# Simulation of Gacs’ Automaton

Atsushi Masumori*1, Lana Sinapayen*1 Takashi Ikegami1

1The University of Tokyo
<EMAIL_ADDRESS>

**These authors contributed equally to this work

###### Abstract

Peter Gacs proposed a one-dimensional cellular automaton capable of a robust
self-reproduction. Because the automaton is exceptionally large and
complicated, very few people have ever succeeded in simulating it on a
computer or analyzing its behavior. Here we demonstrate a partial simulation
of Gacs’ automaton (of Gray’s version), discussing its robustness. We also
discuss the potential applications of Gacs’ framework.

## Introduction

Robust self-replication in living systems is still an unsolved problem. Von
Neumann developed a theory of self-reproduction by building a self-replicating
two-dimensional cellular automaton (von Neumann, (1966)). This automaton,
however, is very unstable against noise that changes the state of the
automaton. Cellular automata, such as Conway’s Game of Life, are generally
vulnerable to noise.

Regarding this stability problem, Peter Gacs showed that there exists a one-
dimensional cellular automaton that can be stabilized against any degree of
noise (Gacs, (2001)). In this automaton, the dynamic evolution of states is
discussed as self-regeneration rather than self-replication, which means that
the system continues to regenerate itself through self-simulation. A one-
dimensional automaton is a system in which cells are linearly arranged, and
each cell determines its next state from its current state and the state of
its neighbors. For one-dimensional cellular automata with a finite interaction
distance (i.e. the number of neighbors), it was considered that even with
slight noise, the system becomes unstable and it is not possible to maintain a
specific pattern without convergence to one stationary measure. This is called
the Positive Rate Conjecture. Gacs falsified the conjecture by building a very
special cellular automaton that becomes stable against any amount of noise.

However, since this automaton is large and the rules are complicated, it has
been considered difficult to implement and run the automaton on a computer, or
analyze it (de Sá and Maes, (1992)). With the help of recent advances of
computer, we implement and analyze Gacs’ automaton, we report the progress of
the simulation and quantitative results.

## Gray’s automaton

We used Gray’s automaton, a simple version of Gacs’ automaton. Gray’s example
was introduced as a “reader’s guide” to Gacs’ work in the same journal (Gray,
(2001)). The automaton consists of colonies which each has $Q$ cells.
Although, in Gacs’ automaton one cell can only interact with its nearest
neighbors, in the Gray’s automaton, one cell can interact with five-range
neighbors; e.g. the cell at the site $x$ can interact with five left and five
right neighbors: $L(x)=\\{x-1,x-2,x-3,x-4,x-5\\}$,
$R(x)=\\{x+1,x+2,x+3,x+4,x+5\\}$ (Fig. 1).

Figure 1: Interaction distance in Gray’s automaton.

Each cell consists of six fields; Age, Address, Flags, SimBit, WorkSpace and
MailBox. The first two fields are called the local structure, while last three
fields are called the simulation structure. The Address field is an integer
between 0 and $Q-1$ and the value of the Addess field at the $i$ th cell is
$i$ (mod $Q$) in the initial configuration. The automaton should maintain this
initial configuration. The Age field counts elapsed time steps, and is reset
to 0 when it reaches the value of $U-1$, with $U=128*Q$. The Flag field is
composed of Flag1 and Flag2, and both have two states (true or false). The
Flags are used to control a process of error correction in the local
structure. The SimBit consists of five bits. The set of the first of these
bits at all cells in a colony represents the state of a single simulated cell
in the upper layer (Fig. 2). The simulated cell also has a SimBit, thus the
set of the first bit of the SimBit in a simulated colony represents a
simulated cell in further upper layer (Fig. 3). The other four bits in SimBit
are used for error correction. The WorkSpace stores bits for self-simulation;
the MailBox has a bit for communication with other cells, and all of the bits
are set to 0 in the initial configuration. In our simulation, each cell is
composed of 293 bits and has $2^{293}$ possible states. The one-dimensional
space is divided into $N$ colonies, each composed of $Q$ cell sites.

Figure 2: Each cell consists of six fields; Age, Address, Flags, SimBit,
WorkSpace and MailBox. The set of the first bit of SimBit at all cells in a
colony represents the state of a single simulated cell in upper layer. Figure
3: The set of the first bit of SimBit in a simulated colony represents a
simulated cell in further upper layer.

The summary of the simulation steps is: first, check the consistency of
patterns in each colony, and repair the colony if it is broken. If the broken
part is too large to be fixed, a coarse-graining process will be useful to
correct the large error: each colony corresponds to a single cell state, and
these simulated cells will be treated as if they are the normal cell sites and
will be corrected. The information in the upper layer trickles down to the
lower layer. To correct larger errors, further coarse-graining will be
required and information from the simulated layer further above this one will
be used for the error correction.

Small errors (level-1 or less) can be corrected by transition rules for only
local structure without the self-simulation (Gray, (2001)). In this paper, we
report preliminary results without implementing the self-simulation part.
Below, we explain the transition rules for error-correction in the local
structure except for some transition rules or conditions required for self-
simulation.

Figure 4: Examples of dynamics of error correction in the local structure. A:
Errors only happen in the first step. B: Errors ocur even after the first
step.

