# A Geometric Picture of Peturbative QFT

Rory O’Dwyer  Department of Physics, Stanford University

###### Abstract.

In previous work, a lattice scalar propagator was rigorously defined in $d=1$
flat space and shown to equal the known Klein-Gordon propagator of QFT. This
work generalizes this lattice propagator to manifolds whose universal cover is
the hyperbolic half plane as well as a broad class of higher dimensional
manifolds. We motivate a conjecture for the power spectrum of these curved
space propagators. Afterwards, an analogous definition of the Dirac spinor
propagator is defined. The formulation of these propagators are unified by the
understanding of the object as a Fourier transform of the volume of path space
of paths with the same length $I$ to mass $m$; the main theorem of this work
will be to demonstrate that the point correlators of scalar peturbative QFT
can be understood as a Fourier transform of the volume of path space of paths,
which realize a Feynman diagram and have some total length I. After
demonstrating this theorem, the author goes on to conjecture the point
correlators of the Abelian Higgs Model in this geometric formulation.

##  1\. Introduction

Previous work defined the relativistic lattice scalar propagator in $d=1$ and
showed that it converged the the result expected from QFT [15]. It is the
purpose of this work to first extend the relativistic lattice scalar
propagator to higher dimensions and to manifold which have $\mathbb{H}$ as
their universal cover. We will also show how our definition of the propagator
may be defined over notions of connected sums and surgeries [4], extending the
definition further.

The lattice scalar propagator is defined on graphs that look locally like
$\mathbb{Z}^{d}\times\mathbb{Z}$. To define it, we first would count discrete
paths between points in this graph, and partition them by a special minkowski
metric like function. The partition function would be a path indexed sum over
elements of the unit circle whose phase was determined by the length of said
path. Then, a special continuum limit was taken, obtained first in [2] and
[20], and this was shown to converge to the known form of the KG propagator in
$d=1$.

If we have some discretization of a 2d manifold, this definition may normally
be extended to said context. In the instance that [15] defined the discrete
propagator on $\mathbb{T}^{2}$, it used the fundamental domain of
$\mathbb{T}^{2}$, a square, to tile $\mathbb{Z}^{2}$. It could then sum over
discrete paths from an initial point to all lifts of the final point in
$\mathbb{Z}^{2}$ to define the discrete propagator given a definition of it in
$\mathbb{Z}^{d}\times\mathbb{Z}$. We not only aim to perform the same
calculation here but to also do so for the full variety of 2d manifolds sans
the two sphere and to obtain the continuum limit of these objects.

In this paper, we will make small adaptations to the proof of [15] so that it
defines a discrete propagator over $\mathbb{Z}^{d}\times\mathbb{Z}$ and a
conintuum one over $\mathbb{R}^{d}\times\mathbb{R}$. We will show two
alternative ways to then define a discretization of general 2d manifolds. The
first consists of using Teichmuller curves; these curves are topologically
isomorphic to variuous $g\geq 2$ curves. They have a natural flat metric on
them and are important in the study of billiards [14]; their structure is not
necessarily conformally or diffeomorphically isomorphic to the 2d manifolds
they describe as they have singular discontinuities in curvature. How we would
extend a propagator over a surgery cut or a connected sum will become apparent
in this section (Section 5).

Our second approach is more nuanced and consists of discretizing the
hyperboloid model of hyperbolic geometry [18]. By the uniformization theorem
of Riemann Surfaces [1], all compact Riemann surfaces with genus $g\geq 2$
have the half plane as their universal cover. This means that their
fundamental domain will tile hyperbolic space. This will be covered in Section
4.

After this, the author will show that the Fermionic propagator has a very
similar geometric interpretation to the bosonic one (Section 6). Namely, he
will show that it is a fourier transform of paths avoiding a direction
dictated by the initial and final spinor states. An almost identical proof to
the geometric propagator will proceed thereafter, and the author is confident
that in this manner one may define the lattice fermionic propagator in a
similarly wide context.

With these tools in hand, the jump to a geometric model of QFT can be made
(Section 7). The author will show that the Feynman rules of peturbative QFT
have an interpretation in this geometric picture. Specifically, the
contribution from some Feynman diagram is equal to the Fourier transform of
the volume of paths that realize that Feynman diagram through space; i.e.
Feynman diagrams aren’t just means of counting point correlator contributions
but represent genuine particle trajectories akin to the original insight of
Feynman.

##  2\. Notations and Statement of Theorems

We will employ the definition of a discrete space $X$ and its continuum
partner $X^{cont}$ as defined in [15]. Namely, let $d\in\\{1,...,\\}$ and
$\\{L_{i}\\}_{i=1}^{d}\subset\mathbb{N}\cup\\{\infty\\}$. I define
$X=(\times_{i}\\{-L_{i},-L_{i}+1,...,L_{i}\\})\times\mathbb{Z}$. The first $d$
copies of $\mathbb{Z}$ are defined as the spatial coordinates and the last
with time. I note that $X\subset\mathbb{Z}^{d}$, so it has a boundary
$\partial X$ in $\mathbb{Z}^{d}$. I will constrain ourselves to $X$ such that
$2|\left|\partial X\right|)$. I can therefore give $\partial X$ an arbitrary
equivalence relation $\sim$ which partitions $\partial X$ into pairs. I shall
leave $\sim$ completely general. In [15], I only defined the polygonal metric
for $d=1$; I define it in the upcoming section on all above X for arbitrary
dimension d.

To define polygonal metrics on X, its appropriate to first define them on the
square lattice. Let
$\vec{x}_{1},\vec{x}_{2}\in\mathbb{Z}^{d}\times\mathbb{Z}$. I define on the
following functions
$d_{l_{2}},d_{l_{2}^{*}}:\mathbb{Z}^{d}\times\mathbb{Z}\rightarrow\mathbb{C}$

(1)
$d_{l_{2}}(\vec{x}_{1},\vec{x}_{2})=\sqrt{proj_{t}(\vec{x}_{1}-\vec{x}_{2})^{2}+\sum_{i=1}^{d}proj(\vec{x}_{1}-\vec{x}_{2})^{2}}\\\
d_{l_{2}^{*}}(\vec{x}_{1},\vec{x}_{2})=\sqrt{proj_{t}(\vec{x}_{1}-\vec{x}_{2})^{2}-\sum_{i=1}^{d}proj(\vec{x}_{1}-\vec{x}_{2})^{2}}$

For all our purposes, I will only be using $d_{l_{2}^{*}}$ on
$d_{l_{2}^{*}}^{-1}(\\{a|a\in\mathbb{R},a\geq 0\\})$. The first equation in
Equation 1 is clearly the $l_{2}$ metric on $X$. The subscript $l_{2}^{*}$ in
the second is so recognize its similarity with the $l_{2}$ metric; I will
refer to the second part of Equation 1 as the minkowski metric. Let
$n\in\\{1,2,...\\}$. Consider the set of scaled primitive pythagorean tupples
with hypotenuse below n in Equation 2.

(2)
$\mathcal{A}_{n}=\\{(\frac{x_{1}}{I},...\frac{x_{d}}{I},\frac{t}{I})\in\mathbb{Q}^{d+1}|\sum_{i=1}^{d}x_{i}^{2}+I^{2}=t^{2},(x_{1},...,x_{d},I,t)\in\mathbb{Z}^{d+2},0\leq
t\leq n,I\neq 0\\}\\\
\cup\\{\frac{1}{gcd(x_{1},...,x_{d},t)}(x_{1},..,x_{d},t)\in\mathbb{Q}^{d+1}|\sum_{i=1}^{d}x_{i}^{2}+t^{2}=0,(x_{1},...,x_{d},t)\in\mathbb{Z}^{d+1},0\leq
t\leq n\\}$

The word ‘primitive’ places an extra constraint on $\mathcal{A}_{n}$ that if
we consider the equivalence class placed upon these vectors by parallelism,
then we only keep the representative with lowest hypotenuse. If we want a
generalization of the polygonal metric developed in [15], we note that the
equation $\\{\vec{x}\in\mathbb{R}^{d+1}|d_{l_{2}^{*}}(0,\vec{x})=1\\}$ is a
hyper-surface of dimension d, so we will want a polyhedron of dimension d
approximating this quadratic surface to be the unit sphere of our polygonal
metric. Let $\vec{v}\in\mathcal{A}_{d}$. Then, we define a neighborhood
$\mathcal{N}_{\vec{v}}$ of $\vec{v}$ to be one choice of the $d$ closest
linearly independent vectors in $\mathcal{A}_{d}$ to $\vec{v}$. We specify
this choice to allow for the case that there are multiple d tupples of vectors
satisfying this requirement. If we have some $\mathcal{N}_{\vec{v}}$, let
$\vec{v}^{\perp}$ be a unit vector (as defined by $d_{l_{2}}$) which
represents the (atmost) one dimensional space
$\\{\vec{w}-\vec{v}|\vec{w}\in\mathcal{N}_{v}\\}^{\perp}$. Then, we define the
half-space function
$\mathcal{H}_{\mathcal{N}_{\vec{v}}}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ as
in Equation 3.

(3)
$\mathcal{H}_{\mathcal{N}_{\vec{v}}}(\vec{x})=\frac{\sum_{i=1}^{d}(proj_{x_{i}}(\vec{v}^{\perp})proj_{x_{i}}(\vec{x}))+proj_{t}(\vec{v}^{\perp})proj_{t}(\vec{x})}{\sum_{i=1}^{d}(proj_{x_{i}}(\vec{v}^{\perp})proj_{x_{i}}(\vec{v}))+proj_{t}(\vec{v}^{\perp})proj_{t}(\vec{v})}$

This function is so defined because it will trace out a $d$ dimensional half-
space (the intersection of which would give us a d dimensional polyhedron); it
agrees with $d_{l_{2}^{*}}$ on setting $\vec{v}$ to length 1, and its roughly
tangent to the point $\vec{v}$. It yields the candidate expression for the d
dimensional polygonal minkowski metric in Equation 4.

(4)
$d_{n}(\vec{x}_{1},\vec{x}_{2})=min(\mathcal{H}_{\mathcal{N}_{\vec{v}}}(\vec{x}_{2}-\vec{x}_{1})|\vec{v}\in\mathcal{A}_{n})$

One can see that these functions approximate the minkowski metric in $d$
dimensional space in Theorem 10 and via Figure 1.

Figure 1. The set $\\{d_{n}(0,\vec{x})=1\\}$ for $d=2$ and $n=1$ and $n=3$

As you let $n\rightarrow\infty$, you see the unit circles for $d_{n}$ in
Figure 1 approximate a hyperboloid of two sheets, which is the unit circle for
$d_{l_{2}^{*}}$ in this dimension. The definition obtained for $d_{n}$ in this
setting, as opposed to [15], generalizes to arbitrary finite dimension d. Now,
for $d=2$ (and in higher dimensional generalizations of discrete orbifold
spaces), the extension of our metric to $X$ is rather simple. Ignoring the
temporal component of $X$, we note that $P$ lifts via the boundary tiling to
the covering lattice $\mathbb{Z}^{2}$. If we want to find the distance between
two points in X, we first find the lifts of each point which have the smallest
graph distance in $\mathbb{Z}^{2}$. Then, we define the distance between our
original points in $X$ as the distance $d_{n}$ on these closest lifted points
in $\mathbb{Z}^{2}$.

I shall return back to Equation 1 and Equation 4 for a moment. Let’s define
$\mathcal{A}^{all}=\\{\vec{x}\in\mathbb{Z}^{d}\times\mathbb{Z}|d_{l_{2}^{*}}(0,\vec{x})=d_{n}(0,\vec{x})\geq
0,proj_{t}(\vec{x})>0\\}$. A subset
$\mathcal{A}^{gen}\subset\mathcal{A}^{all}$ is called generating iff for all
$a\in\mathcal{A}^{all}$ there is
$\\{a_{i}\\}_{i=1}^{N}\subset\mathcal{A}^{gen}$ such that $\sum_{i=1}^{N}=a$.
Then, an axes of symmetry of $d_{n}$, denoted $\mathcal{A}_{n}$, is some
$\mathcal{A}_{gen}$ such that if $\vec{x}_{1},\vec{x}_{2}\in\mathcal{A}_{n}$
and $\vec{x}_{1}\neq c\vec{x}_{2}$ then $c=1$. So, every direction has a
unique representative. The double usage of $\mathcal{A}_{n}$ for scaled
Pythagorean tuples and as an axes of symmetry of $d_{n}$ will be shown to not
be an abuse of notation in Theorem 10 (except the axes of symmetry of $d_{n}$
will include also light paths). Our language in this paper will suggest that
this set is unique to each $d_{n};$ we will not prove this for we do not
require it for the next concepts.

I define as a path as set $\gamma=\\{\vec{x}_{i}\\}_{i=1}^{n}$ such that
$\\{x_{i}x_{i+1}\\}_{i=1}^{n-1}\subset\mathcal{E}$. Then, our achronal, local
paths between $\vec{x}$ and $\vec{y}\in X$ as the following:

(5)
$\Gamma^{\vec{x},\vec{y}}=\\{\gamma|\gamma|_{0}=\vec{x},\gamma|_{n}=\vec{y},d_{l_{2}^{*}}(\vec{x}_{i},\vec{x}_{i+1})\geq
0,d_{l_{2}^{*}}(\vec{x},\vec{y})\in\mathbb{Z},proj_{t}(\vec{x}_{i+1},\vec{x}_{i})\geq
0\\}$

I define our achronal, local $n$ polygonal paths between $\vec{x}$ and
$\vec{y}\in X$ as the following:

(6)
$\Gamma_{n}^{\vec{x},\vec{y}}=\\{\gamma|\gamma|_{0}=\vec{x},\gamma|_{n}=\vec{y},d_{l_{2}^{*}}(\vec{x}_{i},\vec{x}_{i+1})=d_{n}(\vec{x}_{i},\vec{x}_{i+1})\geq
0,d_{n}(\vec{x}_{i},\vec{x}_{i+1})\in\mathbb{Z},proj_{t}(\vec{x}_{i+1},\vec{x}_{i})\geq
0\\}$

On $\Gamma_{n}$ and $\Gamma$ we place an equivalence relation on paths; two
paths are equivalent if their indexing set trace out the same piecewise linear
paths in $X$. The use of an axes of symmetry is made clear in this context;
each equivalence class in $\Gamma_{n}^{\vec{x},\vec{y}}$ has a unique
representative with difference sequence drawn from $\mathcal{A}_{n}$ (Theorem
13). This paper will sometimes refer to sets of paths $\Gamma$ as path spaces.

From Equation 5 and Equation 6 I have
$\Gamma^{\vec{x},\vec{y}}_{n}\subset\Gamma^{\vec{x},\vec{y}}$. Let
$\gamma\in\Gamma^{\vec{x},\vec{y}}$. Then, I define the polygonal and
minkowski proper time in Equation 7.

