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, Hoboken, NJ 07030, USA
44email<EMAIL_ADDRESS>
# Discrete phase space and continuous time relativistic quantum mechanics II:
Peano circles, hyper-tori phase cells, and fibre bundles

Anadijiban Das  Rupak Chatterjee

(Received: date / Accepted: date)

###### Abstract

The discrete phase space and continuous time representation of relativistic
quantum mechanics is further investigated here as a continuation of paper I
DasRCI . The main mathematical construct used here will be that of an area-
filling Peano curve. We show that the limit of a sequence of a class of Peano
curves is a Peano circle denoted as $\bar{S}^{1}_{n}$, a circle of radius
$\sqrt{2n+1}$ where $n\in\\{0,1,\cdots\\}$. We interpret this two-dimensional
Peano circle in our framework as a phase cell inside our two-dimensional
discrete phase plane. We postulate that a first quantized Planck oscillator,
being very light, and small beyond current experimental detection, occupies
this phase cell $\bar{S}^{1}_{n}$. The time evolution of this Peano circle
sweeps out a two-dimensional vertical cylinder analogous to the world-sheet of
string theory. Extending this to three dimensional space, we introduce a
$(2+2+2)$-dimensional phase space hyper-tori
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ as the
appropriate phase cell in the physical dimensional discrete phase space. A
geometric interpretation of this structure in state space is given in terms of
product fibre bundles.

We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional
discrete phase space and continuous time state space using the relativistic
partial difference-differential Klein-Gordon equation. The second quantized
field quantas of this system can cohabit with the tiny Planck oscillators
inside the
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ phase
cells for eternity. Finally, a generalized free second quantized Klein-Gordon
equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is
explored. The resulting discrete phase space dimension is compared to the
significant spatial dimensions of some of the popular models of string theory.

###### Keywords:

Discrete phase space Peano curves Partial difference-differential equations
Fibre bundles

###### pacs:

11.10Ef 11.15Ha 02.70Bf 03.65Fd

##  1 Introduction

We begin in section 2 by introducing the concept of a Peano curve Clark ;
Gelbaum . In 1890, G. Peano startled the mathematical world with the
introduction of an area-filling curve. After that, many area-filling curves
have been discovered including those by David Hilbert. To our knowledge, very
few papers in the mathematical physics arena have used this concept DasHaldar
.

We begin by discussing the original Peano curve that fills a square $\bar{D}$
of unit area inside $\mathbb{R}^{2}$. In section 3, we derive other Peano
curves filling a sequence $\\{\bar{D}_{M}\\}^{\infty}_{M=1}$ of unit areas
each in the shape of a rectangle. Next, we introduce a double sequence
$\\{\bar{D}_{Mn}\\}^{\infty}_{M=1}$ for $n\in\\{0,1,2,...\\}$. Each of closed
regions $\bar{D}_{Mn}$ for a fixed $n$ is endowed with a unit area and a
rectangular shape. Moreover, each of these regions $\bar{D}_{Mn}$ is covered
by a Peano curve. Next, we discuss another double sequence
$\\{\bar{A}_{Mn}\\}$ of closed regions where each region $\bar{A}_{Mn}$ is
annular in shape and possesses a unit area. This annular region $\bar{A}_{Mn}$
is also covered by a Peano curve. Finally, in the limiting process of
$M\rightarrow\infty$ for a fixed $n$, the annular region $\bar{A}_{Mn}$
collapses into a circle $\bar{S}^{1}_{n}$ of unit area. Since this cirlce
contains a ’squashed’ Peano curve, one calls $\bar{S}^{1}_{n}$ a Peano circle.
Section 4 details more analysis on this Peano circle.

The simplified Klein-Gordon equation in the background of a
$[(1+1)+1]$-dimensional discrete phase space and continuous time is discussed
in section 5. Some physically relevant results are also derived.

In section 6, we introduce the abstract concept of fibre bundles Choquet  and
discuss their applications to certain physical problems. We apply this concept
in section 7 to the physically important $[(2+2+2)+1]$-dimensional phase
space. This discrete phase space and continuous time arena has been
investigated with respect to the second quantization of many free relativistic
field theories DasI ; DasII . The corresponding formulations of interacting
relativistic second quantized fields yielding an $S^{\\#}$-matrix series were
given in DasIII . Second order expansion terms of this $S^{\\#}$-matrix
involving Møller scattering of quantum electrodynamics Jauch ; Peskin
produced a non-singular Coulomb potential in DasBen ; DasRC .

In section 7, we define a new set-theoretic mapping Goldberg  by the direct or
Cartesian product Lightstone  as:

$\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\to\infty}}\\\
\\\ \left([g^{M}_{n^{1}}\times h^{M}_{n^{1}}\times
h^{M}_{1}]\times[g^{M}_{n^{2}}\times h^{M}_{n^{2}}\times
h^{M}_{2}]\times[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times
h^{M}_{3}]\right)\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right)\\\
\\\ \subset\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}.\end{array}$
(1)

The left hand side of this equation
($\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$) is a
hyper-torus Massey  of physical dimension
$\left[\dfrac{ML^{2}}{T}\right]^{3}$. It will be shown that these hyper-tori
can represent a phase cell in the usual $(2+2+2)$-dimensional discrete phase
space.

In section 8, the second quantization of the scalar field
$\Phi(n^{1},n^{2},n^{3};t)=:\Phi(\mathbf{n};t)$ is discussed in the arena of
$[(2+2+2)+1]$-dimensional discrete phase space and continuous time using the
relativistic Klein-Gordon operator equation. A partial difference-differential
version of the Klein-Gordon equation is solved for a special class of
solutions involving Fourier-Hermite transforms DasII . The total energy
$\mathcal{H}$, momentum component $\mathcal{P}_{j}$ and electric charge
$\mathcal{Q}$ are worked out for the second quantized Klein-Gordon scalar
field operator $\Phi(\mathbf{n};t)$.

