Two boson loss · KeldyshContraction.jl

Two Body Loss

using KeldyshContraction
using KeldyshContraction: Regularisation.Plus as Plus
using KeldyshContraction: Regularisation.Minus as Minus

System and regularization

The interaction action of two body loss, is defined as

\[S_\mathrm{int} = i\Gamma \int d^d x \, [\frac{1}{2}(\bar{\phi}_+\phi_+)^2 +\frac{1}{2} (\bar{\phi}_-\phi_-)^2 -\bar{\phi}_-^2\phi_+^2]\]

In the RAK basis, this gives

\[S_\mathrm{int} = i\Gamma\int d^d x \, [\frac{1}{2}\bar{\phi}_c\bar{\phi}_q(\phi_c^2+\phi_q^2)-\frac{1}{2}\phi_c\phi_q(\bar{\phi}_c^2+\bar{\phi}_q^2)+2\bar{\phi}_c\bar{\phi}_q\phi_c\phi_q)]\]

A good check if the interaction Lagrangian is a valid physical process, is to check if the normalization identity $Z=1$ holds. We can do this perturbatively in $g$ by expanding $\exp(i S_\mathrm{int})$ and showing the average of the linear part of the system is zero

\[\langle S_\mathrm{int}\rangle = \langle S_\mathrm{int}^2\rangle =\ldots = 0\]

In computing the average, one performs Wick contractions to describe the average in terms of the two-point correlators of the linear part of the system. However, in this case we don't find that the vacuum expectation value of the interaction Lagrangian is zero:

@qfields c::Destroy(Classical) q::Destroy(Quantum)

loss2boson_unregular =
    im * (
        0.5 * c' * q' * (c^2 + q^2) - 0.5 * c * q * ((c')^2 + (q')^2) +
        c' * q' * (c * q + c * q)
    )

KeldyshContraction._wick_contraction(loss2boson_unregular)

\[\left( -0.0 - 1.0 \cdot im \right) \cdot G^K\left( y_1, y_1 \right) \cdot G^R\left( y_1, y_1 \right) + \left( 0.0 + 1.0 \cdot im \right) \cdot G^K\left( y_1, y_1 \right) \cdot G^A\left( y_1, y_1 \right)\]

To make the interaction Lagrangian physically meaningful, we must regularize it by properly handling equal-space-time propagators. These equal-time arguments emerge from the continuum limit of the field theory but are naturally absent in discrete-time formulations (Gerbino et al, 2024). In the discrete picture, operators act on coherent states at adjacent (but distinct) time slices during path integral construction via Trotter decomposition. To ensure all disconnected diagrams vanish identically, we introduce a finite time-shift regularization ε > 0 for the quantum jump operators, motivated by the underlying discrete Trotter structure. One gets:

\[\bar{L}_+(t)L_+(t) \to \bar{L}_+(t)L_+(t-\epsilon) \quad, \bar{L}_-(t)L_-(t) \to \bar{L}_-(t)L_-(t+\epsilon) \quad \text{and} \quad \bar{L}_-(t)L_+(t) \to \frac{1}{2}(\bar{L}_-(t)L_+(t+\epsilon) + \bar{L}_-(t)L_+(t-\epsilon))\]

Applying this regularization to the interaction Lagrangian, we get:

loss2boson =
    im * (
        0.5 * c' * q' * (c(Minus) * c(Minus) + q(Minus) * q(Minus)) -
        0.5 * c(Plus) * q(Plus) * (c' * c' + q' * q') +
        c' * q' * (c(Plus) * q(Plus) + c(Minus) * q(Minus))
    )
L_int = InteractionLagrangian(loss2boson)

\[0.5 i c^{-} c^{-} \bar{q} \bar{c} + 0.5 i q^{-} q^{-} \bar{q} \bar{c} -0.5 i q^+ c^+ \bar{c} \bar{c} -0.5 i q^+ c^+ \bar{q} \bar{q} + 1.0 i c^+ q^+ \bar{c} \bar{q} + 1.0 i c^{-} q^{-} \bar{c} \bar{q}\]

