∎

11institutetext: Anadijiban Das 22institutetext: Department of Mathematics,
Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
22email<EMAIL_ADDRESS>33institutetext: Rupak Chatterjee 44institutetext: Center
for Quantum Science and Engineering
Department of Physics, Stevens Institute of Technology, Castle Point on the
Hudson, Hoboken, NJ 07030, USA
44email<EMAIL_ADDRESS>
# Discrete phase space and continuous time relativistic quantum mechanics I:
Planck oscillators and closed string-like circular orbits

Anadijiban Das  Rupak Chatterjee

(Received: date / Accepted: date)

###### Abstract

The discrete phase space continuous time representation of relativistic
quantum mechanics involving a characteristic length $l$ is investigated.
Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for
most sections of the paper. The energy eigenvalue problem for the Planck
oscillator is solved exactly in this framework. Discrete concircular orbits of
constant energy are shown to be circles $S^{1}_{n}$ of radii
$2E_{n}=\sqrt{2n+1}$ within the discrete $(1+1)$-dimensional phase plane.
Moreover, the time evolution of these orbits sweep out world-sheet like
geometrical entities $S^{1}_{n}\times\mathbb{R}\subset\mathbb{R}^{2}$ and
therefore appear as closed string-like geometrical configurations. The
physical interpretation for these discrete orbits in phase space as
degenerate, string-like phase cells is shown in a mathematically rigorous way.
The existence of these closed concircular orbits in the arena of discrete
phase space quantum mechanics, known for the non-singular nature of lower
order expansion $S^{\\#}$ matrix terms, was known to exist but has not been
fully explored until now. Finally, the discrete partial difference-
differential Klein-Gordon equation is shown to be invariant under the
continuous inhomogeneous orthogonal group $\mathcal{I}\left[O(3,1)\right]$.

###### Keywords:

Discrete phase space Planck oscillators closed string-like configurations

###### pacs:

11.10Ef 11.15Ha 02.70Bf 03.65Fd

##  1 Introduction

There have been many attempts in the past to quantize space-time in order to
remove the divergences found in quantum field theory. In 1960, a divergence
free quantum field theory was presented in the arena of lattice space-time
DasI . Therein, the mathematics of partial difference field equations was used
but unfortunately relativistic covariance was not yet possible.

Traditionally, quantum mechanics has been formulated in a space-time continuum
Greiner . It is well known that quantum mechanics may be represented in terms
of phase space variables within a phase space continuum Weyl ; WignerI  often
involving a characteristic length scale $l>0$. The analog of the “wave
function” in this formulation is the Wigner function being a probability
distribution function in phase space. Furthemore, a charactersitic length
scale has also been utilized by Heisenberg in his pioneering work on matrix
mechanics Heisenberg . After many decades, partial difference operators
involving a characteristic length scale $l>0$ were utilized in an exact
quantum mechanical wave mechanics formulation that maintained exact
relativistic covariance DasII ; DasIII . Following this was the formulation of
the discrete space and continuous time $S^{\\#}$-matrix theory of interacting
fields DasIV ; DasV ; DasVI  whereby the special relativistic
$S^{\\#}$-matrices were found to be non-singular for many lower order
expansion terms.

In the year 2015, three-dimensional Planck oscillators DasVII  were introduced
in a $(2+2+2)$-dimensional phase space continuum (i.e.
$\\{(p_{x},q_{x}),(p_{y},q_{y}),\cdots\\}$) formulation of quantum field
theory. In this paper, we concentrate on partial difference equations in a
discrete phase space being a subset of the continuous phase space. In the wave
mechanical version of the Planck oscillator, discrete orbits
$S^{1}_{n^{1}}\times S^{1}_{n^{3}}\times S^{1}_{n^{3}}$ emerge inside the
$(2+2+2)$-dimensional phase space where $S^{1}_{n}$ indicates a one-
dimensional circle of radius $\sqrt{2n+1}$ with its center at the origin of a
two-dimensional phase plane. We study these concircular orbits and the direct
products of these circles in sections 3,4, and 5. Relativistic second
quantized waves fields are shown to have quanta inside the $(2+2+2)$ discrete
phase space characterized by these direct product orbits. Moreover, use of
circular hyper-cylinders $S^{1}_{n^{1}}\times S^{1}_{n^{3}}\times
S^{1}_{n^{3}}\times\mathbb{R}$ of discrete phase space and continuous time are
explored in sections 5, 6, and 7.

In a previous work DasVI , quantum electrodynamics involving a
$S^{\\#}$-matrix was derived in the arena of $[(2+2+2)+1]$-dimensional
discrete phase space and continuous time. Many of the lower order expansion
terms of the $S^{\\#}$-matrix turned out to be non-singular as opposed to the
usual case of divergences seen in the traditional continuum methods of quantum
field theory. Furthermore, a physically verifiable non-singular Coulomb
potential was derived from a discrete quantum electrodynamics model DasVIII .

The circle $S^{1}_{n}$ inside the $(1+1)$ dimensional discrete phase space has
been shown to resemble a string-like object DasIX ; Zwiebach ; Polchinski
while the direct product $S^{1}_{n^{1}}\times S^{1}_{n^{3}}\times
S^{1}_{n^{3}}$ inside the $(2+2+2)$-dimensional discrete phase space resembles
a hyper-spherical torus Zwiebach ; Polchinski . This topic will be treated in
greater detail in a follow up (part II) to the current work.

The discrete phase space and continuous time model can incorporate the second
quantized form of both free and interacting relativistic fields. Similar
formulations can also include Einstein’s linearized gravitational theory
Einstein .

