From the Wavefunction of the Universe to In-In-Correlators: A Perturbative Map to All Orders

TL;DR

The paper develops an explicit, order-by-order map between diagrams in the Wavefunction of the Universe and the Schwinger–Keldysh in–in formalism. By formulating the wavefunction path integral with an ε-prescription, introducing a generating functional, and using a Green’s-function-based field redefinition, the author constructs bulk-to-bulk and bulk-to-boundary propagators that express wavefunction coefficients ψ_n. A key innovation is the decomposition into conjugate coefficients (black/white vertices), the introduction of composite propagators, and a color-grouping procedure that collapses wavefunction diagrams into standard Schwinger–Keldysh diagrams, valid to all orders and able to handle loops. The results unify the two frameworks, enable systematic diagrammatic translations, and lay groundwork for extending to more complex theories and bootstrap approaches in cosmology.

Abstract

Both the Wavefunction of the Universe and the Schwinger-Keldysh in-in formalism are central tools for analyzing primordial cosmological observables, such as equal-time correlation functions. While their conceptual equivalence is well established, a systematic and explicit map between their diagrammatic expansions has remained elusive. In this article, I construct such a map by analyzing the relation between the two frameworks at the diagrammatic level. I show that diagrams contributing to correlation functions in the Wavefunction of the Universe approach can be uniquely reorganized into Schwinger-Keldysh diagrams. This correspondence holds to all orders in perturbation theory, including arbitrary numbers of interaction vertices and loops.

Figures (7)

• Figure 1: A graph with $V=12$ vertices and $n=8$ external legs partitioned into $P=3$ groups. Each group is labeled by the number of enclosed vertices and the number of legs intersecting the corresponding partition boundary.
• Figure 2: The same graph as in Fig. \ref{['figure_1']}, but now allowing the partition to cross internal legs connecting vertices within the same group. This changes the labeling of the corresponding partition from $n_p=5$ to $n_p=7$.
• Figure 3: A possible coloring of the partition examined in Fig. \ref{['figure_1']}.
• Figure 4: The edge belonging to the partition $\psi_7^{(5)}$, crossing the boundary twice, must be denoted with a double solid--dashed line.
• Figure 5: The same coloring scheme considered in Figs. \ref{['figure_3']} and \ref{['figure_4']}, but with a different partitioning.