Quantum Fields in de Sitter Space

The Bunch–Davies state is distinguished by combining de Sitter symmetry with the local Hadamard short-distance condition and a regular Euclidean continuation. For a free scalar with admissible positive effective mass, these requirements select the familiar invariant quasifree state. They do not give an unqualified vacuum theorem for every mass and coupling: the massless minimally coupled scalar has a zero-mode and infrared obstruction that is a decisive exception.

Required background. Compactly, a quasifree state is fixed by a positive bisolution whose antisymmetric part is the causal propagator; the Hadamard condition fixes its short-distance wavefront structure while allowing smooth state-dependent terms. Partial Differential Equations and Green Functions supplies bisolutions, causal propagators, and distributional boundary values. Helpful background. Flat-Spacetime Quantum Field Theory provides Fock modes and analytic continuation; Causality and Global Structure establishes de Sitter patches and horizons; Quantum Fields in Curved Spacetime provides Hadamard states, KMS restriction, and local observables.

Geometry, field, and hypotheses

Work in four-dimensional de Sitter spacetime with radius $H^{-1}$ and curvature $R=12H^2$. On the expanding Poincaré patch,

$$ds^2=a^2(\eta)(-d\eta^2+d\mathbf x^2), \qquad a(\eta)=-\frac{1}{H\eta}, \qquad -\infty\lt\eta\lt0.$$

For $(\Box-m^2-\xi R)\phi=0$, define the effective mass

$$M^2=m^2+12\xi H^2, \qquad \nu^2=\frac94-\frac{M^2}{H^2}.$$

The basic selection statement used here assumes a free scalar, the full de Sitter geometry or a specified restriction of its state, the standard canonical commutator, and $M^2>0$. Under these hypotheses there is a de Sitter-invariant quasifree state with Hadamard short-distance form obtained by continuation from the Euclidean sphere. Within the invariant alpha-family, it is the Hadamard member. The statement does not cover tachyonic exceptional values, arbitrary noninvariant states, or the massless minimally coupled case $m=\xi=0$.

Poincaré modes and short-distance selection

For the canonical variable $v_k=a\phi_k$, the mode equation is

$$v_k''+\left[k^2-\frac{\nu^2-\tfrac14}{\eta^2}\right]v_k=0.$$

The Wronskian-normalized solution that approaches positive-frequency Minkowski form at short physical wavelengths is

$$v_k(\eta) =\frac{\sqrt{-\pi\eta}}{2} \exp\left[\frac{i\pi}{4}(2\nu+1)\right] H_\nu^{(1)}(-k\eta),$$

because

$$v_k(\eta)\sim\frac{e^{-ik\eta}}{\sqrt{2k}} \qquad\text{as}\qquad -k\eta\to\infty.$$

This limit fixes the phase convention associated with $e^{-i\omega t}$. It also supplies the local Hadamard singularity: sufficiently short wavelengths do not resolve the curvature. A general Bogoliubov rotation of these modes can remain normalized, but symmetry alone is not enough to make it Hadamard.

The construction architecture has two equivalent faces. In mode language, choose the Hankel branch by short-wavelength positive frequency and verify its Wronskian. In invariant language, solve for a two-point function of the de Sitter invariant $Z(x,x')$, select the Euclidean boundary value, and require the standard coincidence singularity. Hollands develops the free massive construction in Hollands 2013, §3 and the perturbative de Sitter construction and its large-separation behavior in §§4–5 and §6. Those perturbative results retain their massive-field and order-by-order hypotheses; they are not a theorem about all infrared sectors.

Invariant two-point function and alpha-states

Define the de Sitter invariant on this patch by $Z=1+[(\eta-\eta')^2-|\mathbf x-\mathbf x'|^2]/(2\eta\eta')$. Coincidence has $Z=1$. For the Wightman ordering $G^+(x,x')=\langle\phi(x)\phi(x')\rangle$, a time-oriented boundary value is

$$Z_\epsilon=1+ \frac{(\eta-\eta'-i\epsilon)^2-|\mathbf x-\mathbf x'|^2} {2\eta\eta'},\qquad \epsilon\to0^+.$$

The Bunch–Davies two-point function may then be written

$$G^+(x,x') =\frac{H^2}{16\pi^2} \Gamma\left(\frac32+\nu\right) \Gamma\left(\frac32-\nu\right) {}_2F_1\left( \frac32+\nu,\frac32-\nu;2; \frac{1+Z_\epsilon}{2} \right).$$

The limit is distributional. A constant $Z-i0$ independent of the ordering of $x,x'$ would give the wrong boundary value for one time ordering. The time-oriented prescription ensures $G^+(x',x)=G^+(x,x')^*$. Away from the cut, dependence on $Z$ expresses de Sitter invariance. For $M^2/H^2\gt9/4$, choose $\nu=i\sqrt{M^2/H^2-9/4}$ (principal series); for $0\lt M^2/H^2\lt9/4$, choose real positive $\nu$ (complementary series). Long-distance decay changes with this classification.

