[Paper Review] Quantum Mechanical Clock and Classical Relativistic Clock

Hitoshi Kitada|ArXiv.org|Feb 13, 2001

Quantum Mechanics and Applications6 references3 citations

TL;DR

This paper proposes a quantum mechanical clock as a fundamental timekeeping mechanism within local quantum systems, showing it is operationally equivalent to classical relativistic clocks like Einstein's light clock. By defining time via the internal quantum motion of particles and imposing relativistic invariance through energy-shell constraints, the model derives time dilation exactly as in special relativity, unifying quantum mechanics and relativity at the level of time measurement.

ABSTRACT

A cyclic nature of quantum mechanical clock is discussed as ``quantization of time." Quantum mechanical clock is seen to be equivalent to the relativistic classical clock.

Motivation & Objective

• To resolve foundational inconsistencies between quantum mechanics and general relativity in time measurement, particularly the problem of finite-sized clocks in curved spacetime.
• To establish a consistent operational definition of time that is both quantum mechanical and relativistically covariant.
• To unify quantum mechanics and special relativity by showing that quantum clocks naturally reproduce relativistic time dilation.
• To provide a physical basis for the Planck time and the concept of a minimum time period (LPT) from quantum dynamics.

Proposed method

• Introduces a quantum mechanical clock as a local system where time is measured by the internal quantum motion of a particle, defined via the evolution of its wave function.
• Imposes two axioms: (1) the sum of squared internal and external velocities equals c², and (2) internal momentum is constant at m₀c, linking quantum and relativistic kinematics.
• Defines the clock period p(v) as the time for one full cycle of the quantum evolution, derived from the Hamiltonian H = λ/m.
• Uses the spectral representation of Hamiltonians in the N-particle case to generalize the one-particle results to clusters of particles.
• Derives time dilation via the relation p(v) = p(0)/√(1−v²/c²), showing agreement with special relativity.
• Establishes that the least period of time (LPT) is 2t_P, linking the quantum clock to the Planck time.

Experimental results

Research questions

• RQ1 Can a quantum mechanical clock be defined such that its timekeeping behavior matches that of a classical relativistic clock?
• RQ2 How can time be consistently measured in quantum systems while preserving relativistic covariance?
• RQ3 Does the quantum mechanical evolution of a local system naturally reproduce relativistic time dilation?
• RQ4 What is the physical origin of the Planck time within a quantum-relativistic framework?
• RQ5 How do the internal quantum dynamics of a particle relate to its observed relativistic mass and velocity?

Key findings

• The quantum mechanical clock's period p(v) for a system moving with velocity v relative to an observer is given by p(v) = p(0)/√(1−v²/c²), exactly matching the relativistic time dilation formula.
• The least period of time (LPT) for a system at rest is p(0) = 2h/(m₀c²), which equals 2 times the Planck time t_P.
• The model reproduces the relativistic mass-energy relation m = m₀/√(1−v²/c²) from quantum mechanical postulates, without assuming it a priori.
• In the N-particle case, each scattered cluster behaves as an independent local system with time dilation governed by the same relativistic rule.
• The quantum clock is operationally equivalent to the classical relativistic clock, resolving a long-standing mystery of why they agree so precisely.
• The internal quantum motion, interpreted as zitterbewegung, provides a physical basis for relativistic effects like mass increase and time dilation.