(7)
$\rho_{n}(\gamma)=\sum_{i=0}^{n-1}d_{n}(x_{i+1},x_{i}),\rho_{l_{2}^{*}}(\gamma)=\sum_{i=0}^{n-1}d_{l_{2}^{*}}(x_{i+1},x_{i})$

From this definition, I immediately obtain Theorem 10. These geometrical
results are developed because I will be viewing the path integral as a
geometrical object. For this purpose, I define the following functions
$K_{n}:X\times X\rightarrow\mathbb{C}$ and $K_{l_{2}^{*}}:X\times
X\rightarrow\mathbb{C}$. Let $m\in\\{a|a\in\mathbb{R},a>0\\}$.

(8)
$K_{n}(\vec{x},\vec{y})=\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}}e^{im\rho_{n}(\gamma)},K_{l_{2}^{*}}(\vec{x},\vec{y})=\sum_{\gamma\in\Gamma^{\vec{x},\vec{y}}}e^{im\rho_{l_{2}^{*}}(\gamma)}$

These functions are our discrete propagators, as initially defined in [15]. In
this original paper, we introduced the discrete Feynman Propagator
$K^{Feyn}_{n}$ as Equation 8 except that we would allow $\rho_{n}(\gamma)$ to
include difference sequence steps with both $\pm d_{n}(x_{i+1},x_{i})$ as
their contribution to the Minkowski length. We will again make this
definition.

These are all our basic definitions in the discrete context. I will devote
some of our paper to the continuous setting. For this, I must define for our
original X a natural domain $X^{cont}$ which is the result of taking
infinitely fine lattices. We note that the continuum spaces in general are the
tiechmuller curves mentioned in Section 1; curves that are homeomorphic but
not conformally isomorphic to some 2d manifold [14].
$X^{cont}=(\times_{i}-[L_{i},L_{i}])\times\mathbb{R}$ with an associated
equivalence relation $\sim$ on pairs of points in $\partial X^{cont}$ as a
subspace of $\mathbb{R}^{d}\times\mathbb{R}$. I need a generalization of the
closest integer function $[*]:X^{cont}\rightarrow X$. For
$L\in\mathbb{N}\cup\\{\infty\\}$ I let
$[*]^{1d}:[-L,L]\rightarrow\\{-L,-L+1,...,L\\}$ be the closest point in the
range to the domain under the $l_{2}$ metric.

Then our generalization is defined as follows:

(9) $[\vec{x}]\textrm{ is the element of }X\textrm{ such that }\\\
proj_{x_{i}}([\vec{x}])=[proj_{x_{i}}(\vec{x})]^{1d}\\\
proj_{x_{i}}([\vec{x}])=[proj_{t}(\vec{x})]^{1d}$

Finally, for those $X$ having a finite element $L_{i}$ in their definition, I
must also change their dimensions to approach $X^{cont}$ as a limiting space
for infinitely fine lattices. Therefore, when I write
$\Gamma_{n}|_{m},\Gamma|_{m},K_{n}|_{m}$, or $K_{l_{2}^{*}}|_{m}$, I am
referring to those objects for
$X|_{m}=(\times_{i}\\{-mL_{i},-mL_{i}+1,...,mL_{i}\\})\times\mathbb{Z}$ when
$X=(\times_{i}\\{-L_{i},-L_{i}+1,...,L_{i}\\})\times\mathbb{Z}$.

Counting lattice paths are naturally connected to the multinomial coefficient;
therefore, continuum propagators seem to require the development of a
continuous multinomial coefficient. As defined by Cano and Diaz in [2], and
later generalized by Wakhare,Vignat,Le, and Robins in [20], we consider the
continuous multinomial coefficient in Equation 11 for l variables. Let
$\\{x_{i}\\}_{i=1}^{l}\subset\mathbb{R}_{+}$, $\mu$ denote the Lebesgue
measure on $\mathbb{R}^{n}$ [9], and for $n,N\in\mathbb{N}$ denote by $D(n,N)$
the number of Smirnov words of length n and N [20]. Furthermore, let $c\in
D(n,N)$, let $\\{c_{k}\\}_{k=1}^{n}$ denote singular letters of our Smirnov
word taking elements among some collection of d dimensional vectors, and let
$\\{e_{k}\\}_{k=1}^{l}$ denote these vectors. For $q\in\mathbb{R}_{+}^{d}$ we
define the path polytope $P(q,c)$ in Equation 10.

(10)
$P(q,c)=\\{(\lambda_{1},...,\lambda_{n})\in\mathbb{R}_{+}^{n}|\sum_{k=0}^{n}\lambda_{k}e_{c_{k}}=q\\}$

Then, we have our desired definition of the continuous multinomial in Equation
11:

(11) $\begin{Bmatrix}\sum x_{i}\\\
x_{1},x_{2},...,x_{l}\end{Bmatrix}=\sum_{n=0}^{\infty}\sum_{c\in
D(n,l)}\mu(P(q,c))\textrm{ where }q=\sum x_{i}e_{i}$

Problematically, the continuous multinomial was never shown by either [2] or
[20] to converge except in special cases. The necessary convergence results
are developed in Section 9; we will also show more rigorously in what sense
they are continuous analogues of the discrete multinomial coefficients.
Namely, for $m\in\mathbb{N}$, we introduce the operator
$\mathcal{T}^{m}_{cont}$ in Equation 13. This operator acts on sums indexed
over paths in $\Gamma^{\vec{y},\vec{x}}$. Let
$f:\Gamma^{\vec{y},\vec{x}}\rightarrow\mathbb{C}$ be some path indexed complex
valued function, and let
$r,\theta:\Gamma^{\vec{y},\vec{x}}\rightarrow\mathbb{R}$ be its radial and
angular components. Let $|\gamma|$ denote the number of distinct linear
segments in $\gamma$. Then, we can rearrange any general sum over
$\gamma\in\Gamma^{\vec{y},\vec{x}}$ as done in Equation 12.

(12)
$\sum_{\gamma\in\Gamma^{\vec{x},\vec{y}}}f(\gamma)=\sum_{n=1}^{\infty}\sum_{\gamma\in\Gamma^{\vec{x},\vec{y}},|\gamma|=n}f(\gamma)$

Then, we may write $\mathcal{T}_{cont}^{m}$ easily in this context (in
Equation 13)

(13)
$\mathcal{T}^{m}_{cont}(\sum_{\gamma\in\Gamma^{\vec{x},\vec{y}}}f(\gamma))=\sum_{n=d}^{\infty}(\sum_{\gamma\in\Gamma^{\vec{x},\vec{y}},|\gamma|=n}r(\gamma)m^{d-n}e^{i\frac{\theta(\gamma)}{m}})$

As demonstrated in Theorem 8, this operator is required to connect a natural
estimate of counting paths to a notion of volume, as one would expect in the
continuous setting. With all the relevant concepts defined, the following
continuum propagators are defined in Equation (in the event they exist).

(14)
$K_{n}^{cont}(\vec{x},\vec{y})=lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}K_{n}([m*\vec{x}],[m*\vec{y}])|_{m}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}\\\
K_{l_{2}^{*}}^{cont}(\vec{x},\vec{y})=lim_{p\rightarrow\infty}K_{p}^{cont}(\vec{X},\vec{y})$

In Equation 14, we will be satisfied if the sequence of $K_{n}^{cont}$ has a
convergent subsequence towards some function in a pointwise or sup norm sense,
and therefore that this limit is unique w.r.t limits. We define
$K_{n}^{Feyn,cont}$ to be Equation 14 but with each instance of $K_{n}$
replaced with $K_{n}^{feyn}$.

These are all the definitions we require for Theorem 1 and Theorem 3. For our
path integral on hyperbolic space, we define an indexed set of hyperboloids in
Equation 15.

(15) $\mathcal{H}_{t_{0}}=\\{(t,x,y)|t-t_{0}+1=\sqrt{1+x^{2}+y^{2}}\\}$

Each $\mathcal{H}_{t_{0}}$ inherits the minkowski $d_{l_{2}^{*}}$ and
polygonal $d_{n}$ metrics from $\mathbb{R}^{2}\times\mathbb{R}$. With this
metric defined, we can consider the group of isometries on
$\mathcal{H}_{t_{0}}$. That group is known to be $PSL(2,R)$[18]. If $G\subset
PSL(2,R)$ is a subgroup, and if $\forall\vec{x}\in\mathcal{H}_{t_{0}}$ the
orbit of x $Orb_{G}(x)=\\{y\in\mathcal{H}_{t_{0}}|y=gx,g\in G\\}$ has no
cluster points under our metric this group is Fuschian [1]. It is a
fundamental theorem of the geometrization of 2d manifolds that any 2d manifold
whose uniform cover is not $\mathbb{S}^{2}$ or $\mathbb{R}^{2}$ has an
isometric embedding into $\mathcal{H}_{t_{0}}$ where its homotopy group
$\pi_{1}(X)$ is realized as a Fuschian group[1]. This allows us to define
another $X$ and $X^{cont}$ which will extend the lattice scalar path integral
to all 2d manifolds.

For what comes next, we want to redefine how $[*]$ works as the closest point
in $\mathbb{Z}^{d}\times\mathbb{Z}$ to some point in
$\mathbb{R}^{d}\times\mathbb{R}$.

We let
$X^{cont}=\\{\mathcal{H}_{t}|t\in\mathbb{R}\\}=\mathbb{R}^{2}\times\mathbb{R}$
with an associated Fuschian group $G$ and equivalence equation $\sim$ such
that $(t_{1},x_{1},y_{2})\sim(t_{1},x_{2},y_{2})$ if $(t_{2},x_{1},y_{2})\in
Orb_{G}(t_{2},x_{2},y_{2})$. X, the discrete space, will be a
$\mathbb{Z}^{2}\times\mathbb{Z}$ with a similar quotient structure. We would
like to give $X$ an equivalence relation $\sim$ such that the closest point
function $[*]$ would always find for any point in its preimage the entire
equivalence class for said point in the same preimage. Say
$x,y\in\mathbb{Z}^{2}\times\mathbb{Z}$ both are the image of $[*]$ of two
points in the same equivalence class $\sim$ for $X^{cont}$, and let
$z\in\mathbb{Z}^{2}\times\mathbb{Z}$ have this same property w.r.t. y. The
Fuschian group of $X^{cont}$ is a subgroup of the group of isometries of
hyperbolic space. So, if there is a point in a geodesic ball about $y$ and one
about $x$ that are mapped to each-other by $g\in G$, and another point about
$z$ to which y maps to by $g_{2}$, then we can consider the image of the ball
about x under $g_{2}g_{1}$. Within this image we will find a point in the ball
around z that is the image of $g_{1}g_{2}$ of a point around a ball in x. If
we consider $\sim$ on $\mathbb{Z}^{2}\times\mathbb{Z}$ defined st $x\sim y$
iff there is $x^{cont}\sim y^{cont}\in\mathbb{R}^{2}\times\mathbb{R}$ such
that $[x^{cont}]=x$ and $[y^{cont}]=y$, then this relation is transitive.
Furthermore, it is reflexive and symmetric, so this would define a valid
equivalence relation on $\mathbb{Z}^{2}\times\mathbb{Z}$.

That this equivalence relation has the desired property that it contains the
whole orbit of $\vec{x}\in\mathbb{R}^{2}\times\mathbb{R}$ follows immediately
from our argument above, and from the fact that the group action are
isometries of this closest point metric.

###  2.1. Fermionic Case

Before moving to define the fermionic path integral in the lattice path
integral picture, one must recognize some resemblance to Feynman’s
checkerboard. The Feynman checkerboard was an initial attempt to obtain the
Fermion propagator as a genuine sum over paths done by Feynman in [8] and
later generalized to other contexts in [3].

One of the nice properties the method presents here and [15] shares with
Feynman’s original calculation is a continuum procedure very similar to that
instituted by $\mathcal{T}_{cont}$. Namely, Feynman also divided the set of
lattice light paths into subsets indexed by their number of linear segments
and weighed each linear segment just as is done by $\mathcal{T}_{cont}$. The
procedure we adopted in [15], while fundamentally different from Feynman’s, is
incidentally and encouragingly a ‘natural’ way of approaching the Feynman path
integral. This calculation had some peculiarities; namely, it assumed the
Fermion traveled exclusively along light paths. We lose these peculiarities;
our method uses light paths for a null set of trajectories and could extend to
a definition of all peturbative QFT.

Let us move to define the fermionic propagator in the same geometric picture
as we have for scalar particles. We define $\gamma^{\mu}$ as the $d+1$ many
matrices constituting the algebra basis of the spin representation of the
Clifford Algebra [16] $Cl_{1,d}(\mathbb{R})$. The fermionic propagator always
includes some two vectors $v,w\in\mathbb{C}^{2}$ describing the initial state
of the fermion waveform; we define from these the special initial frame vector
$\vec{V}=(v^{\dagger}\gamma^{\mu}w)\hat{e}_{\mu}-v^{\dagger}w\hat{I}$.

Now, consider the following argument. In previous work [15], it was shown that
the Klein-Gordon Propagator was the Fourier transform $I\rightarrow m$ of the
number of directed paths with the same action $I$. This number of direct paths
is the multinomial coefficient you will see in Theorem 1. Let us assert that
the fermionic path integral corresponds heuristically to the number of
fermionic paths with some other (as of yet undetermined) action I. We now may
introduce an action for our fermions in the form of Equation 16 for some path
$\gamma$ in space time.

(16) $\mathcal{L}_{\vec{v},\vec{w}}=\int_{\gamma}(md\tau+d\eta)\\\
=\sum_{\vec{a}\in\gamma}(md_{l_{2}}^{*}(\vec{a})+\eta_{\vec{a}})$

Here $\eta_{\vec{a}}$ denotes the rapidity of $\vec{a}$ w.r.t to the last
frame. We expand upon its definition in Equation 17.

(17)
$i>0\implies\eta_{\vec{a}_{i}}=arccosh(\frac{a_{i-1}*a_{i}}{d_{n}(0,a_{i-1})d_{n}(0,a_{i})})\\\
i=0\implies\eta_{\vec{a}_{i}}=arccosh(\frac{\vec{V}*a_{0}}{d_{n}(0,\vec{V})d_{n}(0,\vec{a}_{0})})$

Then, our assertion is the following. We denote as lattice fermionic
propagator $K_{n,\frac{1}{2}}(\vec{x},\vec{y})^{\vec{v},\vec{W}}$ the same
path indexed sum over $\Gamma_{n}^{\vec{x},\vec{y}}$ as Equation 8. We denote
the limit under $\mathcal{T}_{cont}$ of this discrete object as
$K_{n,\frac{1}{2}}(\vec{x},\vec{y})^{cont,\vec{v},\vec{W}}$. We denote the
limit of these continuous objects (regulated as in Equation 14) to be
$K_{\frac{1}{2}}(\vec{x},\vec{y})^{Feyn,cont,\vec{v},\vec{w}}$. This is the
object that we assert will converge to the Dirac fermion propagator.

