<figure><img src="image_1.png" /></figure>

HAL
open science

# Discretization of space and time in wave mechanics: the validity limit

Luca Roatta

► To cite this version:

Luca Roatta. Discretization of space and time in wave mechanics: the validity limit. 2017. hal-01476650

HAL Id: hal-01476650

https://hal.science/hal-01476650

Preprint submitted on 27 Feb 2017

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Discretization of space and time in wave mechanics: the validity limit

Luca Roatta

E-mail: lucaroa@gmail.com

## Abstract

Assuming that space and time can only have discrete values, it is shown that wave mechanics must necessarily have a specific applicability limit: in a discrete context, unlike in a continuous one, frequencies can not have arbitrarily high values.

## 1. Introduction

Let's assume, as work hypothesis, the existence of both discrete space and discrete time, namely spatial and temporal intervals not further divisible; this assumption leads to some interesting consequences. Here, as a first result, we find the limit of applicability of wave mechanics (and consequently also of quantum mechanics).

So, if we suppose that neither space nor time are continuous, but that instead both are discrete, it will not be possible to establish arbitrary values for a spatial coordinate or the time: any length must be an integer multiple of the fundamental length and any time interval must be an integer multiple of the fundamental time. Let's name $l_0$ the fundamental length and $t_0$ the fundamental time.

## 2. The applicability limit of wave mechanics

The values of $l_0$ and $t_0$ are not essential, at least for the moment: what matters is to establish the principle that $l_0 > 0$ and $t_0 > 0$.

Of course, if $l_0$ is the minimum length, neither a wavelength can be less than it; so $l_0$ is also the minimum wavelength. For an electromagnetic wave, as shown by the relation $\lambda v = c$, at the minimum wavelength $\lambda = l_0$ corresponds the maximum frequency $v_{max}$, if we want to keep $c$ constant. Then, if $v$ is the frequency of an electromagnetic wave, it must be $v \le v_{max}$; if we consider $\lambda = l_0$ (and consequently $v = v_{max}$) we have $l_0 v_{max} = c$, from which we obtain:

$$v_{max} = \frac{c}{l_0} \qquad (1)$$

It is evident that in a continuous context ($l_0 \to 0$) there is no limit for the frequency.
The frequency can be expressed as $v = 1/T$ where T is the period; the maximum value for the frequency is reached when T assumes the minimum value $t_0$.

So from Eq. (1) we obtain:

$$ \frac{l_0}{t_0} = c \qquad (2) $$

that indicates the relationship between $l_0$ and $t_0$.

The energy associated to a wave is $E = hv$, so for Eq. (1) there is also an upper limit for the energy a wave can have. In particular:

$$ E_{max} = h v_{max} = \frac{hc}{l_0} \qquad (3) $$

Also for energy, in a continuous context ($l_0 \to 0$) there is no upper limit.

The relation energy-mass $E = mc^2$ allows us to write

$$ E_{max} = h v_{max} = m_{max} c^2 = \frac{hc}{l_0} \qquad (4) $$

obtaining

$$ m_{max} = \frac{h}{c l_0} \qquad (5) $$

that represents the maximum value for the mass that in a discrete context can be treated from the point of view of wave mechanics. Again, also for mass, in a continuous context ($l_0 \to 0$) there is no upper limit.

We can obtain the same result starting from the expression[1][2] of the Compton wavelength $\lambda_c = h/mc$ and imposing $\lambda_c = \lambda_{min} = l_0$ and $m = m_{max}$.

### 3. Conclusion

The assumption that both space and time are discrete has led to find the applicability limit of wave and quantum mechanics: in a discrete context (contrary to what happens in a continuous context) no object, having mass greater than $m_{max}$, as expressed by Eq. (5), can be treated from the point of view of wave or quantum mechanics.

### References

[1] Paul S. Wesson and James M. Overduin “A scalar field and the Einstein vacuum in modern Kaluza-Klein theory” [2012]. Available at https://arxiv.org/pdf/1209.5703.pdf

[2] Subir Ghosh “Planck Scale Effect in the Entropic Force Law” [2011]. Available at https://arxiv.org/pdf/1003.0285.pdf