The conformal case provides a normalization check without a special-function calculation. At $\nu=1/2$, ${}_2F_1(2,1;2;z)=(1-z)^{-1}$, so

$$G^+(x,x')=\frac{1}{4\pi^2a(\eta)a(\eta')} \frac{1}{|\mathbf x-\mathbf x'|^2-(\eta-\eta'-i0)^2}.$$

This is precisely the conformally rescaled Minkowski Wightman distribution. It checks the coefficient, short-distance singularity, and time-ordering sign at once.

An alpha-state mixes the Euclidean mode with its complex conjugate using a $k$-independent Bogoliubov parameter. Such states can preserve formal de Sitter invariance, but generic members acquire an additional antipodal singularity and the wrong local wavefront structure. They are not Hadamard and do not support the usual local renormalization of interacting observables. Thus “de Sitter invariant” is not sufficient; the short-distance condition performs essential work.

Static-patch KMS thermality

Inside one static patch,

$$ds^2=-(1-H^2r^2)d\tau^2 +\frac{dr^2}{1-H^2r^2} +r^2d\Omega_2^2, \qquad 0\le r\lt H^{-1}.$$

The Killing field $\partial_\tau$ is normalized to unit norm at $r=0$. Restricting the Bunch–Davies state to the patch gives a KMS state for this flow at

$$T_{\mathrm{dS}}=\frac{H}{2\pi}.$$

A static observer at radius $r$ has proper time $d\tau_{\mathrm{prop}}=\sqrt{1-H^2r^2}\,d\tau$ and measures the Tolman temperature

$$T_{\mathrm{loc}}(r) =\frac{H}{2\pi\sqrt{1-H^2r^2}}.$$

This is a statement about the restricted algebra and its automorphism, not a claim that the global pure state is an ordinary finite-dimensional thermal density matrix. The divergence of $T_{\mathrm{loc}}$ near the horizon reflects the infinite acceleration of a static trajectory, not a curvature singularity.

The massless minimally coupled obstruction

Set $m=\xi=0$, so $M^2=0$ and $\nu=3/2$. The Poincaré mode amplitude behaves as $|\phi_k|^2\sim H^2/(2k^3)$ at small $k$, producing the logarithmic integral

$$\int_0 \frac{dk}{k}.$$

On global de Sitter space the constant spatial mode has no restoring force. Its canonical pair cannot be assigned the ordinary invariant Fock vacuum while preserving all desired properties. Allen proves the absence of a de Sitter-invariant Fock vacuum for the massless minimally coupled scalar in Allen 1985, complete article, pp. 3136–3149. This does not say that no useful state or observable exists: one may choose a noninvariant infrared prescription, work with derivatives or the shift-invariant algebra, or treat the zero mode separately. Each changes the precise claim.

By contrast, the massless conformally coupled scalar has $M^2=2H^2$ and $\nu=1/2$. Its rescaled modes are plane waves, and the conformal state is regular. “Massless” alone therefore does not diagnose the infrared problem; curvature coupling is decisive.

Independent checks and limitations

As $H\to0$ with physical separations fixed, the Euclidean two-point function tends to the Minkowski vacuum two-point function for fixed positive $m$. At coincidence, its leading singularity is the universal Hadamard one. In the static patch, the imaginary-time periodicity is $2\pi/H$, matching the stated KMS temperature. These tests compare normalization, short-distance behavior, and horizon thermality without identifying them as the same argument.

The construction does not determine every admissible excited state, solve metric backreaction, or settle interacting massless infrared dynamics. A Poincaré-patch prescription can also hide a global zero-mode question. Every claim must therefore record dimension, patch, mass, coupling, algebra of observables, and infrared prescription.

Diagnostic: test an invariant-vacuum claim

A calculation asserts: “Every free massless scalar on de Sitter has a unique invariant Bunch–Davies vacuum because its high-$k$ modes approach Minkowski plane waves.” Identify the missing hypothesis and give one valid massless case and one counterexample.

Solution

The high-$k$ condition fixes ultraviolet behavior but does not control the zero mode. The claim must specify the curvature coupling and require a globally admissible positive two-point function. A massless conformally coupled scalar, $\xi=1/6$, has $M^2=2H^2$, $\nu=1/2$, and a regular conformal Bunch–Davies state. The massless minimally coupled scalar, $\xi=0$, has $\nu=3/2$ and an infrared logarithm plus a global constant-mode obstruction, so no de Sitter-invariant Fock vacuum of the asserted kind exists. One must instead break invariance, alter the observable algebra, or treat the zero mode separately.

This counterexample isolates why a local short-distance condition cannot prove a global existence claim. It also shows that replacing “minimally coupled” by “conformally coupled” changes the field equation, rather than merely choosing another name for the same vacuum.

References

• Allen, Bruce. “Vacuum States in de Sitter Space.” Physical Review D 32 (1985): 3136–3149. DOI.
• Hollands, Stefan. “Correlators, Feynman Diagrams, and Quantum No-Hair in de Sitter Spacetime.” Communications in Mathematical Physics 319 (2013): 1–68. DOI.