Finally, in section 7, we consider world sheet like geometrical figures given
by the Cartesian direct product $S^{1}_{n^{1}}\times S^{1}_{n^{3}}\times
S^{1}_{n^{3}}\times\mathbb{R}$ in the arena of $[(2+2+2)+1]$-dimensional
discrete phase space and continuous time. Specifically, we discuss the
relativistic Klein-Gordon wave equation in such a space time. The relativistic
invariance of the partial difference-differential discretization of the Klein-
Gordon equation is proved rigorously. The Klein-Gordon wave function will be
denoted as $\phi(n^{1},n^{2},n^{3},t)=:\phi(\mathbf{n},t)$. The second
quantized version of this wave function will be denoted as
$\Phi(\mathbf{n},t)$. We assume that the massive particle quanta of
$\Phi(\mathbf{n},t)$ can cohabit with the quantized Planck oscillators inside
the geometrical configuration of $S^{1}_{n^{1}}\times S^{1}_{n^{3}}\times
S^{1}_{n^{3}}\times\mathbb{R}$ temporarily or forever.

##  2 Classical and Quantum Planck Oscillators in a (1+1)-dimensional Phase
Plane Continuum $\mathbb{R}^{2}$

We will explicitly use physical units
$\hbar=\left[\dfrac{ML^{2}}{T}\right],c=\left[\dfrac{L}{T}\right]$, and
$l={L}$ here where the characteristic length $l>0$ is to be set by future
experimental results. However, we expect $l$ to be close to the Planck length
$\sqrt{\dfrac{\hbar G}{c^{3}}}$. The classical Planck oscillator DasVII  in
the $(1+1)$-dimensional phase plane continuum is provided by the physically
dimensionless Hamiltonian

$H(q,p):=\left(\dfrac{1}{2}\right)\left[\left(\dfrac{l}{\hbar}\right)^{2}p^{2}+\left(\dfrac{1}{l}\right)^{2}q^{2}\right]=\left[\dfrac{E}{\hbar\nu}\right]$
(1)

where $E$ and $\nu$ stand for the total energy and circular frequency of the
classical Planck oscillator respectively.

The graph of the classical motion in the phase plane $\mathbb{R}^{2}$ is
depicted by the ellipse in figure 1. Note that in the limit $l\rightarrow
0^{+}$, the ellipse collapses into the $p$-axis.

Figure 1: Ellipse Representing the Classical Trajectory of the Planck
Oscillator

The elliptical world-tube Synge  in figure 2 represents the Planck oscillator
in the $[(1+1)+1]$-dimensional state space ($(p,q,t)$) Lanczos .

Figure 2: Elliptical Vertical Cylinder inside the $[(1+1)+1]$-dimensional
State Space of the Planck Oscillator

In figure 1, the elliptical trajectory resembles a closed string-like
configuration. On the other hand, the elliptical vertical cylinder in figure 2
resembles an open world sheet like object Zwiebach ; Polchinski  inside the
$[(1+1)+1]$-dimensional phase plane plus time continuum
$\mathbb{R}^{2}\times\mathbb{R}$.

The quantum mechanical solution of the Planck oscillator is straightforward
and is as follows. The state vector $\overrightarrow{\mathbf{\Psi}}$ and
linear operators $\mathbf{Q}$ and $\mathbf{P}$ in a separable Hilbert space
Lorch  renders the classical Hamiltonian (1) into the quantum eigenvalue
problem

$H(\mathbf{Q},\mathbf{P})\overrightarrow{\mathbf{\Psi}}=\left(\dfrac{1}{2}\right)\left[\left(\dfrac{l}{\hbar}\right)^{2}\mathbf{P}^{2}+\left(\dfrac{1}{l}\right)^{2}\mathbf{Q}^{2}\right]\overrightarrow{\mathbf{\Psi}}=\left[\dfrac{E}{\hbar\nu}\right]\overrightarrow{\mathbf{\Psi}}.$
(2)

The well-known Schrödinger representation Schiff  is furnished by

$\overrightarrow{\mathbf{\Psi}}:=\psi(q),\,\,\,\mathbf{Q}\overrightarrow{\mathbf{\Psi}}=q\psi(q),\,\,\,\mathbf{P}\overrightarrow{\mathbf{\Psi}}:=-i\hbar\dfrac{d}{dq}\psi(q).$
(3)

Using the above, the Schrödinger wave equation follows as

$\dfrac{d^{2}\psi(q)}{dq^{2}}+\left[\left(\dfrac{2E}{\hbar\nu}\right)-\left(\dfrac{q}{l}\right)^{2}\right]\psi(q)=0.$
(4)

with a solution set given by Schiff

$\begin{array}[]{c}E_{N}=\left(N+\dfrac{1}{2}\right)\hbar\nu,\,\,\,N\in\\{0,1,2,...\\};\\\
\\\
\psi_{N}(q/l)=\dfrac{\exp[(-q/l)^{2}/2]}{\pi^{1/4}2^{N/2}\sqrt{N!l}}H_{N}(q/l),\\\
\\\
\overrightarrow{\mathbf{\Psi}}_{N}\cdot\overrightarrow{\mathbf{\Psi}}_{M}=\int_{\mathbb{R}}\bar{\psi}_{N}(q/l){\psi}_{M}(q/l)dq=\delta_{NM}.\end{array}$
(5)

where $H_{N}(q/l)$ are Hermite polynomials Schiff ; Gradshteyn . Note that the
last expression in (5) is physically dimensionless.