The error correction in the local structure is accomplished based on majority
vote. If $C(x)$ exists, and Flag1 = 0, the majority vote is executed in
$R(x)$. Otherwise, the majority vote is executed in $L(x)$. Here, $C(x)$
represents an apparent colony. If at least three cells of $R(x)$ have values
in the Address field that are consistent with each other, an apparent colony
$C(x)$ is said to exist. $C(x)$ is defined as the interval of $Q$ cells that
includes $x$. If there are no such three cells in $R(x)$, $C(x)$ does not
exist (e.g. if the Address values are as $R(x)=\\{0,1,29,3,81\\}$, then $C(x)$
exists. If the Address values are as $R(x)=\\{3,4,6,5,2\\}$, then $C(x)$ does
not exist.)

The transition rules for Address and Age at a site $x$ are determined by the
majority votes at certain cells following the above conditions, and the Age
value determined by the vote is incremented by 1 (mod $U$). The original rules
for computing the Address, and the rules of transition for the Age have some
problems and we modified the rules (see Appendix).

Here, it is defined that there is inconsistency at x, if one of the following
four conditions is satisfied: (i) $C(x)$ does not exist. (ii) $C(x)$ exists,
and at least three cells in $L(x)\cap C(x)$ have errors in the Address. (iii)
at least three cells in $R(x)$ have different Age values. (iv) at least three
cells in $R(x)$ have the same Age value $a$ and at least three cells in
$L(x)\cap C(x)$ have different Age values from $a$.

The transition rules for Flag1are defined: the value goes from 0 to1, if one
or more of the following three conditions are satisfied: (i) $x$ has an
inconsistency. (ii) Flag1 = 1 in at least three cells in $R(x)$. The value
goes from 1 to 0, if the following conditions are satisfied: (a) (i) does not
hold. (b) no more than one cell in $R(x)\cap C(x)$ has Flag1 = 1.

The transition rules for Flag2 are defined: the value goes from 0 to 1 if at
least one of the following four conditions is satisfied: (i) at least four
cells in $L(x)\cap C(x)$ have Flag2 equal to 1, or (ii) the computed value of
Flag1 at $x$ is 1 and at least four cells in $L(x)$ have Flag2 equal to 1, or
(iii) $C(x)$ does not exist and the computed value of Age at $x$ is divisible
by 16. the value goes from 1 to 0, if condition (iii) is not satisfied and at
least one of the following conditions is satisfied: (a) the computed value of
Flag1 = 0 at $x$ and there is no cell in $L(x)\cap C(x)$ has Flag2 = 0, or (b)
the computed value of Flag1 = 1 at $x$ and no cell in $L(x)$ has Flag2 = 1.

According to these rules, small errors can be corrected like the examples in
Fig. 4. Here, red cells represents damaged cells and green frames represents
cells with Flag1 = 1. Fig. 4A shows the case in which errors only happen in
the first step. In such a case, the error island does not expand and shrinks
from the left side, and the wave of Flag1 = 1 flows to the left side until it
reaches the border between colonies. The Flag1 = 1 island then shrinks from
the right. On the other hand, if other errors have occurred during error
correction, then sometimes the error expands like in Fig. 4B, but it does not
exceed the border between colonies. If the error is not very large, and is
stuck at the right border for long enough, then the error correction process
from the left side catches up with the right side. The dynamics of the Flag1
are as in the first example.

## Results

We report preliminary results without implementing the self-simulation part,
where $Q$ is set to 271. We demonstrate that the automaton can correct errors
with high probability even without self-simulation. In these experiments, at
each time step, a number of cells are destroyed according to the error rate.
The entire contents of the cell is replaced with a random bit string. .

Fig. 5 is a snapshot of spatio-temporal pattern showing how the errors are
getting corrected. The horizontal axis is time, the vertical axis is the state
space. Fig. 5A shows the state without noise and Fig. 5B shows the behavior of
error correction. As you can see from this figure, even if the error rate is
high, the automaton can correct them well. Fig. 6 shows the recovery rate for
different error rates.The recovery rate is high even for high error rates, but
at some point (error rate is more than 60%) the error correction fails and the
automaton cannot restore its original state. The simulation result shows that
the recovery rate is quite better than we expected, even without self-
simulation.

Figure 5: The horizontal axis is time, the vertical axis is a state space (Age
bit). A: no noise. B: self-correction of bits damaged by noise. Figure 6:
Recovery rate against errors. The horizontal axis shows the error rate and the
vertical axis shows the recovery rate.

We also conducted the other simulation experiments to see the dynamics of
error-correction where we input noise to each cell with a certain probability
during 500 time steps and remove the noise input after 500 steps. Fig. 7 shows
the examples of error correction for each error rate. Here, a black dot
represents a damaged cell site, and a gray dot represents a cell with Flag1 =
1, broadly implying that an error was detected in the cell’s state or in the
neighboring cells.

As we can see from the simulation with an error rate of 0.05 (Fig. 7A), for
small errors, the error correction process is almost always completed in 1
step by the majority vote with the neighbors.The information of Flag1 = 1
propagates to the cell on the left side, and if there is no inconsistency at
the border of colony, the wave of Flag1 = 1 stops, and the original state will
be completely restored.

Figure 7: Error correcting dynamics. A black dot represents a damaged cell,
and a gray dot represents a cell with Flag1 = 1. There is noise with specified
probabiliy during 500 time steps and no noise after 500 steps. The dynamics of
colonies 0 to 3 are shown. A: Error rate = 0.05. B: Error rate = 0.5. C: Error
rate = 0.6.