Finally, in section 9, we study discrete $(2+2+2)N$-dimensional phase space
many-particle systems and the corresponding Peano hyper-tori
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\cdots\bar{S}^{1}_{n^{N}}$.
Each of these Peano hyper-tori has the physical dimension of
$\left[\dfrac{ML^{2}}{T}\right]^{N}$ and represent appropriate phase cells
inside the discrete $(2+2+2)N$-dimensional phase space. The generalized Klein-
Gordon equation within this discrete $[(2+2+2)N+1]$-dimensional phase space
and continuous time arena is also discussed. As our $2N$-dimensional Peano
hyper-tori is a phase-cell inside a discrete $6N$-dimensional phase space, we
compare our structure with that of the spatial dimension of string theory
Green ; PolchinskiI ; PolchinskiII .

##  2 The area filling curve of Peano

Consider a continuous, piece-wise linear, oriented, parametrized curve Clark
$f^{1}(u)$ inside a $(1+1)$-dimensional plane $\mathbb{R}^{2}$ as depicted in
figure 1.

Figure 1: The graph of the parametrized curve $f^{1}(u)$.

Note that the curve $f^{1}(u)$ is defined over nine closed intervals
$\left[\dfrac{k-1}{9},\dfrac{k}{9}\right]$ for
$k\in\\{1,2,\cdots,9\\}\subset\mathbb{R}$. The image of the curve of figure 1
is a continuous, piece-wise zigzag, oriented, parametrized curve over a closed
square $\bar{D}\subset\mathbb{R}$ of unit area inside the $x-y$ plane
$\mathbb{R}^{2}$. The image of the function $f^{1}(u)$ in figure 1 has for the
closed domain$[0,1/9]$ the closed range marked as the first linear segment
with an arrow and denoted with a ”1”. Similarly, the closed domain $[1/9,2/9]$
is mapped to the linear segment denoted ”2”. The ten corners of the zig-zag
curve are situated at $f^{1}(k/9),k=0,1,...,9$.

Figure 2: The graph of the parametrized curve $f^{2}(u)$.

The image of the next parametrized curve $f^{2}(u)$, depicted in figure 2, is
continuous, oriented, and piece-wise linear in the $x-y$ plane
$\mathbb{R}^{2}$. It has $9^{2}=81$ linear segments inside the unit square
$\bar{D}$ within $\mathbb{R}^{2}$. This figure is constructed by replicating
figure 1 nine times within the unit square area resulting in $81$ oriented
linear pieces.

Let us define a distance function $d(f^{1}(u),f^{2}(u))$ abstractly as

$d(f^{1}(u),f^{2}(u)):=\displaystyle{\textit{sup}_{u\in[0,1]}}||f^{1}(u)-f^{2}(u)||\leq\dfrac{\sqrt{2}}{3}.$
(2)

Here, the number $\dfrac{\sqrt{2}}{3}$ is the diameter of a square with each
side being $\dfrac{1}{3}$. The sequence of pre-Peano parametrized curves
$\\{f^{j}(u)\\}^{\infty}_{j=1}$ is obtained by replicating the above process
over and over again. Therefore, one may obtain the result that

$d(f^{j}(u),f^{j+1}(u))\leq\dfrac{\sqrt{2}}{3^{j}}.$ (3)

It follows by the uniform Cauchy criterion Clark  that for each $\epsilon>0$,
there exists an integer $N>0$ such that $d(f^{j}(u),f^{m}(u))<\epsilon$ for
all $j$ and $m\geq N$. Therefore, there exists a uniformly continuous function
$f(u)$ such that $\displaystyle{\lim_{j\rightarrow\infty}}f^{j}(u)=f(u)$.
Moreover, on can prove that this parametrized curve $f(u)$ passes through
every point of the closed unit square $\bar{D}$. This is the first example of
a Peano curve filling in the area of $\bar{D}$ Clark .

##  3 Other Peano curves filling a $(1+1)$-dimensional physical phase space

We can interpret physically the areas $\bar{D}$ in both figures 1 and 2 as
phase cells inside a $(1+1)$-dimensional phase plane. Furthermore, the
uncertainty of quantum physics Dirac  $|\Delta x||\Delta y|\geq 1$
$(i.e.\,|\Delta q||\Delta p|\geq 1,\hbar=1)$ automatically holds inside a
phase cell $\bar{D}$ for a quantized particle occupying $\bar{D}$ (here,
$\Delta$ stands for an increment and not a finite difference operator). Any
possible movement of the quantized particle along the Peano curve $f(u)$
covering $\bar{D}$ is physically unobservable.

Now, we shall define an infinite sequence of functions
$\\{h^{M}\\}^{\infty}_{M=1}$ from the region $\bar{D}\subset\mathbb{R}^{2}$
into regions $\\{\bar{D}_{M}\\}^{\infty}_{M=1}$ as depicted in figure 3.

Figure 3: The graph of the function $h^{M}$ for fixed $M$.

The function $h^{M}$ is defined for a specific $M$ as the following linear
transformation:

$\begin{array}[]{c}\rho=\left(\dfrac{1}{2M\pi}\right)x+\dfrac{1}{2},\\\ \\\
\alpha=(2M\pi)y-M\pi,\\\ \\\ M\geq 1.\end{array}$ (4)

The Jacobian of these transformations is

$\dfrac{\partial(\rho,\alpha)}{\partial(x,y)}=1,$ (5)

making these transformations canonical transformations of Hamiltonian
mechanics Lanczos ; Goldstein . Therefore, the area of each of the regions
$\\{\bar{D}_{M}\\}$ is furnished by the double integral of the following
differential 2-form Spivak ; DasTA

$\begin{array}[]{c}Area(\bar{D}^{M}):=\displaystyle{{\int\int}_{\bar{D}^{M}}}d\rho\wedge
d\alpha=\int_{\frac{1}{2}}^{\frac{1}{2M\pi}+\frac{1}{2}}\int_{-M\pi}^{M\pi}d\rho
d\alpha=1,\\\ \\\ M\in\\{1,2,3,...\\}.\end{array}$ (6)

Remarks:

(i) In the limiting process $M\rightarrow\infty$, the sequence of regions
$\\{\bar{D}^{M}\\}^{\infty}_{M=1}$ collapses into the open vertical line given
by $\\{(\rho,\alpha)\in\mathbb{R}^{2}:\rho=1/2,\alpha\in\mathbb{R}\\}$.

(ii) This infinite vertical line is an open string-like phase cell of unit
area as seen by using (6) above.