Note that the a superscript feyn was added to this propagator, meaning path
differences include spontaneous switches $m\rightarrow-m$ throughout the path.
Just as with the $d>1$ scalar propagator, much of the arguments required to
show Equation 41 was developed in previous work, and we need only mention
where new arguments are required.

###  2.2. Discussion on Interactions

In this section, we introduce the manner in which peturbative QFT emerges from
this picture and how it may lend itself to more analytical expressions for
interactions. Consider the following interaction Lagrangian (Equation 18).

(18) $\mathcal{L}=\int
dx^{d}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-m^{2}\phi^{2}+\lambda\phi^{3})$

Methods by Feynman [16],[19] and others showed that this nonlinear interaction
could be approximated by summing over convolutions of free propagators; these
convolutions being indexed by Feynman diagrams (as in Figure 2).

Figure 2. The 3 tree Level Contribution to the Four Field Correlator in
$\lambda^{3}$ Theory [19]

The contribution you obtain from Figure 2 is $\lambda^{2}$ times the
convolution of three propagators. Namely, the propagator in with momentum
$\vec{k}$, and integrating in the center over momenta $\vec{q}$ where the two
paths in the loop take on $\vec{q}$ and $\vec{k}-\vec{q}$. This convolution is
the Fourier transform of a product of free propagators; for instance, the
leftmost diagram in Figure 2 would give Equation 19 as its contribution to the
interaction propagator.

(19) $(i\lambda)^{2}(\frac{1}{i})^{5}\int
d^{d}yd^{d}zK_{l_{2}^{*}}^{Feyn,cont}(y,z)(K_{l_{2}^{x}}^{Feyn,cont}(\vec{x}_{1},y)K_{l_{2}^{x}}^{Feyn,cont}(\vec{x}_{2},y)K_{l_{2}^{x}}^{Feyn,cont}(y,\vec{x}_{3})K_{l_{2}^{x}}^{Feyn,cont}(y,\vec{x}_{4}))$

If we take the inverse Fourier transform of Equation 19, it passes inside the
integrals as the transform is not in terms of any spatial coefficients. Each
propagator’s inverse Fourier transform is the volume of directed paths between
its endpoints (i.e. it is the continuum multinomial coefficients), and we know
that the original expression involves convolutions across points adjoining
propagators; by the relation between convolutions and products via the fourier
transform we would obtain for our expression a product of the volumes of path
spaces. This is what we would obtain, however, if we wanted to measure the
volume of directed paths that arrange themselves to ‘look like’ the Feynman
diagram in Figure 2. That is, if we want to see the contribution to some
correlator for this theory associated to the first power of $\lambda$, then
that will be a sum over direct path-space volumes who agree with Feynman
diagrams having $1$ split in them. This would show how some geometric model of
the path integral can give rise to any peturbative QFT, should we be able to
define spin $>.5$ propagators in this geometric setting. A more in-depth
theory relating to this result is proven in Section 7.

###  2.3. Statement of Theorems

Theorem 1 is a extension of the work of [15] to $d>1$:

######  Theorem 1.

Let $d\in\mathbb{N}$. Furthermore, say that the distribution of rational
points on $S^{d}\in\mathbb{R}^{d+1}$ is equidistributed and that the
asymptotic density of $\mathcal{A}_{n}$ as $t\rightarrow\infty$ is
$\Theta(t^{d})$. Then, for $\vec{x},\vec{y}\in\mathbb{R}^{d}\times\mathbb{R}$
we know $K_{d_{n}}^{cont}(\vec{x},\vec{y})$ exists as does its limit over $n$.
That limit becomes the function below.

$K_{l_{2}^{*}}^{Feyn,cont}(\vec{x},\vec{y})=\mathcal{F}|_{m}^{I}((1-\frac{\sum_{i=1}^{d}x_{i}^{2}+I^{2}}{t^{2}})^{\frac{d-2}{2}})=Cm^{\frac{d-1}{2}}H_{\frac{d-1}{2}}^{(2)}(md_{l_{2}^{*}}(0,\vec{x}))$

were $C$ is allowed to contain a complex phase. amd $\mathcal{F}$ denotes the
Fourier transform.

As shown in [10], this is the Klein Gordon propagator for a d-dimensional
scalar particle. Now, we have a Theorem for 2d Manifolds embedded on the
hyperboloid model of hyperbolic space.

######  Theorem 2.

Let $\vec{x},\vec{y}\in X^{cont}$ with associated Fuschian group $G$. We
denote the flat space propagator on $\mathbb{Z}^{2}\times\mathbb{Z}$ as
$K_{n}^{free,cont}(\vec{x},\vec{y})$. We have the following:

$K_{n}^{cont}(\vec{x},\vec{y})=\frac{\sum_{y^{\prime}\in
Orb_{G}(y),proj_{t}(y^{\prime}-\vec{x})\leq
proj_{x}(y^{\prime}-x)}K_{n}^{free,cont}(\vec{x},\vec{y}^{\prime})}{|\\{y^{\prime}\in
Orb(y)|proj_{x}(\vec{x}-\vec{y}^{\prime})\leq
proj_{t}(\vec{x}-\vec{y}^{\prime})\\}|}$

and therefore

$K_{l_{2}}^{cont}(\vec{x},\vec{y})=\frac{\sum_{y^{\prime}\in
Orb_{G}(y),proj_{t}(y^{\prime}-\vec{x})\leq
proj_{x}(y^{\prime}-x)}\mathcal{F}|_{I}^{m}((1-\frac{proj_{x}(\vec{x}-\vec{y^{\prime}})^{2}+proj_{y}(\vec{x}-\vec{y}^{\prime})^{2}+I^{2}}{proj_{t}(\vec{x}-\vec{y}^{\prime})^{2}})^{-\frac{1}{2}})}{|\\{y^{\prime}\in
Orb(y)|proj_{x}(\vec{x}-\vec{y}^{\prime})\leq
proj_{t}(\vec{x}-\vec{y}^{\prime})\\}|}$

This yields a simple conjecture, which we will explain further in Section 4,
concerning the Kallen-Lehmann form [19] of the manifold propagator.

######  Conjecture 1.

For some manifold M whose universal cover is $\mathbb{H}^{d}$ or
$\mathbb{R}^{d}$ consider the set $\\{\tau_{i}|y_{i}\in Orb(y)\\}$ in the
hyperboloid model (or pacman model from [15]. These deck transformations can
have their proper times $\tau_{i}$ partitioned into disjoint periodic spacings
collection that we denote by $\alpha_{i}\mathbb{N}+b_{i}$. This whole
partition we denote by $W_{M}$.

Then free scalar propagator defined on manifolds whose universal cover is
$\mathbb{H}^{d}$ or $\mathbb{R}^{d}$ have a Kallen-Lehmann power spectra
$\rho(m)$ equivalent to
$\delta(m)+\sum_{i=1}^{\infty}\frac{a_{i}\delta(m-m_{i})}{m}$ where
$\\{m_{i}\\}_{i=1}^{\infty}\in\mathbb{R}$ is a discrete set whose only cluster
point is zero but if $m_{i}\rightarrow 0$ then $a_{i}\rightarrow 0$. For each
$m_{i}$ there is some $\alpha_{i}$ such that $\alpha_{i}m_{i}\in
4\pi\mathbb{N}$ where $\alpha_{i}$ is among the set $W_{M}$. This is a theorem
for manifolds whose universal cover is $\mathbb{R}^{d}$.

Now, we have a theorem concerning Tiechmuller curves. This could be applicable
towards billiard problems. We present the theorem here:

######  Theorem 3.

Let $\vec{x},\vec{y}\in\partial X,\vec{x}\neq\vec{y}$ for X a Tiechmuller
curve (as defined in Section 9). Let

$\Gamma_{n,int,\vec{a}_{1},\vec{a}_{2}}^{\vec{x},\vec{y}}=\Gamma_{n}^{\vec{x},\vec{y}}\cap\\{\gamma|\gamma\cap\partial
X\subset\partial\gamma,\Delta\gamma_{i}=\vec{a}_{1},\Delta\gamma_{f}=\vec{a}_{2}\\}$

where $\Delta\gamma_{i},\Delta\gamma_{f}$ denotes the slope of $\gamma$ at the
initial and final points. If we put $0$ in either direction argument of
$\Gamma^{\vec{x},\vec{y}}_{n,int,\vec{a}_{1},\vec{a}_{2}}$ we mean that the
direction at that point is unconstrained. Let
$K_{n,int,\vec{a}_{1},\vec{a}_{2}}^{feyn,cont}$ denote the path integral
propagator taken over this latter collection of directed paths (and
$K_{n,int}^{feyn,cont}$ denote that which is taken over the former). Then
these propagators exist, $K_{n}^{Feyn,cont}(\vec{x},\vec{y})$ exists, and
satisfies Equation 20

(20) $\small
K_{n}^{Feyn,cont}(\vec{x},\vec{y})=K_{n,int,0,0}^{Feyn,cont}(\vec{x},\vec{y})\\\
+\sum_{M=1}^{\infty}\mathcal{F}|_{m}^{I}(\int...\int\int_{\partial
X}...\int_{\partial
X}\sum_{\vec{a}_{1},...,\vec{a}_{M-1}\in\mathcal{A}_{n}}\Pi_{j=1}^{j=M-1}\mathcal{F}^{-1}|_{m}^{I_{0}}(K_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{Feyn,cont}(w_{j},w_{j+1}))\\\
\mathcal{F}^{-1}|_{m}^{I_{j}}(K_{n,int,\vec{a}_{M-1},0}^{Feyn,cont}(\vec{x},\omega_{1}))\mathcal{F}^{-1}|_{m}^{I_{M-1}}(K_{n,int,\vec{a}_{M-1},0}^{Feyn,cont}(w_{M-1},\vec{y})\delta(I-\sum_{j=1}^{M-1}I_{j}))\Pi_{j=1}^{M-1}dw_{j}dI_{j})\normalsize$

Finally, we find the fermion propagator in $d=1$

######  Theorem 4.

Then for $\vec{x},\vec{y}\in\mathbb{Z}^{d}\times\mathbb{Z}$ we know
$K_{n,\frac{1}{2}}(\vec{x},\vec{y})^{\mu,\nu}(\vec{x},\vec{y})$ exists and

$K_{l_{2}^{*},\frac{1}{2}}^{Feyn,cont}(\vec{x},\vec{y})=(i\gamma_{\mu}\partial^{\mu}-m)\mathcal{F}|_{m}^{I}((1-\frac{\sum_{i=1}^{d}x_{i}^{2}+I^{2}}{t^{2}})^{\frac{d-2}{2}})$

This, as it is the operator $\gamma_{u}p^{u}-m$ operating on the scalar
propagator in [10], is the Dirac propagator for a d-dimensional Dirac spinor
particle. The theorems and conjectures related to a general model of QFT are
posed in Section 7.

##  3\. Proof of the $d>1$ Flat Space Theorem

First, let us obtain the proof of Theorem 1. This is a direct extension of the
work in [15], and therefore I will only be including the minimum necessary
details to said proof.

###### Proof.

We let $\vec{x},\vec{y}\in X=\mathbb{Z}\times\mathbb{Z}$ such that
$proj_{t}(\vec{y}-\vec{x})>0$. Let $\mathcal{A}_{n}$ be as derived in Theorem
10. For $\gamma\in\Gamma_{n}$ we know by Theorem 13 that $\gamma$’s difference
sequence may be drawn from $\mathcal{A}_{n}$; let $I_{\vec{a}}$ be the number
of elements of the difference sequence of $\gamma$ equal to $\vec{a}$. Then,
we note that for any path the following properties hold:

  * •

$\mathcal{I}_{x_{\mu}}:\sum_{a\in\mathcal{A}_{n}}I_{a}proj_{x_{\mu}}(\vec{a})=proj_{x_{\mu}}(\vec{y}-\vec{x})$

  * •

$\mathcal{II}:I=\sum_{a\in\mathcal{A}_{n}}I_{a}d_{n}(0,\vec{a})$

Here, we are denoting $\rho_{n}(\gamma)$ by I. Using the freedom to swap
difference sequences of some $\gamma$ as done in the analogous step in [15],
we have $K_{n}(\vec{y},\vec{x})$ takes on the form in Equation 21.

(21)
$K_{n}(\vec{y},\vec{x})=\sum_{I}(\sum_{\\{I_{a}\\}_{\vec{a}\in\mathcal{A}_{n}}\in\mathcal{I},\mathcal{II}}\begin{pmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}I_{a}\\\
\Pi_{a\in\mathcal{A}_{n}}(I_{\vec{a}})\end{pmatrix})e^{mI}$

Just as in [15], any solution to $\mathcal{I}$ and $\mathcal{II}$ can be
obtained by doing the following. We let
$I_{(0,1)}=I-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1)\\}}I_{\vec{a}}d_{n}(0,\vec{a})$.
We constraint $I_{(0,1)}$ because it’s the element of $\mathcal{A}_{n}$ that
necessarily doesn’t have a position coordinate and can’t be used to remove the
other constraints in $\mathcal{I}$. For the other d+1 constraints, we first
note that $\mathcal{A}_{n}$ contains at least $2d$ many simple null vectors
$\pm\vec{e}_{i}+\hat{t}$ such that $d_{n}(0,\pm\vec{e}_{i}+\hat{t})=0$. We
remove all remaining spatial constraints in $\mathcal{I}$ with these because
they don’t mix with condition $\mathcal{II}$, letting
$I_{(e_{i},1)}=proj_{x_{i}}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{I}}(\vec{a})$.
We note that the equations we obtain to remove all conditions so far
constrained have the property that each variable only features in the equation
that constrains it, meaning that we removed the implicit nature of condition
$\mathcal{II}$ and the spatial parts of $\mathcal{I}$. Now, we can remove the
last constraint in Equation 22

(22)
$I_{-\vec{e}_{1}}=proj_{t}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}}I_{\vec{a}}proj_{t}(\vec{a})\\\
=proj_{t}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1),(\vec{e}_{i},1)\\}}proj_{t}(\vec{a})-\sum_{i=1}^{d}I_{(e_{i},1)}-I_{(0,1)}\\\
=proj_{t}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1),(\vec{e}_{i},1)\\}}I_{\vec{a}}proj_{t}(\vec{a})-\sum_{i=1}^{d}(proj_{x_{i}}(\vec{y}-\vec{x})\\\
-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{i}}(\vec{a}))-(I-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1)\\}}I_{\vec{a}}d_{n}(0,\vec{a}))$

With these replacements, the discrete propagator of Equation 21 becomes
Equation 23.