As we can see from the case of the error rate of 0.5 (Fig. 7B), the error
correction is not completed in 1 step and rather, the island of error
temporarily becomes large. However, the islands gradually shrinks and finally
fades away while flowing to the right. The island of error goes to the right
side as a result of the transition rules of the automaton.

Even in the case of error rate of 0.5, if the noise input is removed after 500
time steps, the automaton is restored to its original state. On the other
hand, when the error rate is 0.6 or more, error correction does not work well,
and the original state can not be restored even when noise input is removed
after 500 time steps (Fig. 7C).

Interestingly, as shown in Fig. 8 , in some cases with an error rate of 0.6,
although different from the original state, the automaton can be restored in a
state where the Address of each cell is shifted. Here, the Address values of
all cells (0-270) are displayed in gray scale. After 500 time steps, all
addresses are in the right order but they are shifted to the right. This seems
to be a phase transition.

Figure 8: Phase shifting observed in the case of error rate = 0.6. The Address
values of all cells (0 to 270) are displayed in gray scale. There is noise
with error rate = 0.6 during 500 time steps and no noise after 500 steps.

## Discussion

The concept of robustness is necessary to construct artificial life systems in
the real world. Without robustness, cellular automata models are just
conceptual ideas that never stand up in the harsh world. The simulation here
focuses on a one-dimensional cellular automaton that holds a simple static
pattern, but this research can be developed in two directions.

First, it can be developed to focus more on the self-simulation aspect of
living systems. From the cell to brain systems, systems can self-reproduce in
a closed form. This is called autopoiesis (Francisco J. Varela, (1979)). To
update the discussion of autopoiesis and discuss a sort of open-autopoiesis
idea, we might get insight from the structures of Gacs’ large and complex
automaton. Gacs’ idea can be applicable to brain systems. Self-consciousness
in the brain is constantly monitoring and simulating its own state, like Gacs’
self-simulation. We consider that Gacs’ system can be translated into neural
circuits. There are several neural network models that simulate environments
to predict and to select actions, but there are few models that simulate
themsleves (without falling into infinite regression).

Secondly, Gacs’ idea can be connected with robust pattern recognition where we
can extract meaning patterns by removing noise. Recently, robust pattern
recognition has been developed by deep learning. If we consider robust visual
perception as a problem of how to restore a pattern from a noisy pattern,
self-simulation can lead to the same results as predictive coding.

The theory of robust perception can also lead to a model of perception in
living systems. Data massively flowing to complex systems can lead to make the
system life-like Ikegami and Oka, (2013). We believe Gacs’ automaton will also
contribute to this research theme.

## Acknowledgements

This work was partially supported by MEXT as Post-K Computer Exploratory
Challenges—Construction of models for interaction among multiple socioeconomic
phenomena (hp170272).

##  Appendix A Appendix: Problems in Gray’s automaton.

We found some errors in the Reader’s guide (Gray, (2001)) and corrected or
added rules as follows.

### Problem in computing Address value

Grays explains how to use a cell’s neighbors to find what that cell’s address
should be and check this ideal address value against the address value that
the cell currently holds.

#### Original rule

p.22: “if the Address field at some site y is involved in a majority vote that
is being used to determine the new Address value at a site x, then the Address
value at y needs to be adjusted by adding $(x-y)~{}mod~{}Q$” can be summarized
as: $adjusted\\_address\\_y=(address\\_y+x-y)~{}mod~{}Q$.

#### Issue

This requires to know $x$ and $y$ (the position of the cell and its neighbor)
with certainty, but if we knew these values we wouldn’t need to calculate
addresses in the first place. We could just say $address\\_x=x$. We do not
know $x$ and $y$ so this rule cannot be used.

#### Modified rule

$adjusted\\_address\\_y=(address\\_y-i\\_y)~{}mod~{}Q$,
where $i$ is the index of the neighbor relative to this cell ($i$ in
$\\{-5..-1\\}\cup\\{1..5\\}$). The only data we know with certainty is which
neighbor we are accessing ($i\\_y$), so we use this information.

For example, in a case where there are no errors, if
$address\\_x=8$, $address\\_y=1$, $Q=10$
(colonies aligned as $[0,1,2,3,4,5,6,7,x,9][0,y,z,...]$), then
$i\\_y=3$ ($y$ is the 3rd neighbor to the right of $x$),
$adjusted\\_address\\_y=(1-3)~{}mod~{}10$ $=-2~{}mod~{}10=8$.
We can do the same with $z$:
$i\\_z=4$
$adjusted\\_address\\_z=(2-4)~{}mod~{}10=-2mod10=8$.

The adjusted addresses equal the correct address for $x$, $address\\_x$. If
$x$ held an erroneous address, for example $address\\_x=6$ instead of the
expected value 8,
$adjusted\\_address\\_y=(1-3)~{}mod~{}10=-2~{}mod~{}Q=8$
$adjusted\\_address\\_z=(2-4)~{}mod~{}10=-2~{}mod~{}Q=8$.

The rule is still working. The final corrected $address\\_x$ is obtained by
taking the majority in the 5 neighbor’s $adjusted\\_address$ values, so that
even if some neighbors hold wrong addresses, $x$ can recover.