(iii) The infinite vertical line in remark (i) is consistent with the
uncertainty principle $|\Delta\rho||\Delta\alpha|\geq 1$.

(iv) The string-like phase cell may or may not contain any quanta of a second
quantized relativistic wave field (see section 5).

(v) Moreover, unlike in conventional string theory Green ; PolchinskiI ;
PolchinskiII , in our approach of string-like phase cells, an open string-like
phase cell cannot be of finite length.

Consider another sequence of transformations $\\{h^{M}_{n}\\}^{\infty}_{M=1}$
where $n$ is chosen to be a fixed non-negative integer. A typical mapping
$h^{M}_{n}$ maps the closed domain $\bar{D}_{M}$ into the closed region
$\bar{D}_{Mn}$ by the linear transformation

$\begin{array}[]{c}r=\rho+n,\\\ \\\ \theta=\alpha,\\\ \\\
\dfrac{\partial(r,\theta)}{\partial(\rho,\alpha)}=1.\end{array}$ (7)

This transformation is exhibited in figure 4.

Figure 4: The mapping $h^{M}_{n}$ from $\bar{D}_{M}$ to $\bar{D}_{Mn}$.

The area of each domain $\\{\bar{D}_{Mn}\\}$ is given by

$\begin{array}[]{c}Area(\bar{D}_{Mn}):=\displaystyle{{\int\int}_{\bar{D}_{Mn}}}dr\wedge
d\theta=\int_{n+\frac{1}{2}}^{n+\frac{1}{2M\pi}+\frac{1}{2}}\int_{-M\pi}^{M\pi}drd\theta=1,\\\
\\\ M\in\\{1,2,3,...\\},\,\,\,\ n\in\\{0,1,2,....\\}.\end{array}$ (8)

Now, consider a third sequence of canonical transformations
$\\{g^{M}_{n}\\}^{\infty}_{M=1}$ for a fixed $n$ as follows,

$\begin{array}[]{c}q=\sqrt{2\rho}\cos\theta,\\\ \\\
p=\sqrt{2\rho}\sin\theta,\\\ \\\
\dfrac{\partial(q,p)}{\partial(\rho,\theta)}=1,\\\ \\\ q^{2}+p^{2}=2r>0,\\\
\\\ \left(\dfrac{p}{q}\right)=\tan\theta,\,\,\,q\neq 0.\end{array}$ (9)

The closed domain of each of the mappings of $\\{g^{M}_{n}\\}^{\infty}_{M=1}$
is furnished by

$\bar{D}_{Mn}:=\left\\{(r,\theta)\in\mathbb{R}^{2}:\left(n+\frac{1}{2}\right)\leq
r\leq\left(n+\frac{1}{2M\pi}+\frac{1}{2}\right),-M\pi\leq\theta\leq
M\pi\right\\}.$ (10)

The corresponding co-domain $\bar{A}_{Mn}$ in the $q-p$ phase plane is
provided by the annular region:

$\begin{array}[]{c}\bar{A}_{Mn}:=\left\\{(q,p)\in\mathbb{R}^{2}:2n+1\leq
q^{2}+p^{2}\leq\left(2n+1+\frac{1}{M\pi}\right)\right\\},\\\ \\\
Area(\bar{A}_{Mn}):=\displaystyle{\int\int_{\bar{A}_{Mn}}}dp\wedge
dq=1.\end{array}$ (11)

The mappings $\\{g^{M}_{n}\\}^{\infty}_{M=1}$ are depicted in the figure 5.

Figure 5: The mapping $g^{M}_{n}$ with topological identifications of the two
horizontal sides of $\bar{D}_{Mn}$.

Note that the annular region $\bar{A}_{Mn}$ is not a simply connected region
of the circular disk $q^{2}+p^{2}=2n+1+\frac{1}{M\pi}$. Furthermore, the
winding number of each boundary of the annular region $\bar{A}_{Mn}$ is
exactly $M$.

##  4 The Peano circle $\bar{S}^{1}_{n}$ in the $(1+1)$-dimensional phase
plane

We shall now investigate the closed domain $\bar{A}_{Mn}$ of the mapping
$g^{M}_{n}$ for a fixed $n$ as exhibited in figure 5. Using (9) and (11), the
inner and outer boundaries of the closed co-domain $\bar{A}_{Mn}$ are found to
be

$\begin{array}[]{c}\partial_{-}[\bar{A}_{Mn}]:=\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1\\},\\\
\\\
\partial_{+}[\bar{A}_{Mn}]:=\left\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1+\frac{1}{M\pi}\right\\}.\end{array}$
(12)

Consider the limiting map
$\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}_{M=1}^{\infty}$.
Clearly, one has for this limiting map

$\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1+\frac{1}{M\pi}\right\\}\right)=\partial_{-}[\bar{A}_{Mn}]$
(13)

which is depicted in figure 6.

Figure 6: The limiting map
$\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}$ and the closed range
as the Peano circle $\bar{S}^{1}_{n}$.

Remarks:

(i) Note that by (11), $Area(\bar{A}_{Mn})=1$ for all $M\in\\{1,2,3,...\\}$.

(ii) Since the closed range of the mapping
$\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}=\bar{S}^{1}_{n}$, the
Peano circle $\bar{S}^{1}_{n}$ itself is endowed with the unit
$(1+1)$-dimensional area.

(iii) The Peano circle can act as a string-like Green ; PolchinskiI ;
PolchinskiII  phase-cell.

(iv) The quantum mechanical uncertainty principle $|\Delta q||\Delta p|\geq 1$
holds for each quanta inside the phase cell $\bar{S}^{1}_{n}$.

(v) The original Peano-curve
$\displaystyle{\lim_{j\rightarrow\infty}}f^{j}(u)=f(u)$ is squashed inside the
Peano circle $\bar{S}^{1}_{n}$.

(vi) The first quantized oscillator introduced in part I of this work DasRCI
follows a zig-zag motion or zitter-bewengung Greiner  along the squashed Peano
curve inside $\bar{S}^{1}_{n}$.

(vii) Unlike the Peano circle $\bar{S}^{1}_{n}$ here, the one-dimensional
circle of figure 2 in part I DasRCI  cannot act as a two-dimensional phase
cell within the $q-p$ phase plane.