(23)
$K_{n}(\vec{y},\vec{x})=\sum_{I}(\sum_{\Omega}\begin{pmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}I_{a}\\\
\Pi_{a\in\mathcal{A}_{n}\setminus\\{(0,1),(e_{i},1),(-e_{1},1)\\}}(I_{\vec{a}}),f_{x_{0}},f_{I},\Pi_{i=1}^{d}f_{x_{i}}\end{pmatrix})e^{mI}\\\
\textrm{ where
}f_{x_{i}}=(proj_{x_{i}}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{I}}(\vec{a})),f_{I}=I-\sum_{\vec{a}\in\mathcal{A}_{n}}I_{\vec{a}}d_{n}(0,\vec{a})\\\
\textrm{,
}f_{x_{0}}=proj_{t}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1),(\vec{e}_{i},1)\\}}I_{\vec{a}}proj_{t}(\vec{a})-\sum_{i=1}^{d}(proj_{x_{i}}(\vec{y}-\vec{x})\\\
-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{i}}(\vec{a}))-(I-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1)\\}}I_{\vec{a}}d_{n}(0,\vec{a}))\\\
\textrm{, }\sum_{\Omega}(*)=\sum_{I_{\vec{a}_{1}}=0}^{C_{1}}\textrm{ ...
}\vec{a}\in\mathcal{A}\setminus\\{(0,1),(e_{i},1),(-e_{1},1)\\}\textrm{ ...
}\sum_{I_{\vec{a}_{|\mathcal{A}_{n}|-d-2}}=0}^{C_{|\mathcal{A}_{k}|-d-2}}(*)\\\
\textrm{, and
}C_{i}=min(\frac{proj_{t}(\vec{y}-\vec{x})-\sum_{j=1}^{i-1}I_{\vec{a}_{j}}proj_{t}(\vec{a}_{j})}{proj_{t}(\vec{a}_{i})},\frac{I-\sum_{j=1}^{i-1}I_{\vec{a}_{j}}d_{n}(0,\vec{a}_{j})}{proj_{t}d_{n}(0,\vec{a}_{i})},l.c.\textrm{
constraint })$

The Equation 23 is intricate, so I will endeavor to explain each part. The
multinomial coefficient term from Equation 21 has some of the $I_{\vec{a}}$
replaced with the expressions we derived before Equation 23. The other
$I_{\vec{a}}$ are completely unconstrained, other than the constraints placed
upon them in the sum $\sum_{\Omega}(*)$. This is a simple sum over all
possible values each $I_{\vec{a}_{i}}$ can take with the upper bound $C_{j}$
being determined by noting each step is monotonic in the time and phase
components. Our expression in Equation 23 must necessarily have non-negative
integer $f_{I}$ (as $I-\sum I_{\vec{a}}d_{n}(0,\vec{a})$ is granted to be non-
negative from the bounds on $\sum_{\Omega}$) and $f_{x_{0}}$. So long as our
steps never take us outside of the inverse light cone of $\vec{y}$,
$f_{x_{i}}$ will also be positive. This is labelled as l.c. constraint in
Equation 23 as is another simple linear upper bound like the other three in
$C_{j}$. With the discrete propagator specified, Theorem 8 allows us to
immediately write the continuum version $K_{n}^{cont}$.

(24)
$K_{n}^{cont}(\vec{y},\vec{x})=\mathcal{F}|_{I}^{m}(\int_{\Omega}\begin{Bmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}I_{a}\\\
\Pi_{a\in\mathcal{A}_{n}\setminus\\{(0,1),(e_{i},1),(-e_{1},1)\\}}(I_{\vec{a}}),f_{x_{0}},f_{I},\Pi_{i=1}^{d}f_{x_{i}}\end{Bmatrix})\\\
\textrm{ where
}f_{x_{i}}=(proj_{x_{i}}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{I}}(\vec{a})),f_{I}=I-\sum_{\vec{a}\in\mathcal{A}_{n}}I_{\vec{a}}d_{n}(0,\vec{a})\\\
\textrm{,
}f_{x_{0}}=proj_{t}(\vec{y}-\vec{x})-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1),(\vec{e}_{i},1)\\}}I_{\vec{a}}proj_{t}(\vec{a})-\sum_{i=1}^{d}(proj_{x_{i}}(\vec{y}-\vec{x})\\\
-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(e_{i},1)\\}}I_{\vec{a}}proj_{x_{i}}(\vec{a}))-(I-\sum_{\vec{a}\in\mathcal{A}_{n}\setminus\\{(0,1)\\}}I_{\vec{a}}d_{n}(0,\vec{a}))\\\
\textrm{, }\int_{\Omega}(*)=\int_{I_{\vec{a}_{1}}=0}^{C_{1}}\textrm{ ...
}\vec{a}\in\mathcal{A}\setminus\\{(0,1),(e_{i},1),(-e_{1},1)\\}\textrm{ ...
}\int_{I_{\vec{a}_{|\mathcal{A}_{n}|-d-2}}=0}^{C_{|\mathcal{A}_{k}|-d-2}}(*)\\\
\textrm{, and
}C_{i}=min(\frac{proj_{t}(\vec{y}-\vec{x})-\sum_{j=1}^{i-1}I_{\vec{a}_{j}}proj_{t}(\vec{a}_{j})}{proj_{t}(\vec{a}_{i})},\frac{I-\sum_{j=1}^{i-1}I_{\vec{a}_{j}}d_{n}(0,\vec{a}_{j})}{proj_{t}d_{n}(0,\vec{a}_{i})},l.c.\textrm{
constraint })$

The next part of the proof from [15] involves showing that $K_{n}^{cont}$ is a
Cauchy sequence in the $sup$ norm on functions. Having the spherical
equidistribution and density results for pythagorean tupples, the exact same
scaling arguments would work; $d_{l_{2}^{*}}$ is still absolutely continuous
w.r.t. the Euclidean norm on time-like paths in higher dimensions. This fact,
which is necessary to demonstrate the sequence of functions $K_{n}^{cont}$ is
Cauchy, holds in this case as well. Therefore, we may assume $K_{l_{2}^{*}}$
exists. In the case of $d=1$ [15], the final step after existence relied on
demonstrating that the limiting object is invariant w.r.t rotations in $x$ and
$I$ and boosts along $t$. For this purpose, we would work with a set of
vectors $\mathcal{A}_{p}$ obeying a discrete rotational and hyperbolic
symmetry that converged to a continuous in the limit of p and showed that the
corresponding sequence of $K_{p}$ would converge to the same limit as that
obtained from the pythagorean triples. This process is explained in detail in
[15]; the only portion of the proof that do not generalize is the last step.
In the last step, we treat the continuum multinomial coefficients as pdfs, and
use Radon Nykodym theorem to assert that if the law of
$\theta=sin(\frac{I}{\tau})$ is uniform then that of I is
$(\tau^{2}-I^{2})^{-.5}$. In this case, the desired law of $I$ is
$(\tau^{2}-I^{2})^{\frac{d-2}{2}}$, or the marginal distribution of points
located on the $d+1$ sphere. That is, the distribution of $I$ is as though it
were a single coordinate on a sphere of radius $t$ in $d+1$ dimensions; this
is exactly the distribution you must have if the set $\\{x_{i},I,t\\}$ is
invariant under rotations as obtained from the asymptotic rotational
invariance of the pythagorean triples. This obtains the desired result.

∎

##  4\. Hyperboloid Model Path Integral

The proof of Theorem 2 is an excercise in definitions; the hard part is
Theorem 1 for $d=2$. Even so, let $X$ and $X^{cont}$ be
$\mathbb{Z}\times\mathbb{Z}^{2}$ and $\mathbb{R}\times\mathbb{R}^{2}$ with
equivalence relation $\sim$ as defined in the context of Theorem 2. Then, we
begin the proof.

###### Proof.

First, lets write
$K_{n}(\vec{x},\vec{y})=\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}}e^{im\rho_{n}(\gamma)}$.
We note that a path from $\vec{x}\rightarrow\vec{y}$ is uniquely specified by
what homotopy group elements it ‘enacts’, or its final point upon lifting to
the uniform hyperboloid cover and performing the same steps. This takes paths
in $\mathbb{R}^{2}\times\mathbb{R}$ with our equivalence relation structure
into those same paths $\mathbb{R}^{2}\times\mathbb{R}$ without said structure.
Therefore $\Gamma_{n}^{\vec{x},\vec{y}}=\bigsqcup_{y^{\prime}\in
Orb(y)}\Gamma_{n}^{\vec{x},\vec{y}^{\prime},free}$ and we have Equation 25 and
$\Gamma_{n}^{\vec{x},\vec{y}^{\prime},free}$ are paths in
$\mathbb{Z}^{2}\times\mathbb{Z}$ without the equivalence structure.

(25) $K_{n}(\vec{x},\vec{y})=\sum_{y^{\prime}\in
Orb(y)}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y^{\prime}},free}}e^{im\rho_{n}(\gamma)}$

Now, the operator $\mathcal{T}_{m}^{cont}$, when applied to
$K_{n}(\vec{x},\vec{y})$, will commute past the first sum. This is because
$\mathcal{T}_{m}^{cont}$ only cares about the number of linear segments in a
path in $\Gamma_{n}^{\vec{x},\vec{y}}$ and that would be independent w.r.t.
what it translates into as you lift it into its corresponding path in
$\mathbb{Z}\times\mathbb{Z}^{2}$. So, we obtain Equation 26

(26)
$K_{n}^{cont}(\vec{x},\vec{y})=lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}K_{n}([m*\vec{x}],[m*\vec{y}])|_{m}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}\\\
lim_{m\rightarrow\infty}\frac{\sum_{y^{\prime}\in
Orb(y)}\mathcal{T}_{cont}^{m}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y^{\prime}},free}}e^{im\rho_{n}(\gamma)}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}\\\
\sum_{y^{\prime}\in
Orb(y)}lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y^{\prime}},free}}e^{im\rho_{n}(\gamma)}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}$

Now, we can consider multiplying and dividing by some factor related to the
maximum paths in the free case, and obtain Equation 27.

(27) $K_{n}^{cont}(\vec{x},\vec{y})=\sum_{y^{\prime}\in
Orb(y)}lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}},free}|_{m}\right|))}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}\\\
\frac{\mathcal{T}_{cont}^{m}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y^{\prime}},free}}e^{im\rho_{n}(\gamma)}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}},free}|_{m}\right|))}$

We note that the maximum number of paths for both the equivalence class and
the free case grow at the same rate w.r.t $t$, and along with that that the
denominator (equivalence class) maxima is a maxima over the sum of disjoint
elements in the upper maxima. Since all the elements are positive (and the
maxima achieved), we expect this ratio to converge to inverse of the number of
deck lifts you have of the point $y^{\prime}$, or $|Orb(y)|$, that have the
property that they lie within the light cone of x. This obtains Equation 28.

(28) $K_{n}^{cont}(\vec{x},\vec{y})=\frac{\sum_{y^{\prime}\in
Orb(y)}K_{n}^{cont,free}(\vec{x},\vec{y}^{\prime})}{|y^{\prime}\in
Orb(y)|proj_{x}(\vec{x}-\vec{y}^{\prime})\leq
proj_{t}(\vec{x}-\vec{y}^{\prime})|}$

Now, as in the proof of Theorem 1, we include negative steps in $I$ to move to
$K_{n}^{cont,feyn}$. This will not affect any of the steps previously
mentioned. Furthermore, allowing more directions by increasing $m$ in the
calculation of the continuous propagator (when thought of as a sum over path
polytopes) does nothing to the discrete sum in Equation 28. Therefore, the
limit over m passes through this sum and applies directly to
$K_{n}^{cont,free}$ in Equation 28. Since Theorem 1 holds in $d=2$, this
proves the desired result.

∎

Let’s do a really simple non-compact Riemann surface example, the 3 branched
cylinder. This entails finding an explicit equation for its deck
transformations on some model of hyperbolic space and mapping it to the
Hyperboloid (where our path integral is defined). We note any automorphism of
the unit disk has the form $\frac{az+b}{\bar{b}z+\bar{a}}$ where
$|a|^{2}-|b|^{2}=1$. The universal cover of the 3 branched cylinder is shown
in Figure 3

Figure 3. Universal Cover Mapping of the 3 Branched Cylinder

We can note that we should be able to generate this tessellation with just
three transformations; automorphisms of the unit disk that take the inner-most
triangle into its adjacent 3 other triangles. We could then use compositions
of these automorphisms to generate all of $\pi_{1}$ for the 3 branched
cylinder. We note that the 3 points on the ends of this interior triangle are
$e^{i(\\{\frac{\pi}{2},\frac{\pi}{2}+\frac{2\pi}{3},\frac{\pi}{2}+\frac{4\pi}{3}\\})}$.
Let’s first construct $\mathfrak{d}_{1}$, the deck transformation which takes
the central triangle to its left neighbor. We know it fixes
$e^{i(\frac{\pi}{2}+\frac{2\pi}{3})}$ and should take
$e^{i\frac{\pi}{2}}\rightarrow e^{i(\frac{\pi}{2}+\frac{\pi}{3})}$ and
$e^{i(\frac{\pi}{2}+\frac{4\pi}{3})}\rightarrow e^{i(\frac{\pi}{2})}$. This
should completely constrain a and b, and indeed we obtain Equation 29.

(29)
$\mathfrak{d}_{1}(z)=-\frac{((2-i)+\sqrt{3})(i+2z)}{((2+i)+\sqrt{3})(2i+z)}$

If we call $\mathfrak{d}_{2}(z)$ and $\mathfrak{d}_{3}(z)$ the right and up
triangles, respectively, then it should be equal to
$e^{-i\frac{2\pi}{3}}\mathfrak{d}_{1}(z)$ and
$e^{-i\frac{4\pi}{3}}\mathfrak{d}_{1}(z)$ as their image is just rotated a
bit. Composition by these three maps gives the entire homotopy group. The
conversion between points on the Poincare Model and the hyperboloid model is
well known, and if $z$ in the unit disk is written $re^{i\theta}$, then the
conversion is in Equation 30.