#### Sample code (Java)

:

⬇

//calculate adjusted addresses

int adj_addr[] = new int[5];

for(int i=0; i<s;i++){

//the adjusted address

adj_addr[i] = neighbors[i].getAddr() - (i + 1)*direction;

// java does not have a modulo

// operator that works with negative

// numbers.

if(adj_addr[i]<0) {

adj_addr[i]+=Constants.Q;

}

if(adj_addr[i]>=Constants.Q){

adj_addr[i]-=Constants.Q;

}

}

### Problem in transition rules for Flag2 and Age

#### Original rule

p.21: Partial transition rule for Flag2: “0$\to$1 if at least one of the
following four conditions is satisfied: (i) at least four sites in $L(x)\cap
C(x)$ have Flag2 equal to 1, or (ii) the computed value of Flag1 at $x$ is 1
and at least four sites in $L(x)$ have Flag2 equal to 1, or (iii) $C(x)$ does
not exist and the computed value of Age at $x$ is divisible by 16, or (iv) at
least three sites in $N(x)\cap C(x)$ have Workspace.Flag2 equal to 1.”

p.22: Transition rule for Age: “If $C(x)$ exists and either the computed value
of Flag1 at $x$ is 0 or the computed value of Flag2 at $x$ is 1, take the
majority vote in $R(x)$.”

#### Issue

There are circular conditions to compute the Age and Flag2 fields. Condition
(iii) for computing Flag2 depends on a computed value of Age, and the computed
value of Age depends on the computed value of Flag2.

#### Modified rule

We choose to prioritize the age calculation, and change the Flag2 transition
rule to use it with an uncorrected value of age as:
if $\lnot C(x)$ and $age~{}mod~{}16==0$
then $conditionFlag2iii=true$.

#### Sample code (Java)

:

⬇

//If C(x) exists and either the computed

//value of Flag1 at x is 0 or the computed

//value of Flag2 at x is 1,

//take the majority vote in R(x)

/** computed [age, address] */

int[] ageAndAddr;

if(hasApparentColony &

(!nextState.flag1() | nextState.flag2())){

ageAndAddr = majorities(r_neigh);

nextState.setAge(ageAndAddr[0]);

nextState.setAddr(ageAndAddr[1]);

}else{

ageAndAddr = majorities(l_neigh);

nextState.setAge(ageAndAddr[0]);

nextState.setAddr(ageAndAddr[1]);

}

//increment age

nextState.ageThisCell();

//Replaced computed value of age

//by actual value of age to

//avoid circular conditions.

if(!hasApparentColony &

(this.getAge()%16)==0){

cond_flag2iii = true;

}

### Problem in transition rules for Workspace.Flag1 and Workspace.Flag2

#### Original rule

p.41: Transition rule for Workspace.Flag1 (not used in this paper, but
necessary for self-simulation) “(iii) the computed value of SimBit at the site
$(x-A(x))+(Q-3)$ is 1…”

#### Issue

The computed value of SimBit at that site is definitely unaccessible
information. Example:
if $Q=15$, $[0,1,x,3,4,5,6,7,8,9,10,11,12,13,14]$
and $computed\\_address\\_x=address\\_x=2$,
then site $(x-A(x))+(Q-3)=(2-2)+(15-3)=12$. Information in the 12th cell is
not accessible to cell $x$, which can only read from its five closest
neighbors. The same issue as Workspace.Flag1 occurs for Workspace.Flag2.

#### Modified Rule:

There does not seem to be an easy solution to this issue in Gray’s version.
Gacs’ work does not suffer from this issue, as the Workspace computation only
uses information from its own cell and 2 neighbors.

## References

  * de Sá and Maes, (1992) de Sá, P. G. and Maes, C. (1992).  The Gacs-Kurdyumov-Levin automaton revisited.  Journal of Statistical Physics, 67(3-4):507–522.
  * Francisco J. Varela, (1979) Francisco J. Varela (1979).  Principles of Biological Autonomy.  Elsevier North‐Holland, New York.
  * Gacs, (2001) Gacs, P. (2001).  Reliable Cellular Automata with Self-Organization.  Journal of Statistical Physics, 103(1-2):45–267.
  * Gray, (2001) Gray, L. F. (2001).  A Reader ’ s Guide to Gacs ’ s “ Positive Rates ” Paper.  pages 1–44.
  * Ikegami and Oka, (2013) Ikegami, T. and Oka, M. (2013).  Massive data flows: Self-organization of energy, material, and information flows.  In Proceedings of the 6th International Conference on Agents and Artificial Intelligence., pages 237–242.
  * von Neumann, (1966) von Neumann, J. (1966).  A Theory of Self-Reproduction.  University of Illinois Press.


# Klein-Gordon equations for energy-momentum of relativistic particle in
rapidity space

Robert M. Yamaleev
Joint Institute for Nuclear Research, LIT,Dubna, Russia
Universidad Nacional Autonoma de México, México.
<EMAIL_ADDRESS>
###### Abstract

The notion of four- rapidity is defined as a four-vector with one time-like
and three space-like coordinates. It is proved, the energy and momentum
defined in the space of four-rapidity obey Klein-Gordon equations constrained
by the classical trajectory of a relativistic particle. It is shown, for small
values of a proper mass influence of the constraint is weakened and the
classical motion gains features of a wave motion.

