[Paper Review] A Local Resolution of the Problem of Time

Edward Anderson|arXiv (Cornell University)|Sep 6, 2018

Algebraic and Geometric Analysis16 references21 citations

TL;DR

This paper presents a local resolution to the Problem of Time in quantum gravity by unifying seven interrelated facets—temporal, configurational, and spacetime relationalism, constraint and generator closure, foliation independence, and quantum observables—through a novel framework of Temporal Relationalism in a Dynamics (TRiPoD), TRiFol (Temporal Relationalism for foliations), and TRiCQT (canonical quantum theory). The key contribution is a consistent, order-preserving, and fully relational approach that resolves the Problem of Time locally across classical and quantum gravity, with full compatibility across all facets and a foundation for semiclassical time and observables.

ABSTRACT

We here announce and outline a solution of this major and longstanding foundational problem, dealing with all seven of its heavily-interrelated local facets.

Motivation & Objective

• To resolve the longstanding Problem of Time in quantum gravity by addressing its seven interrelated local facets within a unified relational framework.
• To establish a consistent, order-preserving resolution that avoids the pitfalls of prior partial or isolated solutions to individual facets.
• To develop a quantum theory (TRiCQT) that 'finds observables afresh' at the quantum level, avoiding the limitations of promoting classical observables via commutators.
• To ensure compatibility between temporal, configurational, and spacetime relationalism through iterative, consistent implementation of symmetry corrections and invariance principles.
• To lay the groundwork for a global resolution and address open frontiers such as topological Background Independence and multiple quantization choices.

Proposed method

• Implementing Temporal Relationalism via a TRiPoD (Principles of Dynamics) framework, using a homogeneous quadratic action (e.g., Jacobi action) to derive primary constraints and an emergent generalized local ephemeris time (GLET).
• Correcting configuration changes via Lie derivatives with auxiliary cyclic variables $\textrm{d}\alpha$, ensuring joint implementation of Temporal and Configurational Relationalism while preserving TRi homogeneity.
• Using group-averaging and $\mathfrak{g}$-invariant operations to construct $\mathfrak{g}$-invariant objects, including constraints like the Hamiltonian $\mathcal{H}$ and momentum constraints $\mathcal{M}_i$ in GR.
• Introducing TRiFol (Temporal Relationalism for foliations) to ensure Refoliation Invariance and compatibility with Spacetime Relationalism under $Diff(\mathfrak{M})$, with the Dirac algebroid encoding this invariance.
• Constructing a canonical quantum theory (TRiCQT) via kinematical quantization, followed by solving the Wheeler–DeWitt equation $\widehat{\mathcal{H}}\Psi = 0$ with a semiclassical ansatz $\Psi = e^{iS(h)}|\chi(h,l)\rangle$.
• Defining quantum observables as those that commute with the generators of $Diff(\mathfrak{M})$, ensuring relationalism at the quantum level and avoiding reliance on classical-to-quantum promotion.

Experimental results

Research questions

• RQ1 How can all seven local facets of the Problem of Time be consistently resolved within a single, unified framework?
• RQ2 Can a relational dynamics (TRiPoD) simultaneously implement Temporal and Configurational Relationalism without breaking homogeneity or consistency?
• RQ3 How can Refoliation Invariance and Spacetime Relationalism be consistently incorporated into a relational quantum gravity framework?
• RQ4 What is the correct quantum formulation of time and observables that 'finds them afresh' rather than promoting classical structures?
• RQ5 How can the order of addressing facets—starting from temporal relationalism and progressing through configurational, generator, and foliation closure—lead to a consistent global resolution?

Key findings

• The paper achieves a joint local resolution of the Problem of Time by unifying all seven facets through a consistent, iterative application of TRiPoD, TRiFol, and TRiCQT, avoiding prior failures of isolated solutions.
• Generalized Local Ephemeris Time (GLET) emerges as a Machian time derived from a sufficient totality of locally relevant change (STLRC), providing a more robust and relational time than previous notions.
• The Dirac algebroid of GR is shown to be a local algebraic formulation of Refoliation Invariance, ensuring that TRiFol maintains compatibility with Spacetime Relationalism.
• In the quantum regime, the wavefunction $\Psi = e^{iS(h)}|\chi(h,l)\rangle$ supports a semiclassical time $t^{\rm sem}$ with a Machian interpretation, derived afresh from quantum change $\textrm{d}|\chi\rangle$.
• Quantum observables are defined via Lie-bracket commutativity with $Diff(\mathfrak{M})$ generators, ensuring relationalism at the quantum level and avoiding the pitfalls of classical-to-quantum promotion.
• The framework is universally applicable beyond GR, with a full program of over 120 research problems outlined in the companion work [38], including global extensions and topological Background Independence.