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 Incoming links: Time integral

The time integral of torque about a specified origin equals the change in angular momentum about that origin under the usual fixed-origin inertial frame balance. Its units are

N m s

, equivalently

kg m^{2} s^{- 1}

. For rotation about a fixed principal axis, the magnitude is

I Δ ω

τ_{0}

⟨ O (τ_{0}) ⟩_{λ} = i \int_{- \infty (1 - i ϵ)}^{τ_{0}} d τ ⟨[H_{int} (τ), O_{I} (τ_{0})]⟩ .

Fourier transforming the three derivatives gives

H_{int} (τ) = - iλ a^{4} \int_{p_{1} \dots p_{4}} (2 π)^{3} δ^{(3)} (p_{1} + \dots + p_{4}) [p_{2} \cdot (p_{3} \times p_{4})] \prod_{a = 1}^{4} ϕ_{a} (p_{a}, τ),

where

\int_{p} = \int d^{3} p / (2 π)^{3}

. Because the four species are distinct, Wick contraction pairs each vertex field with the external field of the same species and introduces no permutation factor. If

E = k_{2} \cdot (k_{3} \times k_{4}), k_{T} = k_{1} + k_{2} + k_{3} + k_{4},

then

⟨ H_{int} (τ) O (τ_{0})⟩ = iλ a^{4} (τ) (2 π)^{3} δ^{(3)} (\sum_{a} k_{a}) E \prod_{a = 1}^{4} f (k_{a}, τ) f^{*} (k_{a}, τ_{0}) .

Combining this with part i gives the requested time integral:

⟨ O (τ_{0}) ⟩_{λ} = 2 iλ (2 π)^{3} δ^{(3)} (a \sum k_{a}) E \int_{- \infty (1 - i ϵ)}^{τ_{0}} d τ a^{4} (τ) Re [i a = 1 \prod 4 f (k_{a}, τ) f^{*} (k_{a}, τ_{0})] .

The factor

E

is a pseudoscalar, so the resulting primordial trispectrum is parity odd and purely imaginary in this momentum-space convention.