##  3 Planck Oscillators in a new coordinate system of the (1+1)-dimensional
Phase Plane Continuum $\mathbb{R}^{2}$

Let us introduce a new coordinate system of physically dimensionless
quantities for the $(1+1)$-dimensional phase plane continuum in
$\mathbb{R}^{2}$ as follows (hatted quantities refer to transformed versions
of the corresponding unhatted quantities)

$\begin{array}[]{c}y=\left(\dfrac{1}{2}\right)\left(\dfrac{q}{l}\right)^{2}>0,\\\
\\\ \left(\dfrac{q}{l}\right)=+\sqrt{2y}>0.\end{array}$ (6)

This leads to the following transformation of (3),

$\begin{array}[]{c}\overrightarrow{\mathbf{\Psi}}=\hat{\psi}(y),\\\ \\\
\mathbf{Q}\overrightarrow{\mathbf{\Psi}}=(\sqrt{2y}l)\hat{\psi}(y),\\\ \\\
\mathbf{P}\overrightarrow{\mathbf{\Psi}}:=-i\left(\dfrac{\hbar}{l}\right)\sqrt{2y}\dfrac{d}{dy}\hat{\psi}(y)\\\
\\\
=-i\left(\dfrac{\hbar}{l}\right)\sqrt{\dfrac{y}{2}}\left\\{\displaystyle{\lim_{a\to
0_{+}}}\left[\dfrac{\hat{\psi}(y+a)-\hat{\psi}(y)}{a}\right]+\displaystyle{\lim_{a\to
0_{-}}}\left[\dfrac{\hat{\psi}(y+a)-\hat{\psi}(y)}{a}\right]\right\\}.\end{array}$
(7)

The Schrödinger wave equation is furnished by

$\sqrt{2y}\dfrac{d}{dy}\left[\sqrt{2y}\dfrac{d}{dy}\hat{\psi}(y)\right]+\left[\left(\dfrac{2E}{\hbar\nu}\right)-2y\right]\hat{\psi}(y)=0,$
(8)

with eigenvalue solutions of

$\begin{array}[]{c}E_{N}=\left(N+\dfrac{1}{2}\right)\hbar\nu,\\\ \\\
\hat{\psi}_{N}(y)=\dfrac{e^{-y}H_{N}(\sqrt{2y})}{\pi^{1/4}2^{N/2}\sqrt{N!}}.\end{array}$
(9)

The real valued wave function $\hat{\psi}_{N}(y)$ is physically dimensionless
as desired.

##  4 Quantum Planck Oscillators in a (1+1)-dimensional Discrete Phase Plane

Let us begin by defining various finite difference operations DasIV ; DasVII ;
DasIX  on a real or complex-valued wave function $\phi(n)$ as

$\begin{array}[]{c}\Delta\phi(n):=\phi(n+1)-\phi(n),\\\ \\\
\Delta^{{}^{\prime}}\phi(n):=\phi(n)-\phi(n-1),\\\ \\\
\Delta^{o}\phi(n):=\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n+1}\phi(n+1)+\sqrt{n}\phi(n-1)\right],\\\
\\\
\Delta^{\\#}\phi(n):=\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n+1}\phi(n+1)-\sqrt{n}\phi(n-1)\right],\\\
\\\ n\in\\{0,1,2,...\\}.\end{array}$ (10)

There exists many more variations of difference operators (see Jordan ).
$\Delta^{\\#}$ has beed used several times before DasIV ; DasVII ; DasVIII ;
DasIX  whereas $\Delta^{o}$ may be found in DasVII ; DasIX . Mathematicians
generally use finite difference operators for numerical analysis. They are
generally not concerned with the physical dimensions of various mathematical
entities. Yet, for our purposes, we must be careful at being able to return to
physical dimensions for experimental verification once we go dimensionless.

Consider a scale $l>0$ without concern for the physical dimensionality such
that the mathematical analysis reads as follows,

$\begin{array}[]{c}y_{n}:=nl,\,\,\,y_{n+1}-y_{n}=l,\\\ \\\
\chi(nl)\equiv\chi(y_{n}),\\\ \\\ \chi(\dfrac{y_{n}}{l})=:\phi(n),\\\ \\\
\Delta_{l}\left[\chi(\dfrac{y_{n}}{l})\right]=:\Delta\phi(n),\\\ \\\
\Delta^{{}^{\prime}}_{l}\left[\chi(\dfrac{y_{n}}{l})\right]=:\Delta^{{}^{\prime}}\phi(n),\\\
\\\
\Delta^{o}_{l}\left[\chi(\dfrac{y_{n}}{l})\right]=:\left(\dfrac{l}{2}\right)^{1/2}\left[\sqrt{(n+1)l}\phi(n+1)+\sqrt{nl}\phi(n-1)\right]=l\Delta^{o}\phi(n),\\\
\\\
(-i\hbar)\Delta^{\\#}_{l}\left[\chi(\dfrac{y_{n}}{l})\right]=:(-i\hbar)\left(\dfrac{1}{\sqrt{2}l^{3/2}}\right)\left[\sqrt{n+1}\phi(n+1)-\sqrt{n}\phi(n-1)\right]\\\
\\\ =(-i\hbar)\left(\dfrac{1}{l}\right)\Delta^{\\#}\phi(n).\end{array}$ (11)

Returning to the physically admissible first quanitized Planck oscillator
model, let the state vector $\overrightarrow{\mathbf{\Psi}}$ and linear
operators $\mathbf{Q}$ and $\mathbf{P}$ be defined in the following manner:

