Bunch–Davies, Euclidean, and Alpha-State Diagnostics

In de Sitter space, the Bunch–Davies state is selected by short-distance positive frequency in the expanding patch and agrees with analytic continuation from the regular Euclidean sphere. Formal alpha vacua retain de Sitter invariance at the free-field level but add negative-frequency and antipodal singularities; they must not be confused with UV-soft finite initial-state excitations.

Required background. Vacuum choice fixes state data; the microlocal spectrum condition tests singularities; and state-selection failure modes fixes the claim ceiling. Helpful background. Review FLRW modes and ground/KMS selection.

Euclidean and expanding-patch modes

For de Sitter $a=-1/(H\eta)$, $\eta<0$, the site scalar mode equation is

$$v_k''+\left[k^2-\frac{\nu^2-\frac14}{\eta^2}\right]v_k=0, \qquad \nu^2=\frac94+12\xi-\frac{m^2}{H^2}.$$

The Bunch–Davies mode is

$$v_k^{\rm BD}(\eta) =\frac{\sqrt{-\pi\eta}}{2} e^{i\pi(2\nu+1)/4}H_\nu^{(1)}(-k\eta),$$

whose phase gives $e^{-ik\eta}/\sqrt{2k}$ as $-k\eta\to\infty$. The same two-point function follows by continuing the regular Green function on the Euclidean four-sphere, provided the mass/coupling admits the inverse and zero modes are treated. This equivalence is an analytic and geometric statement, not a generic rule for an arbitrary FLRW history.

Allen classifies de Sitter-invariant scalar vacua and their analytic properties Allen 1985, §§II–III, pp. 3138–3144.

First application: an alpha transform

A formal alpha mode has

$$v_k^\alpha=N_\alpha \left(v_k^{\rm BD}+e^\alpha v_k^{{\rm BD}*}\right), \qquad N_\alpha=\left(1-e^{\alpha+\alpha^*}\right)^{-1/2}, \quad \operatorname{Re}\alpha<0.$$

The Wronskian is normalized, but the two-point function contains the usual short-distance singularity with altered frequency content and additional antipodal singularities. Except for the Bunch–Davies limit, the state fails the standard Hadamard condition. In conventional interacting perturbation theory these extra singularities lead to pinched products or counterterms that are not the local state-independent ones of ordinary curved-space QFT Einhorn and Larsen 2003, §§2–4.

This does not rule out every excited initial state. A Bogoliubov coefficient that decays sufficiently at high physical momentum, encoded by a finite boundary EFT, can preserve admissibility while breaking exact de Sitter invariance. That construction is not an alpha vacuum with constant ultraviolet mixing.

Zero modes and distinct state classes

The Euclidean argument needs a separate qualification for the massless minimally coupled scalar. At $m=0$ and $\xi=0$, the Euclidean operator on the compact sphere has a constant zero mode, so its inverse is not defined until that mode is treated. In Lorentzian de Sitter there is no ordinary de Sitter-invariant Fock vacuum for the unsmeared field $\phi$ with a positive two-point function Allen 1985, §IV, pp. 3144–3147. This obstruction is not evidence for choosing an alpha vacuum. One may instead break de Sitter invariance, regulate the zero mode, or restrict to derivative/shift-invariant observables, but the algebra and state claim then change.

For positive effective mass, the Euclidean prescription determines the regular Green function and its Lorentzian boundary value under the stated continuation. The resulting Bunch–Davies state is Hadamard. Formal alpha states are a different, constant Bogoliubov family; their additional antipodal singularity persists at arbitrarily short wavelengths. By contrast, a finite excitation with $\beta_k$ vanishing rapidly above a preparation scale changes only the smooth part of the two-point function and can remain Hadamard.

This classification prevents three distinct issues from being collapsed: an operator zero mode, an inadmissible ultraviolet singularity, and an allowed state-dependent infrared excitation. Each requires a different remedy and supports a different set of observables.

The structure map shows Euclidean analyticity and Hadamard singularity as separate filters before state-dependent predictions.

Bunch–Davies/Euclidean agreement uses de Sitter analyticity; formal alpha states pass free normalization but fail the standard short-distance and interaction tests. Schematic; not to scale.

Domain and failure conditions

See the chapter’s canonical domain table. State the de Sitter patch, zero-mode treatment, mass/coupling range, analytic continuation, and whether the claim concerns a free two-point function or interacting composites.

Adversarial test. Insert BD and alpha two-point functions into a one-loop local composite. BD singularities are removed by the standard local covariant counterterms. Constant alpha mixing introduces additional singular products; if their removal requires state-dependent or antipodal nonlocal counterterms, the candidate is not an admissible interacting vacuum in this framework.

The failure map downgrades exact free de Sitter invariance when it conflicts with local interacting renormalization.

Symmetry and Wronskian normalization do not replace the Hadamard and interaction tests; UV-soft finite boundary excitations are a distinct construction. Schematic; not to scale.

References

• Allen, B., “Vacuum States in de Sitter Space,” Physical Review D 32, 3136–3149 (1985), doi:10.1103/PhysRevD.32.3136.
• Einhorn, M. B., and F. Larsen, “Interacting Quantum Field Theory in de Sitter Vacua,” Physical Review D 67, 024001 (2003), doi:10.1103/PhysRevD.67.024001.