(30)
$t=\frac{1+r^{2}}{1-r^{2}},x=\frac{rcos(\theta)}{1-r^{2}},y=\frac{rsin(\theta)}{1-r^{2}}\\\
r=\frac{\sqrt{x^{2}+y^{2}}}{(1+t)},\theta=arctan(\frac{x}{y}),$

This gives Equation 31 for the value of $K_{l_{2}^{*}}(\vec{x},\vec{y})$ for
$\vec{x}$ and $\vec{y}$ on the 3 branched cylinder:

(31)
$\frac{\sum_{y^{\prime}=\mathfrak{d}_{i}^{(n_{i})}(\frac{proj_{x}(\vec{y})+iproj_{y}(\vec{y})}{1+proj_{t}(\vec{y})}),proj_{t}(\vec{y^{\prime}}-\vec{x})\leq
proj_{x}(\vec{y^{\prime}}-\vec{x})}\mathcal{F}|_{I}^{m}(\sqrt{1-\frac{\sum_{i\in\\{x,y\\}}proj_{i}(\vec{x}-\vec{y}^{\prime})^{2}+I^{2}}{proj_{t}(\vec{x}-\vec{y}^{\prime})^{2}}})}{|\\{y^{\prime}=\mathfrak{d}_{i}^{(n_{i})}(\frac{proj_{x}(\vec{y})+iproj_{y}(\vec{y})}{1+proj_{t}(\vec{y})}),proj_{t}(\vec{y^{\prime}}-\vec{x})\leq
proj_{x}(\vec{y^{\prime}}-\vec{x})\\}|}$

where $\vec{n}$ is a Smirnov word of arbitrary length composed from three
letters and $n_{i}$ denotes letter frequency (this encodes arbitrary
composition of $\mathfrak{d}_{i}$). This derivation demonstrates how the most
difficult part in constructing the propagator for arbitrary 2 manifold is to
find an explicit construction for the manifold’s deck transformations; once
this is complete, one may simply plug it into this discrete sum. One can
envision that an alternate way to obtain the KG propagator on manifolds could
then be used with this formula to probe deck transformations.

###  4.1. Comments on the 1$\times$ 3 Case

The results of Theorem 1 and Theorem 1,2 seem immediately extendable to $d=3$.
If the pythagorean triples have the desired properties defined in the
hypothesis of Theorem 1 for 5 tupples of pythagorean triples, then our
procedure to generate the $K_{l_{2}}^{*}$ propagator in $\mathbb{R}^{2}$ and
$\mathbb{H}$ extend to analogous procedures to define propagators for
$\mathbb{R}^{3}\textrm{ and }\mathbb{H}^{3}$. These constant curvature spaces
are clearly embeddable with minkowski signature in $4$ space (they would
inherit Minkowski and de-Sitter structures as $4d$ manifolds). The same idea
of covering $4$ space in slides of these three manifolds (hyperplanes and 1
sheet hyperboloids resp.) would immediately extend Theorem 2. Similarly, if we
take the product of $\mathbb{H}^{2}$ embedding in $1\times 2$ space and
$\mathbb{R}$ in itself, we get an analogous immersion of
$\mathbb{H}^{2}\times\mathbb{R}$. Our work applies to these three Thurston
Model geometries.

If we wanted to be exhaustive, then according to the geometrization theorem of
3 dimensional, compact manifolds [4], we would need to define our procedure on
5 other simple, prime manifolds, and then define the propagator over the
spherical cuts used to glue prime manifolds together. The procedure to glue
manifolds together in geometrization is defined as the connected sum; one puts
an equivalence relation on a copy of $\mathbb{S}^{d}$ inside each manifold
thereby ‘glue-ing’ them together. In Section 5, we introduce the method by
which we would rigorously define the propagator over the gluing and that is by
integrating over the boundary of the ball all tupples of path to a boundary
point and then away from it (and possibly passing through the point multiple
times). The author is convinced that any more rigorous treatment of gluing
would be a tedious, but straightforward application of the work in Section 5.

The Nil geometry and the universal cover of $SL(2,R)$ are described as
‘twisted’ versions of $\mathbb{R}^{2}\times\mathbb{R}$ (i.e. something like a
vector bundle over $\mathbb{R}$) and $\mathbb{H}^{2}\times\mathbb{R}$
respectively suggesting an extension to this setting may not be too difficult.
Therefore, the difficult simply connected prime manifolds for which our
definitions may not have an definition are $Sol,\mathbb{S}^{3}\textrm{ and
}\mathbb{S}^{2}\times\mathbb{R}$. Sol is rather complicated, so the next
domain to move to is $\mathbb{S}^{2}$. Despite it being the universal cover
for only one manifold in $d=2$, defining it for this manifold could extend
$\mathbb{S}^{3}$ and ultimately cover the definition for all but Sol. Then,
every compact 3 manifold could have its propagator defined by our deck
extension procedure on a simply connected 3 manifold and for simply connected
3 manifolds via stitching together our discrete paths between copies of the 8
simply connected prime manifolds. The author also would like to note that in
the case of high dimensional manifolds, the continuum multinomial should be
integrable across the surgery cuts used for their classification. This may
yield the scalar path integral in a much more general regime.

###  4.2. Kallen-Lehmann Form

The Kallen-Lehmenn form [19] of the QFT propagator takes the form of Equation
32. It realizes the two point propagator of a complicated non-linear QFT as a
sum over many different free particle propagators with different masses; any
sharp prominence in $\rho(m)$ normally correspond to resonances or particles
that arise naturally from the theory. It is a central question of QFT to
demonstrate that certain 2 point correlators have no sharp prominence in
$\rho(m)$ within some neighborhood of $m=0$ [19].

(32) $\int dm\rho(m)H_{d}(x,m)=\pm\int
dm\rho(m)H_{d}(0-m)=\pm\mathcal{F}|_{I}^{0}(\rho(I)(\tau^{2}-I^{2})^{\frac{d-2}{2}})$

In this section, the path integral on manifolds with $\mathbb{H}^{d}$ as their
universal cover was demonstrated to be the discrete sum over flat space
propagators originating from the origin to points in the image of deck
transformations of our destination point. That is, the propagator (instead of
being the marginal distribution of a single sphere centered at the origin) was
the sum of marginal distributions from multiple spheres originating at the
deck transformation images of the starting point. From simple multiplication
this implies $\rho(I)=\frac{1}{N}\sum_{y^{\prime}\in
Orb(y)}(\frac{\tau_{i}^{2}-I^{2}}{\tau^{2}-I^{2}})^{\frac{d-2}{2}}rect(\frac{I}{\tau_{i}})$.
Now in $d=2$ this simplifies to $\rho(I)=\frac{1}{N}\sum_{y^{\prime}\in
Orb(y)}rect(\frac{I}{\tau_{i}})$ and in that case
$\rho(m)=\frac{1}{N}\sum_{y^{\prime}\in
Orb(y)}\tau_{i}sinc(\frac{\tau_{i}m}{2})$. It is the purpose of this section
to gather evidence that this converges in large N limit as a distribution to
the distribution $\delta(m)-\sum_{i=1}^{\infty}\frac{a_{i}\delta(m-m_{i})}{m}$
where $\\{m_{i}\\}_{i=1}^{\infty}$ are the discrete spectrum of this
propagator and while $0$ is necessarily a cluster point of this set for all
$m_{i}\rightarrow 0$, we have $a_{i}\rightarrow 0$ (this last condition is
somewhat like the existence of a mass gap but unfortunately we seemingly dont
have a true mass gap).

To obtain this result, consider for $m\in(-\infty,\infty)$ the set

$M_{N}=\\{sin(\frac{\tau_{i}m}{2})|\tau_{i}\in Orb(y),proj_{t}(y)\geq N\\}$

Up to order $N$ the series $\sum_{m\in M_{N}}m$ is either $o(N)$ or $aN+o(N)$
for $a\in[-1,1]$ as the range of $sin$ is restricted to $[-1,1]$. In the first
case let $\sum_{m\in M_{N}}m=f(n)+o(f(n))$, then the sequence of running
averages converges (by the squeeze thoerem) to $0$ as it is bounded above and
below asymptotically by $\pm 2f(N)N^{-1}\rightarrow 0$. The same reasoning
asserts that the running average in the other case converges pointwise to $a$.

We now need to see when $\frac{\tau_{i}m}{2}$ is uniform w.r.t $2\pi$; on this
collection of $m$ our pointwise limit of $\rho(m)$ becomes zero. The nature of
$\tau_{i}$, the proper time between the origin and deck transformations of the
target point, suggests that there is some collection
$\\{(\alpha_{i},\beta_{i})\\}_{i=1}^{N}$ such that
$\tau^{\prime}_{i}\in\cup_{i}\\{\alpha_{i}\mathbb{N}+\beta_{i}\\}$ (or that
this is approximately true). Assume there is some subset of
$\\{\tau^{\prime}_{i}\\}=\alpha\mathbb{N}+\beta$ and that on this subset
$\frac{\tau^{\prime}_{i}m}{2}$ is equidistributed mod $2\pi$. This occurs iff
(by Weyl’s equidistribution theorem [21])
$\alpha\in\mathbb{R}\setminus\frac{4\pi}{m}\mathbb{Q}$. Returning back to our
set $\tau_{i}$, the set of $m$ which satisfies
$\frac{\alpha_{i}m}{4\pi}\in\mathbb{Q}$ for some $\alpha_{i}$ is countable,
and therefore so are the masses $m$ for which $\rho(m)$ converges to something
non-zero.

Now, we may also consider those sets of $m$ such that
$\frac{\alpha_{i}m}{4\pi}\in\mathbb{Q}$. While these aren’t equidistributed,
if we map them into $\mathbb{S}$ via the exponential map (so that the image
under sin would be the imaginary portion of this mapping) they may also be
radially symmetric on this circle. This implies their image under sin would
destructively interfere; therefore the only $m$ which does not necessarily
destructively decohere are those such that $\alpha_{i}m=4\pi\mathbb{N}$
(because these give a constant constructive value for $sin$. This gives us
that the set $m_{i}$ where the above running average doesn’t vanish is
necessarily countable and discrete except possibly at $m=0$. The only way to
ensure that $m=0$ is not a cluster point is if long $\alpha_{i}$ are
regulated. However if $\alpha_{i}$ is some $\alpha$ at which we enjoy
constructive interference, then so is any $N\alpha_{i}$. These higher modes
will be farther spaced out in our series for $\rho(m)$ and regulated by $N$.
Therefore, any sequence of masses $m_{i}\rightarrow m$ will have their $a_{i}$
reduced by this spacing and tend towards zero.

That this running average would converge to a sum of Dirac delta functions
(regulated by $m^{-1}$) is obtained as it is the limit of distributions that
concentrate non-zero ’mass’ around a discrete and countable set (where
$m^{-1}$ is obtained from the $m^{-1}$ in the $sinc$ function). If we multiply
any test function $\psi\in\mathbb{C}^{\infty}$ times
$\frac{1}{N}\sum_{y^{\prime}\in Orb(b)}\tau_{i}sinc(\frac{\tau_{i}m}{2})$ and
integrate, one may separate the integral into open sets about
$\\{\mathbb{B}_{N^{-.5}}(m_{i})\\}\cup\\{\mathbb{B}_{N^{-.5}}(0)\\}$ and its
complement. The integral of the complement is bounded by $N^{-1}$. Meanwhile
its value at $m_{i}$ is approximately $\frac{a_{i}}{m}\phi(m_{i})$ where
$a_{i}$ is independent of $m$ and test function. Here we note that in our
definitions we should have divided by $\sqrt{N}$ rather than $N$, so that the
integral at the point $m_{i}$ does not diminish as $N\rightarrow 0$.

Now, we state in what manner the periodicity of $\tau_{i}$ can be said to be
true. In the pacman universal cover of $\mathbb{T}^{2}$ by $\mathbb{R}^{2}$,
our deck transformations arrange themselves in a regular grid and the
$\tau_{i}$ would be minkowski distances between regular points on a grid and
some target point. Since the minkowski distance scales linearly with distance,
it is immediate that the minkowski distances (like euclidean distances) for
these points would occur along periodic distances from the target point. In
manifolds with $\mathbb{H}^{2}$ as their cover, we note that the orbits of our
target points will not be arranged along lattice points but along sheets of a
hyperbola. However, as the target point moves farther out from the origin the
local curvature of the hyperbola becomes asymptotically flat and lends
credence to the idea that $\tau_{i}$ may still be arranged (approximately) in
periodic orbits.

##  5\. Tiechmuller Curve Theorem

In this section, we prove existence and a nice relation for the propagators on
Tiechmuller Curves $X$ given in Theorem 3. It should become obvious how one
would define propagators over connected sum manifolds (as mentioned in Section
4) because this proof will involve integrating the propagator over boundaries.

###### Proof.

Consider the decomposition in Equation 33. $\bigotimes$ denotes a cartesian
product. This expression denotes the fact that if you have some
$\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}$ is must intersect (i.e. cross over)
$\partial X$ some i number of times. We can therefore equate $\Gamma_{n}$ into
a disjoint union over i of the set of paths that cross i times. If we then
look at the set of paths that cross over $i$ times, that is equal to some
cartesian $i$ dimensional product where the first entry is a path from you
origin point to the boundary (hitting the boundary with some angle) you have
$n-1$ paths between points on the boundary, and then a path from your boundary
to the last point originating with some initial angle. Once we take the
disjoint union over possible directions these junctures can take we get
Equation 33.

(33)
$\Gamma_{n}^{\vec{x},\vec{y}}=\bigsqcup_{i=1}^{N}(\bigsqcup_{a_{1},...,a_{N-1}\in\mathcal{A}_{n}}\Gamma_{n,int,0,\vec{a}_{1}}^{\vec{x},\partial\gamma}\bigotimes_{j=1}^{N-1}\Gamma_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{\partial\gamma,\partial\gamma}\bigotimes\Gamma_{n,int,\vec{a}_{N-1},0}^{\partial\gamma,\vec{y}})$

In this equation, $N$ is some finite number. If $D=diam(X)$, then
$N=\lceil\frac{D}{proj_{t}(\vec{y}-\vec{x})}\rceil$ is sufficient. We shall
let it be this value for now. Using Equation 33, and specifically the disjoint
union of the path sets, we obtain Equation 34 for $K_{n}^{feyn}$.