$\begin{array}[]{c}\overrightarrow{\mathbf{\Psi}}:=\chi(\dfrac{y_{n}}{l}):=\phi(n),\\\
\\\
\mathbf{Q}\overrightarrow{\mathbf{\Psi}}:=\Delta^{o}_{l}\chi(\dfrac{y_{n}}{l})=l\Delta^{o}\phi(n),\\\
\\\
\mathbf{P}\overrightarrow{\mathbf{\Psi}}:=-i\hbar\Delta^{\\#}_{l}\chi(\dfrac{y_{n}}{l})=\left(\dfrac{-i\hbar}{l}\right)\Delta^{\\#}\phi(n).\end{array}$
(12)

The first equation of (12) is physically dimensionless whereas the second and
third equations are of physical dimensions of length and momentum
respectively. From these, we can directly compute the fundamental commutation
relation,

$[\mathbf{P},\mathbf{Q}]\overrightarrow{\mathbf{\Psi}}:=(\mathbf{P}\mathbf{Q}-\mathbf{Q}\mathbf{P})\overrightarrow{\mathbf{\Psi}}=(-i\hbar)(\Delta^{\\#}\Delta^{o}-\Delta^{o}\Delta^{\\#})\phi(n)=-i\hbar\phi(n)=-i\hbar\overrightarrow{\mathbf{\Psi}}.$
(13)

The quantum Hamiltonian equation (2) reduces to

$\begin{array}[]{c}\left(\dfrac{1}{2}\right)\left[\left(\dfrac{l}{\hbar}\right)^{2}\mathbf{P}^{2}+\left(\dfrac{1}{l}\right)^{2}\mathbf{Q}^{2}\right]\overrightarrow{\mathbf{\Psi}}=\left[\dfrac{E}{\hbar\nu}\right]\overrightarrow{\mathbf{\Psi}},\\\
or\\\
\left[-(\Delta^{\\#})^{2}+(\Delta^{o})^{2}\right]\phi(n)=\left(\dfrac{2E}{\hbar\nu}\right)\phi(n).\end{array}$
(14)

The corresponding eigenvalue problem is exactly solvable with the following
solution

$\begin{array}[]{c}E_{N}=\left(N+\dfrac{1}{2}\right)\hbar\nu,\,\,\,E_{0}=\dfrac{\hbar\nu}{2},\\\
\\\ N\in\\{0,1,2,...\\},\\\ \\\
\overrightarrow{\mathbf{\Psi}}_{N}=\phi_{N}(n)=\delta_{Nn},\\\ \\\
\overrightarrow{\mathbf{\Psi}}_{N}^{\dagger}\cdot\overrightarrow{\mathbf{\Psi}}_{M}:=\displaystyle{\sum_{n=0}^{\infty}}\delta_{Nn}\delta_{Mn}=\delta_{NM},\\\
\\\ ||\overrightarrow{\mathbf{\Psi}}_{N}||^{2}=1.\end{array}$ (15)

Equations (13) and (15) prove that the present finite difference
representations of quantum mechanics are exact (with the simplest possible
eigenfunction ever of $\phi_{N}(n)=\delta_{Nn}$). By (14) and (15), the
discretization of the phase plane via the relation
$\left(\dfrac{1}{2}\right)\left[\left(\dfrac{l}{\hbar}\right)^{2}p^{2}+\left(\dfrac{1}{l}\right)^{2}q^{2}\right]=\left(n+\dfrac{1}{2}\right),\,\,\,n\in\\{0,1,2,...\\}$
is achieved. This relation represents denumerably infinite numbers of confocal
ellipses in the background of the continuous $q-p$ phase plane DasVII .
Therefore, the particle’s accessible phase space is discrete in this case
(’phase cell’) in such a way that the relation between $q$ and $p$ is integer
valued.

Finally, from (11) and (12), we have

$\begin{array}[]{c}\mathbf{Q}\overrightarrow{\mathbf{\Psi}}:=\sqrt{2y_{n}l}\,\,\chi\left(\dfrac{y_{n}}{l}\right)+\left[\Delta\left(\sqrt{\dfrac{y_{n}l}{2}}\,\chi\left(\dfrac{y_{n}}{l}\right)\right)-\sqrt{\dfrac{y_{n}l}{2}}\Delta^{{}^{\prime}}\left(\chi\left(\dfrac{y_{n}}{l}\right)\right)\right],\\\
\\\
\mathbf{P}\overrightarrow{\mathbf{\Psi}}:=-i\hbar\left(\dfrac{1}{l}\right)\left[\Delta\left(\sqrt{\dfrac{y_{n}l}{2}}\,\chi\left(\dfrac{y_{n}}{l}\right)\right)+\sqrt{\dfrac{y_{n}l}{2}}\Delta^{{}^{\prime}}\left(\chi\left(\dfrac{y_{n}}{l}\right)\right)\right].\end{array}$
(16)

With respect to the first quantized Planck oscillator equations in the unusual
coordinate system of section 3, a mathematician conversed in standard
numerical analysis Jordan ; Pettofrezzo  will the finite difference equations
involving operators $\Delta$ and $\Delta^{{}^{\prime}}$ only to obtain
approximate behaviors of the unknown functions
$\phi(n)\equiv\chi\left(\dfrac{y_{n}}{l}\right)$. Yet, our representation of
quantum mechanics using the operators $\Delta^{\\#}$ and $\Delta^{o}$ provide
the exact quantum conditions (13)-(15). Note also the striking similarity
between the differential relations found in (7) and the difference relations
found above in (16).

