Bunch–Davies, Euclidean, and Alpha-State Diagnostics

In exact de Sitter space and the stable scalar sector described below, the Bunch–Davies state can be characterized in two equivalent ways: by an analytic positive-frequency condition at the infinite past of the expanding patch, or by continuation of the regular Euclidean Green function on the four-sphere. The equivalence has hypotheses. Euclidean invertibility alone is not enough: the continued two-point function must also be positive and infrared controlled. The construction can fail when the Euclidean operator has a zero or negative mode, and it does not turn an arbitrary FLRW positive-frequency prescription into a preferred vacuum. Constant alpha mixing preserves free de Sitter invariance but fails the standard Hadamard ultraviolet test.

Required background. Vacuum choice and initial states fixes normalized mode data; the microlocal spectrum condition tests short-distance singularities; and state-selection failure modes fixes the claim ceiling. Helpful background. Review FLRW mode quantization and initial density matrices.

Bunch–Davies modes in the expanding patch

In spatially flat de Sitter coordinates,

$$ds^2=a(\eta)^2(d\eta^2-d\boldsymbol x^2), \qquad a(\eta)=-\frac1{H\eta}, \qquad \eta<0.$$

For the canonically rescaled scalar mode $v_k=a\phi_k$, the site conventions give

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

It is useful to define $M_{\mathrm{eff}}^2=m^2-12\xi H^2$, so that $\nu^2=9/4-M_{\mathrm{eff}}^2/H^2$. The conformally coupled massless case in these conventions is $\xi=-1/6$, hence $\nu=1/2$.

The Bunch–Davies mode is

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

The large-argument Hankel asymptotic gives

$$v_k^{\mathrm{BD}}(\eta) \longrightarrow \frac{e^{-ik\eta}}{\sqrt{2k}} \qquad(-k\eta\to\infty).$$

This is an asymptotic condition at $\eta\to-\infty$ with the analytic $i\epsilon$ boundary prescription. It is stronger and more specific than saying that a mode looks locally Minkowskian at one finite time. Bunch and Davies obtain the corresponding de Sitter Green function by Euclidean continuation in Bunch and Davies 1978, §§2–3, pp. 119–126.

Euclidean continuation and the invariant Green function

Embed de Sitter space as a hyperboloid and define the invariant

$$Z(x,x')=H^2X(x)\mathbin{\cdot}X(x').$$

Coincident points have $Z=1$; antipodal points $x_A$ have $Z(x_A,x')=-Z(x,x')$. In the flat patch,

$$Z(x,x') =1+\frac{(\eta-\eta')^2-|\boldsymbol x-\boldsymbol x'|^2} {2\eta\eta'}.$$

Set $h_\pm=3/2\pm\nu$. In the simple stable sector $M_{\mathrm{eff}}^2>0$, the Euclidean Klein–Gordon operator on $S^4$ is positive and invertible, and the regular Euclidean solution continues to the Lorentzian Bunch–Davies Wightman function

$$G_{\mathrm{BD}}^+(Z) =\frac{H^2}{16\pi^2} \Gamma(h_+)\Gamma(h_-) {}_2F_1\!\left( h_+,h_-;2;\frac{1+Z_\epsilon}{2} \right),$$

where $Z_\epsilon=Z-i0\,\operatorname{sgn}(\eta-\eta')$ fixes the Wightman boundary value in this patch. For $\nu=1/2$, the hypergeometric function reduces to the conformal result proportional to $(1-Z_\epsilon)^{-1}$.

This calculation explains the word Euclidean: regularity on the compact sphere selects one inverse, and analytic continuation selects its Lorentzian boundary value. It also exposes the hypotheses—exact de Sitter geometry, the chosen analytic continuation, a positive Euclidean operator, infrared convergence, and a specified observable algebra. Outside the stated stable sector the displayed hypergeometric expression can still exist by analytic continuation in its parameters, but that fact alone does not establish a positive Wightman state. Allen analyzes the invariant scalar state family and these analytic properties in Allen 1985, §§II–III, pp. 3138–3144.

