Local Time and the Unification of Physics
| Scientific Paper | |
|---|---|
| Title | Local Time and the Unification of Physics |
| Read in full | Link to paper |
| Author(s) | Lance R Fletcher |
| Keywords | Clocks, Time, Quantum, Relativity, Rigid bodies, Frame of reference |
| Published | 1996 |
| No. of pages | 14 |
Read the full paper here
Abstract
The notions of time in the theories of Newton and Einstein are reviewed so that the difficulty which impedes the unification of quantum mechanics (QM) and general relativity (GR) is clarified. It is seen that GR by itself contains an intrinsic difficulty relating to the definition of local clocks, as well as that GR still requires a kind of absolute that can serve as an objective reference standard. We present a new understanding of time, which gives a consistent definition of a local time associated with each local system in a quantum mechanical way, so that it serves the requirements of both GR as well as QM. As a consequence, QM and GR are reconciled while preserving the current mathematical formulations of both theories.
Overview
This is Part I of a two-part paper by Hitoshi Kitada, of the Department of Mathematical Sciences at the University of Tokyo, and Lancelot R. Fletcher, published in Apeiron in April 1996 and circulated as gr-qc/9408027. It is the expository companion to Kitada's technical "Theory of local times" (Il Nuovo Cimento 109 B, 1994). Part I sets out the philosophical problem and defines the central notion; Part II, promised in the same journal, supplies the mathematical relationship between local time and the observer's time.
The claim is unusually modest in one respect and radical in another. Modest, because nothing in the existing mathematics of either theory is to be altered: "QM and GR are reconciled while preserving the current mathematical formulations of both theories". Radical, because the reconciliation is achieved by inverting the usual relationship between clocks and time. Time is not a parameter given in advance which clocks then measure; a clock is a time, and "clocking" is a necessary activity of every existing thing. The universe as a whole, having no motion outside itself, is on this account not in time at all.
The argument
Orthogonalizing the two geometries
Kitada's earlier proposal was to remove the apparent mathematical incompatibility of the two theories — quantum mechanics being framed in a Euclidean geometry, general relativity in a curved Riemannian one — by making them factors of a direct product rather than rivals for the same space. The total space is written X × R6, where X is the Riemannian manifold of general relativity and R6 is the Euclidean phase space of positions and velocities (x, v) of non-relativistic quantum mechanics. As orthogonal components, the two are "compatible without contradiction". This device establishes only that the theories may be held apart; the substantive work of coordinating them falls to the notion of time.
The coordination proceeds through the local system: a finite collection of particles whose positions are all referred to the same frame. Considered internally, such a system has only quantum-mechanical properties. Each classical point (t, x) of the manifold X is then correlated with the centre of mass of some local system, and it is these centres of mass, not the quantum particles themselves, that are the classical particles of general relativity. The obvious objection — that a local system contains sub-local systems whose centres of mass would then be classical particles inside a quantum system — is met by insisting that the distinction is one of reference frame, not of inclusion: described in the observer's time a system appears classical, described in its own time it must be treated quantum-mechanically.
Why the problem is a problem about time
The authors trace the incompatibility to the two theories' divergent inheritance from Newton. They quote the Principia at length on absolute time, which "flows equably without relation to anything external", and note that Newton establishes it by a distinction: absolute time is the ideal standard by which the relative, apparent, common time read off actual motions is corrected. Einstein discards the absolute and keeps only clock readings. Quantum mechanics does the reverse: in the Schrödinger equation, with its solution ψ(x, t) = exp[−itH/ℏ]ψ0, the parameter t is given a priori and externally, exactly in Newton's manner, which is also why the equation is not invariant under relativistic coordinate transformations.
The paper's sharpest passage argues that Einstein did not in fact eliminate the Newtonian absolutes but disguised them. Their epistemological function was to supply "something which is the same for all observers"; in special relativity that role is taken over by rigid reference bodies and standard clocks, and Einstein himself is quoted declaring continuum spatii et temporis est absolutum. The disguise is exposed by comparing Weyl's definition of a clock — an absolutely isolated physical system that returns to an earlier state, so that its cycles mark equal durations — with Newton's absolute time. Weyl's clock has precisely the same defect: "that which allows it to be accurate makes it at the same time unobservable", and strictly speaking the only absolutely isolated system imaginable is the universe itself. Only with general relativity, the authors contend, is the absolute genuinely abandoned, and at that moment two needs arise together: a consistent definition of a local clock, and an objective reference standard. Their resolution is that the standard cannot be a global frame containing the local system; it must be the local system itself, taken as the system.
