[Paper Review] TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)

Edward Anderson|arXiv (Cornell University)|Jan 30, 2015

Quantum Mechanics and Applications10 references16 citations

TL;DR

This paper introduces TRiPoD (Temporal Relationalism incorporating Principles of Dynamics), a reformulation of classical and quantum mechanics that eliminates any fundamental notion of time by replacing time-dependent structures with relational, parameter-free dynamics based on changes (dQ^A). It introduces the d-anti-Routhian and d-almost Hamiltonian formalism, redefines momentum and actions via Jacobi arc elements, and preserves constraints and Hamilton–Jacobi theory while resolving the Problem of Time in background-independent theories.

ABSTRACT

Temporal Relationalism is that there is no time for the universe as a whole at the primary level. Time emerges rather at a secondary level; one compelling idea for this is Mach's: that time is to be abstracted from change. Temporal Relationalism leads to, and better explains, the well-known Frozen Formalism Problem encountered in GR and other background-independent theories at the quantum level. Abstraction from change is then a type of emergent time resolution of this. Moreover, the Frozen Formalism Problem is but one of the many Problem of Time facets, which are notoriously interconnected. The current article concerns modifications of physical formalism which ensure that once Temporal Relationalism is resolved, it stays incorporated. At the classical level, this involves modifying much of the Principles of Dynamics. I first introduce the anti-Routhian to complete the Legendre square of Lagrangian, Hamiltonian and Routhian. I next pass from velocities $\dot{Q}\mbox{}^{A}$ to changes d$Q^{A}$. Then Lagrangians are supplanted by Jacobi arc elements, Euler--Lagrange equations by Jacobi--Mach ones, and momentum requires redefining but actions remain unchanged. A differential (d) version of the Hamiltonian is required, giving rise to a variant of the Dirac approach based on a d-almost Hamiltonian subcase of the d-anti Routhian. On the other hand, the forms of the constraints themselves, and of Hamilton--Jacobi theory, remain unaltered.

Motivation & Objective

• To resolve the Problem of Time in background-independent theories by fully embedding Temporal Relationalism into the Principles of Dynamics.
• To eliminate any fundamental or auxiliary time parameter in classical and quantum formalisms, replacing it with relational change (dQ^A) and geometric structures.
• To reformulate Lagrangian, Hamiltonian, and Routhian mechanics using differential forms and Jacobi arc elements, preserving physical content while achieving manifest parametrization irrelevance.
• To develop a d-almost Hamiltonian formalism that is both classically consistent and quantum-mechanically viable, ensuring temporal relationalism is preserved through quantization.
• To extend the framework to include Kuchař observables and semiclassical approximations, enabling application to quantum cosmology and general relativity

Proposed method

• Introduces the anti-Routhian to complete the Legendre square, enabling a duality between Lagrangian, Hamiltonian, and Routhian formulations in a relational context.
• Replaces velocities (dQ^A/dλ) with differential changes (dQ^A), leading to a manifestly parametrization-irrelevant formulation based on configuration space geometry.
• Supplants Lagrangians with Jacobi arc elements (ds = ||dQ||_M), and replaces Euler–Lagrange equations with Jacobi–Mach equations to describe dynamics without time.
• Redefines momentum via the d-anti-Routhian and introduces a differential (d) version of the Hamiltonian, forming a d-almost Hamiltonian subcase that preserves constraint structure.
• Preserves the form of constraints and Hamilton–Jacobi theory, ensuring consistency with standard approaches while embedding relationalism at the foundational level.
• Applies the framework to finite models (e.g., minisuperspace GR) and extends it to canonical quantum mechanics and semiclassical quantum cosmology via geometrical quantization

Experimental results

Research questions

• RQ1 How can the Problem of Time in quantum gravity be resolved by fully embedding Temporal Relationalism into the Principles of Dynamics?
• RQ2 What is the correct relational replacement for time-dependent Lagrangians and Hamiltonians in background-independent theories?
• RQ3 How can the Legendre transformation be completed in a relational framework, and what role does the anti-Routhian play in this?
• RQ4 Can a differential (d) formulation of the Hamiltonian preserve the structure of constraints and allow for consistent quantization?
• RQ5 How do Kuchař observables and the Hamilton–Jacobi formulation adapt in a fully relational, parameter-free dynamics?

Key findings

• The d-almost Hamiltonian formalism provides a time-free, relational alternative to the standard Hamiltonian, preserving constraint structure and enabling consistent quantization.
• Jacobi arc elements (ds) replace Lagrangians, and the resulting Jacobi–Mach equations describe dynamics without reference to any time parameter or labeling variable.
• The momentum is redefined via the d-anti-Routhian, but the action remains invariant under the new formalism, ensuring physical consistency.
• Constraints and Hamilton–Jacobi theory retain their standard forms, demonstrating that relationalism can be implemented without altering the core structure of these formalisms.
• The framework successfully extends to minisuperspace gravity and supports the construction of Kuchař observables, enabling relational quantum cosmology.
• The method allows for a consistent semiclassical approximation to quantum cosmology (TRiSQC), bridging classical relational dynamics and quantum theory