##  5 String-like Phase Plane Orbits and Fundamental Units

The fundamental physical units in this section will be chosen to be
$\hbar=1\left[\dfrac{ML^{2}}{T}\right],c=1\left[\dfrac{L}{T}\right],l=1[L]$.
In these units, the discrete phase plane ’phase cell’ is denoted by
$S^{1}_{n}$. The mathematical equations of the previous sections reduce to
much simpler forms if expressed in these fundamental units. The one-
dimensional circular orbits $S^{1}_{N}$ with discrete energy $2E_{N}=(2N+1)$
are expressed as, following (15),

$\begin{array}[]{c}S^{1}_{N}:=\left\\{(q,p)\in\mathbb{R}^{2}:q^{2}+p^{2}=2N+1\in\mathbb{R}\right\\},\\\
\\\ S^{1}_{n}\equiv S^{1}_{N},\\\ \\\ N\in\\{0,1,2,\cdots\\}.\end{array}$ (17)

These circular orbits are depicted in figure 3.

Figure 3: The String-like Closed Circular Orbit $S^{1}_{N}$ for the First
Quantized Planck Oscillator

The corresponding $(1+1)$-dimensional world-sheet like circular cylinder
Zwiebach  in the $[(1+1)+1]$-dimensional discrete phase plane and continuous
time is exhibited in figure 4.

Figure 4: The World-Sheet like Circular Cylinder $S^{1}_{N}\times\mathbb{R}$
embedded inside the $[(1+1)+1]$-dimensional State Space (discrete phase plane
and continuous time)

These two figures may be compared and contrasted to those of standard string
theory Zwiebach ; Polchinski . The discrete one-dimensional orbit $S^{1}_{N}$
of figure 3 closely resembles closed strings. There exists a denumerable
infinite number of concircular orbits of Planck oscillators in the discrete
phase plane. Furthermore, the hollow circular cylinders in state space
$S^{1}_{N}\times\mathbb{R}$ resemble the two-dimensional world sheets of usual
string theory. Clearly, a denumerable infinite number of vertical co-axial
world-sheet like objects $S^{1}_{N}\times\mathbb{R}$ accommodate the first
quantized Planck oscillators for all time.

##  6 Partial Difference and Partial Difference-Differential versions of the
Relativistic Klein-Gordon Equation

The same fundamental physical units of the previous section are used here.
Greek indices will take values from $\\{1,2,3,4\\}$ whereas the Roman indices
take values from $\\{1,2,3\\}$. Hatted quantities refer to transformed
versions of the corresponding unhatted quantities. The flat space-time metric
with signature $+2$ is denoted as $\eta_{\mu\nu}:=[1,1,1,-1]$. An element of
the relativistic discrete phase space will be denoted as
$(n^{1},n^{2},n^{3},n^{4})\in\mathbb{N}^{4}$ where $n^{\mu}$ is a non-negative
integer $\mathbb{N}=\\{0,1,2,...\\}$. Moreover, an element of discrete phase
space and continuous time will be denoted as
$(n^{1},n^{2},n^{3},t)=:(\mathbf{n},t)$. Here $n^{j}$ is a non-negative
integer and $t$ is a real number. The complex or real valued wave function of
the relativistic Klein-Gordon equation is denoted as either
$\phi(n^{1},n^{2},n^{3},n^{4})$ or
$\phi(n^{1},n^{2},n^{3},t)=:\phi(\mathbf{n},t)$.

The various partial difference and differential operations from equation (10)
are defined on the relativistic wave function as follows DasIV ; DasVII ;
DasVIII ; DasIX :

$\begin{array}[]{c}\Delta_{\mu}\phi(n^{1},n^{2},n^{3},n^{4}):=\phi(\cdots,n^{\mu}+1,\cdots)-\phi(\cdots,n^{\mu},\cdots),\\\
\\\
\Delta^{{}^{\prime}}_{\mu}\phi(n^{1},n^{2},n^{3},n^{4}):=\phi(\cdots,n^{\mu},\cdots)-\phi(\cdots,n^{\mu}-1,\cdots)),\\\
\\\ \Delta^{o}_{\mu}\phi(n^{1},n^{2},n^{3},n^{4}):=\\\
\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n^{\mu}+1}\phi(\cdots,n^{\mu}+1,\cdots)+\sqrt{n^{\mu}}\phi(\cdots,n^{\mu}-1,\cdots)\right],\\\
\\\ \Delta^{\\#}_{\mu}\phi(n^{1},n^{2},n^{3},n^{4}):=\\\
\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n^{\mu}+1}\phi(\cdots,n^{\mu}+1,\cdots)-\sqrt{n^{\mu}}\phi(\cdots,n^{\mu}-1,\cdots)\right].\\\
\end{array}$ (18)

Similarly, we define

$\begin{array}[]{c}\Delta_{j}\phi(n^{1},n^{2},n^{3};t):=\phi(\cdots,n^{j}+1,\cdots;t)-\phi(\cdots,n^{j},\cdots;y),\\\
\\\
\Delta^{{}^{\prime}}_{j}\phi(n^{1},n^{2},n^{3};t):=\phi(\cdots,n^{j},\cdots;t)-\phi(\cdots,n^{j}-1,\cdots;t)),\\\
\\\ \Delta^{o}_{j}\phi(n^{1},n^{2},n^{3};t):=\\\
\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n^{j}+1}\phi(\cdots,n^{j}+1,\cdots;t)+\sqrt{n^{j}}\phi(\cdots,n^{j}-1,\cdots;t)\right],\\\
\\\ \Delta^{\\#}_{j}\phi(n^{1},n^{2},n^{3};t):=\\\
\left(\dfrac{1}{\sqrt{2}}\right)\left[\sqrt{n^{j}+1}\phi(\cdots,n^{j}+1,\cdots;t)-\sqrt{n^{j}}\phi(\cdots,n^{j}-1,\cdots;t)\right],\\\
\\\
\partial_{t}\phi(n^{1},n^{2},n^{3};t)\equiv\partial_{t}\phi(\mathbf{n},t):=\dfrac{\partial}{\partial
t}\left[\phi(n^{1},n^{2},n^{3};t)\right].\end{array}$ (19)

