Initial Density Matrices and Boundary EFT

A cosmological initial condition is most generally a density operator inserted at an initial hypersurface, not a mode-by-mode slogan about positive frequency. In a Schwinger–Keldysh path integral it becomes a boundary functional coupling the forward and backward fields; locality, Hermiticity, normalization, positivity, and cutoff power counting constrain its kernels.

Required background. Vacuum choice and initial states supplies Gaussian data; initial density matrices on real-time contours supplies the two branches; and controlled EFT expansions supplies power counting. Helpful background. Review curvature counterterms and local covariance with boundaries.

Initial kernels on the closed-time path

At $\eta=\eta_0$, write

$$\langle\varphi_+|\rho_0|\varphi_-\rangle =\mathcal N\,e^{iS_0[\varphi_+,\varphi_-]}.$$

Hermiticity requires

$$S_0[\varphi_+,\varphi_-]^* =-S_0[\varphi_-,\varphi_+],$$

and $\operatorname{Tr}\rho_0=1$ fixes $\mathcal N$. A homogeneous Gaussian state is equivalently described by $N_k$ and $C_k$ with

$$N_k\ge0,\qquad |C_k|^2\le N_k(N_k+1).$$

These inequalities are the one-mode positivity condition. The boundary action packages the same covariance into quadratic kernels $A_k$ and $B_k$. Its insertion changes the propagator through a boundary self-energy,

$$G^{ab}=G_0^{ab} +G_0^{ac}\,\Sigma^{cd}_0\,G_0^{db}+\cdots,$$

where $\Sigma_0$ is supported at $\eta_0$ and $a,b=\pm$. The retarded, advanced, and statistical components must satisfy the canonical jump and reality conditions.

First application: a Gaussian excited scalar

Begin with normalized reference modes $v_k$ and choose a finite excitation

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

with $B_k$ suppressed for physical momentum $k/a_0$ near the EFT cutoff $\Lambda$. The quadratic $S_0$ that enforces the corresponding Robin kernel modifies the free propagator by terms proportional to $\widetilde v_k\widetilde v_k^*-v_kv_k^*$. Check the Wronskian, $\operatorname{Tr}\rho_0=1$, and positivity before using the correlator.

Loops separate into ordinary bulk divergences and divergences localized on $\eta_0$. The latter require boundary operators built from the induced metric, extrinsic curvature, fields, and tangential derivatives. Collins and Holman derive this boundary renormalization and the running of initial conditions Collins and Holman 2005, §§II–IV, Eqs. (2.20)–(4.14). A state with UV deviations that do not admit the boundary EFT expansion is outside the prediction.

Boundary power counting and causal data

Write the Schwinger–Keldysh variables as $\varphi_r=(\varphi_++\varphi_-)/2$ and $\varphi_a=\varphi_+-\varphi_-$. Normalization requires the influence of the initial insertion to vanish when the two histories coincide, while Hermiticity fixes the reality properties of terms odd and even in $\varphi_a$. The quadratic $\varphi_a\varphi_r$ kernel changes deterministic initial conditions; the imaginary $\varphi_a\varphi_a$ kernel supplies initial statistical noise and must have the sign required by positivity.

For a free field the retarded commutator is fixed by the equation and canonical jump condition, whereas the statistical two-point function carries $N_k$ and $C_k$. This gives a practical diagnostic: modifying the initial density matrix may change the Keldysh correlator, but it must not alter the causal jump or move support outside the light cone. If a numerical boundary insertion changes both arbitrarily, it has modified the dynamics rather than only the state.

Power counting uses physical tangential momentum $k/a_0$, not comoving $k$ by itself. Operators related by boundary integration by parts or lower-order boundary conditions should not be double counted. At a fixed truncation, changing the boundary renormalization scale shifts Wilson coefficients so low-energy correlators remain invariant up to omitted powers. This scale test separates a calculable EFT uncertainty from an unrenormalized dependence on the arbitrary initial slice.

The structure map places the density matrix before propagation and adds a boundary-counterterm branch absent for a purely bulk vacuum calculation.

Mixed or excited cosmological data are encoded at the initial boundary; their propagator corrections and divergences obey boundary EFT constraints. Schematic; not to scale.

Domain and failure conditions

Use the chapter’s canonical domain table. Specify the initial hypersurface, physical cutoff, boundary operator basis, density-kernel positivity, and bulk versus boundary renormalization conditions.

Adversarial test. Raise the comoving cutoff at fixed $a_0$ while holding unsuppressed $B_k$ to arbitrarily high $k$. The excitation energy and boundary divergences grow beyond the EFT hierarchy. Alternatively choose kernels violating $|C_k|^2\le N_k(N_k+1)$; the “density matrix” is not positive. In either case no cosmological prediction survives until the state is replaced by a bounded, power-counted boundary functional.

The failure map distinguishes an inadmissible state from a legitimate but coefficient-dependent EFT correction.

Boundary EFT controls initial-state sensitivity only when density positivity, cutoff scaling, and localized counterterms are all satisfied. Schematic; not to scale.

References

• Collins, H., and R. Holman, “Renormalization of Initial Conditions and the Trans-Planckian Problem of Inflation,” Physical Review D 71, 085009 (2005), doi:10.1103/PhysRevD.71.085009.