[Paper Review] Classical geodesics from the canonical quantisation of spacetime coordinates

Iñaki Garay, Salvador Robles-Pérez|arXiv (Cornell University)|Jan 16, 2019

Noncommutative and Quantum Gravity Theories10 references4 citations

TL;DR

This paper proposes a canonical quantization of spacetime coordinates in general relativity, showing that classical geodesics emerge from the quantum Klein-Gordon equation derived via this procedure. By generalizing the Maupertuis principle to relate geodesics in a curved spacetime to non-affine geodesics in a conformally related spacetime with an effective potential, the authors recover both the classical geodesic equation and the Schrödinger equation in the Newtonian limit, unifying quantum field theory, general relativity, and non-relativistic quantum mechanics under a single framework.

ABSTRACT

A canonical quantisation of the coordinates of the spacetime within the general relativity theory is proposed. This quantisation will depend on the observer but it provides an interesting perspective on the problem of relating the non-relativistic and classical limits of a possible quantum gravity theory. In this sense, within this formalism, it is possible to recover from the quantum equation satisfied by a test field the classical geodesics of the corresponding spacetime. On the other hand, the Schrödinger equation is recovered in the Newtonian limit. A key ingredient for this procedure is a generalization of the Maupertuis principle, that is, the possibility of describing the geodesics of a given metric as the non-affine geodesics of other conformal metric under the action of a potential.

Motivation & Objective

• To explore a canonical quantization of spacetime coordinates in general relativity that is observer-dependent but provides a bridge to quantum gravity.
• To demonstrate how classical geodesics can be recovered from the quantum Klein-Gordon equation obtained via this quantization procedure.
• To unify the classical, quantum, and non-relativistic limits of gravity and matter fields within a single formalism.
• To show that the Newtonian limit of the quantized spacetime yields the Schrödinger equation with a potential derived from the metric's temporal component.

Proposed method

• Apply canonical quantization to spacetime coordinates using a reparametrization-invariant action, leading to a Hamiltonian constraint that generates a Klein-Gordon-type equation.
• Utilize a generalized Maupertuis principle to map geodesics in a spacetime with metric $ g_{\mu\nu} $ to non-affine geodesics in a conformally related spacetime with a potential $ \tilde{V}(x) $.
• Construct a quantum wave function $ \phi $ satisfying the Klein-Gordon equation with $ \hbar $-dependent coefficients, derived from the canonical quantization of the spacetime coordinates.
• Perform a semiclassical expansion in powers of $ \hbar $, showing that the zeroth-order term reproduces the classical geodesic equation of the spacetime.
• Derive the Newtonian limit by taking $ \hbar \to 0 $ and identifying the effective potential $ \tilde{V}(x) $, leading to a Schrödinger equation in curved space.
• Apply the formalism to de Sitter spacetime in static coordinates, showing that the geodesics correspond to motion under a central potential in a curved 3D space.

Experimental results

Research questions

• RQ1 Can classical geodesics in a curved spacetime be derived from a quantum equation obtained via canonical quantization of spacetime coordinates?
• RQ2 How does the generalized Maupertuis principle relate geodesics in a given spacetime to trajectories in a conformally related spacetime with an effective potential?
• RQ3 Does the canonical quantization of spacetime coordinates lead to the Schrödinger equation in the non-relativistic limit?
• RQ4 Can the Bunch-Davies vacuum state in de Sitter spacetime be used to recover the classical geodesic equation?
• RQ5 How does the inclusion of $ \hbar $ in the Klein-Gordon equation enable the recovery of classical and non-relativistic limits?

Key findings

• The zeroth-order term in the $ \hbar $-expansion of the quantum wave function satisfies the classical geodesic equation of the spacetime, demonstrating that classical trajectories emerge from quantum spacetime coordinates.
• The geodesic equation in de Sitter spacetime in static coordinates is recovered from the Bunch-Davies vacuum state of the quantized spacetime coordinates, confirming the consistency of the formalism.
• The Newtonian limit of the quantized spacetime yields a Schrödinger equation with a potential $ \tilde{V}(x) $ derived from the metric's temporal component, matching the expected non-relativistic dynamics.
• The generalized Maupertuis principle allows the description of geodesics in a curved spacetime as non-affine geodesics in a conformally related spacetime under a potential, providing a geometric unification of classical and quantum descriptions.
• The Klein-Gordon equation derived from canonical quantization of spacetime coordinates contains information about both the matter field and the spacetime geometry, as the geodesic equation is encoded in the $ \hbar $-dependent coefficients.
• The formalism successfully unifies the classical limit (geodesics), the quantum limit (Klein-Gordon), and the non-relativistic limit (Schrödinger equation), showing consistency across all regimes.