Consider now the infinite $[(1+1)+1]$-dimensional hyper-circular cylinder
$\bar{S}^{1}_{n}\times\mathbb{R}$ inside the $[(1+1)+1]$-dimensional discrete
phase plane plus continuous time $t\in\mathbb{R}$ called the state space
Lanczos . Within this state space is a world-sheet like object Green ;
PolchinskiI ; PolchinskiII  depicted in figure 7.

Figure 7: The $[(1+1)+1]$-dimensional infinite hyper-cylinder
$\bar{S}^{1}_{n}\times\mathbb{R}$ inside the $[(1+1)+1]$-dimensional discrete
phase plane plus continuous time.

The $[(1+1)+1]$-dimensional circular cylinder $S^{1}_{n}\times\mathbb{R}$
inside the state space in figure 4 in part I DasRCI  cannot represent an
evolving phase cell unlike the hyper-circular cylinder
$\bar{S}^{1}_{n}\times\mathbb{R}$ depicted in figure 7.

Let us summarize what we have achieved so far. Figures 1 and 2 have described
the area filling curve of Peano, which completely fills the area of a unit
square. We identify this unit area as a possible phase cell within a
$(p,q)$-phase plane. The area filling Peano curve is identified as the
trajectory of one quanta of certain quantized simple harmonic oscillators
introduced in DasRCI . The trajectory of the Peano curve is not observable due
to the uncertainty principle of quantum mechanics Greiner ; Bethe . Equations
(4), (5), (7), and (9) indicate sequences of area preserving mathematical
mappings physically representing canonical transformations of Hamiltonian
mechanics Lanczos ; Goldstein . Therefore, in figure 5, the original unit
square has yielded a sequence of circular, annular domains each of unit area.
In figure 6, this sequence of circular annular domains collapses into a Peano
circle $\bar{S}^{1}_{n}$ of unit area. The quanta of the Planck oscillator
DasRCI  still goes around the Peano circle with constant energy $\sqrt{2n+1}$
in a zig-zag pattern but is undetectable by external observations.

Note that a Peano circle $\bar{S}^{1}_{n}$ of unit area in the $(1+1)$-phase
plane is analogous to a closed string of prototypical string theory Green ;
PolchinskiI ; PolchinskiII . Figure 7 depicts the $[(1+1)+1]$-dimensional
hyper-circular-cylinder inside the three-dimensional state space that is
analogous to the two-dimensional world-sheet evolution of closed strings.

##  5 The Klein-Gordon equation in a $[(1+1)+1]$-dimensional discrete phase-
plane and continuous time

Now, we investigate the second quantization of the Klein-Gordon equation in a
$[(1+1)+1]$-dimensional discrete phase-plane and continuous time scenario. The
relativistic Klein-Gordon equation in a $[(2+2+2)+1]$-dimensional discrete
phase-plane and continuous time was discussed in section 7 of part I of this
paper DasRCI . The second quantized scalar field linear operator, denoted by
$\Phi(n,t)$, acts on a Hilbert space bundle Choquet ; DasTA . The
corresponding Klein-Gordon equation is

$\begin{array}[]{c}[\mathbf{P}\cdot\mathbf{P}-(\mathbf{P_{t}})^{2}+m^{2}\mathbf{I}]\overrightarrow{\mathbf{\Psi}}=\overrightarrow{\mathbf{0}},\\\
or\\\
(\Delta^{\\#})^{2}\Phi(n,t)-(\partial_{t})^{2}\Phi(n,t)-m^{2}\Phi(n,t)=0.\end{array}$
(14)

Consider the complex valued functions $\xi_{n}(k)$ involving Hermite
polynomials $H_{n}(k)$ Olver ,

$\begin{array}[]{c}\xi_{n}(k):=\dfrac{(i)^{n}e^{-k^{2}/2}H_{n}(k)}{\pi^{1/4}2^{n/2}\sqrt{n!}},\\\
\\\ \xi_{n}(-k)=\overline{\xi_{n}(k)},\\\ \xi_{2n+1}(0)=0,\\\
\displaystyle{\int_{\mathbb{R}}}\overline{\xi_{m}(k)}\xi_{n}(k)dk=\delta_{mn},\\\
\displaystyle{\sum_{n=0}^{\infty}}\overline{\xi_{n}(k)}\xi_{n}(\hat{k})=\delta(k-\hat{k}),\\\
\Delta^{\\#}\xi_{n}(k)=ik\xi_{n}(k).\end{array}$ (15)

Here, $\delta(k-\hat{k})$ indicates the Dirac delta distribution function
Zemanian . A special class of exact solutions to the partial difference-
differential equations of (14) are given by

$\begin{array}[]{c}\Phi^{-}(n,t):=\displaystyle{\int_{\mathbb{R}}}\dfrac{1}{\sqrt{2\omega(k)}}\left[A(k)\xi_{n}(k)e^{-i\omega
t}\right]dk,\\\ \\\
\Phi^{+}(n,t):=\displaystyle{\int_{\mathbb{R}}}\dfrac{1}{\sqrt{2\omega(k)}}\left[B^{\dagger}(k)\overline{\xi_{n}(k)}e^{i\omega
t}\right]dk,\\\ \\\ \Phi(n,t):=\Phi^{-}(n,t)+\Phi^{+}(n,t),\\\ \\\
\omega=\omega(k):=+\sqrt{k^{2}+m^{2}}>0.\end{array}$ (16)

The Fourier-Hermite integrals in (16) are supposed to be uniformly convergent
Buck . Moreover, the linear operators $A(k),A^{\dagger}(k),B(k)$ and
$B^{\dagger}(k)$ (creation and annihilation operators in momentum space)
acting linearly on a Hilbert space bundle must satisfy the following
commutation relations:

$\begin{array}[]{c}[A(k),A^{\dagger}(\hat{k})]=[B(k),B^{\dagger}(\hat{k})]=\delta(k-\hat{k})I(k),\\\
\\\
\,[A(k),A(\hat{k})]=[A^{\dagger}(k),A^{\dagger}(\hat{k})]=[B(k),B(\hat{k})]=[B^{\dagger}(k),B^{\dagger}(\hat{k})]=0,\\\
\\\ N^{+}(k):=A^{\dagger}(k)A(k),\\\ \\\
N^{-}(k):=B^{\dagger}(k)B(k),\end{array}$ (17)

where the eigenvalues of the number operators $N^{\pm}(k)$ are the set
$\\{0,1,2,...\\}$. Physically speaking, the scalar field operator $\Phi(n,t)$
represents a collection of massive, spin-less, electrically charged, second
quantized scalar particle excitations.

