Vacuum Choice and Initial-State Effects

An FLRW geometry rarely selects a unique vacuum. A physically admissible cosmological state must satisfy canonical positivity and ultraviolet singularity conditions; symmetry, energy minimization, Euclidean analyticity, or a preparation history may then narrow the family without turning local regularity into uniqueness.

Required background. FLRW mode quantization fixes Gaussian data; vacuum ambiguity separates state from particle basis; and the Hadamard parametrix fixes local singularities. Helpful background. Review constructing Hadamard states and state-selection failure modes.

Gaussian initial data and ultraviolet admissibility

A pure homogeneous isotropic Gaussian state is specified by normalized modes $v_k$ on one Cauchy slice. A mixed Gaussian state also needs occupation $N_k$ and pairing $C_k$, constrained by positivity. Its equal-time two-point function contains

$$\langle\chi_{\mathbf k}\chi_{\mathbf q}\rangle =\delta^3(\mathbf k+\mathbf q) \left[(2N_k+1)|v_k|^2 +2\operatorname{Re}(C_kv_k^2)\right],$$

with $|C_k|^2\le N_k(N_k+1)$. The Wronskian fixes the commutator but not $N_k,C_k$.

A Hadamard state has the universal microlocal short-distance singularity. An all-orders adiabatic construction supplies it under appropriate hypotheses; finite adiabatic order controls only a corresponding finite set of composite observables. Fourth-order ultraviolet data are the conventional minimum for a four-dimensional adiabatically renormalized stress calculation, not a proof that two states are identical.

Junker proves the relation between adiabatic constructions and Hadamard states on Robertson–Walker spacetimes under stated regularity assumptions Junker 1996, §3.5, pp. 1092–1099.

First application: two fourth-order states

Choose two mode sets related at $\eta_0$ by

$$\widetilde v_k=A_kv_k+B_kv_k^*, \qquad |A_k|^2-|B_k|^2=1,$$

with $B_k$ falling rapidly enough at large $k$ to preserve fourth-order ultraviolet behavior, but nonzero for $k\lesssim k_\star$. Evolve both with the same equation and canonical convention. Their dimensionless field spectra differ by

$$\Delta\mathcal P_\phi(k,\eta) =\frac{k^3}{2\pi^2a^2} \left(|\widetilde v_k|^2-|v_k|^2\right),$$

and their renormalized stresses differ by a finite smooth state-dependent mode integral. The same local adiabatic subtraction applies to both because their short-distance singularities agree to the required order.

The construction separates admissibility from selection. One may prefer a state because it is prepared by an asymptotic past, minimizes a smeared energy, or admits Euclidean continuation, but the criterion and its domain must be stated. Fulling’s nonuniqueness result is the foundational warning against inferring a vacuum from the metric alone Fulling 1973, §§II–III, pp. 2852–2858.

A hierarchy of selection criteria

State selection is most transparent as a sequence of non-equivalent filters. Canonical normalization and covariance positivity first define a state. Hadamard or sufficiently high adiabatic ultraviolet behavior then licenses the desired local composites. Spatial homogeneity and isotropy reduce the data to functions of $k$, but do not determine those functions. A preparation rule—an asymptotic past, Euclidean continuation, a boundary density matrix, or minimization of a smeared energy—can finally select a member within the remaining class.

Each filter has a different failure mode. Instantaneous Hamiltonian minimization depends on the canonical variable and chosen time. Minimizing energy at one slice can control a finite adiabatic order yet fail to define an all-orders Hadamard state. Euclidean continuation requires an appropriate analytic geometry and a treatment of zero modes. An asymptotic prescription requires that the asymptotic region actually exist. Combining conclusions from incompatible filters can create an apparent uniqueness that no single construction possesses.

For mixed states, positivity is not a cosmetic condition. The bound $\lvert C_k\rvert^2\le N_k(N_k+1)$ says that pairing cannot exceed the noise needed by the uncertainty relation. Saturation gives a pure squeezed covariance; strict inequality leaves entropy. Evolving the modes symplectically preserves this inequality, so a later violation identifies numerical or initial-data error rather than exotic cosmological dynamics.

The structure map shows ultraviolet admissibility feeding multiple allowed initial states before observables are computed.

Hadamard or adiabatic ultraviolet control defines an admissible class; preparation and symmetry criteria select within it. Schematic; not to scale.

Domain and failure conditions

See the chapter’s canonical domain table. State comparisons require the same field, scale factor, canonical variable, subtraction, and initial slice—or an explicit transport between slices.

Adversarial test. Choose two states with identical large-$k$ asymptotics but different low-$k$ $B_k$. Both pass local ultraviolet tests, yet their long-wavelength spectra and finite stress differ. Therefore Hadamard regularity does not select one vacuum. Conversely, an ultraviolet excitation that decays too slowly can invalidate local stress renormalization even if its infrared spectrum looks harmless.

The failure map separates a failed admissibility test from legitimate state dependence.

Local singularity control is necessary for renormalized observables but does not manufacture a unique cosmological state. Schematic; not to scale.

References

• Fulling, S. A., “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time,” Physical Review D 7, 2850–2862 (1973), doi:10.1103/PhysRevD.7.2850.
• Junker, W., “Adiabatic Vacua and Hadamard States for Scalar Quantum Fields on Curved Spacetime,” Reviews in Mathematical Physics 8, 1091–1159 (1996), doi:10.1142/S0129055X96000422.