dS/CFT Wave-Function Proposals

The most concrete dS/CFT statement identifies the late-time de Sitter wave function with a Euclidean generating functional, not with a thermal partition function. It can organize perturbative wave-function coefficients and their conformal covariance, but its probability interpretation requires $|\Psi|^2$, a contour, and an inner product. This page fixes $(d+1)$-dimensional de Sitter space, the Bunch–Davies state selected by a Euclidean past contour, and a free scalar of mass $m$ before discussing interactions. A conventional unitary CFT dual of complete de Sitter quantum gravity remains a proposal, not an established result, as of 2026-08-20.

Required background. Gauge Theory and String-Theory Preparation provides generating functionals, sources, and analytic continuation.

Helpful background. Quantum Gravity frames the duality claim; Quantum Information supplies state and probability language; Causality and Global Structure supplies spacelike conformal infinity; and Semiclassical Gravity supplies saddle-point wave functions.

The late-time scalar data

In conformal time $\eta\lt0$, planar de Sitter has

$$ds^2=\frac{L^2}{\eta^2}\left(-d\eta^2+d\mathbf x^2\right).$$

Near $\eta=0^-$, solutions of $(\Box-m^2)\phi=0$ have two behaviors

$$\phi(\eta,\mathbf x)\sim(-\eta)^{\Delta_-}\phi_-(\mathbf x) +(-\eta)^{\Delta_+}\phi_+(\mathbf x), \qquad \Delta_\pm=\frac d2\pm\nu,$$

where

$$\nu=\sqrt{\frac{d^2}{4}-m^2L^2}.$$

For $0\lt m^2L^2\lt d^2/4$, the complementary-series weights are real. Above the threshold, $\nu=i\mu$ and the weights $d/2\pm i\mu$ are complex conjugates. At the threshold $\nu=0$, the two roots coincide and the independent behaviors are $(-\eta)^{d/2}$ and $(-\eta)^{d/2}\log(-\eta/L)$; the two-power formula must be replaced. These facts follow by substituting $\phi\propto(-\eta)^\Delta$ into the zero-momentum equation, which gives $\Delta(\Delta-d)+m^2L^2=0$. Strominger 2001, §§ 5–8 discusses the scalar weights and their interpretation.

The minimally coupled massless endpoint needs separate infrared data. Its formal roots are $0,d$, but its homogeneous mode does not have the ordinary normalizable de Sitter-invariant Bunch–Davies Fock vacuum Allen 1985, p. 3136. The Euclidean problem makes the issue visible: a constant scalar on the sphere has zero gradient action and no mass term, leaving an undamped integral over its constant value. One may regulate this zero mode or specify a restricted observable algebra; one must not silently import the massive Gaussian vacuum into that endpoint. The ordinary Gaussian discussion below assumes that this infrared issue is absent or explicitly treated.

Wave function, generating functional, and probability

With late-time boundary value $\phi_0$, the semiclassical wave function is

$$\Psi_{\mathrm{BD}}[\phi_0] \simeq \exp\!\left(iS_{\mathrm{ren,on\text{-}shell}}[\phi_0]\right).$$

The proposal schematically writes

$$\Psi_{\mathrm{dS}}[\phi_0]=Z_{\mathrm{EFT}}[\phi_0],$$

where the right side is a Euclidean generating functional whose sources and operator dimensions are fixed by the asymptotic modes. Functional derivatives of $\log\Psi$ produce wave-function coefficients. They are not automatically expectation values in a positive-norm boundary Hilbert space.

For a Gaussian mode,

$$\log\Psi=-\frac12\int\frac{d^dk}{(2\pi)^d} \,\phi_0(\mathbf k)K(k)\phi_0(-\mathbf k)+\cdots .$$

The physical late-time distribution is proportional to

$$|\Psi|^2=\exp\!\left(2\operatorname{Re}\log\Psi\right).$$

Consequently, a local imaginary phase in $K$ can affect momentum correlations while dropping out of equal-time field probabilities. For one real regulated mode with $\Psi(x)\propto e^{-Kx^2/2}$ and $\operatorname{Re}K\gt0$, direct Gaussian integration gives $\langle x^2\rangle=1/(2\operatorname{Re}K)$. The probability depends on the real part, not on $1/K$. Calling $\Psi$ a thermal partition function would lose this distinction: thermal traces and pure-state amplitudes obey different conjugation, normalization, and observable rules.

Results and limitations through 2026-08-20

The formulation axes are vacuum, Lorentzian contour, scalar mass, asymptotic branch, boundary condition, renormalization phase, and probability prescription.