(34)
$K_{n}^{feyn}(\vec{x},\vec{y})=\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}}e^{im\rho_{n}(\gamma)}=\sum_{i=1}^{N}\sum_{a_{1},...,a_{N-1}\in\mathcal{A}_{n}}\\\
\sum_{\gamma\in\Gamma_{n,int,0,\vec{a}_{1}}^{\vec{x},\partial\gamma}\bigotimes_{j=1}^{i-1}\Gamma_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{\partial\gamma,\partial\gamma}\bigotimes\Gamma_{n,int,\vec{a}_{N-1},0}^{\partial\gamma,\vec{y}}}e^{im\rho_{n}(\gamma)}$

Now, we want to demonstrate the existence of the continuum object upon
applying $\mathcal{T}^{m}$ to Equation 34 and normalizing it. One can see that
this is a corollary of the work in Section 3 via the following argument. The
directed paths that lie in $\Gamma_{n,int,0,0}^{\vec{x},\vec{y},cont}$ may
still be partitioned into collections by the number of linear segments in said
paths. These collections are still finite dimensional spaces, and each has
less volume than it would have were it composed of any directed path as they
are confined to not cross $\partial X$. Since we have demonstrated the sum of
finite volumes of parameter spaces of $\Gamma_{n}^{\vec{x},\vec{y}}$ for
$\vec{x},\vec{y}\in\partial\gamma$ is absolutely convergent in Section 3, the
same must be true of the sum of volumes of directed path spaces for our
constrained case. Now consider
$\Gamma_{n,int,\vec{a}_{1},\vec{a}_{2}}^{\vec{x},\vec{y},cont}$. The
expression for the measure of $\Gamma_{n}^{\vec{x},\vec{y}}$ as a continuum
multinomial is composed of a sum over dyke words which capture the order of
directions one uses along a path. To obtain the measure of
$\Gamma_{n,int,\vec{a}_{1},\vec{a}_{2}}^{\vec{x},\vec{y},cont}$, we only sum
over the dyke words that begin with letter corresponding to $\vec{a}_{1}$ and
end with that corresponding to $\vec{a}_{2}$. This measure, for the same
reasons as $\Gamma_{n,\int,0,0}^{\vec{x},\vec{y},cont}$, will converge; it is
a series that is composed of reordering and diminishing of terms of the series
representing the measure of $\Gamma_{n}^{\vec{x},\vec{y}}$ and therefore will
also converge absolutely. The propagators over these path spaces will converge
as they represent changing the sign of terms in an absolutely convergent
series. Finally, $K_{n}^{fen,cont}$ must exist as it will be a finite sum (the
sum $\sum_{i=1}^{N}\sum_{a_{1},...,a_{N-1}\in\mathcal{A}_{n}}$ persists
through the application of $\mathcal{A}_{n}$) of the convergent propagators
over the continuum version of path spaces specified in Equation 33. Now, we
show the nice relation for $K_{n}^{feyn,cont}$.

In Section 1 we introduced the notation $\\#_{B}^{I}$ to denote the measure of
paths between two points as a function of their minkowski lengths $I$ and
showed in Section 3 demonstrated that $K_{n}^{feyn,cont}$ can be rigorously
treated as the fourier transform of $\\#_{B}^{I}$ from $I$ to m. As
$\\#_{B}^{I}$ doesn’t have enough indices to to capture the index $n$ from
$\mathcal{A}_{n}$ as well as many other nuances to our path spaces in Equation
33, we use our work from Section 3 to denote the measure of
$\Gamma_{n,int,\vec{a}_{N-1},0}^{\partial\gamma,\vec{y}}$ by
$\mathcal{F}^{-1}|_{m}^{I}(K_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{feyn,cont}(\vec{x},\vec{y}))$.
If we fix the points in $\partial\gamma$ in the expression
$\Gamma_{n,int,0,\vec{a}_{1},cont}^{\vec{x},\partial\gamma}\bigotimes_{j=1}^{i-1}\Gamma_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{\partial\gamma,\partial\gamma,cont}\bigotimes\Gamma_{n,int,\vec{a}_{N-1},0}^{\partial\gamma,\vec{y},cont}$,
then as a cartesian product is measure will be the product of measures of each
individual part. Namely we have that is would be Equation 35

(35)
$\mathcal{F}^{-1}|_{m}^{I_{0}}(K_{n,int,0,\vec{a}_{1}}^{feyn,cont}(\vec{x},\partial\gamma))\Pi_{j=1}^{i-1}\mathcal{F}^{-1}(K_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{feyn,cont}(\partial\gamma,\partial\gamma))\mathcal{F}^{-1}|_{m}^{I_{N}}(K_{n,int,\vec{a}_{N-1},0}^{feyn,cont}(\partial\gamma,\vec{y}))$

Our propagator is a sum over every possible collection of our boundary points
in the discrete setting; this becomes an integral under the limit by
$\mathcal{T}_{cont}$ and we obtain the Equation 36 for the measure of the
continuum paths that cross the boundary $N$ times.

(36) $\int_{\partial X}...\int_{\partial
X}\mathcal{F}^{-1}|_{m}^{I_{0}}(K_{n,int,0,\vec{a}_{1}}^{feyn,cont}(\vec{x},\omega_{1}))\Pi_{j=1}^{i-1}\mathcal{F}^{-1}|_{m}^{I_{j}}(K_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{feyn,cont}(\omega_{j},\omega_{j+1}))\\\
\mathcal{F}^{-1}|_{m}^{I_{N}}(K_{n,int,\vec{a}_{N-1},0}^{feyn,cont}(\omega_{N-1},\vec{y}))\Pi_{j=1}^{N-1}d\omega_{j}$

We make the observation that we have many distinct paths with lengths
$I_{1},...,I_{N-1}$ in Equation 36. Ultimately we only want to collect paths
that path through the boundary $N$ times and have the total length $I$. This
introduces the further integral and $\delta$ we observe in Equation 37

(37) $\small\int_{\mathbb{R}}...\int_{\mathbb{R}}\int_{\partial
X}...\int_{\partial
X}\mathcal{F}^{-1}|_{m}^{I_{0}}(K_{n,int,0,\vec{a}_{1}}^{feyn,cont}(\vec{x},\omega_{1}))\Pi_{j=1}^{i-1}\mathcal{F}^{-1}|_{m}^{I_{j}}(K_{n,int,\vec{a}_{j},\vec{a}_{j+1}}^{feyn,cont}(\omega_{j},\omega_{j+1}))\\\
\mathcal{F}^{-1}|_{m}^{I_{N}}(K_{n,int,\vec{a}_{N-1},0}^{feyn,cont}(\omega_{N-1},\vec{y}))\delta(I-\sum_{j=1}^{N-1}I_{j})\Pi_{h=1}^{N-1}d\omega_{j}dI_{j}\normalsize$

We pass the sum $\sum_{i=1}^{N}\sum_{a_{1},...,a_{N-1}\in\mathcal{A}_{n}}$
from Equation 33 into this expression. Finally we take the fourier transfrom
back from $I$ to $m$ to obtain the expression in Theorem 3

∎

##  6\. Fermion Path Integral

In this section we will prove Theorem 4.

###### Proof.

We begin by manipulating the Dirac propagator’s analytic form until it may be
realized as a fourier transform of some volume of directed paths. It is well-
known that we have the following Equation 38 in general relating the Dirac
propagator to the bosonic one [16]. Here, we use $\\#^{b}_{I}$ and
$\\#^{f}_{I}$ as shorthand for the volume of paths with $I$ as their bosonic
or fermionic action.

(38)
$\mathcal{F}|_{I}^{m}(\\#^{f}_{I})=(i\gamma^{\mu}\partial_{\mu}+m)\mathcal{F}|_{I}^{m}(\\#^{b}_{I})\\\
\mathcal{F}|_{I}^{m}(\\#^{f}_{I})=(i\gamma^{\mu}\partial_{\mu}-i(im))\mathcal{F}|_{I}^{m}(\\#^{b}_{I})\\\
\mathcal{F}|_{I}^{m}(\\#^{f}_{I})=\mathcal{F}|_{I}^{m}((i\gamma^{\mu}\partial_{\mu}-i\partial_{I})\\#^{b}_{I})\\\
\\#^{f}_{I}=i(\gamma^{\mu}\partial_{\mu}-\partial_{I})\\#^{b}_{I}$

Next, we note that from taking the inverse Fourier transform of the KG
propagator (as well as my previous work) we would obtain
$\\#^{b}_{I}\sim(t^{2}-x^{2}-I^{2})^{\frac{d-2}{2}}$. This and Equation 38
implies Equation 39.

(39)
$\\#^{f}_{I}=(\frac{d-2}{2})i(\gamma^{\mu}x_{\mu}-\mathbbm{1}I)(t^{2}-x^{2}-I^{2})^{\frac{d-4}{2}}\\\
=\frac{d-2}{2}i(\frac{\gamma^{\mu}x_{\mu}-\mathbbm{1}I}{\sqrt{t^{2}-x^{2}-I^{2}}})(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}$

Let’s consider the following. We will evaluate both sides of Equation 39 by
Dirac spinors $\vec{v},\vec{w}\in\mathbbm{C}^{2}$ (where evaluation is
performed by multiplying by $v^{\dagger}$ on the left and $\vec{w}$ on the
right. Keep in mind the initial frame vector
$\vec{V}=(v^{\dagger}\gamma^{\mu}w)\hat{e}_{\mu}-v^{\dagger}w\hat{I}$ as we
now denote the unit vector in along $(\vec{e}_{\mu},I)$ as $\vec{e}_{r}$ (i.e.
a sort of a radial vector). Then, Equation 39 remarkably becomes Equation 40

(40)
$\\#^{f}_{I}=\frac{d-2}{2}i(\vec{V}*\vec{e}_{r})(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}\\\
=C(cosh\eta_{\vec{V}})(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}\\\
=C(e^{\eta_{V}}+e^{-\eta_{V}})(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}$

where $\eta_{V}$ is the rapidity between the radial direction and $\vec{V}$
(and C is some constant dependent only on the magnitude of $\vec{v}$ and
$\vec{w}$).

It is a regular exercise in special relativity to show that the relationship
between inner products and cosine of an angle translates to minkowski inner
products and the cosh of rapidity; refer to [13] for further work on this.
Namely, $(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}$ is the marginal distribution
of a spherical distribution, which was shown in [15] and for scalar particles
to arise as volume of a rotationally uniform space of directed paths (except
in this case we have one less dimension than usual). The last equation of
Equation 40 was included as it is what we can massage to get the particle
lagrangian for the fermion as was down in Equation 8 (we want to get some
volume of paths times a complex phase factor). I now reintroduce the Fourier
term to further make this comparison and to obtain Equation 41.

(41)
$v^{\dagger}K^{\frac{1}{2}}(t,\vec{x})w=C\int_{-\infty}^{\infty}e^{imI}(e^{\eta_{V}}+e^{-\eta_{V}})(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}dI\\\
=C\int_{-\infty}^{\infty}e^{i(mI\pm
i\eta_{V})}(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}dI$

It is at this point that we move from manipulating the Dirac propagator to
equating it to a fourier transform of a volume of directed paths. We will do
so by proving Equation 42.

(42)
$lim_{n\rightarrow\infty}(lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}}e^{i(\sum_{\vec{a}\in\gamma}(md_{l_{2}}^{*}(\vec{a})+i\eta_{\vec{a}_{i}}))}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))})\\\
=C\sum_{b\in\\{\pm
1\\}}\int_{-\infty}^{\infty}e^{i(mI+ib\eta_{V})}(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}dI$

where the LHS is the continuum object defined as $K_{l_{2}}^{feyn,v,w}$ in
Section 6. The first easy step is to use the Einstein summation formula [13]
to obtain $\eta_{V}=\sum_{i}\eta_{\vec{a}_{i}}$. This, and Equation 21 from
Section 3, obtains for us Equation 43.

(43)
$lim_{m\rightarrow\infty}\frac{\mathcal{T}_{cont}^{m}\sum_{\gamma\in\Gamma_{n}^{\vec{x},\vec{y}}}e^{i(\sum_{\vec{a}\in\gamma}(md_{l_{2}}^{*}(\vec{a})-i\eta_{\vec{a}}))}}{\mathcal{T}_{cont}^{m}(max_{\vec{x^{\prime}},\vec{y^{\prime}}\in
X|_{m},proj_{t}(\vec{y}^{\prime}-\vec{x}^{\prime})=[nt]}(\left|\Gamma_{n}^{\vec{x^{\prime}},\vec{y^{\prime}}}|_{m}\right|))}\\\
=lim_{m\rightarrow\infty}C\mathcal{T}_{cont}^{m}\sum_{I}(\sum_{\\{I_{a}\\}_{\vec{a}\in\mathcal{A}_{n}}\in\mathcal{I},\mathcal{II}}\begin{pmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}I_{a}\\\
\Pi_{a\in\mathcal{A}_{n}}(I_{\vec{a}})\end{pmatrix})e^{imI+\eta_{\vec{V}}}\\\
$

Here, we have grouped together those paths with length $I$, giving us a single
factor in the argument of the exponential with C used to capture our
normalizing factor. The continuum limit of this expression becomes a carbon
copy of Equation 24 with a factor $e^{\eta_{\vec{V}}}$ due to the work of
Section 3. We now acknowledge that the alteration we introduced in the scalar
proof (that is, moving from $K^{cont}$ to $K^{cont,Feyn}$) of allowing each
step to take $\pm I$ increments would imply that our expression would be
necessarily even in $\eta_{V}$ immediately giving as our result Equation 24
but times $cosh(\eta_{\vec{V}})$. At this point, it’s almost definitional. The
same proof mentioned in Section 3 and done in its full extent in [15] for the
existence of the lattice scalar propagator works for the lattice Dirac
propagator; showing or sequence of $K_{\frac{1}{2},n}^{cont,Feyn}$ is Cauchy
and has some limit just requires the same real analysis steps. The only thing
we must demonstrate is that our symmetry arguments for the scalar propagator
actually yield one less dimension in this instance (we don’t get the marginal
distribution of the $d$ sphere but the $d-1$ sphere). Just as in the scalar
propagator argument, we would fix the final distribution using symmetry
arguments. Every lorentz transformation, just like every rotation, is indexed
by the choice of two vectors in an orthonormal basis of $\mathbb{R}^{d+1}$
(you can think of its as a rotation/boost about a plane in higher dimensions).
If we choose one of the axes to be $\vec{V}$ (and make a basis after), then
our transformations are invariant every transformation using a pairs of two
vectors (none of which is $\vec{V}$). This means our resulting distribution
$(t,\vec{x},I)$ will have be distributed as
$(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}$ when projected on the hyperspace
orthogonal to $\vec{V}$.

The decomposition of $(t,\vec{x},I)$ into its projection along $\vec{V}$ and
the hyperplane orthogonal is unique. Therefore, using the additivity property
of probability ($P(x,y)=\sum_{y\in\Omega}P(x|y)P(y)$), we know that the
probability that $(t,\vec{x},I)$ takes some value is the likelihood that we
have a given hyperplane projection (which is
$(t^{2}-x^{2}-I^{2})^{\frac{(d-1)-2}{2}}$) times the probability we then draw
the some projection along $\vec{V}$. But, we already know that the probability
of drawing vectors along $\vec{V}$ in terms of the likelihood of drawing some
$\eta_{V}$, and therefore we simply multiply the pdf of $\eta_{V}$ to this
original expression, and we get the law of $(t,\vec{x},I)$. That law is
$\\#_{I}^{f}$, and we are done. ∎

###  6.1. Fermionic Path Integral On Curved Spaces

The work of Section 4 extends immediately to the fermionic case, but we will
not elaborate further on this. There ought to be a relationship between the
power spectrum of the fermion and purely geometric quantities ala Subsection
4.2 as well; in this section we will demonstrate something unique to the
fermion: rapidity based cancellation. Let $d=3$. Then, our fermionic
propagator in Euqation 42 simplifies to Equation 44

(44) $\sum_{b\in\\{\pm 1\\}}\int e^{i(mI+ib\eta_{V})}rect(\frac{I}{\tau})dI$

In the curved space setting, this would become (without normalization) the
Equation 45.