One can show the following relations for total energy, total momentum and
total electric charge respectively DasI ; DasII ,

$\begin{array}[]{c}\mathcal{H}:=\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)+N^{-}(k)+\delta(0)I(k)]\omega(k)dk,\\\
\\\ \mathcal{P}:=\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)+N^{-}(k)]kdk,\\\
\\\
\mathcal{Q}:=e\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)-N^{-}(k)]dk.\end{array}$
(18)

The divergent null point energy term $\delta(0)I(k)$ may be ignored for
physical interpretations but cannot be rectified directly.

##  6 Fibre Bundles

Let $M$ and $M^{\\#}$ be two topological manifolds Choquet ; DasTA . The
Cartesian product $M\times M^{\\#}$ and the projection mapping $\Pi$ from
$M\times M^{\\#}$ into $M$ constitute a product or trivial bundle $(M\times
M^{\\#},M,\Pi)$ depicted in figure 8. The vertical linear segment inside
$M\times M^{\\#}$ is called a fibre Choquet ; DasTA .

Figure 8: The trivial or product bundle $(M\times M^{\\#},M,\Pi)$.

Figure 9 provides an explicit example of a product bundle using our Peano
cirle $\bar{S}^{1}_{n}$ to create a vertical circular cylinder
$\bar{S}^{1}_{n}\times I_{t}$,

Figure 9: The product bundle $(\bar{S}^{1}_{n}\times
I_{t},\bar{S}^{1}_{n},\Pi_{t})$.

The closed interval $I_{t}:=[0,T]\subset\mathbb{R}$ and $(\chi,[-\pi,\pi])$ is
a coordinate chart DasTA  in figure 9. The second quantized scalar field
operators defined in (16) are restricted for fixed numbers $n$ and $T$ of the
product bundle in figure 9.

Another product bundle associated with the linear operators (16) is created by
using our Peano circle $\bar{S}^{1}_{n}$ and the closed linear line segment
$I_{k}:=[K_{1},K_{2}]\subset\mathbb{R}$ as in figure 10.

Figure 10: The product bundle $(\bar{S}^{1}_{n}\times
I_{k},\bar{S}^{1}_{n},\Pi_{k})$.

Remarks:

(i) Figure 9 represents geometrically the occupation of a tiny first quantized
Planck oscillator of figures 6 and 7 in the interval $[0,T]\subset\mathbb{R}$
(the full actual physics takes place in the time interval $-\infty<t<\infty$).

(ii) Moreover, in figure 9, the second quantized scalar field $\Phi(n,t)$
quanta can cohabit with a tiny first quantized Planck oscillator during the
time interval $I_{t}:=[0,T]\subset\mathbb{R}$.

(iii) Furthermore, in figure 10, the second quantized scalar field quanta with
a range of momentum $K_{1}<k<K_{2}$ can cohabit with a tiny first quantized
Planck oscillator.

(iv) The complete mathematical treatment of this phenomena involves the
product bundles $(\bar{S}^{1}_{n}\times
I_{t},\bar{S}^{1}_{n},\Pi_{t})\times(\bar{S}^{1}_{n}\times
I_{k},\bar{S}^{1}_{n},\Pi_{k})$ (which is difficult to depict).

##  7 Discrete phase space and hyper-tori like phase cells

Here, we shall extend the various mappings of figures 3,4, and 5 to the
$(2+2+2)$-dimensional discrete phase space arena. Choosing the fixed indices
$a\in\\{1,2,3\\}$ and $M\in\\{1,2,3,\cdots\\}$, the relevant mappings are
illustrated in figure 11. It depicts the composite map $g^{M}_{n^{a}}\circ
h^{M}_{n^{a}}\circ h^{M}_{a}$ for fixed indices $a$ and $M$.

Figure 11: The discrete phase space mappings $h^{M}_{a},h^{M}_{n^{a}}$, and
$g^{M}_{n^{a}}$.

The closed domains and the closed range of the composite map
$g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}$ are provided by

$\begin{array}[]{c}closed\,\,\ domain\,\left[g^{M}_{n^{a}}\circ
h^{M}_{n^{a}}\circ h^{M}_{a}\right]=\bar{D}_{a}\subset\mathbb{R}^{2},\\\ \\\
closed\,\,\ range\,\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ
h^{M}_{a}\right]=\bar{A}_{Mn^{a}}\subset\mathbb{R}^{2},\\\ or\\\
\bar{A}_{Mn^{a}}=\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ
h^{M}_{a}\right](\bar{D}_{a}).\end{array}$ (19)

where the last relation illustrates a set theoretic mapping Goldberg .

Consider the sequence of composite mappings $\left\\{g^{M}_{n^{a}}\circ
h^{M}_{n^{a}}\circ h^{M}_{a}\right\\}_{M=1}^{\infty}$ for a fixed $a$ and
$n^{a}$. The limiting set-theoretic mapping from figure 6 and the last
relation of (19) is given by

$\bar{S}^{1}_{n}=\displaystyle{\lim_{M\rightarrow\infty}}\left\\{\left[g^{M}_{n^{a}}\circ
h^{M}_{n^{a}}\circ h^{M}_{a}\right](\bar{D}_{a})\right\\}$ (20)

From this, one can derive the set-theoretic Cartesian product mapping Goldberg
; Lightstone :

$\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{\bar{A}_{Mn^{1}}\right\\}\times\left\\{\bar{A}_{Mn^{2}}\right\\}\times\left\\{\bar{A}_{Mn^{3}}\right\\}\right),\\\
or\\\
\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{\left[g^{M}_{n^{1}}\circ
h^{M}_{n^{1}}\circ
h^{M}_{1}\right](\bar{D}_{1})\right\\}\times\left\\{\left[g^{M}_{n^{2}}\circ
h^{M}_{n^{2}}\circ h^{M}_{2}\right](\bar{D}_{2})\right\\}\right.\\\
\left.\times\left\\{\left[g^{M}_{n^{3}}\circ h^{M}_{n^{3}}\circ
h^{M}_{3}\right](\bar{D}_{3})\right\\}\right).\end{array}$ (21)

