Bases of Wormholes in Quantum Cosmology

TL;DR

This paper demonstrates that the space of wormhole wave functions in quantum cosmology can be structured as a Hilbert space, where basis functions are eigenfunctions of compatible observables, exemplified in a Friedmann-Robertson-Walker model.

Contribution

It establishes a Hilbert space framework for wormhole wave functions and identifies their basis as eigenfunctions of a complete set of observables in quantum cosmology.

Findings

Hilbert space of wormhole states coincides with that of the gravitational system

Wormhole basis functions are eigenfunctions of compatible observables

In the FRW model, all statements about wormhole bases are validated

Abstract

We show that if the space of physical states spanned by the wormhole wave functions can be equipped with a Hilbert structure, such a Hilbert space must coincide with that of the Lorentzian gravitational system under consideration. The physical inner product can then be determined by imposing a set of Lorentzian reality conditions. The Hilbert space of the gravitational model admits in this case a basis of wormhole solutions, and every proper quantum state can be interpreted as a superposition of wormholes. We also argue that the wave functions that form the basis of wormholes must be eigenfunctions of a complete set of compatible observables. The associated eigenvalues provide a set of well-defined wormhole parameters, in the sense that they can be employed to designate the different elements of the basis of wormholes. We analyse in detail the case of a Friedmann-Robertson-Walker spacetime minimally coupled to a massless scalar field. For this minisuperspace, we prove the validity of all the above statements and discuss various admissible choices of bases of wormhole wave functions.