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Journal of Physics Conference Series

Which came first, spacetime or clocks?

Garnet N. Ord

Published Apr 14, 2014• 3 citations
Publisher

Abstract

Emergent quantum mechanics seeks a deeper level theory, anticipating that such a theory will provide a clearer picture of the relation between the quantum and classical worlds. In this work we show that the quantum-classical divide is a manifestation of the transition from Newton's absolute time to relativity's path-dependent time. The prior theory in this case is that particles are intrinsic clocks . The emergence of separate classical and quantum behaviour is seen by considering different continuum limits in a single digital clock model. A continuum limit that constructs a continuous worldline provides a simple basis for Minkowski spacetime. An alternative limit in which the clock itself contains boost information leads to the Dirac equation.

DOI
10.1088/1742-6596/504/1/012011

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Emergent quantum mechanics seeks a deeper level theory, anticipating that such a theory will provide a clearer picture of the relation between the quantum and classical worlds. In this work we show that the quantum-classical divide is a manifestation of the transition from Newton's absolute time to relativity's path-dependent time. The prior theory in this case is that particles are intrinsic clocks . The emergence of separate classical and quantum behaviour is seen by considering different continuum limits in a single digital clock model. A continuum limit that constructs a continuous worldline provides a simple basis for Minkowski spacetime. An alternative limit in which the clock itself contains boost information leads to the Dirac equation.
The modern concept of spacetime usually emerges from the consideration of moving clocks on the assumption that world-lines are continuous. In this paper we start with the assumption that natural clocks are digital and that events are discrete. By taking different continuum limits we show that the phase of non-relativistic quantum mechanics and the odd metric of spacetime both emerge from the consideration of discrete clocks in relative motion. From this perspective, the continuum limit that manifests itself in 'spacetime' is an infinite mass limit. The continuum limit that gives rise to the Schrödinger equation retains a finite mass as a beat frequency superimposed on the 'Zitterbewegung' at the Compton frequency. We illustrate this in a simple model in which a Poisson process drives a relativistic clock that gives rise to a Feynman path integral, where the phase is a manifestation of the twin paradox. The example shows that the non-Euclidean character of spacetime and the wave-particle duality of quantum mechanics share a common origin. They both emerge from the necessity that clocks age at rates that are path dependent.
We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. In this way, a physical clock with discrete states is introduced, which presently is still treated as decoupled from the system. This is motivated by the recent discussion of ``timeless'' reparametrization invariant models, where discrete physical time has been constructed based on quasi-local observables. Employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. We derive the emergent quantum Hamiltonian in terms of the underlying classical one. Such Hamiltonians typically need a regularization - here performed by discretization - in order to arrive at models with a stable groundstate in the continuum limit. This is demonstrated in several examples, recovering and generalizing a model advanced by 't Hooft.
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical principles underlying the theory, we rigorously construct a discrete model of quantum mechanics. Kets are names, or labels which categorise states of matter but do not directly describe them. The principle of superposition is seen as a definitional truism in the categorisation of states. This resolves the measurement problem of quantum mechanics by attributing the collapse of the wave function to information. The probability interpretation has a natural meaning in which the configuration of interacting particles plays the role of a hidden variable. The model supports a form of relativistic quantum field theory which does not depend on quantisation or second quantisation from classical mechanics. Continuous laws of wave mechanics are found in a discrete metaphysic which does not involve waves. Classical law is the expected behaviour of many elementary particles. The constraints on the theory are sufficient to establish that particles are point-like entities with specific properties found in nature. Newton's first law and conservation of momentum are established from the principle of homogeneity. Maxwell's equations are derived from the simple interaction in which a Dirac particle emits or absorbs a photon. Feynman rules are calculated for the discrete theory and differ from the standard rules by the removal of the ultraviolet divergence and the use of proper loop integrals. They give finite and unambiguous results.
The state-of-the-art physics consists of two irreconcilable branches, i.e., the quantum theory and the general relativity, which work well in their own territories, independently. However, what are quantum and spacetime after all? The key question was never addressed, satisfactorily. In this work, we describe a possibility to reformulate the quantum theory in the Minkowski spacetime from the viewpoint of classic physics in the Euclidean spacetime, i.e., classic mechanics and stochastic process theory. We show that quantum theory and Minkowski spacetime may connect with each other and emerge from a single fundamental entity.
In the standard formulation of quantum mechanics, time appears as an externalevolution parameter rather than as an operator. In this work we analyze an alterna-tive perspective: time as an emergent quantity resulting from relations between twoquantum states of the same system. A pair of wave functions defines a point in the six-dimensional bilocal configuration space R3,3 , in which a natural “quantum distance”operator leads to an invariant with Minkowski structure ds2 = dt2 − dr2 .Constructing a basis adapted to this quantity yields a metric equivalent to M4 + S 2 —four-dimensional Minkowski spacetime coupled to an internal sphere of radius equal tothe quantum distance t. On this space we analyze the scalar field equations and showthat the spherical harmonics correspond to Klein–Gordon equations with geometric,time-dependent masses. The lowest modes l = 0, 1 form a four-component multipletwhich can be linearized in the spirit of Dirac, producing a first-order equation withadditional terms arising from the time-varying geometric mass.
Jacobi's action principle is known to lead to a problem of time. For example, the timelessness of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi's principle to define the dynamics of 3-geometries through superspace. In addition, using Jacobi's principle for nonrelativistic particles is equivalent classically to Newton's theory but leads to a time-independent Schr\"odinger equation upon Dirac quantization. In this paper, we study the mechanism for the disappearance of time as a result of using Jacobi's principle in these simple particle models. We find that the path integral quantization very clearly elucidates the physical mechanism for the timeless of the quantum theory as well as the emergence of duration at the classical level. Physically, this is the result of a superposition of clocks, which occurs in the quantum theory due to a sum over all histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the boundary conditions in the path integral.