Heisenberg and Quantization

WikiMechanics is outright quantum mechanics. Particle attributes are always quantized because we use a finite categorical scheme of binary distinctions to describe sensation. Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. Time is quantized too because the time coordinate depends on a chain's event index. Indices are always integers so t changes in steps. Motion is discontinuous in principle, and sometimes this is even observed as quantum leaping and tunnelling. Because of this quantization, we are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So WikiMechanics does not require calculus; calculations are designed to be implemented on digital computers in a finite number of discrete steps.

Werner Heisenberg, 1901—1976.

Another consequence of unmitigated quantum mechanics is that there are lower limits to the uncertainty for some measurements. For example, consider a particle P that is described by the historically ordered chain of events

$\mbox{\fontsize{14}{18}\selectfont $ \Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2} \ \ldots \ \sf{P}_{\it{i}} \ \ldots \ \sf{P}_{\it{f}} \ \ldots \ \right) $}$

If P is isolated then the elapsed time between events P_i and P_f is

$\mbox{\fontsize{14}{18}\selectfont $ \Delta t = \left( f-i \right) \hat{\tau} $}$

where $\mbox{\fontsize{18}{18}\selectfont$ \overset{ \hat{\tau} }{ \textcolor{white}{\text{\_}} } $}$ is the period of P. Consider finding the elapsed time from observations of event indices and periods. By the usual rules for assessing the propagation of experimental errors, the uncertainty in elapsed time is

$\mbox{\fontsize{14}{18}\selectfont $ \delta t = \left( \delta \! f + \delta \! i \right) \hat{\tau} + \left( f-i \right) \delta \hat{\tau} $}$

because the event indices and the lifetime are logically independent quantities. The uncertainty in the period is bounded by

$\mbox{\fontsize{14}{18}\selectfont $ \delta \! \hat{\tau} \ge \hat{\tau} k_{S} $}$

And some uncertainty is also associated with event indices; they are required to be integers, so

$\mbox{\fontsize{14}{18}\selectfont $ \delta \! i \ge 1/2 \ \ \ \ \ \ \ \text{\large{\sf{and}}} \ \ \ \ \ \ \ \delta \! f \ge 1/2 $}$

Then

$\mbox{\fontsize{14}{18}\selectfont $ \delta t \ge \hat{\tau} + k_{S} \left( f-i \right) \hat{\tau} $}$

And since $\mbox{\fontsize{18}{18}\selectfont$ \overset{ i < f }{ \textcolor{white}{\text{\_}} } $}$ we know that $\mbox{\fontsize{18}{18}\selectfont$ \overset{ f - i \, \ge \, 1 }{ \textcolor{white}{\text{\_}} } $}$ so

$\mbox{\fontsize{14}{18}\selectfont $ \delta t \ge \left( 1 + k_{S} \right) \hat{\tau} $}$

or in terms of the energy

$\mbox{\fontsize{14}{18}\selectfont $ \delta t \ge h \! \left( 1 + k_{S} \right) /E $}$

The uncertainty in a time measurement can be decreased by working with larger particles. In contrast, the hypothesis of temporal homogeneity contends that

$\mbox{\fontsize{14}{18}\selectfont $ \delta \! E \ge k_{S} E \ \ \ \ \ \ \ \text{\large{\sf{where}}} \ \ \ \ \ \ \ \it{ k_{S} }= \frac{ \sqrt{ 1 + 1 / \pi } - 1}{2} $}$