Furthermore, one has

$\begin{array}[]{c}\left[g^{M}_{n^{1}}\times h^{M}_{n^{1}}\times
h^{M}_{1}\right]\times\left[g^{M}_{n^{2}}\times h^{M}_{n^{2}}\times
h^{M}_{2}\right]\times\left[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times
h^{M}_{3}\right]\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right):=\\\
\\\ \left\\{\left[g^{M}_{n^{1}}\circ h^{M}_{n^{1}}\circ
h^{M}_{1}\right](\bar{D}_{1})\right\\}\times\left\\{\left[g^{M}_{n^{2}}\circ
h^{M}_{n^{2}}\circ
h^{M}_{2}\right](\bar{D}_{2})\right\\}\times\left\\{\left[g^{M}_{n^{3}}\circ
h^{M}_{n^{3}}\circ h^{M}_{3}\right](\bar{D}_{3})\right\\}\end{array}$ (22)

From the above two relations, a new set-theoretic mapping is furnished by

$\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left\\{\left[g^{M}_{n^{1}}\times
h^{M}_{n^{1}}\times h^{M}_{1}\right]\times\left[g^{M}_{n^{2}}\times
h^{M}_{n^{2}}\times h^{M}_{2}\right]\times\right.\\\ \\\
\left.\left[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times
h^{M}_{3}\right]\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right)\right\\}\subset\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}.\end{array}$
(23)

Remarks:

(i) The $(2+2+2)$-dimensional hyper-sphere or hyper-torus Massey
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ for
three fixed integers $(n^{1},n^{2},n^{3})$ denotes a region where a tiny first
quantized Planck oscillator inhabits this region for all time.

(ii) The whole of the $(2+2+2)$-dimensional discrete phase space contains a
denumerably infinite number of concentric hyper-tori each containing a single
tiny first quantized Planck oscillator.

(iii) Each
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$
constitutes a phase cell of physical dimension
$\left[\dfrac{ML^{2}}{T}\right]^{3}$. Therefore, in the fundamental physical
units, each of these phase cells is endowed with a hyper-volume of
$1\left[\dfrac{ML^{2}}{T}\right]^{3}$.

(iv) Each of these phase-cells bears a resemblance to a $D$-dimensional hyper-
torus in standard string theory Green ; PolchinskiI ; PolchinskiII .

(v) In the arena of $[(2+2+2)+1]$-dimensional discrete phase space and
continuous time, the hyper-torus
$\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ sweeps
out a world-sheet like vertical, circular hyper-cylinder. Such a vertical,
circular cylinder always contains just one first quantized tiny Planck
oscillator for all time.

(vi) Finally, one or more second quantized scalar field $\Phi(\mathbf{n};t)$
quanta may cohabit the tiny first quantized Planck oscillator temporarily or
forever.

##  8 Second quantized, relativistic Klein-Gordon field equations in
$[(2+2+2)+1]$-dimensional discrete phase space and continuous time

From part I of this work DasRCI , the second quantized, relativistic Klein-
Gordon field equations are given by

$\begin{array}[]{c}a,b\in\\{1,2,3\\},\\\ \\\ n^{a}\in\\{0,1,2,...\\},\\\ \\\
\mathbf{n}:=(n^{1},n^{2},n^{3}),\\\ \\\
\left[(\delta^{ab}\Delta^{\\#}_{a}\Delta^{\\#}_{b})-(\partial_{t})^{2}-m^{2}\right]\Phi(\mathbf{n};t)=\mathbf{0}.\end{array}$
(24)

The three dimensional extensions of the functions $\xi_{n}(k)$ in (15) imply
that Olver

$\begin{array}[]{c}\xi_{n^{a}}(k_{a}):=\dfrac{(i)^{n^{a}}e^{-k_{a}^{2}/2}H_{n^{a}}(k_{a})}{\pi^{1/4}2^{n^{a}/2}\sqrt{n^{a}!}},\\\
\\\ \mathbf{k}:=(k_{1},k_{2},k_{3}),\\\ \\\
\Delta^{\\#}_{a}\xi_{n^{a}}(k_{a})=ik_{a}\xi_{n^{a}}(k_{a}),\\\ \\\
\displaystyle{\sum_{n^{1}=0}^{\infty}\sum_{n^{2}=0}^{\infty}\sum_{n^{3}=0}^{\infty}}\left[\overline{\xi_{n^{1}}}(k_{1})\xi_{n^{1}}(\hat{k}_{1})\right]\left[\overline{\xi_{n^{2}}}(k_{2})\xi_{n^{2}}(\hat{k}_{2})\right]\left[\overline{\xi_{n^{3}}}(k_{3})\xi_{n^{3}}(\hat{k}_{3})\right]\\\
=\delta({k_{1}}-{\hat{k}_{1}})\delta({k_{2}}-{\hat{k}_{2}})\delta({k_{3}}-{\hat{k}_{3}})=:\delta^{3}(\mathbf{k}-\mathbf{\hat{k}}).\end{array}$
(25)

A special class of exact solutions for these relativistic partial difference-
differential operator equations (24) are given by

$\begin{array}[]{c}\Phi^{-}(\mathbf{n};t):=\displaystyle{\int_{\mathbb{R}^{3}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[A(\mathbf{k})\xi_{n^{1}}(k_{1})\xi_{n^{2}}(k_{2})\xi_{n^{3}}(k_{3})e^{-i\omega
t}\right]dk_{1}dk_{2}dk_{3},\\\ \\\
\Phi^{+}(\mathbf{n};t):=\displaystyle{\int_{\mathbb{R}^{3}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[B^{\dagger}(\mathbf{k})\overline{\xi_{n^{1}}}(k_{1})\overline{\xi_{n^{2}}}(k_{2})\overline{\xi_{n^{3}}}(k_{3})e^{i\omega
t}\right]dk_{1}dk_{2}dk_{3},\\\ \\\
\Phi(\mathbf{n};t):=\Phi^{-}(\mathbf{n};t)+\Phi^{+}(\mathbf{n};t),\\\ \\\
\omega=\omega(\mathbf{k}):=+\sqrt{k_{1}^{2}+k_{3}^{2}+k_{3}^{2}+m^{2}}>0.\end{array}$
(26)