##  1 Introduction

The goal of this paper is to prove that energy and momentum defined in the
space of four- rapidity obey the differential equation similar the Klein-
Gordon equation. This method is based on a key-formula connecting the fraction
with an exponential function the argument of which is proportional the
difference between numerator and denominator. The energy and momentum can be
defined either as functions of the hyperbolic angle, or as functions of the
circular angle. In a covariant formulation we arrive to concept of rapidity
expressed as a four-vector the time-like part of which is presented by the
hyperbolic angle. The circular angle is extended into three quantities
functionally depending of the hyperbolic angle. It is shown, the energy and
momentum defined in a such way satisfy the Klein-Gordon equations written in
four-dimensional space with Minkowskii signature.

##  2 Key-formulae linking an exponential function with ratio of two
quantities

2.1 Parametrization of relativistic evolution with respect to hyperbolic
angle.

The dynamical variables of the relativistic particle, the energy $p_{0}$, the
momentum $p$ and the proper mass satisfy a mass-shell equation [1]

$p_{0}^{2}=p^{2}+m^{2},$ $None$

where the speed of light $c$ taken in unit $c=1$.

Our construction is based on the Key-formula which establishes some natural
interrelation between the ratio of a pair of quantities and an exponential
function.

Consider general form of the complex number given by unit ${\mathbf{e}}$
obeying the quadratic equation [2]:

${\mathbf{e}}^{2}-2p_{0}{\mathbf{e}}+p^{2}=0,$ $None$

with distinct positive real roots $x_{1},x_{2}$, so that,

$2p_{0}=x_{1}+x_{2},~{}~{}p^{2}=x_{1}x_{2}.$ $None$

The coefficients $p_{0},p^{2}$ are real numbers and $p_{0}^{2}>p^{2}$. The
solutions of equation (2.1) are defined by

$x_{1}=p_{0}+m,~{}~{}x_{2}=p_{0}-m,~{}~{}m=+\sqrt{p_{0}^{2}-p^{2}}.$ $None$

The normal form of the matrix obeying equation (2.1) is given by

$E=\left(\begin{array}[]{cc}0&-p^{2}\\\ 1&2p_{0}\end{array}\right).$ $None$

Consider an evolution generated by matrix $E$. Write the Euler formula

$\exp(E\phi)=E~{}g_{1}(\phi)+I~{}g_{0}(\phi),$ $None$

$I$-is a unit matrix. Diagonal form of this matrix equation consists of two
equations

$\exp(x_{2}\phi)=x_{2}~{}g_{1}(\phi)+g_{0}(\phi),~{}~{}\exp(x_{1}\phi)=x_{1}~{}g_{1}(\phi)+g_{0}(\phi).$
$None$

Notice, the modified cosine-sine functions $g_{0}(\phi),~{}g_{1}(\phi)$ depend
also of coefficients $p_{0},p^{2}$. Form the following ratio

$\exp((x_{2}-x_{1})\phi)=\frac{x_{2}~{}g_{1}(\phi)+g_{0}(\phi)}{x_{1}~{}g_{1}(\phi)+g_{0}(\phi)}=\frac{x_{2}+D}{x_{1}+D},$
$None$

where

$D=\frac{g_{0}(\phi)}{g_{1}(\phi)}.$ $None$

Let $\phi=\phi_{0}$ be the point where $g_{0}(\phi_{0})=0$. Then,

$\exp((x_{2}-x_{1})\phi_{0})=\frac{x_{2}}{x_{1}}.$ $None$

From (2.3) it follows

$m=\frac{1}{2}(x_{2}-x_{1}),~{}~{}p_{0}=\frac{1}{2}(x_{1}+x_{2}),~{}p^{2}=x_{1}x_{2}.$
$None$

Let $\phi=\phi_{0}$ be an initial point. Then according with (2.9) we conclude
that the roots and the coefficients of equation (2.1) $x_{1},x_{2}$ and
$p_{0},p$ are functions of $\phi_{0}$, however the difference $2m=x_{2}-x_{1}$
does not depend of $\phi_{0}$:

$x_{2}(\phi_{0})=\exp(m\phi_{0})\frac{m}{\sinh(m\phi_{0})},~{}x_{1}(\phi_{0})=\exp(-m\phi_{0})\frac{m}{\sinh(m\phi_{0})},$
$None$

Use these formulae in (2.9). Then,

$\exp((x_{2}-x_{1})\phi_{0})=\frac{p_{0}+m}{p_{0}-m}.$ $None$

Consequently, we have the following dependence $p_{0},p$ of $\phi_{0}$:

$p_{0}(\phi_{0})=m\coth(m\phi_{0}),~{}~{}p(\phi_{0})=\frac{m}{\sinh(m\phi_{0})}.$
$None$

In Refs.[3, 4, 5] formula (2.9) has been denominated as Key-formula.