Indeed, the vacuum expectation value of the interaction Lagrangian is now zero:

KeldyshContraction._wick_contraction(loss2boson)

\[\]

First order Green's function

GF = DressedPropagator(L_int)

Dressed Propagator:
keldysh:  Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴬ(y₁⁻,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴷ(y₁⁻,x₂) + Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂)
retarded: Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,x₂)
advanced: -1.0*Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂) + Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,x₂)

Σ = SelfEnergy(GF)

Self Energy:
keldysh:  Gᴷ(y₁⁻,y₁) + Gᴷ(y₁⁺,y₁) + -1.0*Gᴿ(y₁⁺,y₁) + Gᴬ(y₁⁻,y₁)
retarded: Gᴷ(y₁⁻,y₁) + Gᴬ(y₁⁻,y₁)
advanced: -1.0*Gᴷ(y₁⁺,y₁) + Gᴿ(y₁⁺,y₁)

The following indeed corresponds with what is reported in (Gerbino et al, 2024).

Second order Green's function

GF = DressedPropagator(L_int, 2)

Dressed Propagator:
keldysh:  Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴷ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴷ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴷ(y₂⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴷ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴷ(y₂⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴷ(y₁⁻,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁻,x₂) + -0.5*Gᴷ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴷ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,x₂) + Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴷ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₂⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₂⁻,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -0.5*Gᴷ(x₁,y₁)*Gᴬ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₂⁻,x₂) + -1.0*Gᴷ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂)
retarded: Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴿ(y₂⁻,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₂)*Gᴿ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴿ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴬ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴿ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴿ(y₁⁻,x₂) + -1.0*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₁)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₂)*Gᴿ(y₂⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴿ(y₁⁻,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴿ(y₂⁻,x₂) + -0.5*Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,x₂) + Gᴿ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴿ(y₂⁻,x₂)
advanced: Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)*Gᴬ(y₁⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₁)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₂⁺,x₂) + Gᴬ(x₁,y₁)*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂) + -0.5*Gᴬ(x₁,y₁)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -0.5*Gᴬ(x₁,y₁)*Gᴬ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁)*Gᴬ(y₂⁺,x₂) + -1.0*Gᴬ(x₁,y₁)*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂)*Gᴬ(y₁⁺,x₂)

Σ = SelfEnergy(GF, 2)

Self Energy:
keldysh:  Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + -1.0*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + -0.5*Gᴿ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁) + Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₁) + -1.0*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + -0.5*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁) + Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁) + -1.0*Gᴷ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁) + -0.5*Gᴬ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴷ(y₂⁺,y₁) + -1.0*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + -1.0*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₁) + Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)
retarded: Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + -0.5*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁) + Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + -1.0*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴷ(y₂⁻,y₁) + -0.5*Gᴿ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁) + -1.0*Gᴷ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁺,y₁) + Gᴷ(y₁⁻,y₂)*Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁) + Gᴿ(y₁⁻,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂)
advanced: -0.5*Gᴷ(y₁⁻,y₂)*Gᴷ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁) + -1.0*Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴿ(y₂⁺,y₂) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + Gᴷ(y₁⁺,y₂)*Gᴬ(y₂⁺,y₁)*Gᴷ(y₂⁺,y₂) + -1.0*Gᴷ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + -1.0*Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴿ(y₂⁺,y₁) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴷ(y₂⁻,y₁)*Gᴬ(y₂⁻,y₂) + -0.5*Gᴬ(y₁⁻,y₂)*Gᴬ(y₁⁻,y₂)*Gᴿ(y₂⁺,y₁) + -1.0*Gᴿ(y₁⁺,y₂)*Gᴬ(y₂⁻,y₁)*Gᴷ(y₂⁻,y₂) + Gᴷ(y₁⁺,y₂)*Gᴬ(y₁⁺,y₂)*Gᴷ(y₂⁺,y₁)

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