Coordinate- and spacetime-independent quantum physics

TL;DR

This work tackles the ambiguity of quantum particles in curved spacetime by constructing a covariant scalar solution $sol_K(y)$ that remains a zero-rank tensor under general coordinate transformations and is executable across AdS, dS, closed ESU, and open ESU spacetimes. The authors derive a Klein-Gordon framework with covariant variables and demonstrate non-perturbative-in-curvature solutions that reduce to Minkowski plane waves in the flat limit, enabling standard particle-physics applications in strong gravity. They develop two covariant solutions $sol_{K}^{(\alpha)}(y)$, connect them to Wightman functions in multiple spacetimes, and show that for $\alpha\in\{0,1\}$ a single, consistent solution exists, compatible with known states such as Bunch–Davies and Chernikov–Tagirov. The results provide a coordinate-frame-independent particle notion in curved backgrounds and open pathways to simulate strong-gravity quantum effects with tabletop analogues, advancing both foundational understanding and potential experimental exploration of quantum fields in curved spacetime.

Abstract

The concept of a particle is ambiguous in quantum field theory. It is generally agreed that particles depend not only on spacetime, but also on coordinates used to parametrise spacetime points. One of us has in contrast proposed a coordinate-frame-independent model of quantum particles within the framework of quantum field theory in curved spacetime. The aim of this article is to present a scalar-field-equation solution that is not only a zero-rank tensor under general coordinate transformations, but also common for anti-de-Sitter, de-Sitter, closed and open Einstein static universes. Moreover, it locally reduces to a Minkowski plane-wave solution and is non-perturbative in curvature. The former property makes it suitable for the standard applications of quantum theory in particle physics, while the latter allows then to gain insights into quantum physics in the strong-gravity regime.

Figures (3)

• Figure 1: Left: A closed (C) Einstein static universe (CESU) can be mapped onto an open (O) Einstein static universe (OESU) through analytic continuation ($\mathcal{AC}$) of $\sqrt{-R}$ to $i\sqrt{-R}$, where $R$ is the Ricci scalar. The map is invertible and works for the pair of a dS and an AdS as well. Besides, a CESU can also be mapped into a de-Sitter spacetime through dimensional reduction and analytic continuation ($\mathcal{DR}\circ\mathcal{AC}$). This is accomplished by reducing the CESU to its spatial section and by promoting one of its spatial (Riemann normal) coordinates to an imaginary variable. Its imaginary part turns into a time variable in the dS. This procedure also works for the map from an OESU to an AdS. Right: The commutative diagram is represented through the maps of isometry groups of the corresponding spacetimes.
• Figure 2: A complex $p$-plane with poles in the integrand of \ref{['eq:phi-i']} shown by cross and empty-dot marks originating from $\varphi_{ip-\alpha}^{(\gamma)}(\eta)$ and $\varphi_{\alpha-ip}^{(\gamma)}(\eta)$, respectively. After the integration over momentum, a pair of extra first-order poles at $p = 0$ and $p = -2i\alpha$ emerge, being marked by solid dots. We then choose a rectangular contour with the lower side at $\Im(p) = -2\alpha + 0$ in order to evaluate the integral over $p$. Left: For the integer values of $\alpha$, e.g. $\alpha = 1$, the cross and empty-dot poles lying on a line to pass through $p = -i\alpha$ give residues which cancel each other. Right: For the half-integer values of $\alpha$, e.g. $\alpha = \frac{3}{2}$, we set $c_{\alpha - ip}^{(\alpha)} = 0$ at $p = \pm \Im(\gamma)$ to avoid the residues at these poles.
• Figure 3: Left: A complex $p$-plane with poles in the integrand of \ref{['eq:phi-ii']} shown by cross marks. By use of the residue theorem, the solid contour is chosen for the integrand part involving $\tilde{\varphi}_{ip-\alpha}^{(\gamma)}(\eta)$ as this vanishes for $\Im(p) > 0$ faster than any exponential function in the limit $R \to \infty$. For this property to hold for the integrand part in \ref{['eq:phi-ii']} involving $\tilde{\varphi}_{\alpha - ip}^{(\gamma)}(\eta)$, the dotted contour is used. Right: The pole structure of \ref{['eq:phi-ii']} alters after the integration over the angles in momentum space. The cross-marked first-order poles disappear, while solid-dot-marked first-order poles emerge at $p = 0$ and $p = -2i\alpha$, which yield the result \ref{['eq:wf-ii']} by making use of the residue theorem.