Explicit finite-time illustration of improper unitary evolution for the Klein--Gordon field in de Sitter space

TL;DR

The paper investigates quantum field theory of a free Klein–Gordon field in de Sitter space, highlighting that states on different spatial slices cannot be connected by a proper unitary time evolution due to explicit time dependence of the Hamiltonian. Through a canonical quantisation in a flat, conformal-time slicing and the construction of a Fock space in the interaction picture, it demonstrates that the vacuum overlap between conformal times vanishes in the infinite-volume limit, signaling unitary inequivalence and a finite-time realization of Haag's theorem. The analysis employs Bogoliubov transformations, a Schrödinger-like picture, and coherent-state methods to compute the vacuum persistence amplitude and its convergence properties. The results have implications for Hadamard states and the foundational understanding of QFT in curved spacetime, particularly in cosmological contexts.

Abstract

It is known that quantum field theories in curved spacetime suffer from a number of pathologies, including the inability to relate states on different spatial slices by proper unitary time-evolution operators. In this article, we illustrate this issue by describing the canonical quantisation of a free scalar field in de Sitter space and explicitly demonstrating that the vacuum at a given time slice is unitarily inequivalent to that at any other time. In particular, we find that, if both background and Hamiltonian dynamics are taken into account, this inequivalence holds even for infinitesimally small time steps and not only in the asymptotic time limits.

Figures (3)

• Figure 1: Depiction of the function $\Delta^2(\bar{\tau},\delta\tau)$ for $\mathfrak{m} =1$, illustrating that $\Delta^2\geq 0$ for $\tau\geq \tau_0$ ($\bar{\tau}<-\delta\tau/2$).
• Figure 2: Depictions of the function $\aleph(\bar{\tau})$ for different values of $\mathfrak{m}$ and $\delta\tau$; all considered cases lead to the same qualitative behaviours as shown here. (a): $\mathfrak{m} =\delta\tau =1$, (b): $\mathfrak{m} = 10^4$, $\delta\tau =1$
• Figure 3: Depiction of the function $\beth(\bar{\tau})$ for $\mathfrak{m} =\delta\tau =1$; changing the values of $\mathfrak{m}$ or $\delta\tau$ appears to only shift the figure along the abscissa but does not alter its qualitative features