##  7 The Relativistic Klein-Gordon Equation as a Finite Difference or Finite
Difference-Differential Equation

We will know define the quantum or wave version of the relativistic Klein-
Gordon equation in the covariant discrete phase space of $(4+4)$-dimension as
follows,

$\begin{array}[]{c}\overrightarrow{\mathbf{\Psi}}:=\phi(n^{1},n^{2},n^{3};t),\\\
\\\
\mathbf{Q}^{\mu}\overrightarrow{\mathbf{\Psi}}:=\eta^{\mu\nu}\Delta^{o}_{\nu}\phi(n^{1},n^{2},n^{3},n^{4}),\\\
\\\
\mathbf{P}_{\nu}\overrightarrow{\mathbf{\Psi}}:=-i\Delta^{\\#}_{\nu}\phi(n^{1},n^{2},n^{3},n^{4}),\\\
\\\
\left[\left(\eta^{\mu\nu}\mathbf{P}_{\mu}\mathbf{P}_{\nu}\right)+m^{2}\mathbf{I}\right]\overrightarrow{\mathbf{\Psi}}=\overrightarrow{\mathbf{0}},\\\
or\\\
\left[\left(\eta^{\mu\nu}\Delta^{\\#}_{\mu}\Delta^{\\#}_{\nu}\right)-m^{2}\right]\phi(n^{1},n^{2},n^{3},n^{4})=0.\end{array}$
(20)

The final equation above is the operator version of the relativistic Klein-
Gordon equation in our discrete $(4+4)$-dimension phase space. It is a partial
difference equation using the covariant partial difference operator
$\Delta^{\\#}_{\mu}$.

We shall now propose a partial difference-differential equation for the
relativistic Klein-Gordon equation in a discrete phase space continuous time
state space of dimension $[(2+2+2)+1]$,

$\begin{array}[]{c}\overrightarrow{\mathbf{\Psi}}:=\phi(n^{1},n^{2},n^{3};t)=:\phi(\mathbf{n};t),\\\
\\\
\mathbf{Q}^{j}\overrightarrow{\mathbf{\Psi}}:=\delta^{ij}\Delta^{o}_{k}\phi(\mathbf{n};t),\,\,\,\mathbf{Q}^{4}\overrightarrow{\mathbf{\Psi}}:=t\phi(\mathbf{n};t)\\\
\\\
\mathbf{P}_{k}\overrightarrow{\mathbf{\Psi}}:=-i\Delta^{\\#}_{k}\phi(\mathbf{n};t),\,\,\,\mathbf{P}_{4}\overrightarrow{\mathbf{\Psi}}:=-i\partial_{t}\phi(\mathbf{n};t),\\\
\\\
\left[\left(\delta^{jk}\mathbf{P}_{j}\mathbf{P}_{k}\right)-\left(\mathbf{P}_{4}\right)^{2}+m^{2}\mathbf{I}\right]\overrightarrow{\mathbf{\Psi}}=\overrightarrow{\mathbf{0}},\\\
or\\\
\left[\left(\delta^{jk}\Delta^{\\#}_{j}\Delta^{\\#}_{k}\right)-\partial^{2}_{t}-m^{2}\right]\phi(\mathbf{n};t)=0.\end{array}$
(21)

The proof of the exact relativistic invariance of both (20) and (21) is as
follows. First, consider the proper orthochronous ten-parameter Poincaré group
$\mathcal{I}\left[O(3,1)\right]^{+}_{+}$ WignerII ; Weinberg  given by the
following equations:

$\begin{array}[]{c}\hat{q}^{\mu}=c^{\mu}+l^{\mu}_{\nu}q^{\nu},\\\ \\\
\eta_{\mu\nu}l^{\mu}_{\alpha}l^{\nu}_{\beta}=\eta_{\alpha\beta},\\\ \\\
det[l^{\alpha}_{\beta}]=+1,\\\ \\\ l^{4}_{4}\geq 1.\end{array}$ (22)

Note that the scalar field $\phi(n^{1},n^{2},n^{3},n^{4})$ transforms
according to the usual tensor analysis DasTA  as
$\phi(\hat{q}^{1},\hat{q}^{2},\hat{q}^{3},\hat{q}^{4})=\phi(q^{1},q^{2},q^{3},q^{4})$.

Consider the following unitary operator DasIV  $U[c^{\mu},l^{\alpha}_{\beta}]$
which depends upon the ten-parameters of
$\mathcal{I}\left[O(3,1)\right]^{+}_{+}$:

$\begin{array}[]{c}U[c^{\mu},l^{\alpha}_{\beta}]:=\exp\left\\{-ic^{\mu}\mathbf{P}_{\mu}+\dfrac{i}{4}\omega^{\alpha\beta}\left[\mathbf{Q}_{\alpha}\mathbf{P}_{\beta}-\mathbf{Q}_{\beta}\mathbf{P}_{\alpha}+\mathbf{P}_{\beta}\mathbf{Q}_{\alpha}-\mathbf{P}_{\alpha}\mathbf{Q}_{\beta}\right]\right\\},\\\
\omega^{\beta\alpha}=-\omega^{\alpha\beta}.\\\ \end{array}$ (23)

In the Schrödinger picture, transformations on the state vector
$\overrightarrow{\mathbf{\Psi}}$ and operators $\mathbf{Q}^{\mu}$ and
$\mathbf{P}^{\nu}$ are induced by the operator
$U[c^{\gamma},l^{\alpha}_{\beta}]$ defined above as follows,

