[Paper Review] A note on the foundation of relativistic mechanics. I: Relativistic observables and relativistic states

Carlo Rovelli|arXiv (Cornell University)|Nov 13, 2001

Relativity and Gravitational Theory15 citations

TL;DR

This paper proposes a covariant formulation of relativistic mechanics by redefining 'state' and 'observable' in terms of correlations between partial observables—such as time and position—without privileging any one variable as time. It introduces a relativistic phase space (Heisenberg phase space) where states label relations between observables, leading to a fully covariant dynamics via an evolution equation f(α,t;A,φ)=0, which simplifies the formalism and resolves foundational issues in quantum gravity.

ABSTRACT

Is there a version of the notions of "state" and "observable" wide enough to apply naturally and in a covariant manner to relativistic systems? I discuss here a tentative answer.

Motivation & Objective

• To resolve conceptual difficulties in relativistic and quantum mechanics arising from the noncovariant use of time in standard formulations.
• To develop a foundation for mechanics that is inherently relativistic, avoiding the privileging of time as an independent variable.
• To provide a framework for quantum gravity by formulating states and observables in a generally covariant, background-independent manner.
• To show that Hamiltonian mechanics simplifies when expressed in terms of relativistic states and partial observables.
• To lay the groundwork for a covariant quantum theory where time plays no special role, applicable to general relativity and field theory.

Proposed method

• Define 'partial observables' as measurable quantities (e.g., clock reading t and pendulum angle α), forming a relational configuration space C.
• Introduce the concept of a 'motion' as a curve in C, defined by a relation f(α,t;A,φ)=0, representing physical correlations between observables.
• Define the Heisenberg phase space Γ as the two-dimensional space of all such possible motions, parameterized by state variables (A,φ).
• Formulate the evolution equation as f(α,t;A,φ)=0, where each state (A,φ) determines a specific correlation curve in the extended configuration space.
• Extend the formalism to quantum mechanics by defining states as vectors in a Hilbert space over the extended configuration space, with transition amplitudes given by integrals over correlation regions.
• Use the propagator W(x,y) to compute probability amplitudes between regions R and R′ in configuration space, leading to a covariant interpretation of the wave function.

Experimental results

Research questions

• RQ1 How can the notions of 'state' and 'observable' be redefined to be manifestly covariant in relativistic theories?
• RQ2 What is the role of time in a generally covariant mechanical system, and can it be eliminated as a privileged variable?
• RQ3 How can quantum mechanics be formulated in a way that respects spacetime covariance and diffeomorphism invariance?
• RQ4 Can the formalism of Hamiltonian mechanics be simplified and made more natural when expressed in terms of relational correlations between observables?
• RQ5 What is the physical meaning of the wave function in a background-independent quantum theory of gravity?

Key findings

• The relativistic notion of state as a point in a Heisenberg phase space Γ, parameterized by (A,φ), provides a Lorentz-invariant and generally covariant alternative to the nonrelativistic concept of an instantaneous state.
• The evolution of the system is described by a single equation f(α,t;A,φ)=0, which defines a curve in the extended configuration space C, replacing the traditional time-evolution equation.
• For the harmonic oscillator, the evolution equation takes the explicit form α − A sin(ωt + φ) = 0, showing how the state (A,φ) determines the observable correlation between t and α.
• In quantum mechanics, the probability amplitude to observe a correlation in region R given a prior correlation in R′ is given by A_R,R′ = ⟨R|R′⟩, computed via the propagator W(x,y) as shown in equations (37) and (38).
• The interpretation postulate reduces to the standard Born rule in the nonrelativistic limit, confirming consistency with conventional quantum mechanics.
• The formalism eliminates the need for a preferred time or spacelike slicing, making it suitable for quantum gravity where diffeomorphism invariance forbids such structures.