ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 308–318.
Pleiades Publishing, Inc., 2016.
Original Russian Text
V.A. Malyshev, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 117–127.
The Newton and Coulomb Laws
as Information Transfer by Virtual Particles
V. A. Malyshev
Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University, Moscow, Russia
Received February 3, 2016; in ﬁnal form, May 19, 2016
Abstract—In elementary particle physics the philosophy of virtual particles is widely used.
We use this philosophy to obtain the famous inverse square law of classical physics. We deﬁne a
formal model without ﬁelds or forces but with a virtual (auxiliary) particle, information carrier.
This formal model admits a very simple (school level) interpretation with two classical particles
and one virtual. Then we prove (in a mathematically rigorous way) that the trajectories in our
model converge to standard Newtonian trajectories of classical physics.
The main result is formulated in the title of the paper. Although it has connections with various
ﬁelds of mathematics and mathematical physics, the author could not ﬁnd a unique best framework
for it. Therefore, this needs serious comments, which is done in this introduction.
Dynamical systems theory. We consider here a formal system of recurrent equations (2)–(4),
which deﬁnes a strongly nonlinear dynamical system in the three-dimensional space. Similar iter-
ations of rational functions were studied by many authors and are normally rather diﬃcult to
study . However, our problems are diﬀerent, and we do not use results of this big science. The
goal of this paper is to get completed results for some scaling of parameters.
Classical particle physics. Gravitational and electric forces, which describe so diﬀerent phys-
ical phenomena, surprisingly have the same form, the inverse square law, diﬀering only by constant
factors. These laws are related, of course, with harmonic functions and the Poisson equation.
Moreover, already long ago there existed other—geometric—approaches to the Kepler laws; see,
for example, [2,3]. But more importantly, it appeared possible to deduce the Newton and Coulomb
laws from more general (and more complicated) physical theories, namely general relativity theory
of Einstein and Maxwell’s electrodynamics, respectively, where ﬁelds play a basic role. Here we
show a connection of these laws with quite diﬀerent models.
Numerical methods. Computational methods in physics are quite developed, and it might
seem that the formal system (2)–(4) is just an example of such computational schemes (for Newton’s
equations). However, this is not quite true, for two reasons:
1. One needs the number of steps of order c to reach times of order 1, and the main parameter c
can be too large;
2. But the main diﬀerence is that time steps Δ
t form one recurrent system with coordinates and
On information transmission in classical and quantum physics. In nonrelativistic clas-
sical physics, one has the action-at-a-distance principle. For example, any particle with nonzero