Decoherence and Emergent Classical Histories

A Wheeler–DeWitt wave function can contain several semiclassical geometries at once. Decoherence explains why interference between suitably coarse-grained branches becomes negligible when unobserved perturbations record the background. It can justify probabilities for approximately consistent histories relative to an internal clock; it does not select one outcome or derive the initial wave function.

Helpful background. Flat-Spacetime Quantum Field Theory supplies field modes and Gaussian states; Einstein Equations supplies background histories; Quantum Information supplies reduced states and decoherence; Quantum Gravity supplies scope; Causality and Global Structure supplies histories and horizons; and Quantum Fields in Curved Spacetime supplies perturbations on each branch.

WKB branches and an internal clock

Separate heavy homogeneous variables $Q$ from lighter inhomogeneous modes $x$. A Born–Oppenheimer form is

$$\Psi(Q,x)= \sum_r A_r(Q)e^{iM^2S_r(Q)}\chi_r(Q,x),$$

where $M^2$ is proportional to the inverse gravitational coupling and $r$ labels WKB branches. At leading order, $S_r$ satisfies the gravitational Hamilton–Jacobi equation. Its gradient defines a derivative along the associated classical trajectory,

$$\frac{\partial}{\partial t_r} =G^{AB}(Q)\,\partial_AS_r\,\partial_B,$$

with $G^{AB}$ the inverse minisuperspace metric, up to the chosen lapse normalization. At the next order, $\chi_r$ obeys an approximate Schrödinger equation

$$i\frac{\partial\chi_r}{\partial t_r} =\widehat H_x(Q_r(t_r))\chi_r.$$

Time is therefore branch dependent. Near a turning point where $\partial_AS_r$ vanishes, or when branches interfere strongly, the WKB clock and the Schrödinger approximation can fail.

Tracing perturbative records

Consider two background branches entangled with environmental states:

$$\lvert\Psi\rangle= c_1\lvert Q_1\rangle\lvert E_1\rangle+ c_2\lvert Q_2\rangle\lvert E_2\rangle.$$

Tracing unobserved modes gives

$$\begin{aligned} \rho_Q={}& \lvert c_1\rvert^2\lvert Q_1\rangle\langle Q_1\rvert +\lvert c_2\rvert^2\lvert Q_2\rangle\langle Q_2\rvert\\ &+c_1c_2^*\mathcal D_{12}\lvert Q_1\rangle\langle Q_2\rvert +c_1^*c_2\mathcal D_{12}^*\lvert Q_2\rangle\langle Q_1\rvert. \end{aligned}$$

where $\mathcal D_{12}=\langle E_2|E_1\rangle$. If the environment factorizes approximately into modes, $\mathcal D_{12}=\prod_k d_k$. Even modest distinguishability per mode can make the product exponentially small. In cosmology, scalar, tensor, and matter perturbations can record different expansion histories or field configurations.

This suppression is basis dependent. The stable “pointer” variables are those whose correlations with the environment persist under the interaction. Assuming from the start that the scale factor is the pointer basis would be circular; the coupling should select it. Kiefer and Zeh derive branch decoherence from cosmological perturbations and emphasize its semiclassical domain Kiefer and Zeh 1995, §§ 2–4.

For Gaussian environmental modes, each overlap follows from the two branch-dependent covariance matrices and displacement vectors. Its magnitude is at most one by Cauchy–Schwarz, providing a normalization check. Ultraviolet modes that evolve adiabatically can contribute branch-independent phases rather than genuine records; counting them as independent suppressors can manufacture cutoff-dependent decoherence. A defensible calculation regulates the product, renormalizes background terms, and demonstrates that the remaining suppression is stable as the mode window is enlarged.

Consistent histories and records

A coarse-grained history $\alpha$ is represented by a chain of projectors or class operators $C_\alpha$. Given an initial state $\rho$, the decoherence functional is

$$D(\alpha,\alpha')= \operatorname{Tr}\!\left(C_\alpha\rho C_{\alpha'}^\dagger\right).$$

Probabilities $p(\alpha)=D(\alpha,\alpha)$ obey ordinary sum rules when off-diagonal terms are negligible for the chosen coarse graining. Records are late variables correlated with the alternatives; their near-orthogonality can guarantee decoherence. The same state may decohere for one coarse graining and not another.

In a timeless constraint theory, class operators should be relational—for example, “the scale factor lies in a range when $\phi=\phi_0$”—rather than tied to an external coordinate time. The physical inner product and constraint must be respected. A calculation in gauge-fixed coordinate time must show that its result corresponds to a relational question.

What becomes classical?

Decoherence suppresses observable interference but does not turn a pure global state into a literal classical mixture. The reduced state behaves like a mixture for observables that ignore the environment. Selecting one realized history requires an interpretation or additional dynamics.

Classicality also requires more than diagonality. A branch should remain narrowly peaked, obey approximate Hamilton–Jacobi evolution, and have fluctuations small compared with its coarse-graining scale. Chaotic squeezing or strong particle production can decohere a variable rapidly while also broadening it enough to spoil a classical trajectory.

An independent check compares the decoherence time with the background evolution time and with the time over which backreaction remains negligible. If decoherence occurs only after the WKB approximation fails, it cannot justify that approximation. The result should also be stable when the environmental cutoff and system–environment split are varied within the physical regime.

The broader quantum-cosmology construction, including semiclassical time and decoherence, is reviewed by Kiefer and Sandhöfer 2022, §§ 5–6. Those qualifications matter because tracing over variables before defining the physical constraint inner product need not equal tracing after reduction. A relational observable should be built in the physical state space whenever that construction is available.

Exercise

Suppose $N$ independent environmental modes have overlaps $d_k=e^{-\epsilon}$ with $\epsilon\ll1$. Find the decoherence factor and the number of modes needed for $\lvert\mathcal D_{12}\rvert\le e^{-10}$.

Solution

Factorization gives

$$\lvert\mathcal D_{12}\rvert=\prod_{k=1}^{N}e^{-\epsilon} =e^{-N\epsilon}.$$

The condition $e^{-N\epsilon}\le e^{-10}$ is $N\epsilon\ge10$, hence $N\ge10/\epsilon$. For $\epsilon=10^{-3}$, ten thousand weakly distinguishing modes suffice. This demonstrates exponential suppression, but it assumes independence; correlated modes require evaluating the determinant or covariance matrix of the full Gaussian overlap.

References

• Kiefer, Claus, and Barbara Sandhöfer (2022), “Quantum Cosmology,” Zeitschrift für Naturforschung A 77, 543–559, arXiv:0804.0672v3.
• Kiefer, Claus, and H. Dieter Zeh (1995), “Arrow of Time in a Recollapsing Quantum Universe,” Physical Review D 51, 4145–4153, arXiv:gr-qc/9402036.