Quantum Phase from the Twin Paradox

Department of Mathematics
Ryerson University
Toronto, Canada

The classical Poisson process may be analyzed by either counting events or by adding inter-arrival times. The two pictures are equivalent on the assumption of an absolute time, but inequivalent in a relativistic universe in which space and time are linked. 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. This example suggests that the non-Euclidean character of spacetime and the wave-particle duality of quantum mechanics share a common origin.

Short Bio:
My Ph.D. work considered exactly solvable models of phase transitions in percolating systems. While writing the thesis I became intrigued by the discovery that Fractals lurk beneath the Heisenberg uncertainty principle[1]. The Fractal picture conveniently explained some of the scaling relations that have parallel expression in classical statistical mechanics and quantum mechanics. It strongly suggested that the quantum equations were phenomenological, not fundamental, and that one needed to look underneath the differential operators to make further progress. My research since `Fractal Spacetime' has explored a succession of stochastic models that capture aspects of quantum propagation from a particle perspective. A recent model[2], relevant to this conference, assumes that all particles are clocks that determine their age based on some geometric feature of their world line. This assumption is a convenient alternative to "spacetime tells particles how to move". It leads to an emergent view of the spacetime frame of classical relativity as an infinite mass approximation. It also associates a fundamental (Compton) frequency with massive particles. The existence of mass as a fundamental frequency allows contact with both the Dirac and Schrödinger equation, in appropriate continuum limits, and strongly suggests that quantum propagation is a relativistic effect.
References
[1] G. N. Ord. Fractal space-time a geometric analog of relativistic quantum mechanics. J.Phys.A, 16:1869-1884, 1983.
[2] G.N. Ord and R.B. Mann. How does an electron tell the time? Int. J. Theor. Phys., To Appear, 2011.