(45) $\sum_{y^{\prime}\in Orb(y)}\sum_{b\in\\{\pm 1\\}}\int
e^{i(mI+ib\eta_{V}^{\prime})}rect(\frac{I}{\tau^{\prime}})dI$

where $\tau^{\prime}$ and $\eta^{\prime}$ are defined between our origin and
the deck transformation of our target point. The addition of these rect which
lead to a discrete power spectrum at geometrically defined mass points in
Subsection 4.2 is now complicated by the presence of a hyperbolic cosh which
depends not only on the spacing of the deck transformation points but also
their orientation w.r.t each-other. Therefore, while it is still quite likely
that there is a geometric expression of the Kallen Lehmann power spectrum for
the fermionic path integral on curved spaces, it will encode both information
about the relative spacing of the deck transformations and their orientation
w.r.t each-other (and is not easy to calculate).

##  7\. Interactions

In this section, we demonstrate how peturbative scalar QFT can be encoded in a
geometric formalism. We will be careful, in the statement of our final
theorem, to mention that this formalism makes no promises about the
convergence of peturbative scalar QFT; we simply assert that should our
geometric model converge it would converge to the value expect in QFT as it
obeys Feynman diagram rules. For the same to be said about QFT in general, we
would need an analogous geometric picture for Spin 1 particles. It may be the
case that said picture could be achieved by light cone paths (photons), where
massive Spin 1 particles are really mass-less particles with a Higgs scalar
coupling as realized by Stueckelberg. We will give this as a conjecture in the
end of this section. Let us prove our foundational theorem relating the
geometric interpretation to peturbative QFT. For this purpose, let us restate
the Feynman rules.

If our field Lagrangian has a three point term $\lambda\phi^{3}$, then we can
consider the three ‘first order’ corrections to our four point correlator in
Figure 4. These are not first order in the sense of number of loops but rather
in terms of $\lambda$, the correlation strength.

Figure 4. The 3 tree Level Contribution to the Four Field Correlator in
$\lambda^{3}$ Theory [19]

Focusing on the right-most term in Figure 4, we can follow the following rules
to obtain its contribution to the four point correlator. Let our contribution
be $1$ from the start. We read the figure from left to right, ignoring the
exterior arms. If we encounter a vertex of some degree, we multiply our
contribution by $(i\lambda)$ where $\lambda$ is the coefficient associated
with $\phi$ to that same degree in the action of the lagrangian. We label
these vertices with points, and as we read our diagram left to right, we
include a propagator $-iK$ between every labeled point that has a line in the
diagram. Finally, we integrate over the points in the vertices, hence the
$\int d^{d}y$ terms in Equation 19. This amounts to a full description of the
Feynman rules for scalar QFT; a more eloquent explanation is given in
[19],[16].

Consider taking the inverse Fourier transform of Equation 19. We would obtain
Equation 46.

(46) $(i\lambda)^{2}(\frac{1}{i})^{5}\int
d^{d}yd^{d}zK_{l_{2}^{*}}^{Feyn,cont}(y,z)(K_{l_{2}^{x}}^{Feyn,cont}(\vec{x}_{1},y)K_{l_{2}^{x}}^{Feyn,cont}(\vec{x}_{2},y)K_{l_{2}^{x}}^{Feyn,cont}(z,\vec{x}_{3})K_{l_{2}^{x}}^{Feyn,cont}(z,\vec{x}_{4}))\\\
=(i\lambda)^{2}(\frac{1}{i})^{5}\int
d^{d}yd^{d}z\int_{-\infty}^{\infty}e^{iI_{0}m}(t^{2}-(x-y)^{2}-I^{2})^{\frac{d-2}{2}}dI_{0}\int_{-\infty}^{\infty}e^{iI_{1}m}(t^{2}-(\vec{x}_{1}-y)^{2}-I_{1}^{2})^{\frac{d-2}{2}}dI_{1}\\\
\int_{-\infty}^{\infty}e^{iI_{2}m}(t^{2}-(\vec{x}_{2}-y)^{2}-I_{2}^{2})^{\frac{d-2}{2}}dI_{2}\int_{-\infty}^{\infty}e^{iI_{3}m}(t^{2}-(\vec{x}_{3}-z)^{2}-I_{3}^{2})^{\frac{d-2}{2}}dI_{3}\int_{-\infty}^{\infty}e^{iI_{4}m}(t^{2}-(\vec{x}_{4}-z)^{2}-I_{4}^{2})^{\frac{d-2}{2}}dI_{4}\\\
=(i\lambda)^{2}(\frac{1}{i})^{5}\int d^{d}yd^{d}z\int\textrm{...}\int
e^{im(I_{0}+I_{1}+I_{2}+I_{3}+I_{4})}\Pi_{j=0}^{4}(t^{2}-(X_{j})^{2}-I_{j}^{2})^{\frac{d-2}{2}}dI_{j}\\\
=(i\lambda)^{2}(\frac{1}{i})^{5}\int\textrm{...}\int
e^{im(I_{0}+I_{1}+I_{2}+I_{3}+I_{4})}(\int
d^{d}yd^{d}z\Pi_{j=0}^{4}(t^{2}-(X_{j})^{2}-I_{j}^{2})^{\frac{d-2}{2}}dI_{j})\\\
\textrm{ where
}X_{0}=x-y,X_{1}=\vec{X}_{1}-y,X_{2}=\vec{x}_{2}-y,X_{3}=\vec{x}_{3}-z,\textrm{
and }X_{4}=\vec{x}_{4}-z.$

After the steps in Equation 46, we consider taking the inverse Fourier
transform $\mathcal{F}^{-1}|_{m}^{I}$ to obtain Equation 47.

(47)
$\mathcal{F}^{-1}|_{m}^{I}(i\lambda)^{2}(\frac{1}{i})^{5}\int\textrm{...}\int
e^{im(I_{0}+I_{1}+I_{2}+I_{3}+I_{4})}(\int
d^{d}yd^{d}z\Pi_{j=0}^{4}(t^{2}-(X_{j})^{2}-I_{j}^{2})^{\frac{d-2}{2}}dI_{j})\\\
=(i\lambda)^{2}(\frac{1}{i})^{5}\int\textrm{...}\int\int
dme^{im(I_{0}+I_{1}+I_{2}+I_{3}+I_{4}-I)}(\int
d^{d}yd^{d}z\Pi_{j=0}^{4}(t^{2}-(X_{j})^{2}-I_{j}^{2})^{\frac{d-2}{2}}dI_{j})$

Now, we are working in the setting (Schwartz class functions) where the Dirac
delta function is defined as an operator on functions underneath an integral.
As an operator, we have
$\delta(x-y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dme^{im(x-y)}$, so if we
make this substitution into Equation 47 we obtain Equation 48

(48) $(i\lambda)^{2}(\frac{1}{i})^{5}\int\textrm{...}\int(\int
d^{d}yd^{d}z\Pi_{j=0}^{4}(t^{2}-(X_{j})^{2}-I_{j}^{2})^{\frac{d-2}{2}}\delta(I-\sum_{i=0}^{4}I_{i})dI_{j})$

Now we want to incorporate our lattice propagators into Equation 48, and we
have shown $(t^{2}-(X_{j})^{2}-I_{j})^{2}$ is a point-wise limit of a
multinomial coefficient representing the number of paths from 0 to $X_{j}$
aligned with the axes in $\mathcal{A}_{n}$. Using theorems about limits (the
commutation of a limit with a product), we obtain Equation 49.

(49)
$lim_{n\rightarrow\infty}(i\lambda)^{2}(\frac{1}{i})^{5}\int\textrm{...}\int(\int
d^{d}yd^{d}z\Pi_{j=0}^{4}\int_{\Omega}\begin{Bmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}J_{a}\\\
\Pi_{a\in\mathcal{A}_{n}\setminus\\{(0,1),(e_{i},1),(-e_{1},1)\\}}(J_{\vec{a}}),f_{x_{0}},f_{I_{i}},\Pi_{i=1}^{d}f_{x_{i}}\end{Bmatrix}d\Omega\delta(I-\sum_{i=0}^{4}I_{i})dI_{j})\\\
\textrm{ where }f_{\pm}\textrm{ and }f_{x_{0}}\textrm{ are defined as in
Equation~{}\ref{eq:propStep2}}$

We may interpret Equation 49 more easily if we consider it from the view as
the limit itself of a discrete object, or a number of discrete objects. We
note that the continuous multinomial coefficient with the $\Omega$ integral is
the limit of $|\Gamma|$ under $\mathcal{T}_{cont}$ for paths from $0$ to
$X_{j}$. Under this limit, the integral over $y$ and $z$ arises from sums over
the intermediate points $y,z\in\mathbb{Z}^{d}$. The multinomial coefficients
tell us the number of paths between intermediate points, and the sum over
locations of the intermediate points demonstrate that if we took the limit
under $\mathcal{T}_{cont}$ of the number of lattice paths (with possibly
multiple individual adjoining or splitting paths) arranged to look realize the
left diagram in Figure 4, then we would obtain the contribution of that
Feynman diagram to the interaction four point correlator with a $\psi^{3}$
interaction term. This is done in the exact same way as Theorem 15 was
obtained in [15], so that,with a $\psi^{3}$ term the propagator, it can be
understood as an expansion in $\lambda$ where the contribution for a given
power of $\lambda$ is a limit over the number of paths between the initial and
final points with that number of intersection points. This leads to Theorem 5.

######  Theorem 5.

Let $\mathcal{L}=(\nabla\phi)^{2}+\sum_{i=0}^{n}a_{i}\phi^{i}$ be the
Lagrangian density of a scalar QFT in $d+1$ dimensions. Say the pythagorean
tupples in $d+1$ are asymptotically rotationally symmetric. Then, should they
be convergent, the $p$ point correlators
$\mathbb{E}[T\phi(x_{1})...\phi(x_{p})]$ from perturbation theory are
equivalent to taking the rigorous limit $\mathcal{T}_{m}$ of $\sum_{\sum
n_{j}=1}^{\infty}(\Pi_{j=1}^{m}a_{j}^{n_{j}}\Gamma_{n}(\\{n_{j}\\})$ where
$\Gamma_{n}(\\{n_{j}\\})$ denotes the number of lattice embedding of Feynman
diagrams connecting our p points with $n_{j}$ many intersection points of
degree $j$ with steps in $\mathcal{A}_{n}$. We obtain a continuum object
representing the volume of the space of directed continuous lattice embedding
of Feynman diagrams in $\mathbb{R}^{d}$ with continuum steps in
$\mathcal{A}_{n}$ and total length $I$. We then take the limit as
$n\rightarrow\infty$, allowing more directions to be included in our
measurement of path space. Finally, we take the Fourier transform of the
resulting set of directed paths from $I$ to m.

This should give a geometric version of QFT for all scalar theories. One
interesting consequence of this interpretation is that (should this
$\mathcal{T}_{cont}$ operation be convergent) we evade the divergences
normally associated with computing point correlators in momentum space. The
earlier discussion with the fermionic propagator will, in the same exact
manner, lead to a geometric interpretation of all QFT theories involving only
scalar and Dirac spinor fields. The author believes a geometric interpretation
ought to exist for all peturbative QFT. In the next subsection, he presents
the evidence for a conjecture about for theories with a spin 1 particle.

###  7.1. Massive Spin 1 Particles

First, we make a new definition: the $d_{n}$ photon propagator, as
$K_{n,photon}(\vec{x},\vec{y})^{\mu\nu}=i\frac{\delta(d_{n}(0,\vec{y}-\vec{x}))}{\mu(\\{\vec{x}|d_{n}(0,\vec{y}-\vec{x})=0\\})}g^{\mu\nu}$
where the Lebesgue measure of the $d_{n}$ light cone is well-defined because
it will be a polyhedral fan. We note that this distribution has a
distribution-wise limit as $n\rightarrow\infty$ which is simply Equation 50.

(50)
$K_{photon,l_{2}}(\vec{x},\vec{y})^{\mu\nu}=\frac{i\delta(proj_{t}(\vec{x}-\vec{y})^{2}-\sum_{i}proj_{x_{i}}(\vec{x}-\vec{y})^{2})}{\mu(\\{\vec{x}|d_{l_{2}}(0,\vec{y}-\vec{x})=0\\})}g^{\mu\nu}\\\
=\frac{i\delta(proj_{t}(\vec{x}-\vec{y})^{2}-\sum_{i}proj_{x_{i}}(\vec{x}-\vec{y})^{2})}{Vol(\mathbb{S}^{d-1}(proj_{t}(y)-proj_{t}(x)))}g^{\mu\nu}$

One thing to note is that $K_{n,photon}$ as of yet does not have a necessary
notion of gauge associated with it. It is what you obtain when you fix the
gauge of the photon propagator in the Feynman gauge (the mass-less Spin-1
particle gauge). If you want to reintroduce the gauge of this Spin-1 boson,
you would apply the operator
$g^{\mu,\nu}+(1-\frac{1}{\lambda})\frac{p^{\mu}p^{\nu}}{p^{2}}$ to our
original light cone propagator. The normal interpretation of momentum as
differential operators is a little difficult in this setting; one thing we can
note is that $(1-\frac{1}{\lambda})\frac{p^{\mu}p^{\nu}}{p^{2}}$ in position
space would look like the differential operator
$(1-\frac{1}{\lambda})\partial_{\mu}\partial_{\nu}$ times the position space
transform of $p^{-2}$ which we know is $\delta(\tau^{2})$. This means that the
full position space contribution from the gauge terms would be
$((1-\frac{1}{\lambda})\partial_{\mu}\partial_{\nu}\delta(\tau))*\delta(\tau)$
where $*$ denotes convolution. The resulting contribution to the propagator is
still only supported on the light cone and will likely enforce some jump in
derivatives along paths at junctures with the photon propagator.

With a geometric notion of the zero mass spin 1 propagator defined, we can
move to the massive one. With the Higgs mechanism [19], we consider a mass-
less spin 1 field coupled to a scalar field with non-zero expectation.
Expectation values of $\phi$ give rise to a non-zero mass for the otherwise
mass-less spin 1 boson. Consider the $U(1)$ higgs-coupled lagrangian
$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+|(\partial-
iqA)\phi|^{2}-\lambda(|\phi|^{2}-\Phi^{2})^{2}$. The Feynman rules for
correlators in this theory include your regular photon and scalar terms (which
$\phi^{\prime}s$ mass coupling being $\lambda(1-2\Phi)$), a constant
$\Phi^{4}$ which we will normalize out, and a four vertex $\phi^{4}$ term
times $\lambda$. Noting that the Abelian Higgs Model is peturbative in some
parameter, this necessitates the following theorem for the correlators of the
Abelian Higgs Model.

