Physical Implications of a Fundamental Period of Time

If time is described by a fundamental process rather than a coordinate, it interacts with any physical system that evolves in time. The resulting dynamics is shown here to be consistent provided the fundamental period of the time system is sufficiently small. A strong upper bound $T_C<{10}^{-33}\mathrm{s}$ of the fundamental period of time, several orders of magnitude below any direct time measurement, is obtained from bounds on dynamical variations of the period of a system evolving in time.

Figure 1

Density plots of the wave function for small $\lambda$ (top) and large $\lambda$ (bottom), respectively, using a harmonic-oscillator Hamiltonian $\widehat{H}=\frac{1}{2}({\widehat{p}}^2+{\widehat{q}}^2)$ and a coherent initial state.

Figure 2

Relative standard deviation $\sigma$ of the system period over many system cycles as a function of $\lambda$. The analytical approximation (13) agrees well with the upper limit of a numerical computation of many system periods, both confirming a $1/\lambda$ behavior of $\sigma$.

Figure 3

Radius expectation values as multiples of the Bohr radius $a_0$ for small $\lambda$ (top) and large $\lambda$ (bottom), respectively, using a superposition of hydrogen eigenstates. The example of large $\lambda$ is visually indistinguishable from $\lambda =0$.