The definition of local time
For a local system L of N particles with position vectors xj and momenta pj = mjvj, whose quantum operators satisfy the canonical commutation relations, the local time is defined as the quotient of position by velocity:
tL = |xj| / |vj|
The definition appears to depend on which particle j is chosen. The load-bearing technical claim is that it does not: results of Enss (1986) on asymptotic observables in many-body Schrödinger scattering, as interpreted and extended by Kitada, are said to show the quotient independent of j. This is what allows the quantity to be attached to the system as a whole rather than to a point within it, and it is also why local time "must nonetheless appear as if it were an absolute, Newtonian time". Time is thereby derived from what is actually observed — positions and velocities — instead of being presupposed. The authors support this with an observational point: we never perceive time directly, only coincidences of pointer and scale, so that reading a clock is already a comparison of positions and motions.
The timeless universe
Since time has been defined only for local systems as a measure of their internal motions, and the universe as a whole is not a local system, it follows — this is Kitada's axiom 1 — that no time is associated with the total universe. Mathematically the universe is an eigenstate φ of a total Hamiltonian H of infinite degrees of freedom, Hφ = λφ, hence stationary in the technical sense. Local Hamiltonians of finite degrees of freedom are "no more than convenient approximations to the total Hamiltonian", and it is exactly this approximate character that leaves room for local systems to change while the whole remains stationary: variation inside a local system is compensated by variation outside it. The universe does not change as an existence, yet is not internally frozen.
The authors read this through Spinoza's Definition 8 of eternity — existence following from the definition of the eternal thing alone, which "cannot be explained by duration or time" — and draw the reversal that gives the paper its character: "the local clock does not measure time, but it is time"; "clocks exist and their operation is necessarily expressed by duration"; to exist is to be clocking, "without which there is no interaction". Relative to Newton they claim to have localized absolute time while keeping its absoluteness for the universe, minus the flow; relative to Einstein, to have supplied the consistent local clock his theory required but could not define.
Assessment
The diagnosis is the strongest part of the paper and deserves to be taken seriously quite apart from the proposed cure. The observation that an ideal clock defined as a perfectly isolated system is unobservable for the very reason that makes it ideal, and that this is Newton's difficulty over again in Weyl's language, is a genuine and rarely made criticism. So is the point that the operational rhetoric of relativity conceals a residual absolute in the rigid rod and the standard clock. Grounding time in the internal motions of a bounded system, rather than positing it as a background parameter, is a serious proposal in the same family as later relational and "timeless" approaches, and the authors have the merit of leaning on an actual theorem — Enss's asymptotic completeness results — rather than on intuition alone.
The difficulties are real. The definition tL = |x|/|v| is an asymptotic relation: it is the large-time behaviour of scattering states, for which a particle streams outward so that position divided by speed approaches elapsed time. Actual clocks are bound systems — atoms, pendulums, crystals — whose particles do not stream, and for which the quotient does not track elapsed time at all. The paper concedes only that the relation holds "approximately" and refers the reader elsewhere for the precise version; without that, the very systems the notion is meant to explain are the ones the formula fits worst. The expression also presupposes an origin, presumably the centre of mass, which is not established here.
The framework is throughout non-relativistic quantum mechanics: it is the Schrödinger equation and Schrödinger operators that describe the interior of every local system. Reconciling general relativity with non-relativistic quantum mechanics is a weaker achievement than the abstract's unqualified "QM and GR are reconciled", and it leaves untouched the difficulties — renormalization of gravity, the behaviour of quantum fields in strong curvature — that motivate most work on the problem. The claim that the two geometries are incommensurable is itself loose, since quantum field theory on curved spacetime already places quantum systems on Riemannian manifolds without contradiction.
Most importantly, the paper is Part I. The step on which everything depends — showing that the local time so defined actually reproduces the proper time of general relativity at the centre of mass, and that observer time and local time relate in the required way — is explicitly deferred. And because the programme by design changes neither formalism, it yields no prediction that could distinguish it from orthodoxy: it makes no statement about the rates of transported atomic clocks, gravitational redshift, or any other measurement that would let observation adjudicate. It must therefore be judged as an interpretation, on coherence rather than on evidence. As such it is careful, philosophically literate and honest about its own provisional status — the authors call their exposition "a rather intuitive definition" — but it should not be mistaken for a physical theory that has been tested.