The Fourier-Hermite integrals in (26) are supposed to be uniformly convergent
Buck . Moreover, the linear operators
$A(\mathbf{k}),A^{\dagger}(\mathbf{k}),B(\mathbf{k})$ and
$B^{\dagger}(\mathbf{k})$ (creation and annihilation operators in momentum
space) acting linearly on a Hilbert space bundle Choquet ; DasTA  must satisfy
the following commutation relations:

$\begin{array}[]{c}[A(\mathbf{k}),A^{\dagger}(\hat{\mathbf{k}})]=[B(\mathbf{k}),B^{\dagger}(\hat{\mathbf{k}})]=\delta(\mathbf{k}-\hat{\mathbf{k}})\mathbf{I}(\mathbf{k}),\\\
\\\
\,[A(\mathbf{k}),A(\hat{\mathbf{k}})]=[A^{\dagger}(\mathbf{k}),A^{\dagger}(\hat{\mathbf{k}})]=[B(\mathbf{k}),B(\hat{\mathbf{k}})]=[B^{\dagger}(\mathbf{k}),B^{\dagger}(\hat{\mathbf{k}})]=\mathbf{0},\\\
\\\ N^{+}(\hat{\mathbf{k}}):=A^{\dagger}(\mathbf{k})A(\mathbf{k}),\\\ \\\
N^{-}(\hat{\mathbf{k}}):=B^{\dagger}(\mathbf{k})B(\mathbf{k}),\end{array}$
(27)

where the eigenvalues of the number operators $N^{\pm}(\hat{\mathbf{k}})$ are
the set $\\{0,1,2,...\\}$. The second quantized scalar field operator
$\Phi(\mathbf{n};t)$ represents a collection of massive, spin-less,
electrically charged, physical particles or quantas.

One can show the following relations for total energy, total momentum and
total electric charge respectively DasI ; DasII ,

$\begin{array}[]{c}\mathcal{H}:=\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})+N^{-}(\mathbf{k})+\delta^{3}(\mathbf{0})\mathbf{I}(\mathbf{k})]\omega(\mathbf{k})dk_{1}dk_{2}dk_{3},\\\
\\\
\mathcal{P}_{j}:=\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})+N^{-}(\mathbf{k})]k_{j}dk_{1}dk_{2}dk_{3},\\\
\\\
\mathcal{Q}:=e\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})-N^{-}(\mathbf{k})]dk_{1}dk_{2}dk_{3}.\end{array}$
(28)

The divergent null point energy term
$\delta^{3}(\mathbf{0})\mathbf{I}(\mathbf{k})$ may be ignored for physical
interpretations but cannot be directly remedied.

##  9 Many-particle systems, $(2+2+2)N$-dimensional phase space and
$(2+2+2)N$-dimensional hyper-tori as phase cells

Recall Hamilton’s canonical equations of motion for non-relativistic classical
mechanics Lanczos ; Goldstein

$\begin{array}[]{c}A\in\\{1,2,3,...,N\\},\\\ \\\
\dot{q}^{A}:=\dfrac{d\mathcal{Q}^{A}(t)}{dt},\\\ \\\
\dot{p}_{A}:=\dfrac{d\mathcal{P}_{A}(t)}{dt},\\\ \\\
\dot{q}^{A}:=\dfrac{\partial H}{\partial
p_{A}}(q^{1},\cdots,q^{3N};p_{1},\cdots,p_{3N};t),\\\ \\\
\dot{p}_{A}:=-\dfrac{\partial H}{\partial
q^{A}}(q^{1},\cdots,q^{3N};p_{1},\cdots,p_{3N};t).\end{array}$ (29)

The corresponding Schrödinger wave equation for the first quantized physical
system is furnished by

$\begin{array}[]{c}i\dfrac{\partial}{\partial_{t}}\psi\left(q^{1},\cdots,q^{3N};t\right)=\\\
H\left(q^{1},\cdots,q^{3N};-i\frac{\partial}{\partial
q^{1}},\cdots,-i\frac{\partial}{\partial
q^{3N}};t\right)\psi(q^{1},\cdots,q^{3N};t),\end{array}$ (30)

which has become ubiquitous with standard quantum mechanical systems being
verified experimentally for a multitude of specific systems.

From part I of this paper DasRCI , the $N=1$ relativistic Klein-Gordon
equation in four-dimensional space is given by ($\eta_{\mu\nu}:=[1,1,1,-1]$)

$\begin{array}[]{c}\eta^{\mu\nu}\dfrac{\partial^{2}}{\partial q^{\mu}\partial
q^{\mu}}\psi\left(q^{1},q^{2},q^{3},q^{4}\right)-m^{2}\psi\left(q^{1},q^{2},q^{3},q^{4}\right)=0,\\\
or\\\ \delta^{ab}\dfrac{\partial^{2}}{\partial q^{a}\partial
q^{b}}\psi\left(\mathbf{q};t\right)-(\partial_{t})^{2}\psi\left(\mathbf{q};t\right)-m^{2}\psi\left(\mathbf{q};t\right)=0.\\\
\end{array}$ (31)

The relativistic wave equation for the case of two spin-$\frac{1}{2}$
particles $q_{(1)}^{\mu}$ and $q_{(2)}^{\nu}$ is furnished by the Bethe-
Salpeter equation Bethe

$\begin{array}[]{c}\left\\{\gamma_{(1)}^{\mu}\left[\left(\dfrac{m_{1}}{m_{1}+m_{2}}P_{\mu}+P_{\mu}\right)-im_{1}\right]\right\\}\left\\{\gamma_{(2)}^{\nu}\left[\left(\dfrac{m_{1}}{m_{1}+m_{2}}P_{\nu}+P_{\nu}\right)-im_{2}\right]\right\\}\\\
\\\
\cdot\psi\left(q_{(1)}^{1}-q_{(2)}^{1},q_{(1)}^{2}-q_{(2)}^{2},q_{(1)}^{3}-q_{(2)}^{3},q_{(1)}^{4}-q_{(2)}^{4}\right)=\\\
\\\
i\bar{G}\left(q_{(1)}^{1}-q_{(2)}^{1},q_{(1)}^{2}-q_{(2)}^{2},q_{(1)}^{3}-q_{(2)}^{3},q_{(1)}^{4}-q_{(2)}^{4}\right)\psi\left(q_{(1)}^{1}-q_{(2)}^{1},\cdots\right)\end{array}$
(32)

where $\bar{G}$ is the appropriate Green’s function.