$\begin{array}[]{c}\widehat{\overrightarrow{\mathbf{\Psi}}}=U[c^{\mu},l^{\alpha}_{\beta}]\overrightarrow{\mathbf{\Psi}},\\\
\widehat{\mathbf{Q}}_{\mu}=\mathbf{Q}_{\mu},\\\
\widehat{\mathbf{P}}_{\nu}=\mathbf{P}_{\nu}.\end{array}$ (24)

However, in the Heisenberg picture, transformations on the state vector
$\overrightarrow{\mathbf{\Psi}}$ and operators $\mathbf{Q}^{\mu}$ and
$\mathbf{P}^{\nu}$ are induced by the operator $U[c^{\mu},l^{\alpha}_{\beta}]$
as follows,

$\begin{array}[]{c}\widehat{\overrightarrow{\mathbf{\Psi}}}=\overrightarrow{\mathbf{\Psi}},\\\
\\\
\widehat{\mathbf{Q}}_{\mu}=U^{\dagger}[c^{\gamma},l^{\alpha}_{\beta}]\mathbf{Q}_{\mu}U[c^{\gamma},l^{\alpha}_{\beta}],\\\
\\\
\widehat{\mathbf{P}}_{\nu}=U^{\dagger}[c^{\gamma},l^{\alpha}_{\beta}]\mathbf{P}_{\nu}U[c^{\gamma},l^{\alpha}_{\beta}].\end{array}$
(25)

Below, we use only the Schrödinger picture. The Casimir invariant of
$\mathcal{I}\left[O(3,1)\right]^{+}_{+}$ is given by the operator
$\eta^{\mu\nu}\mathbf{P}_{\mu}\mathbf{P}_{\nu}$. It commutes with all ten
generators $\mathbf{P}_{\mu}$ and
$\left(\mathbf{Q}_{\alpha}\mathbf{P}_{\beta}-\mathbf{Q}_{\beta}\mathbf{P}_{\alpha}+\mathbf{P}_{\beta}\mathbf{Q}_{\alpha}-\mathbf{P}_{\alpha}\mathbf{Q}_{\beta}\right)$.
Therefore, the Klein-Gordon operator equation in (20) transforms via (24) as

$\begin{array}[]{c}\left[\left(\eta^{\mu\nu}\widehat{\mathbf{P}}_{\mu}\widehat{\mathbf{P}}_{\nu}\right)+m^{2}\mathbf{I}\right]\widehat{\overrightarrow{\mathbf{\Psi}}}=\left[\left(\eta^{\mu\nu}\mathbf{P}_{\mu}\mathbf{P}_{\nu}\right)+m^{2}\mathbf{I}\right]U[c^{\gamma},l^{\alpha}_{\beta}]\overrightarrow{\mathbf{\Psi}}\\\
\\\
=U[c^{\gamma},l^{\alpha}_{\beta}]\left[\left(\eta^{\mu\nu}\mathbf{P}_{\mu}\mathbf{P}_{\nu}\right)+m^{2}\mathbf{I}\right]\overrightarrow{\mathbf{\Psi}}=\overrightarrow{\mathbf{0}}.\\\
\end{array}$ (26)

Therefore, the Klein-Gordon operator equation, the fourth relation in (20),
remains invariant under the covariant transformations (24) proving that our
Klein-Gordon operator equation is completely special relativistic.

Now we appeal to the partial difference representation of the quantum
mechanics provided by last the equation of (20),

$\left[\eta^{\mu\nu}\Delta^{\\#}_{\mu}\Delta^{\\#}_{\nu}-m^{2}\right]\phi(n^{1},n^{2},n^{3},n^{4})=0.$
(27)

The unitary operator $U[c^{\mu},l^{\alpha}_{\beta}]$ which depends upon the
ten-parameters of $\mathcal{I}\left[O(3,1)\right]^{+}_{+}$ may be written as

$\begin{array}[]{c}U[c^{\mu},l^{\alpha}_{\beta}]:=\exp\left\\{-c^{\mu}\Delta^{\\#}_{\mu}+\dfrac{1}{4}\omega^{\alpha\beta}\left[\Delta^{o}_{\alpha}\Delta^{\\#}_{\beta}-\Delta^{o}_{\beta}\Delta^{\\#}_{\alpha}+\Delta^{\\#}_{\beta}\Delta^{o}_{\alpha}-\Delta^{\\#}_{\alpha}\Delta^{o}_{\beta}\right]\right\\},\\\
\\\ \hat{\phi}(n^{1},n^{2},n^{3},n^{4}):=\\\
\exp\left\\{-c^{\mu}\Delta^{\\#}_{\mu}+\dfrac{1}{4}\omega^{\alpha\beta}\left[\Delta^{o}_{\alpha}\Delta^{\\#}_{\beta}-\Delta^{o}_{\beta}\Delta^{\\#}_{\alpha}+\Delta^{\\#}_{\beta}\Delta^{o}_{\alpha}-\Delta^{\\#}_{\alpha}\Delta^{o}_{\beta}\right]\right\\}\phi(n^{1},n^{2},n^{3},n^{4}).\end{array}$
(28)

In the $[(2+2+2)+1]$-dimensional discrete phase space and continuous time
(state space), the state vector $\overrightarrow{\mathbf{\Psi}}$ and operators
$\mathbf{Q}^{j},\mathbf{Q}^{4},\mathbf{P}^{j},\mathbf{P}^{4}$ are defined in
(21) with the partial difference-differential Klein-Gordon equation being
given by
$\left[\delta^{jk}\Delta^{\\#}_{j}\Delta^{\\#}_{k}-\partial^{2}_{t}-m^{2}\right]\phi(\mathbf{n};t)=0$.
The Klein -Gordon wave function solution transforms under the
$\mathcal{I}\left[O(3,1)\right]^{+}_{+}$ group as