Constant alpha mixing adds singularities

A formal alpha mode is

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

The Wronskian is normalized, and the free two-point function remains de Sitter invariant. Schematically—and with the phases fixed by the displayed mode convention—it contains

$$\begin{aligned} G_\alpha^+(x,x')=N_\alpha^2\big[& G_{\mathrm{BD}}^+(x,x') +e^{\alpha+\alpha^*}G_{\mathrm{BD}}^+(x',x)\\ &+e^\alpha G_{\mathrm{BD}}^+(x_A,x') +e^{\alpha^*}G_{\mathrm{BD}}^+(x,x'_A) \big]. \end{aligned}$$

The first line changes the positive/negative-frequency mixture at arbitrarily large $k$. The second line copies the ordinary coincidence singularity at $Z=1$ to antipodal separation $Z=-1$. Mottola constructs the invariant family in Mottola 1985, §§III–IV.

These observations give two independent failures for every nontrivial constant alpha state:

1. Its two-point function differs from the Bunch–Davies Hadamard function by a distribution that is not smooth near coincidence. Equivalently, its wavefront set contains both frequency orientations.
2. It has an additional antipodal singularity. Products needed for local composite operators and conventional interacting perturbation theory then acquire singularities that are not removed by the usual local, state-independent counterterms.

Brunetti, Fredenhagen, and Hollands give the sharp Hadamard argument and show divergent fluctuations for averaged local observables in Brunetti, Fredenhagen, and Hollands 2005, §§1–3. Einhorn and Larsen analyze the associated obstruction in conventional interacting perturbation theory in Einhorn and Larsen 2003, §§2–4.

The conclusion is deliberately scoped: nontrivial constant alpha states are not admissible states for standard local, covariantly renormalized interacting QFT. This diagnosis does not by itself prove that every proposed nonlocal or antipodal reformulation is mathematically impossible.

Three state classes that should not be conflated

State prescription	Ultraviolet mixing	Exact de Sitter invariance	Hadamard test	Conventional local interacting QFT
Bunch–Davies/Euclidean	$\beta_k=0$ in the BD basis	Yes	Passes, subject to the zero-mode qualification below	Standard local covariant renormalization applies
Constant alpha state	Nonzero constant mixing as $k\to\infty$	Yes for the free field	Fails; coincidence and antipodal singularities remain	Not admissible in the standard construction
UV-soft excitation	$\beta_k$ decreases rapidly above a preparation scale	Generally no	Passes when its two-point difference from BD is smooth	Standard counterterms apply; finite observables remain state dependent

The last row is the controlled alternative to a constant alpha state. For example, a smooth rapidly decreasing $\beta_k$ changes the state-dependent smooth part of the two-point function without changing its Hadamard singularity. A boundary-EFT truncation describes only a finite physical-momentum window and must be completed smoothly in the ultraviolet; the truncation by itself is not a global state prescription.

The massless minimally coupled zero mode

For $m=0$ and $\xi=0$, one has $\nu=3/2$ and $h_-=0$. The factor $\Gamma(h_-)$ in the invariant Green function diverges because the Euclidean operator on $S^4$ has a constant zero mode. Its inverse therefore does not exist on the full field space.

In Lorentzian de Sitter there is correspondingly no ordinary de Sitter-invariant Fock vacuum for the unsmeared field $\phi$ with a positive two-point function Allen 1985, §IV, pp. 3144–3147. One may choose a non-invariant state, isolate the zero mode, or restrict to derivative or shift-invariant observables, but each choice changes the state or the observable algebra. Allen and Folacci construct the commonly used $O(4)$-invariant treatment in Allen and Folacci 1987, §§II–IV.

The zero-mode obstruction is not evidence in favor of an alpha state. It is an infrared failure of invertibility, whereas the constant-alpha problem is ultraviolet and antipodal. They require different remedies.

Checks and failure conditions

For a claimed de Sitter vacuum, state the patch, mass and coupling, zero-mode treatment, analytic continuation, and whether the claim concerns only a free two-point function or also local composites and interactions. Then check:

• the Wronskian and the exact $\eta\to-\infty$ analytic mode condition;
• the $Z\to1$ coefficient and wavefront orientation;
• the presence or absence of a $Z=-1$ antipodal singularity;
• smoothness of the difference from one known Hadamard two-point function;
• finiteness of smeared local-composite fluctuations with standard local counterterms.

Free de Sitter invariance and canonical normalization are not enough. If a proposed state passes those two tests but fails smooth-difference or local-composite tests, the strongest surviving statement is only that it defines a formal free invariant two-point distribution—not an interacting vacuum of local curved-spacetime QFT.

Common pitfalls

Treating “positive frequency” as a local uniqueness rule. In this example it is the exact past-asymptotic analytic condition of the de Sitter patch. A finite-time adiabatic prescription in a general FLRW spacetime is a different construction.

