Rigorous foundation of Weinberg's tree theorem for the in-in formalism: the classical double-trajectory prescription for tree-level correlations

Steven Weinberg's "A Tree Theorem for Inflation" (Phys. Rev. D 78, 063534 (2008)) shows that the sum of all connected tree graphs of the in-in (Schwinger-Keldysh) formalism for the generating functional of equal-time expectation values equals a classical double-trajectory action: two complex solutions of the classical equations of motion, joined at the observation time with a momentum mismatch proportional to the current, and constrained at the initial time by the state. Its derivation deforms the phase-space path-integral contours off the real axis and takes a stationary-phase limit as a fictitious coupling $g\to 0$ — steps the paper itself flags as not mathematically rigorous — and agreement with the in-in diagrammatic rules is verified only through second order for a single-field inflationary model. Later literature substantially delivered the paper's proposed application program (tree-level conservation theorems for non-Gaussian correlations outside the horizon), but left the foundation open: the analytic-continuation/stationary-phase step is not justified in the literature, and no non-perturbative all-orders proof or refutation of the prescription for general interacting theories and arbitrary initial states exists. The closest later results — Gelis's independent diagrammatic derivation, exact at tree level for one- and two-point functions and for higher points in the strong-field regime, and Chen-Namjoo-Wang's linear-order equivalence proof in multi-field inflation — cover only special cases. The surviving open problem is to establish the tree theorem rigorously, with an explicit domain of validity, or to exhibit a counterexample within its intended scope.

Background

In cosmology one does not compute scattering amplitudes between "in" and "out" states, as in particle physics, but expectation values of Heisenberg-picture operators in a single initial state, usually one resembling the vacuum at very early times before perturbations leave the horizon during inflation. The appropriate framework is the "in-in" formalism (also called the Schwinger-Keldysh or closed-time-path formalism): a diagrammatic expansion in which each interaction vertex sits on one of two time branches, producing time-ordered, anti-time-ordered, and mixed propagators with opposite vertex signs on the two branches.

Tree graphs — Feynman graphs containing no closed circuits — dominate these expectation values in the theories of interest. In general relativity a connected graph with $L$ loops is suppressed relative to trees by roughly $(GH^2)^L$ , where $G$ is Newton's constant and $H$ the inflationary expansion rate, and CMB observations suggest $GH^2\approx 10^{-10}$ ; the same power counting holds in any theory whose coupling $g$ enters the action only as an overall factor $1/g$ . Moreover, in a tree graph the wave number of every internal line is a sum of external wave numbers, so superhorizon conservation statements (time independence of quantities such as the curvature perturbation $\zeta$ while physical wavelengths exceed $H^{-1}$ ) hold tree by tree, whereas loops mix in arbitrarily large virtual wave numbers. Tree-level in-in correlators are therefore the central computational object of primordial cosmology, and their higher-point ("non-Gaussian") members are a primary observational test of inflation.

For ordinary S-matrix elements it is a classical result that the sum of all tree graphs can be obtained by solving the classical field equations in the presence of a c-number external current. In the Appendix to a 2005 paper Weinberg asserted an in-in analog, but the prescription given there was incorrect; the paper "A Tree Theorem for Inflation" states and derives the corrected theorem for a general Hamiltonian system. Introduce the generating functional

W[J,t_1]=\ln\langle 0,\mathrm{in}|\exp\big(\textstyle\sum_a Q_a(t_1)J_a\big)|0,\mathrm{in}\rangle,

whose derivatives with respect to the c-number current $J_a$ give all equal-time expectation values ( $a$ is a compound index labeling field species and spatial position). In the tree approximation $W[J,t_1]$ is exactly

W[J,t_1]_{\mathrm{tree}} = i\int_{t_0}^{t_1} dt\,\big[L(q_R,\dot q_R,t)-L(q_L,\dot q_L,t)\big] + \sum_a q_{R,a}(t_1)J_a,

where $q_L(t)$ and $q_R(t)$ are a pair of complex c-number histories ("double trajectories") satisfying: (A) both solve the classical Euler-Lagrange equations of the Lagrangian $L$ associated with the Hamiltonian; (B) at the observation time $t_1$ they join, $q_L(t_1)=q_R(t_1)$ , and their conjugate momenta differ by the imaginary current, $\partial L/\partial\dot q_{L,a}(t_1)-\partial L/\partial\dot q_{R,a}(t_1)=-iJ_a$ ; (C) at the initial time $t_0$ they obey constraints fixed by the initial state — for a Bunch-Davies-like vacuum (the standard inflationary initial state, in which modes behave as positive-frequency oscillators at early times) at $t_0=-\infty$ , $q_L$ and $q_R$ must be purely positive- and negative-frequency superpositions, respectively. The full in-in tree expansion — normally an ever-growing bookkeeping of nested time integrals and branch signs — is thereby encoded in one classical boundary-value problem; the paper illustrates this for a single inflaton field in a Robertson-Walker (spatially flat, expanding) background.