In $N+1$-dimensional state space, the generalization to the relativistic
Klein-Gordon wave equation is

$\begin{array}[]{c}\left[\delta^{AB}\dfrac{\partial^{2}}{\partial
q^{A}\partial
q^{B}}-(\partial_{t})^{2}-m^{2}\right]\psi\left(q^{1},\cdots,q^{N};t\right)=0,\\\
A,B\in\\{1,\cdots,N\\}.\\\ \end{array}$ (33)

with a group invariance of $\mathcal{I}[O(N,1)]_{+}^{+}$.

Similarly, we can express within a $[(2+2+2)N+1]$-dimensional discrete phase
space and continuous time arena, the first quantized partial differential-
difference Klein-Gordon equation as

$\left[\delta^{AB}\Delta^{\\#}_{A}\Delta^{\\#}_{B}-(\partial_{t})^{2}-m^{2}\right]\phi(n^{1},\cdots,n^{N};t)=0.$
(34)

The second quantized version of this generalized Klein-Gordon equation is
given by

$\left[\delta^{AB}\Delta^{\\#}_{A}\Delta^{\\#}_{B}-(\partial_{t})^{2}-m^{2}\right]\Phi(n^{1},\cdots,n^{N};t)=\mathbf{0}$
(35)

The group invariance of the above equation is provided by
$\mathcal{I}[O(N,1)]_{+}^{+}$ (see DasRCI ).

A special class of exact solutions for these relativistic partial difference-
differential operator equations (35) are given by

$\begin{array}[]{c}\Phi^{-}(n^{1},\cdots,n^{N};t):=\displaystyle{\int_{\mathbb{R}^{N}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[A(\mathbf{k})\xi_{n^{1}}(k_{1})\cdots\xi_{n^{N}}(k_{N})e^{-i\omega
t}\right]dk_{1}\cdots dk_{N},\\\ \\\
\Phi^{+}(n^{1},\cdots,n^{N};t):=\displaystyle{\int_{\mathbb{R}^{N}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[B^{\dagger}(\mathbf{k})\overline{\xi_{n^{1}}}(k_{1})\cdots\overline{\xi_{n^{N}}}(k_{N})e^{i\omega
t}\right]dk_{1}\cdots dk_{N},\\\ \\\
\Phi(n^{1},\cdots,n^{N};t):=\Phi^{-}(n^{1},\cdots,n^{N};t)+\Phi^{+}(n^{1},\cdots,n^{N};t),\\\
\\\
\omega=\omega(k_{1},\cdots,k_{N}):=+\sqrt{\delta^{AB}k_{A}k_{B}+m^{2}}>0.\end{array}$
(36)

The second quantized linear operators
$A(k_{1},\cdots,k_{N}),B^{\dagger}(k_{1},\cdots,k_{N})$, etc. (creation and
annihilation operators in momentum space) acting linearly on a Hilbert space
bundle must satisfy very similar commutation relations as those of (27). The
linear operator $\Phi(n^{1},\cdots,n^{N};t)$ is defined over the geometric
configuration
$\bar{S}_{n^{1}}^{1}\times\bar{S}_{n^{2}}^{1}\times\cdots\times\bar{S}_{n^{N}}^{1}\times\mathbb{R}$
where $\bar{S}_{n^{A}}^{1}$ is a Peano circle of physical dimension
$\left[\dfrac{ML^{2}}{T}\right]$.

Table 1: Discrete Phase Space and Popular String Theory Dimension Comparison

Finally, we would like to make a final comparison with our method to two
popular approaches to quantizing gravity: string theory and loop quantum
gravity. Table 1 consists of a higher dimensional comparison between our
discrete phase space theory put forward here and that of various popular
string theories Green ; PolchinskiI ; PolchinskiII . There appears to be a
striking similarity between the spatial or gauge group dimension of string
theory and the dimension of our discrete phase space. One clear advantage of
our method is that we have gone further in understanding the scattering matrix
of our theory, the $S^{\\#}$-matrix, as delineated in DasI ; DasII ; DasIII .
In DasRC , we have explicitly calculated new Feynman rules for computations of
the $S^{\\#}$-matrix elements and have derived an exact singularity free
Coulomb-type potential within our discrete phase space approach. See Green ;
PolchinskiI ; PolchinskiII  for further information on the actual calculation
abilities of string theory.

Loop quantum gravity Rovelli , and its spin network structure of space-time
composed of extremely fine but finite loops has a similarity to our Peano
circle $\bar{S}^{1}_{n}$ of unit area in the $(1+1)$ dimensional phase plane.
The area filling Peano curve is identified as the trajectory of one quanta as
shown here and in DasRCI . It would be interesting to further investigate this
analogy to possibly find a quantum theory of gravity within our framework or
find Peano type motion within the framework of loop quantum gravity.

##  10 Concluding Remarks

In this paper, we have shown how an area filling Peano curve represents a
possible particle trajectory in the unit phase cell of a discrete phase space
and continuous time relativistic quantum mechanical system. This is one of the
first uses of this fascinating mathematical structure as a physical construct.
Both first quantized Planck oscillators, first explored in part I of this work
DasRCI , and second quantized Klein-Gordon excitations were explored with
respect to the Peano curve formalism. Furthermore, the state space evolution
of our Peano circle was shown to be analogous to the world-sheet evolution of
closed strings. The geometric framework of this evolution was interpreted in
terms of a product fibre bundle structure. Finally, extensions of our model to
higher dimensions and striking similarities to popular dimensions used in
traditional string theory were explored.

## Acknowledgements

A.D. thanks Dr. Jack Gegenberg for some informal discussions.

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