Wavefunction Coefficients versus In-In Correlators

Wavefunction coefficients are vertices in $\log\Psi$; in-in correlators are normalized expectation values obtained from a forward-and-backward contour or, at one late time, from the probability weight $|\Psi|^2$. The conversion takes real parts, inverts the quadratic kernel, joins lower-point coefficients, and cancels vacuum diagrams. A coefficient cannot be renamed a correlator term by term.

Required background. Late-Time Wavefunctions and Boundary Data fixes $\psi_n$; In-In Cosmological Correlators supplies the Schwinger–Keldysh definition.

Helpful background. Initial Density Matrices and Contour Boundary Conditions treats state insertions; Cosmological Loops and Renormalization and Open-System Noise and Dissipation in de Sitter control loop and coarse-grained effects; Wavefunction and Correlator Object Dictionary fixes terminology.

Born-rule conversion at one time

Suppress momentum delta functions and write

$$\Psi[\varphi]=\mathcal N\exp\!\left[-\frac12\psi_2\varphi^2 -\frac1{3!}\psi_3\varphi^3-\frac1{4!}\psi_4\varphi^4-\cdots\right].$$

An equal-time expectation value is

$$\langle F[\varphi]\rangle =\frac{\int\mathcal D\varphi\,\Psi^*[\varphi]F[\varphi]\Psi[\varphi]} {\int\mathcal D\varphi\,|\Psi[\varphi]|^2}.$$

Therefore the probability vertices are $2\operatorname{Re}\psi_n$, and the Gaussian covariance is

$$P(k)=\frac{1}{2\operatorname{Re}\psi_2(k)}.$$

The denominator removes disconnected vacuum contributions. For unequal-time or operator-ordering questions, one must retain separate $+$ and $-$ fields on the full in-in contour; $|\Psi|^2$ at one slice is not sufficient.

First application: derive the tree-level three-point function

Expand the cubic probability vertex once around the Gaussian measure. Wick contraction gives

$$\langle\varphi_{\mathbf k_1}\varphi_{\mathbf k_2}\varphi_{\mathbf k_3}\rangle_c =-2\operatorname{Re}\psi_3(\mathbf k_1,\mathbf k_2,\mathbf k_3) \prod_{i=1}^{3}\frac{1}{2\operatorname{Re}\psi_2(k_i)},$$

with the overall momentum delta function understood. Thus even at tree level the answer uses the real part of the cubic coefficient and three inverse quadratic kernels. At four points, $2\operatorname{Re}\psi_4$ contributes together with exchange contractions of two $\psi_3$ vertices. At loops, coefficient and Born-rule loops mix, and the initial density matrix and contour counterterms are essential.

For the massless Gaussian kernel $\operatorname{Re}\psi_2=k^3/H^2$, the formula reproduces $P(k)=H^2/(2k^3)$. Maldacena’s bispectrum computation implements exactly this logic: a late-time wavefunctional contribution is converted to a normalized expectation value and supplemented by field-redefinition terms Maldacena 2003, §§ 3–4.

Phases, contacts, and cuts

Multiplying the wavefunction by

$$\Psi[\varphi]\mapsto e^{iF_{\mathrm{loc}}[\varphi]}\Psi[\varphi]$$

shifts $\psi_n$ by imaginary local functions but leaves every equal-time field probability unchanged. Such phases can matter for conjugate momenta, unequal-time observables, or gluing to another contour, so “unobservable” must be qualified by the task. Conversely, nonlocal imaginary parts enter cosmological cutting relations even when an equal-time correlator uses a real combination.

The conversion assumes a normalizable state and a regulated functional integral. Infrared divergences, secular growth, and gauge constraints can obstruct the naive late-time limit. Gravitational correlators require gauge-invariant or relational observables, not merely the wavefunction of coordinate perturbations.

Adversarial control: add a pure phase

Add $i c\int d^3x\,\varphi^3$ to $\log\Psi$ and recompute the equal-time three-point function. $\psi_3$ changes, but $2\operatorname{Re}\psi_3$ and the correlator do not. Any proposed dictionary mapping the complex coefficient directly to a measured bispectrum fails. Then calculate a momentum insertion to show why the phase can still carry canonical information.

The evidence ceiling is an exact functional-integral relation and a systematic perturbative conversion between two distinct objects. It does not make every wavefunction coefficient observable, and it does not turn bootstrap consistency into a boundary dual. In-in loop theory, open-system reductions, and holographic interpretation remain separate handoffs.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Maldacena, J. (2003). “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models.” Journal of High Energy Physics 2003(5), 013. DOI.