The derivation writes the doubled in-in path integral over phase-space histories $(q_a(t),p_a(t))$ on the two time branches, introduces a fictitious coupling $g$ rescaling the entire action by $1/g$ (a connected graph with $L$ loops then carries $g^{L-1}$ , so the $g\to 0$ limit of $W[J,t_1,g]$ isolates the sum of tree graphs), and evaluates that limit by stationary phase: the integral is dominated by complex histories at which the coefficient of $1/g$ in the exponent is stationary. Passing to that saddle requires distorting the integration contours of $q_a(t)$ and $p_a(t)$ away from the real axis, and this analytic continuation is not justified — the paper concedes the derivation "was something less than mathematically rigorous". As a check, the paper expands the double-trajectory integral equations in powers of the interaction and confirms agreement with the in-in diagrammatic rules (external-line factors, branch vertex signs, and the four propagator types LL, RR, LR, RL) through second order — the first order containing internal lines — for the single-field inflationary model only. The introduction also expresses the hope that the classical-trajectory formulation "may prove useful in proving general theorems about non-Gaussian correlations in cosmology".

Problem Statement

Establish the tree theorem for the in-in (Schwinger-Keldysh, or closed-time-path) formalism — the diagrammatic framework for expectation values of Heisenberg-picture operators in a single initial state, as opposed to scattering amplitudes between "in" and "out" states — as a theorem, or refute it. The tree theorem asserts that the sum of all connected tree graphs (Feynman graphs containing no closed circuits) of the in-in expansion is generated exactly by one classical boundary-value problem: a pair of complex classical trajectories joined at the observation time.

(a) Rigor of the derivation. Provide a mathematically rigorous derivation of the theorem under explicitly stated hypotheses on the Hamiltonian (Hermiticity and the reality condition $L^*[q]=L[q^*]$ on the Lagrangian), on the interactions, and on the initial state.

(b) Generality of the statement. Prove non-perturbatively — to all orders in the field amplitude, without reliance on an order-by-order perturbative check — that the classical double-trajectory prescription (the pair of complex Euler-Lagrange solutions $q_L$ , $q_R$ satisfying the joining and momentum-matching endpoint constraints at the observation time $t_1$ , the momentum mismatch being $-iJ_a$ , together with the initial-time constraints fixed by the initial state) yields exactly the sum of all connected tree graphs of the in-in expansion for the generating functional $W[J,t_1]=\ln\langle 0,\mathrm{in}|\exp\big(\textstyle\sum_a Q_a(t_1)J_a\big)|0,\mathrm{in}\rangle$ , whose derivatives with respect to the current $J_a$ generate the equal-time expectation values of the Heisenberg operators $Q_a$ , for general interacting quantum field theories and arbitrary initial states; or, failing that, delimit precisely the class of Hamiltonians, interactions, and initial states for which the prescription holds.

Alternatively, refute the theorem by exhibiting a Hamiltonian system and initial state within its intended scope for which the double-trajectory prescription disagrees with the sum of connected tree graphs of the in-in expansion at some finite order. A genuine obstruction to the contour deformation described in Background bears on the validity of that derivation route, not on the theorem's truth, and so does not by itself refute the theorem.

(a) and (b) are two aspects of one scientific question: whether the classical-trajectory evaluation of tree-level in-in correlators is a theorem. Establishing it requires both a rigorous derivation and its validity to all orders, beyond the existing second-order, single-field check.

References

Steven Weinberg, "A Tree Theorem for Inflation", Phys. Rev. D 78, 063534 (2008). DOI: 10.1103/PhysRevD.78.063534, https://doi.org/10.1103/PhysRevD.78.063534 (arXiv:0805.3781, https://arxiv.org/abs/0805.3781)
Steven Weinberg, "Non-Gaussian Correlations Outside the Horizon", Phys. Rev. D 78, 123521 (2008). DOI: 10.1103/PhysRevD.78.123521, https://doi.org/10.1103/PhysRevD.78.123521 (arXiv:0808.2909, https://arxiv.org/abs/0808.2909)
Steven Weinberg, "Non-Gaussian Correlations Outside the Horizon II: The General Case", Phys. Rev. D 79, 043504 (2009). DOI: 10.1103/PhysRevD.79.043504, https://doi.org/10.1103/PhysRevD.79.043504 (arXiv:0810.2831, https://arxiv.org/abs/0810.2831)
François Gelis, "Tree-level correlations in the strong field regime", JHEP 09 (2017) 055. DOI: 10.1007/JHEP09(2017)055, https://doi.org/10.1007/JHEP09(2017)055 (arXiv:1703.01227, https://arxiv.org/abs/1703.01227)
Xingang Chen, Mohammad Hossein Namjoo, Yi Wang, "On the equation-of-motion versus in-in approach in cosmological perturbation theory", JCAP 01 (2016) 022. DOI: 10.1088/1475-7516/2016/01/022, https://doi.org/10.1088/1475-7516/2016/01/022 (arXiv:1505.03955, https://arxiv.org/abs/1505.03955)
Joel Meyers, "Non-Gaussian Correlations Outside the Horizon in Local Thermal Equilibrium" (2012), unpublished preprint. arXiv:1212.4438, https://arxiv.org/abs/1212.4438
Francis Bernardeau, "Mode coupling evolution in arbitrary inflationary backgrounds", JCAP 02 (2011) 017. DOI: 10.1088/1475-7516/2011/02/017, https://doi.org/10.1088/1475-7516/2011/02/017 (arXiv:1003.3575, https://arxiv.org/abs/1003.3575)