[Paper Review] The problem of time and gauge invariance in the quantization of cosmological models. II. Recent developments in the path integral approach

Т. П. Шестакова, Claudio Simeone|ArXiv.org|Sep 30, 2004

Cosmology and Gravitation Theories6 references7 citations

TL;DR

This paper presents two distinct path integral approaches to resolving the problem of time in quantum cosmology. The first, by Simeone et al., maintains gauge invariance through deparametrization and unitary quantization of an effective gauge system, enabling consistent time-dependent evolution. The second, by Savchenko, Shestakova, and Vereshkov, argues that gauge invariance is inherently broken in a closed Universe without asymptotic states, leading to a natural emergence of time via non-unitary, irreversible dynamics in the quantum theory—offering a radical alternative to conventional approaches.

ABSTRACT

The paper is the second part of the work devoted to the problem of time in quantum cosmology. Here we consider in detail two approaches within the scope of Feynman path integration scheme: The first, by Simeone and collaborators, is gauge-invariant and lies within the unitary approach to a consistent quantization of gravity. It is essentially based on the idea of deparametrization (reduction to physical degrees of freedom) as a first step before quantization. The other approach by Savchenko, Shestakova and Vereshkov is rather radical. It is an attempt to take into account peculiarities of the Universe as a system without asymptotic states that leads to the conclusion that quantum geometrodynamics constructed for such a system is, in general, a gauge-noninvariant theory. However, this theory is shown to be mathematically consistent and the problem of time is solved in this theory in a natural way.

Motivation & Objective

• To address the longstanding problem of time in quantum cosmology within the path integral framework.
• To explore whether time can be consistently introduced in quantum gravity without relying on canonical time or asymptotic states.
• To examine the consequences of gauge invariance breakdown in a closed Universe lacking asymptotic states.
• To compare two contrasting approaches: one preserving gauge invariance via deparametrization, and another embracing gauge noninvariance as fundamental.
• To investigate the potential link between spacetime topology, reference frame changes, and the irreversibility of time in quantum geometrodynamics.

Proposed method

• Adapting the Faddeev-Popov procedure to path integrals for constrained systems, using gauge-fixing functions and Jacobian determinants to eliminate redundant paths.
• Applying a canonical transformation to convert the original minisuperspace model into a system with a linear constraint and zero Hamiltonian, enabling deparametrization.
• Introducing a time-dependent gauge condition to define physical time evolution while preserving unitarity in the Simeone approach.
• Constructing a quantum theory in extended phase space for a minisuperspace model, allowing time to emerge dynamically without prior time structure.
• Deriving a non-Hermitian Hamiltonian in a fixed measure space due to perturbations in the gauge-fixing function, leading to non-unitary evolution.
• Analyzing the role of spacetime topology in necessitating multiple reference frames and inducing irreversible dynamics through measure changes in physical degrees of freedom.

Experimental results

Research questions

• RQ1 Can time be consistently introduced in quantum cosmology without prior time structure, using path integral methods?
• RQ2 What are the consequences of breaking gauge invariance in a closed Universe without asymptotic states?
• RQ3 How does the choice of gauge-fixing function affect the unitarity and Hermiticity of the quantum Hamiltonian?
• RQ4 Can irreversibility of time emerge from the nontrivial topology of spacetime and reference frame transitions?
• RQ5 Is it possible to construct a mathematically consistent quantum theory of gravity that is inherently gauge-noninvariant yet physically viable?

Key findings

• The Simeone approach achieves gauge-invariant quantization by deparametrizing the system into a true gauge system with linear constraints, allowing consistent time evolution via unitary path integrals.
• The Savchenko-Shestakova-Vereshkov approach shows that the absence of asymptotic states in a closed Universe naturally breaks gauge invariance, leading to a time-dependent, non-unitary quantum theory where time emerges from the path integral.
• Perturbations in the gauge-fixing function induce an anti-Hermitian part in the effective Hamiltonian when the measure is fixed, implying irreversible transitions between reference frames.
• The theory is mathematically consistent and solves the problem of time by making time a physical, external parameter in the Schrödinger equation, even without a priori time.
• Nontrivial spacetime topology may necessitate multiple reference frames, and transitions between them lead to non-unitary, irreversible changes in the physical wave function, suggesting a geometric origin for time's arrow.
• The paper establishes that gauge invariance is not a necessary requirement for a consistent quantum theory of gravity if the system lacks asymptotic states, challenging conventional assumptions.