Euclidean quantum wormholes

TL;DR

The paper investigates Euclidean quantum wormholes in a Friedman-Robertson-Walker (FRW) minisuperspace by solving the Wheeler-DeWitt equation under Hawking-Page boundary conditions with arbitrary factor ordering parameters $p$ and $q$. It analyzes two matter content scenarios: a minimally coupled scalar field and perfect-fluid sources, deriving and solving the corresponding Wheeler-DeWitt equations for multiple potentials $V(\phi)$ and fluid models; solutions are obtained via separation of variables and expressed through Bessel, Kummer, Heun, and Airy functions, with asymptotic analyses. The results show wavefunctions that are regular at $a\to0$ and exponentially damped as $a\to\infty$, indicating quantum wormhole states, and suggest infinite discrete spectra of wormhole solutions in these minisuperspace models. The work clarifies how matter content and operator ordering influence the existence and form of Euclidean wormholes in quantum cosmology, providing analytic and semi-analytic solutions and visual illustrations. These findings contribute to the understanding of microscopic wormholes and their potential role in the early universe.

Abstract

We study wormhole as the solution of the Wheeler-deWitt (WdW ) equation satisfying Hawking-Page wormhole boundary conditions in Friedmann-Robertson-Walker (FRW) cosmology. The quantum wormholes are formulated with arbitrary factor ordering of the Hamiltonian constraint operators with perfect fluid matter sources as well as minimally coupled scalar fields.

Figures (13)

• Figure 1: The wave function represents the quantum wormhole solution (left panel) for $p=1, k= -1, q=0$, (right panel) for $p=1, k= -1, q=1$ .
• Figure 2: The wave function represents the quantum wormhole solution (left panel) for $p=1, k= -1, q=-2$, (right panel) for $p=1, k= -1, q=-1$ .
• Figure 3: The wave function for scale factor $a \approx 0$ represents the quantum wormhole solution (left panel) for $p=1, q=-2$, (right panel) for $p=1, q=2$ .
• Figure 4: The wave function for large value of scale factor i.e. $a \rightarrow \infty$, represents the quantum wormhole solution (left panel) for $p=1, q=-2$, (right panel) for $p=1, q=2$ .
• Figure 5: The wave function represents the quantum wormhole solution (left panel) for $p=4, q=-2, n=-2, k=-1$, (right panel) for $p=4, q=2, n=-2, k=-1$.