2.1 Parametrization of evolution with respect to periodic angle.

Now, consider general complex algebra with generator ${\mathbf{e}}$ obeying
the quadratic equation

${\mathbf{e}}^{2}-2p{\mathbf{e}}+p_{0}^{2}=0.$ $None$

which differs from (2.1) by transposition of the coefficients $p_{0}$ and $p$.
Since $p_{0}^{2}>p^{2}$, two solutions of equation (2.14) are given by complex
conjugated numbers:

$y_{2}=p+im,~{}~{}y_{1}=p-im,~{}~{}m=+\sqrt{p_{0}^{2}-p^{2}}.$ $None$

Exponential function at solutions of this equation is defined by expansions

$\exp(y_{2}\theta)=y_{2}~{}f_{1}(\theta)+f_{0}(\theta),~{}~{}\exp(y_{1}\theta)=y_{1}~{}f_{1}(\theta)+f_{0}(\theta),$
$None$

where functions $f_{0}(\theta),f_{1}(\theta)$ depend of coefficients
$p,p_{0}^{2}$. Form the following ratio

$\exp(i2m\theta)=\frac{y_{2}~{}f_{1}(\theta)+f_{0}(\theta)}{y_{1}~{}f_{1}(\theta)+f_{0}(\theta)}=\frac{y_{2}+F}{y_{1}+F}$
$=\frac{p+im+F}{p-im+F},$ $None$

where

$F=\frac{f_{0}}{f_{1}}.$ $None$

Let $\theta=\theta_{0}$ be the initial point where $f_{0}(m\theta_{0})=0$.
Then, formula (2.17) is reduced into the following relationship

$\exp(i2m\theta_{0})=\frac{p(\theta_{0})+im}{p(\theta_{0})-im}.$ $None$

From this formula it follows

$p(\theta_{0})=m\cot(m\theta_{0}),~{}~{}p_{0}(\theta_{0})=m\frac{1}{\sin(m\theta_{0})}.$
$None$

The roots $y_{1},y_{2}$ also are functions of $\theta_{0}$,

$y_{1}=\exp(-im\theta_{0})\frac{m}{sin(m\theta_{0})},~{}y_{2}=\exp(im\theta_{0})\frac{m}{sin(m\theta_{0})}.$
$None$

Thus, we obtained two representations for the energy- momentum. The first one
is done via hyperbolic trigonometric functions,

$p_{0}(\phi)=m\coth(m\phi),~{}~{}~{}p(\phi)=m\frac{1}{\sinh(m\phi)},$ $None$

and the other one is defined by ordinary periodic trigonometric functions

$p_{0}(\theta)=m\frac{1}{\sin(m\theta)},~{}~{}~{}p(\chi)=m\cot(m\theta).$
$None$

In the both representations the arguments of the trigonometric functions are
proportional to mass $m$.

Since formulae (2.22) and (2.23) are related to same physical quantities, we
come to the next relationships between hyperbolic and periodic trigonometric
functions

$\tanh(m\phi)=\sin(m\theta),~{}\mbox{or},~{}\sinh(m\phi)=\tan(m\theta).$
$None$

Notice, when $m=0$, $\phi=\theta$.

The relationships between $\phi$ and $\theta$ can be presented also as follows

$\exp(m\phi)=\frac{1+\sin(m\theta)}{1-\sin(m\theta)}=\frac{1+\tan\frac{m\theta}{2}}{1-\tan\frac{m\theta}{2}},$
$None$
$\exp(im\theta)=\frac{1+i\sinh(m\phi)}{1-i\sinh(m\phi)}=\frac{1+i\tanh\frac{m\phi}{2}}{1-i\tanh\frac{m\phi}{2}}.$
$None$

Also, it is important to notice that the differential relationship between
variables $\theta$ and $\phi$ coincides with the definition of the velocity:

$\frac{d\theta}{d\phi}=\frac{dr}{dt}=\frac{p}{p_{0}}.$ $None$

These formulae express a general interrelation between periodic and hyperbolic
trigonometry. Let $\triangle ABC$ be a right-angled triangle with right angle
at $C$. Denote the sides by $a=BC,b=AC$, the hypotenuse $AB$ by $c$. If we
make a geometrical motion by moving the point $A$ along line $AC$, then this
motion changes the sides $c,b$, but remains invariant the side $a$. The angle
$A$ can be used in quality of parameter this evolution. In accordance with the
Key-formula (2.19) we write

$\frac{b+ia}{b-ia}=\exp(2ia\theta),~{}~{}~{}b=a\cot(a\theta),~{}~{}c=a\frac{1}{\sin(a\theta)}.$
$None$

It is easily seen that $a\theta=A$. On the other hand, In accordance with Key-
formula  (2.12) we have

$\frac{c+a}{c-a}=\exp(2a\phi),~{}~{}c=a\coth(a\phi),~{}~{}b=\frac{a}{\sinh(a\phi)}.$
$None$

##  3 Pythagoras theorem and two dimensional Fermi-like oscillator

For the sake of convenience in this section let us use for derivatives short
notations

$\frac{d}{d\phi}=d,~{}~{}~{}\frac{d}{d\theta}=\partial.$

Then differentiating formula (2.22) and (2.23) we come to the following system
of differential equations

$dp_{0}=-p^{2},~{}~{}~{}dp=-pp_{0},~{}~{}~{}\partial
p_{0}=-pp_{0},~{}~{}~{}\partial p=-p^{2}_{0}.$ $None$

The operators $d$ and $\partial$ do not commute. Introduce two dimensional
vector of a state by

$\Phi(p_{0},p)=\left(\begin{array}[]{c}p_{0}\\\ p\end{array}\right).$ $None$

Calculate actions of the operators $d^{2}-\partial^{2}$ and
$d\partial-\partial d$ on this vector:

$(d^{2}-\partial^{2})\Phi(p_{0},p)=m^{2}~{}\Phi(p_{0},p),$ $None$ $(\partial
d-d\partial)\Phi(p_{0},p)=m^{2}~{}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\Phi(p_{0},p).$ $None$