$\begin{array}[]{c}\hat{\phi}(\mathbf{n};t):=\exp\left\\{-c^{j}\Delta^{\\#}_{j}-c^{4}\partial_{t}+\dfrac{1}{4}\left[\omega^{jk}\Delta^{o}_{j}\Delta^{\\#}_{k}-\Delta^{o}_{k}\Delta^{\\#}_{j}+\Delta^{\\#}_{k}\Delta^{o}_{j}-\Delta^{\\#}_{j}\Delta^{o}_{k}\right.\right.\\\
\left.\left.+\omega^{j4}(t\Delta^{\\#}_{j}-\Delta^{o}_{j}\partial_{t})\right]\right\\}\phi(\mathbf{n};t).\end{array}$
(29)

By following similar steps as in (26), one can prove that
$\hat{\phi}(\mathbf{n};t)$ satisfies the partial difference-differential
Klein-Gordon equation of (21), i.e.

$\left[\delta^{jk}\Delta^{\\#}_{j}\Delta^{\\#}_{k}-\partial^{2}_{t}-m^{2}\right]\hat{\phi}(\mathbf{n};t)=0,$
(30)

thereby proving that the partial difference-differential Klein-Gordon equation
is exactly invariant under all possible relativistic transformations of the
ten parameter Poincaré group $\mathcal{I}\left[O(3,1)\right]^{+}_{+}$ .

##  8 Concluding Remarks

In the relativistic quantum field theory, a non-Hermitian scalar field
operator $\Phi(n^{1},n^{2},n^{3},n^{4})$ or else $\Phi(\mathbf{n};t)$ is a
linear operator acting on a Hilbert space vector bundle DasTA . The physical
principle of the micro-causality DasV  demands that
$[\Phi(\mathbf{n};t),\Phi(\hat{\mathbf{n}};\hat{t})]|_{t=\hat{t}}=\mathbf{0}\,\,\,for\,\,\,\mathbf{n}\neq\hat{\mathbf{n}}$.
However, a similar condition
$[\Phi(n^{1},n^{2},n^{3},n^{4}),\Phi(\hat{n}^{1},\hat{n}^{2},\hat{n}^{3},\hat{n}^{4})]|_{n^{4}=\hat{n}^{4}}=\mathbf{0}$
does not imply the physical principle of micro-causality. Just for this
physical consistency, we are justified in working on the more complicated
relativistic quantum field theory in the discrete phase space and continuous
time arena DasV .

We would like to mention the historical fact that Hamilton Hamilton
introduced the propagation of light wave phenomena using a partial difference
equation in lattice space-time as
$[\delta^{jk}\Delta_{j}\Delta^{{}^{\prime}}_{k}-\Delta_{4}\Delta^{{}^{\prime}}_{4}]\phi(n^{1},n^{2},n^{3},n^{4})=0$.
This equation is of course non-relativistic. In previous work DasI , a non-
relativistic divergence-free interacting quantum field theory in lattice
space-time was similarly introduced.

The quantized Planck oscillators in $[(2+2+2)]$-dimensional phase space create
denumerably infinite number of three-dimensional discrete orbits
$S^{1}_{n^{1}}\times S^{1}_{n^{2}}\times S^{1}_{n^{3}}$. The Planck
oscillators are assumed to be extremely small in size and mass beyond present
day experimental detection techniques. However, these oscillators do produce
discrete orbits in $[(2+2+2)]$-dimensional discretized phase space. The second
quantized relativistic Klein-Gordon operator wave field $\Phi(\mathbf{n};t)$
representing other physical particles like mesons, that are distinct from the
first quantized oscillators, can temporarily, or for all time, occupy any of
the geometrical objects $S^{1}_{n^{1}}\times S^{1}_{n^{2}}\times
S^{1}_{n^{3}}$. Bosonic particles can cohabit with a first quantized Planck
oscillator in any one of the orbits $S^{1}_{n^{1}}\times S^{1}_{n^{2}}\times
S^{1}_{n^{3}}$. Furthermore, some of the configurations $S^{1}_{n^{1}}\times
S^{1}_{n^{2}}\times S^{1}_{n^{3}}$ can remain totally empty of Bosons for a
period of time. In the $S^{\\#}$-matrix theory DasVI  of interacting Fermion
and Boson fields, it was assumed that a field quanta can jump from one
configuration $S^{1}_{n^{1}}\times S^{1}_{n^{2}}\times S^{1}_{n^{3}}$ into
another one completely ignoring the Planck oscillators.

In future work, we intend to improve the physical side of this theory by
changing the three dimensional geometrical objects $S^{1}_{n^{1}}\times
S^{1}_{n^{2}}\times S^{1}_{n^{3}}$ into six-dimensional objects
$\bar{S}_{n^{1}}\times\bar{S}_{n^{2}}\times\bar{S}_{n^{3}}$ inside the
$(2+2+2)$-dimensional discrete phase space. Here, $\bar{S}^{1}_{n^{j}}$
denotes a Peano circle Clark  of dimension two inside the
$(2+2+2)$-dimensional discrete phase space. We can identify
$\bar{S}_{n^{1}}\times\bar{S}_{n^{2}}\times\bar{S}_{n^{3}}$ as a six-
dimensional phase cell inside
$\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}$. We shall elaborate
on this topic exhaustively in the succeeding part II of this paper entitled
Discrete phase space and continuous time relativistic quantum mechanics II:
Peano circles and hyper-tori like phase cells.

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