Keldysh formalism

The in-in formalism computes expectation values at a finite time by evolving a state forward and backward along a closed time contour. At first order in the interaction Hamiltonian,An $iϵ$ tilt of the early-time contour projects onto the interacting vacuum.

With ${\zeta }_k(\tau )=u_k(\tau )a_k+u_k^{\ast }(\tau )a_{-k}^{†}$, the operator order external field then vertex gives $⟨{\zeta }_k(0){\zeta }_p^{\prime }(\tau )⟩=(2\pi {)}^3\delta (k+p)u_k(0)u_p^{\prime }{}^{\ast }(\tau )$. For positive-frequency $u\propto e^{-ik\tau }$, these vertex contractions contain $e^{+iK\tau }$ and converge on ${\tau }_i=-\infty (1-i0)$. Conjugating the contraction representation also conjugates the outside in-in prefactor. Mixing the two changes signs and defeats the vacuum prescription.

The Gaussian Bunch-Davies vacuum is selected by tilting the early endpoint to $-\infty (1-i\epsilon )$ for integrands $e^{iK\tau }$ with $K>0$. They then decay at the early boundary. The complex-conjugate integrand requires the opposite tilt. Ordered contractions and their contour must be conjugated together.

For $K=k_1+k_2+k_3$, $s_2=k_1{k}_2+k_1{k}_3+k_2{k}_3$ and $s_3=k_1{k}_2{k}_3$, the convergent in-in formalism integral is $I={\int }_{-\infty (1-i0)}^{0^{-}}{\tau }^{-2}{\prod }_i(1-i{c}_s{k}_i\tau )e^{i{c}_{s}K\tau }d\tau$. Its real endpoint divergence drops out of the expectation value, but the finite imaginary endpoint term $-c_{s}K$ must be kept. The remaining terms contribute $c_s{s}_2/K+c_s{s}_3/K^2$. Reversing the exponential requires conjugating both the contour and the outside prefactor.

The closed time contour of the in-in formalism gives four bulk propagators: time-ordered $G_{RR}$, anti-time-ordered $G_{LL}$, and the two Wightman functions $G_{LR}$ and $G_{RL}$. Vertices on the two contour branches carry opposite factors of $i$.

For a free mode function $f(k,\eta )$ and a correlator evaluated at ${\eta }_0$, the two cosmological bulk-to-boundary propagators are $K_R=f(k,\eta )f^{\ast }(k,{\eta }_0)$ and $K_L=f^{\ast }(k,\eta )f(k,{\eta }_0)$, up to the chosen contour-label convention.

For a free mode $f(k,\eta )$, the right-right bulk propagator is time ordered,whereas the mixed propagators are Wightman functions.

For real couplings and real external data, reflection of every contour label complex conjugates an in-in diagram: $B_{LL}=B_{RR}^{\ast }$ and $B_{RL}=B_{LR}^{\ast }$.

The interaction Hamiltonian generates interaction-picture time evolution. To leading order for many cubic field interactions it is the negative of the cubic interaction Lagrangian, although derivative interactions require the canonical Legendre transform to justify this relation.

For $\mathcal{L}={\dot{\varphi }}^2/2-{\mathcal{V}}_2+{\mathcal{L}}_3(\varphi ,\dot{\varphi })$, where ${\mathcal{L}}_3$ is cubic in perturbation amplitude, the canonical momentum has $\pi =\dot{\varphi }+O({\varphi }^2)$. Put $\dot{\varphi }=\pi +\Delta$ with $\Delta$ second order. Then $\pi \dot{\varphi }-{\dot{\varphi }}^2/2={\pi }^2/2-{\Delta }^2/2$, so through cubic order the Legendre transform in mechanics gives ${\mathcal{H}}_3=-{\mathcal{L}}_3$ evaluated with the free momentum. At fourth order there are extra terms, so derivative interactions do not obey this simple rule at all orders.