Introduce operators

$a^{-}=d-\partial,~{}~{}a^{+}=d+\partial,$

with following commutation and anti-commutation rules

$\frac{1}{2}(a^{-}a^{+}+a^{+}a^{-})=m^{2}~{}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),$ $None$
$a^{-}a^{+}-a^{+}a^{-}=2m^{2}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ $None$

It is seen, we deal with some kind of two dimensional Fermi-like oscillator
with Hamilton operator

$H=\frac{1}{2}(a^{-}a^{+}-a^{+}a^{-}),$ $None$

and with anti-commutation relation given by

$(a^{-}a^{+}+a^{+}a^{-})=2m^{2},$ $None$

acting on the state $\Phi_{0}(p_{0},p)$. The Hamilton operator possesses with
two eigenvalues

$H\Phi_{n}=E_{n}\Phi_{n},~{}n=1,2,$ $None$

where

$E_{1}=+m^{2},~{}~{}E_{2}=-m^{2},$

so that,

$a^{+}\Phi_{1}=\Phi_{2},~{}~{}a^{-}\Phi_{2}=\Phi_{1},$

and, due to anti-commutation relations (3.8),

$a^{\pm}H+Ha^{\pm}=0.$

##  4  Klein-Gordon equations for energy-momentum of classical relativistic
particle in the space of rapidity

It is seen that equation (3.3) is nothing else than two dimensional Klein-
Gordon equation. Comparing this equation with two dimensional Klein-Gordon
equation written in terms of space-time coordinates we come to conclusion that
the parameter $\phi$\- is a time-like parameter, whereas the parameter
$\theta$ is an analogue of a space coordinate. In order to pass to the Klein
-Gordon equation in four-dimensional Minkowski space with signature $(+---)$
we shall extend the parameter $\theta$ till to three dimensional vector. In
this way we arrive to covariant formulation of evolution equations.

The momentum is a spatial part of the four-vector energy-momentum with
components $p_{k},k=1,2,3$. Now, instead of $\phi$ we will use the letter
$\rho_{0}$, and $\theta$ has to be replaced by spatial part of four-vector of
rapidity containing components $\rho_{1},\rho_{2},\rho_{3}$.

In these variables the evolution equations have to be written in a Lorentz-
covariant form. The evolution equations we shall extend as follows. The single
variable $p$ is replaced by the components of three-vector of momentum,
$p_{k},k=1,2,3$. The square $p^{2}$ means $p^{2}=-p^{k}p_{k}$. In this way we
arrive to the following set of equations

$(a)~{}~{}~{}d^{0}~{}p_{0}=~{}-p^{k}p_{k},~{}~{}~{}(b)~{}~{}d^{0}~{}p_{k}=p_{k}p^{0},~{}~{}k=1,2,3.$
$None$
$(a)~{}~{}~{}\partial_{k}~{}p_{0}=p_{k}p_{0},~{}~{}~{}(b)~{}~{}~{}~{}\partial^{k}~{}p_{k}=-p^{2}_{0}.$
$None$

Hereafter we use the following notations for derivatives

$\partial^{k}=\frac{\partial}{\partial\rho_{k}},~{}~{}~{}d^{0}=\frac{d}{d\rho_{0}},$

and adopt, so-called, a summation convention, according to which any repeated
index in one term, once up, once down, implies summation over all its values.

Remember, however, that there exist some functional dependence between
$\rho_{0}$ and $\rho_{k},k=1,2,3$, so that the spatial variables are functions
of the time-like parameter, i.e., $\rho_{k}=\rho_{k}(\rho_{0}),k=1,2,3.$ This
means, the full derivative with respect to $\rho_{0}$ is

$d^{0}p_{0}=-p^{k}p_{k}=~{}\frac{d\rho_{k}}{d\rho_{0}}\frac{\partial}{\partial\rho_{k}}p_{0}.$
$None$

On making use of equations (4.1)-(4.2), we get

$d^{0}p_{0}=p^{2}=-p^{k}p_{k}=~{}p_{k}\frac{d\rho_{k}}{d\rho_{0}}~{}p_{0}.$
$None$

In order to provide this equality we have to take

$~{}p_{k}\frac{d\rho_{k}}{d\rho_{0}}=\frac{p^{2}}{p_{0}}.$ $None$

Our purpose is to complete the evolution equations (4.1)-(4.2) with an
equation containing the following derivative

$~{}~{}\frac{\partial}{\partial\rho_{n}}p_{k}.$

For that reason let us re-write equation (4.1b) as follows

$~{}\frac{d}{d\rho_{0}}~{}p_{k}=~{}\frac{d\rho_{n}}{d\rho_{0}}\frac{\partial}{\partial\rho_{n}}~{}p_{k}=p_{k}p_{0}.$

In order to provide this equality we have to suppose that

$~{}\frac{\partial}{\partial\rho_{n}}p_{k}=p_{k}p^{n}\frac{p_{0}^{2}}{p^{2}}.$
$None$

One may easily check that formula (4.6) is in accordance with (4.1) and (4.2).

Equations with second order derivatives.