Evidence or position	Controlled result	Limitation or adverse discriminator	Date
Scalar dS/CFT, Strominger 2001, §§5–8, pp. 7–14	Asymptotic symmetries and free-field calculations give Euclidean conformal weights and boundary correlator structures	Complex weights for the principal series obstruct an unqualified reflection-positive CFT interpretation	2001
Asymptotic quantum states, Witten 2001, pp. 7–15	Past and future asymptotic data define mathematically meaningful state pairings and “meta-observables”	A single observer cannot operationally measure the global object, and the construction does not provide microscopic dynamics	2001
Euclidean analytic continuation	Continuing a regular Euclidean-AdS saddle can generate Bunch–Davies perturbative wave-function coefficients	Continuing the metric does not by itself continue the integration cycle, inner product, or nonperturbative completion	2026 cutoff
Strong dS/CFT interpretation	Conformal covariance suggests a compact boundary organization of late-time data	No controlled argument establishes a complete conventional unitary Euclidean CFT dual for realistic interacting de Sitter gravity	2026 cutoff

The perturbative wave-function dictionary is controlled in selected free or weakly interacting calculations. Competing positions regard it as a calculational continuation, a Euclidean boundary dual with unusual reality conditions, or part of a complete quantum-gravity theory. The last step remains conjectural. An unresolved limitation is how to define a positive probability rule, complete observable algebra, and nonperturbative contour simultaneously.

Direct and continued calculations

A meaningful continuation test holds fixed the state prescription. The positive-frequency Bunch–Davies mode behaves as $e^{-ik\eta}$ at early real time; its damped continuation lies below the real $\eta$ axis. The classical solution used to prepare the ket wave function with $e^{iS}$ instead uses the conjugate early-time behavior on the usual past contour $\eta\to-\infty(1-i\epsilon)$, above that axis. Confusing these two choices reverses the Gaussian sign. Evaluate the renormalized on-shell action with the declared preparation contour, then compare with a regular Euclidean-AdS solution after continuing coordinates, radius, sources, and counterterms together. In a nondegenerate mass range, the nonlocal $k^{2\nu}$ dependence supplies a check; resonant orders can contain logarithms, and local contact terms depend on subtraction.

Complex conjugation supplies another check. For real boundary fields, $K(-\mathbf k)=K(\mathbf k)$, while $|\Psi|^2$ depends only on its real part. Above the mass threshold, the two weights must be treated together to respect field reality. Selecting a single complex branch without declaring a contour is not a complete boundary condition.

Interactions add higher kernels to $\log\Psi$. Their singularities and conformal Ward identities can be tested perturbatively, but loops introduce infrared and secular issues. Analytic continuation remains useful only within a specified approximation and regulator. It does not prove de Sitter stability or a microscopic state count.

What the dictionary does not yet supply

Even exact knowledge of every late-time coefficient would leave a formulation question: which coefficients are observables for one static observer, and which only describe correlations between asymptotic conditions? The answer cannot be inferred from conformal covariance. One must give the boundary inner product, identify gauge redundancies, and show how probabilities obey normalization and composition. Gravity also makes the future geometry fluctuate, so fixing a boundary metric can select a conditional amplitude rather than an ordinary correlator in one theory.

There is a second distinction between perturbative completeness and microscopic completeness. Reconstructing all tree-level bulk vertices from wave-function singularities would be a major controlled result, but it would not establish a finite nonperturbative state space, horizon microstates, or stability against topology-changing saddles. Conversely, unusual complex dimensions are an adverse test for a conventional Euclidean CFT interpretation, not by themselves a contradiction of every possible boundary formulation. Keeping these alternatives separate prevents a successful calculational interface from being promoted prematurely to a full duality.

What could change these conclusions

The conclusions could change if a theorem supplies a reflection-positive boundary construction with an explicit bulk inverse, if an independent Lorentzian calculation reproduces interacting continued kernels and their probability measure, or if a counterexample exposes contour-dependent observables previously claimed universal. The conclusions also depend on whether a nonperturbative integration cycle, new mass-sector obstruction, or controlled quantum-gravity model changes the status of the state space.

Exercises

Compute $\Delta_\pm$ for a massless scalar and for $m^2L^2\gt d^2/4$. Then explain why a wave-function phase is not a thermal observable.

Solution

For $m=0$, $\nu=d/2$, so the weights are $\Delta_-=0$ and $\Delta_+=d$. Above the threshold write $\nu=i\mu$ with $\mu=\sqrt{m^2L^2-d^2/4}$; therefore $\Delta_\pm=d/2\pm i\mu$ form a conjugate pair. This checks the reality structure but not reflection positivity. A phase contributes to $\operatorname{Im}\log\Psi$ and cancels between $\Psi$ and $\Psi^*$ in $|\Psi|^2$. By contrast, a thermal partition function is a trace used to derive ensemble expectation values. Equating them without specifying conjugation and normalization changes the physical question.

References

• Allen, Bruce. “Vacuum States in de Sitter Space.” Physical Review D 32 (1985): 3136–3149. DOI. Open PDF.
• Strominger, Andrew. “The dS/CFT Correspondence.” Journal of High Energy Physics 10 (2001) 034. arXiv:hep-th/0106113.
• Witten, Edward. “Quantum Gravity in de Sitter Space.” In Strings 2001. arXiv:hep-th/0106109.