[Paper Review] Effective gravity formulation that avoids singularities in quantum FRW cosmologies

Jaume de Haro, E. Elizalde|ArXiv.org|Jan 19, 2009

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper proposes a novel effective gravity formulation based on a Schrödinger equation with time-ordered boundary conditions, avoiding the big bang singularity in FRW cosmologies by ensuring the effective scale factor remains strictly positive through self-adjoint extensions of the Hamiltonian. Unlike loop quantum cosmology, which relies on regularization, this approach avoids singularities purely through the mathematical structure of self-adjoint operators and the physical assumption of absolute time.

ABSTRACT

Assuming that time exists, a new, effective formulation of gravity is introduced, which lies in between the Wheeler-DeWitt approach and ordinary QFT. Remarkably, the Penrose-Hawking singularity of usual Friedman-Robertson-Walker cosmologies is naturally avoided there. The theory is made explicit via specific examples, and compared with loop quantum cosmology. It is argued that it is the regularization of the classical Hamiltonian performed in this last theory what avoid the singularity, rather than quantum effects as in our case.

Motivation & Objective

• To resolve the Penrose-Hawking singularity in Friedmann-Robertson-Walker (FRW) cosmologies by introducing a new effective gravity framework.
• To address the failure of the Heisenberg picture in quantum cosmology by formulating a time-dependent Schrödinger equation with boundary conditions.
• To demonstrate that the avoidance of singularities arises from self-adjoint extensions of the Hamiltonian, not quantum effects per se.
• To show that the effective formulation naturally leads to semi-classical Einstein equations and quantum field theory in curved spacetime.
• To contrast this approach with loop quantum cosmology, arguing that regularization—not quantum effects—prevents singularities in LQC.

Proposed method

• Formulate a time-dependent Schrödinger equation: $ i\hbar\partial_t\Phi(t) = \hat{H}\Phi(t) $, with initial condition $ \Phi(t^*) = \Psi $ and $ \langle\hat{H}\rangle_\Psi = 0 $, assuming absolute time.
• Apply von Neumann's theorem to extend the symmetric Hamiltonian $ \hat{H} $ to a self-adjoint operator, ensuring unitary time evolution and well-defined quantum averages.
• Define the effective scale factor as $ a_{\text{eff}}(t) = \langle \Phi(t) | \hat{a} | \Phi(t) \rangle $, which remains strictly positive due to the domain of the self-adjoint extension.
• Model the singularity avoidance as an infinite potential barrier at $ a = 0 $, preventing the scale factor from reaching zero.
• Use the Heisenberg picture with caution, noting its failure due to ill-defined commutators when the scale factor approaches zero.
• Compare with loop quantum cosmology (LQC), showing that LQC's singularity avoidance stems from regularization of the classical Hamiltonian, not quantum effects.

Experimental results

Research questions

• RQ1 Can a time-ordered effective gravity formulation avoid the big bang singularity in FRW cosmologies without relying on quantum corrections?
• RQ2 What is the role of self-adjoint extensions of the Hamiltonian in ensuring a non-singular evolution of the scale factor?
• RQ3 How does this approach differ from loop quantum cosmology in its mechanism for singularity avoidance?
• RQ4 Can quantum field theory in curved spacetime and the semi-classical Einstein equation be derived from this effective formulation?
• RQ5 Why does the Heisenberg picture fail to describe the effective dynamics when the scale factor approaches zero?

Key findings

• The effective scale factor $ a_{\text{eff}}(t) $ is strictly positive for all time due to the self-adjoint extension of the Hamiltonian, which enforces a physical boundary at $ a = 0 $.
• The singularity is avoided not by quantum effects, but by the mathematical structure of self-adjoint extensions, which model an infinite potential barrier at $ a = 0 $.
• The Heisenberg picture fails to describe the dynamics near $ a = 0 $ because the commutator $ [\hat{H}, \hat{a}] $ becomes ill-defined, invalidating its use.
• In contrast to loop quantum cosmology, where regularization of the classical Hamiltonian prevents the singularity, this model avoids it through the same mechanism, not through quantum corrections.
• The effective formulation allows derivation of the semi-classical Einstein equation via the condition $ \langle \hat{\widetilde{H}} \rangle_\Phi = 0 $, yielding a back-reaction equation in linear order.
• The framework reproduces quantum field theory in curved spacetime by assuming a wave function factorized into a background and matter part, leading to $ i\hbar\partial_\eta\chi = \hat{H}_m(a_c(\eta), \psi)\chi $.