Calling every excited state an alpha vacuum. Constant ultraviolet mixing is the defining danger. A UV-soft Bogoliubov excitation generally breaks exact de Sitter symmetry but can preserve the Hadamard singularity.

Using symmetry to skip composite-operator tests. A two-point function can be de Sitter invariant and Wronskian normalized while still giving ill-defined fluctuations for local observables.

Exercises

Use the large-$z$ asymptotic of $H_\nu^{(1)}(z)$ to verify the phase and normalization of the Bunch–Davies mode.

Solution

For $z\to\infty$,

$$H_\nu^{(1)}(z) \sim\sqrt{\frac{2}{\pi z}} \exp\!\left[i\left(z-\frac{\pi\nu}{2}-\frac\pi4\right)\right].$$

Set $z=-k\eta$. The real prefactors give

$$\frac{\sqrt{-\pi\eta}}2 \sqrt{\frac{2}{\pi(-k\eta)}} =\frac1{\sqrt{2k}}.$$

The remaining phase is

$$\frac{\pi(2\nu+1)}4 +z-\frac{\pi\nu}{2}-\frac\pi4=z=-k\eta.$$

Hence $v_k^{\mathrm{BD}}\sim e^{-ik\eta}/\sqrt{2k}$, with the standard Wronskian normalization.

For real $\nu$, an exact check uses

$$H_\nu^{(1)}(z)\,\partial_zH_\nu^{(2)}(z) -\partial_zH_\nu^{(1)}(z)\,H_\nu^{(2)}(z) =-\frac{4i}{\pi z}.$$

With $z=-k\eta$ and the prefactor in $v_k^{\mathrm{BD}}$, the derivatives of $\sqrt{-\eta}$ cancel between the two terms and give $v_kv_k^{*\prime}-v_k'v_k^*=i$.

For a principal-series field, $\nu=i\mu$ rather than real. Then complex conjugation gives

$$\left[H_{i\mu}^{(1)}(z)\right]^* =H_{-i\mu}^{(2)}(z) =e^{\pi\mu}H_{i\mu}^{(2)}(z).$$

The product of the two mode prefactors contributes $e^{-\pi\mu}$, canceling this order-reflection factor. The same Hankel Wronskian therefore again gives the canonical value $i$.

Explain directly from the alpha two-point function why constant alpha mixing is not a Hadamard perturbation of the Bunch–Davies state, and contrast it with a rapidly decreasing $\beta_k$.

Solution

Two Hadamard two-point functions differ by a smooth bisolution. In the alpha expression, the coefficient of the reversed-frequency term $G_{\mathrm{BD}}^+(x',x)$ is nonzero at every momentum, so the difference retains a coincidence singularity with the opposite frequency orientation. The antipodal terms contain $G_{\mathrm{BD}}^+(x_A,x')$ and $G_{\mathrm{BD}}^+(x,x'_A)$, which become singular when $Z(x,x')=-1$. The difference is therefore not smooth, and the state is not Hadamard.

If instead $\beta_k$ and all derivatives decrease rapidly as $k\to\infty$, the extra mode integrals define a smooth function under the usual regularity assumptions. The two-point function then has the same local singular part as Bunch–Davies, although the state generally loses exact de Sitter invariance.

References

• Allen, B., “Vacuum States in de Sitter Space,” Physical Review D 32, 3136–3149 (1985), doi:10.1103/PhysRevD.32.3136.
• Allen, B., and A. Folacci, “Massless Minimally Coupled Scalar Field in de Sitter Space,” Physical Review D 35, 3771–3778 (1987), doi:10.1103/PhysRevD.35.3771.
• Brunetti, R., K. Fredenhagen, and S. Hollands, “A Remark on Alpha Vacua for Quantum Field Theories on de Sitter Space,” Journal of High Energy Physics 2005, 063 (2005), doi:10.1088/1126-6708/2005/05/063, Open PDF.
• Bunch, T. S., and P. C. W. Davies, “Quantum Field Theory in de Sitter Space: Renormalization by Point-Splitting,” Proceedings of the Royal Society A 360, 117–134 (1978), doi:10.1098/rspa.1978.0060.
• 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.
• Mottola, E., “Particle Creation in de Sitter Space,” Physical Review D 31, 754–766 (1985), doi:10.1103/PhysRevD.31.754.