# How Does Spacetime “Tell an Electron How to Move”?

## Abstract

**:**

## 1. Prolog

provides a clear visualization that, encoded mathematically with strong smoothness assumptions, yields modern relativity. The clarity of the image that Wheeler’s statement in natural language invokes and the closeness to the resulting mathematical models is so evident that, by comparison, quantum mechanics looks like a deliberate cypher. (This is not to impugn the importance or effectiveness of quantum theory. It is arguably the most accurate and useful theory we have, and the mathematical framework is no less elegant than relativity. However, the interpretations of the theory are many and varied and there is no sensible statement in natural language that illustrates what quantum propagation physically represents.)“Spacetime tells matter how to move, matter tells spacetime how to curve.”[2]

“What are the spacetime instructions that tell an electron how to move?”

**signals**on the scale of the Compton length. Classical relativity is obtained by ignoring the signal and retaining only the rotation and stretch of the Lorentz transformation; while the Dirac equation is obtained by keeping the lowest frequencies of the signal. Once this is discovered, it changes the perspective on the relationship between relativity and quantum mechanics. If you start with discrete events under the restrictions of special relativity, classical physics arises only if you deliberately ignore the fact that a discrete spacetime forces the existence of a non-trivial signal in place of the worldline.

## 2. Introduction

## 3. Imaging a Particle at Rest

- What, if anything, does the timestamp have to do with quantum propagation?
- Is there a mapping of the timestamp image onto a domain that admits a smooth continuum limit?

## 4. Building a Transfer Matrix

- The edges of timestamps that enclose causal areas in spacetime have been mapped onto the two pixels corresponding to the initial conditions, introducing a form of spatial stationarity. (This is a qualitatively different approach from pre-relativistic physics where the ‘rest frame’ is special and movement is constructed assuming space and time are independent. Here the rest frame is constructed from the light cones.)
- The pixel intensities of the timestamp that were binary $(\pm 1)$ and could be stored in a single bit, now require a bigger ‘word’ to store the varying intensity values.
- In this diagonal formulation of instruction, the ‘intensities’ are actually complex numbers; not a data type usually used for images in the spatial domain, however the appearance in this context is welcome and instructive.

## 5. A Particle in a Box

## 6. Conclusions

- How do we create a digital image of the timestamp?
- How do we map the digital image onto one that becomes smooth on finer resolution?

## 7. Discussion

taken down to microscopic scales in spacetime implicates ‘zitterbewegung’, not stationarity in space for elementary particles. Classical special relativity, by assuming a smooth worldline, eliminates this feature immediately, having to recapture energy and momentum relations through dynamics. The Dirac equation, in its usual context, transplants features found in non-relativistic quantum mechanics into the relativistic domain. However, in its usual context, it is an overlay of quantum propagation on Minkowski space that is enforced as a recipe rather than discovered as a consequence of the Lorentz transformation.“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Light-like lines through the origin partition spacetime into four distinct regions. The partition exists on all scales. Throughout this article all diagrams assume natural units in which $c=1$.

**Figure 2.**Periodic events on a stationary worldline give rise to sequences of causal areas. We call a succession of two of these a timestamp. The period here is 4. The ‘worldline’ here is assumed to lie on the t-axis.

**Figure 3.**A sequence of timestamps stretched and rotated by Lorentz boosts. Time is vertical and space horizontal. Clockwise from the top. (

**A**) a Red portion of the timestamp intersects the line $t=9$. In (

**B**,

**C**) the relative speed of the observer increases but the Red portion still intersects $t=9$. In (

**D**) velocity is high enough that the earlier Green segment intersects $t=9$. As the relative speed of the observer increases, the earlier the ‘history’ recorded on the space axis.

**Figure 4.**The History-Map of a binary clock is the variable frequency square wave in this figure. The smooth curve is the real part of the Feynman propagator [13]. A single number mass in natural units, registers the two patterns. While the path integral assembles the propagator from an infinite number of Feynman paths, the History-Map is a mapping of a single sequence of timestamps via the Lorentz boosts. The smooth curve is standard quantum mechanics. The History Map is Special Relativity with Timestamps replacing worldlines.

**Figure 5.**The two pixel ‘intensities’ in the digital array image Equation (11). Proceeding from the initial intensity of 1 in the central two pixels, the pixel intensities plotted in the complex plane evolve through discrete spirals traversed in opposite directions.

**Figure 6.**A visual representation of the timestamp made smooth. The result of the filtering process is to map the two halves of the timestamp onto two cylinders that are side by side. The curves touch periodically and mimic the planar figure of the timestamp. The colours here have been retained to compare to the original planar version Figure 2.

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Ord, G.
How Does Spacetime “Tell an *Electron* How to Move”? *Symmetry* **2021**, *13*, 2283.
https://doi.org/10.3390/sym13122283

**AMA Style**

Ord G.
How Does Spacetime “Tell an *Electron* How to Move”? *Symmetry*. 2021; 13(12):2283.
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**Chicago/Turabian Style**

Ord, Garnet.
2021. "How Does Spacetime “Tell an *Electron* How to Move”?" *Symmetry* 13, no. 12: 2283.
https://doi.org/10.3390/sym13122283