Firstly, calculate the second order derivatives of $p_{0}$ and $p$ with
respect to time-like variable $\rho_{0}$. We have,

$\frac{d}{d\rho^{0}}\frac{d}{d\rho_{0}}p_{0}=-2p^{k}p_{k}p_{0}=2p^{2}p_{0}.$
$None$

Secondly, calculate action of the Laplace operator on $p_{0}$. Define the
Laplace operator by

$\Delta=\frac{\partial}{\partial\rho_{k}}\frac{\partial}{\partial\rho^{k}}.$

By taking into account (4.2a) we obtain

$\partial^{k}\partial_{k}~{}p_{0}=-p_{0}^{2}p_{0}+p^{k}p_{k}p_{0}=-p_{0}^{3}-p^{2}p_{0}.$
$None$

Joining this equation with (4.7) we come to Klein-Gordon equation for $p_{0}$:

$d^{0}d_{0}~{}p_{0}+\partial^{k}\partial_{k}~{}p_{0}=-m^{2}~{}p_{0}.$ $None$

Now calculate action of operator $\Delta$ on $p_{k}$ by using formulae (4.6)
and (4.2b).

$\partial^{n}\partial_{n}~{}p_{k}=\partial^{n}~{}(~{}p_{k}p_{n}\frac{p_{0}^{2}}{p^{2}}~{})$
$=(\partial^{n}~{}(~{}p_{k}p_{n}))~{}\frac{p_{0}^{2}}{p^{2}}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=\frac{p_{0}^{2}}{p^{2}}~{}(p^{n}p_{n})~{}\frac{p_{0}^{2}}{p^{2}}+p_{k}(\partial^{n}p_{n})~{}\frac{p_{0}^{2}}{p^{2}}~{}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}~{}+p_{k}p_{n}\partial^{n}(~{}\frac{p_{0}^{2}}{p^{2}})$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}~{}~{}-2p_{k}p^{n}\frac{p_{n}p_{0}^{2}m^{2}}{p^{4}}$
$=-2\frac{p_{0}^{4}}{p^{2}}p_{k}+~{}~{}2p_{k}\frac{p_{0}^{2}m^{2}}{p^{2}}$
$=(~{}~{}-2\frac{p_{0}^{4}}{p^{2}}p_{k}+~{}2p_{k}\frac{p_{0}^{4}}{p^{2}}~{}~{})=-2p_{k}p_{0}^{2}.$

Joining this result with

$\frac{d}{d\rho^{0}}\frac{d}{d\rho_{0}}~{}p_{k}=p_{k}p_{0}^{2}-p_{k}p^{n}p_{n}=p_{k}p_{0}^{2}+p_{k}p^{2},$

we come to analogue of Klein-Gordon equation for $p_{k}$:

$\Delta~{}p_{k}+d^{0}d_{0}~{}p_{k}~{}=-m^{2}~{}p_{k}.$

Comparison with Klein-Gordon equation used in relativistic quantum mechanics.

From formula (4.5)

$~{}p_{k}\frac{d\rho^{k}}{d\rho_{0}}=\frac{p^{2}}{p_{0}},$

we may conclude that

$\frac{d\rho^{k}}{d\rho_{0}}=v^{k}+M^{kl}p_{l},$

where

$v^{k}=\frac{dx^{k}}{dx^{0}}$

is the velocity with respect to coordinate time, $M^{kl}=-M^{lk}$ is an
arbitrary anti-symmetric tensor.

In the relativistic quantum mechanics the Klein-Gordon equation is obtained
simply by using some conventional receipt according to which components of
four-momentum in the mass-shell equation are replaced by corresponding
differential operators as follows [6]

$p_{k}=-i\hbar\frac{\partial}{\partial
x^{k}},~{}p_{0}=i\hbar\frac{\partial}{\partial x^{0}}.$

So, we come to the following correspondence

$\hbar\rho^{\mu}\Rightarrow
x^{\mu},~{}~{}\frac{\partial}{\partial\rho^{\mu}}=\hbar\frac{\partial}{\partial
x^{\mu}}.$

##  5 Concluding remarks

1\. We considered two ways of description an evolution constrained by
Pythagoras formula. The first one is given by the hyperbolic angle, and the
second one, by the periodic angle. The both angles are proportional to the
fixed side of the right angled triangle. In the case of relativistic
mechanics, the hypotenuse is the energy, the fixed side is the mass and the
moving side is the momentum.

2\. The derivative of the periodic angle with respect to the hyperbolic angle
is equal to a ratio of the moving side to the hypotenuse, in the case of
relativistic mechanics, this ratio is the velocity

$\frac{d\theta}{d\phi}=\frac{p}{p_{0}}=\frac{v}{c}.$

3\. This relationship prompts us to conclude that the hyperbolic angle $\phi$
is the time-like parameter, whereas the periodic angle $\theta$ is the space-
like parameter.

4\. The evolution equations admit an extension to the case of three (or more)
dimensions, however, in this case we could not find an explicit expression for
the momenta.

5\. Near the point where the fixed side of the triangle ( mass) becomes
infinitesimal and according to the Klein-Gordon equation it is conjectured
that this motion will display features of the wave motion.

## References

  * [1] A.O. Barut, Electrodynamics and classical theory of fields and particles (Dover Publ., New York, 1980).
  * [2] R.M. Yamaleev, J. Math. Anal. Appl. 340, 1046 (2008),doi:10.1016/j.jmaa.2007.09.018.
  * [3] R.M. Yamaleev, Phys. Atom. Nucl., 74, 1775 (2011).
  * [4] R.M. Yamaleev J. Mod. Phys., 2, 849 (2011).
  * [5] R.M. Yamaleev, hep-th/0905.0234.
  * [6] L.H. Ryder, Quantum Field Theory (Cambridge Univ. Press., Cambridge, 1985).