρ σ} Δ_{0}^{+}, = δ^{μ}_{ν} Δ_{0}^{+}, H_{ϕϕ} H^{c ˉ_{ν} c^{μ}} = Δ_{m}^{+}, = - δ^{μ}_{ν} Δ_{0}^{+} .$
unlabeled · line 477
$Ψ = (h_{μν}, ϕ, c^{μ}, c ˉ_{μ}, β_{μ}) .$
unlabeled · line 513
$P_{hh} = - T □, P_{ϕϕ} = - (□ + m^{2}), P_{β β} = - \frac{1}{2} η,$
unlabeled · line 519
$P_{odd} = (0 - □ □ 0) .$