######  Conjecture 2.

Let $\\{A^{\mu},\phi\\}$ be the particle content of the Abelian Higgs Model.
Then, $\mathbb{E}[A^{\mu}(x_{1})\phi(x_{2})..\phi(x_{n-1})A^{\nu}(x_{n})]$, an
arbitrary point correlator of this model, may be first obtained by the
application of $\mathcal{T}_{cont}$ to $\sum_{\sum
n_{1}+n_{2}=1}^{\infty}q^{n_{1}}\lambda^{n_{2}}\Gamma_{n}(\\{n_{1},n_{2}\\})$
where $\Gamma_{n}$ denotes the number of lattice embedding of Feynman diagrams
connecting the photon and scalar points that contain $n_{1}$ many interaction
terms where a photon path terminates into a $\phi$ and $n_{2}$ many terms
wherein $\phi$ has a four point vertex where each direction is constrained to
lying along $\mathcal{A}_{n}$ for the $\phi$ paths and that intersected with
$\\{\vec{a},d_{n}(0,\vec{a})=0\\}$ for the photon paths. Note, as previously,
that we will make this continuum object a function of $I$ which is the
accumulated length of the scalar paths in minkowski distance. We then take the
limit $n\rightarrow\infty$ and take the Fourier transform from $I$ to
$\lambda(1-2\Phi)$.

##  8\. Conclusion

With the work thus far stated, we capture much of the dynamics required for
general QFT. We generalized the work of a previous paper, which showed how the
$1-1$ scalar propagator can be realized as the Fourier transform of a genuine
sum over paths with their length being the transform variable, to the case of
$1-d$ dimensions. We were then able to show how this definition easily
generalized to all $1-2$ manifolds with a spatial structure that admitted the
hyperbolic plane as its universal cover, and how we could use the
uniformization theorem in $3$ dimensions along with the path integral across a
stitched boundary of manifolds to define our object on a general class of
$1-3$ manifolds.

We then moved onto the Dirac propagator in anticipation of a general statement
of interactions. We were able to show how a small modification of our work
would produce the Dirac fermion propagator in $1-1$ dimension; the author
fully anticipates this generalizing to higher dimensions. The author then
inspected the rules of peturbative QFT. Resulting from a simplistic proof
relating multiplication and convolution through the Fourier transform, the
author demonstrated that the Feynman rules that are used to calculate point
correlators in QFT arise naturally from the assumption that these point
correlators are obtained from a Fourier transform over the number of paths of
length I which realize a Feynman diagram in your manifold. This gives rise to
a purely geometric interpretation of peturbative QFT which the author
conjectures generalizes to all theories including Spin $\frac{1}{2}$ and Spin
1 particles.

In future work, the author aims to obtain a theory applying
$\mathcal{T}_{cont}$ to higher dimensional simplicial complexes in space. The
simplex correlators he would obtain from this theory would arise from the
Fourier transform of world-sheets of volume I to a particle mass $m$. He aims
to show that this theory is rigorously defined, and contains peturbative QFT
as a sub-branch of it. He hopes that non-peturbative QFT and perhaps other
theories will emerge as special limits of simplex correlators for higher
dimensional simplices.

##  9\. Necessary Theorems

This section includes general purpose results required for many portions of
this paper. The proofs of these results, unless explicitly proven or cited
below, are located in [15].

######  Theorem 6.

Let $\\{x_{i}\\}_{i=1}^{l}\subset\mathbb{N}$ such that
$min(x_{i}-\frac{\sum_{i=1}^{l}x_{i}}{l})\geq n$

$\begin{pmatrix}\sum_{i=1}^{l}x_{i}\\\
x_{1},...,x_{l}\end{pmatrix}=\frac{l^{\sum_{i=1}^{l}x_{i}+\frac{l}{2}}}{\sqrt{2\pi\sum_{i=1}^{l}x_{i}}^{l-1}}e^{-\frac{l}{2\sum_{i=1}^{l}x_{i}}(\sum_{i=1}^{l}(x_{i}-\frac{\sum_{i=1}^{l}x_{i}}{l})^{2})+o(1)}$

where $o(1)$ denotes some function that goes to zero as $n\rightarrow\infty$.

This theorem is given in [12]

######  Theorem 7.

The continuous multinomial

$\begin{Bmatrix}\sum x_{i}\\\ x_{1},x_{2},...,x_{l}\end{Bmatrix}<\infty$

is a finite and analytic function from $\mathbb{R}^{l}\rightarrow\mathbb{R}$.

######  Theorem 8.

The Continuous Multinomial is a Limit of the Discrete One

Let $\\{x_{1}\\}_{i=1}^{l}\subset\mathbb{R}_{+}$. Then we have

$lim_{m\rightarrow\infty}\frac{\mathcal{T}^{m}_{cont}\begin{pmatrix}\sum[mx_{i}]\\\
[mx_{1}],...,[mx_{l}]\end{pmatrix}}{\mathcal{T}^{m}_{cont}\begin{pmatrix}\sum_{i}\lfloor
nx_{i}\rfloor\\\
\lfloor\frac{\sum_{i}nx_{i}}{l}\rfloor,...,\lfloor\frac{\sum_{i}nx_{i}}{l}\rfloor\end{pmatrix}}\rightarrow\frac{\begin{Bmatrix}\sum_{i}x_{i}\\\
x_{1},...,x_{l}\end{Bmatrix}}{\begin{Bmatrix}\sum_{i}x_{i}\\\
\frac{\sum_{i}nx_{i}}{l},...,\frac{\sum_{i}nx_{i}}{l}\end{Bmatrix}}$

This will also prove other forms of convergence (here we normalize by the
maxima, we could also normalize by the integral of each expression).

######  Theorem 9.

Let $\\{x_{i}\\}_{i=1}^{l}\in\mathbb{R}$ where
$min(x_{i}-\frac{\sum_{i=1}^{n}x_{i}}{l})>R\in\mathbb{R}_{+}$. Then

$\begin{Bmatrix}\sum_{i=1}^{l}x_{i}\\\
x_{1},...,x_{l}\end{Bmatrix}=\frac{l^{\sum_{i=1}^{l}x_{i}+\frac{l}{2}}}{\sqrt{2\pi\sum_{i=1}^{l}x_{i}}^{l-1}}e^{-\frac{l}{2\sum_{i=1}^{l}x_{i}}(\sum_{i=1}^{l}(x_{i}-\frac{\sum_{i=1}^{l}x_{i}}{l})^{2})+o(1)}$

This implies that, should $\sum_{i=1}^{l}x_{i}$ be some constant large number,
then its Schwartz class [17] so all derivatives of it have Fourier transforms
which are Schwartz class.

######  Theorem 10.

Let $d=2$. Then one $\mathcal{A}_{n}$ is the set
$\\{\frac{1}{gcd(x,I,t)}(I,t)\in\mathbb{N}^{2}|x^{2}+I^{2}=t^{2},(x,I,t)\in\mathbb{Z}^{3},0\leq
t\leq n\\}\cup\\{(\pm 1,1)\\}$ (i.e. a minimal $\mathcal{A}^{gen}$ for $d_{n}$
is this above set).

######  Theorem 11.

Let $d\geq 2$. Then $\mathcal{A}_{1}=\\{(\pm e_{1},1),...,(\pm
e_{d},1),(0,1)\\}$ where $e_{i}$ denote unit directions in $\mathbb{Z}^{d}$

These are theorems necessary for the directions in [15]. For our purposes we
require the same proof but for the $\mathcal{A}_{n}$ which corresponds to our
metric $d_{n}$

######  Theorem 12.

Let $d\geq 2$, Then one $\mathcal{A}_{n}$ for $d_{n}$ is the Equation 51

(51)
$\mathcal{A}_{n}=\\{\frac{1}{gcd(\\{x_{i}\\},t,I)}(\vec{x}_{i},t)\in\mathbb{N}^{d+1}|\sum_{i=1}^{d}x_{i}^{2}+I^{2}=t^{2},(x_{1},...,x_{d},I,t)\in\mathbb{Z}^{d+2},0\leq
t\leq n\\}\\\
\cup\\{\frac{1}{gcd(\\{x_{i}\\},t)}(\vec{x}_{i},t)\in\mathbb{N}^{d+1}|\sum_{i=1}^{d}x_{i}^{2}=t^{2},(x_{1},...,x_{d},t)\in\mathbb{Z}^{d+1},0\leq
t\leq n\\}$

###### Proof.

Consider the sets $\mathcal{C}_{l^{*}_{2}}=\\{d_{l^{*}_{2}}(0,\vec{x})=1\\}$
and $\mathcal{C}_{n}=\\{d_{n}(0,\vec{x})=1\\}$. Let us show that
$\mathcal{C}_{l^{*}_{2}}\cap\mathcal{C}_{n}$ is Equation 2. One may simply
evaluate the points Equation 2 and immediately discern is a subset of
$\mathcal{C}_{l^{*}_{2}}\cap\mathcal{C}_{n}$ ($\mathcal{C}_{n}$ is carefully
designed so this was the case). The first part of the definition of $d_{n}$ is
built from hyperplanes interpolating $\vec{v}$ in Equation 2; these
hyperplanes are the sets $\mathcal{H}_{\mathcal{N}_{\vec{v}}}^{-1}(1)$. The
second part of the definition includes hyperplanes whose defining direction is
a null direction (i.e. along a light path). Using this, we can characterize
the set $\mathcal{C}_{n}$ into sets of hyperplanar sections which are compact
and interpolate points, and into those planes which are non-compact and
orthogonal to a null direction.

The sheets of the hyperboloid $\mathcal{C}_{l_{2}^{*}}$ are individually the
images of convex function to the t coordinate (namely
$f(x)=\sqrt{1-\sum_{i}x_{i}^{2}}$) and by the definition of convexity any
hyperplanar sections formed by adjoining points lying on its surface must all
have greater than or equal t values. The equality only occurs if the hessian
of our convex function is zero, which is not the case in this setting.
Therefore, we immediately have that the compact hyperplanar sections of
$\mathcal{C}_{n}$ lie above $\mathcal{C}_{t^{*}}$. The non-compact sections of
$\mathcal{C}_{n}$, due to their non-compactness, will lie ougtside the non-
compact sections of $\mathcal{C}_{n}$. Their null-direction orthogonal
direction implies that, along any direction, they increase larger than any
timelike curve in the t-direction as one traces along their surfaces radially.
This implies that the non-compact sections of $\mathcal{C}_{n}$ lie properly
above $\mathcal{C}_{l_{2}^{*}}$ as well because its orthogonal directions are
time-like. These fact together imply that
$\mathcal{C}_{l_{2}^{*}}\cap\mathcal{C}_{n}$ is Equation 2. Consider all
directions in $\mathcal{A}^{all}$ with non-zero minkowski metric values; if we
divide these vectors by their norm we find a point in
$\mathcal{C}_{l_{2}^{*}}\cap\mathcal{C}_{n}$ and therefore in Equation 2.
There is some smallest multiplicative number one may multiply elements of
Equation 2 to obtain a vector in $\mathbb{Z}^{d+1}$. The resultant vectors are
precisely the non-null portion of Equation 51 (as you divide them by their gcd
implying they could not have a smaller parallel direction that lies upon the
lattice). Therefore, the non-null portion of Equation 51 generates all non-
null parts of $\mathcal{A}_{all}$ with unique representatives for each
direction; we now only need to show the null portion of Equation 51 does so
for the null directions.

If we now call $\mathcal{B}_{l_{2}^{*}}=\\{d_{l_{2}^{*}}(0,\vec{x})=0\\}$ and
$\mathcal{B}_{n}=\\{d_{n}(0,\vec{x})=0\\}$, then
$\mathcal{A}_{all}\cap\\{\vec{v}|d_{l_{2}^{*}}(0,\vec{v})=0\\}=\mathcal{B}_{l_{2}^{*}}\cap\mathcal{B}_{n}$.
This is a set of rays emanating from the origin; if we show that the null
portion of Equation 51 has a component or can generate a component aligned
with all directions in this set, then we will be done. This is because the
division by the gcd in Equation 51 implies that there are no smaller vectors
parallel to the one in question that also lie upon
$\mathbb{Z}^{d}\times\mathbb{Z}$, and they generate all other vectors of the
same direction lying upon the lattice.

$\mathcal{B}_{n}$ is going to only feature intersection of hyperplanes given
by $\mathcal{H}_{\mathcal{N}_{\vec{a}}}$ for $\vec{a}$ among null directions
in $\mathcal{A}_{n}$. These are the same null directions as defined in
Equation 51. These hyperplanes interpolate between the rays of each direction
in $\mathcal{A}_{n}$. Since our cone $\mathcal{B}_{l_{2}^{*}}$ has some
curvature, convexity implies that the hyperplanes interpolating these rays
must lie above the conical regions of $\mathcal{B}_{l_{2}^{*}}$ (which also
interpolate these rays). Therefore,
$\mathcal{B}_{l_{2}^{*}}\cap\mathcal{B}_{n}$ must be the set of rays aligned
with directions along $\mathcal{A}_{n}$, and therefore with directions along
Equation 51.

∎

######  Theorem 13.

Let $\gamma=\\{x_{i}\\}_{i=1}^{n}\in\Gamma_{n}$. Let $\mathcal{A}_{n}$ denote
some axes of symmetry of $d_{n}$. Then, we can choose our $\\{x_{i}\\}^{n}$
such that $x_{i+1}-x_{i}\in\mathcal{A}_{n}$. This alteration will yield the
same path up to our equivalence relation on $\Gamma$, and this equivalence
class representative is unique.

######  Theorem 14.

Denote the set of primitive pythagorean quadruples with hypotenuse below
$n\in\mathbb{R}$ as $\mathcal{A}_{n}$. Then,
$\mathcal{A}_{n}=\frac{n^{2}}{32G}+O(n)$, and they are equidistributed on the
unit circle when ordered according to hypotenuse (as the hypotenuse goes to
infinity). Here, G denotes Catalan’s constant and has a value approximately
$\sim.9159655$

###### Proof.

The proof of the first part of this theorem is found in [11] and the second in
[5].

∎

######  Theorem 15.

The continuum multinomial has the following property

$\begin{Bmatrix}\sum_{i}x_{i}\\\
x_{1},...,x_{n}\end{Bmatrix}=\int_{-\infty}^{\infty}\begin{Bmatrix}\sum_{i}x_{i}\\\
x_{1},...,I\end{Bmatrix}\begin{Bmatrix}I\\\ x_{n-1},x_{n}\end{Bmatrix}dI$

This implies that should any of the coefficients of the continuum multinomial
coefficient be zero, then the function is also zero.

## Acknowledgement

I want to thank Eviatar Procaccia and Parker Duncan from Technion University
for their help in the development of this paper through work in [7] and [6]. I
thank Hailey Leclerc for her aid in editing this paper, as well as Grigory
Rogachev and Lauren Tompkins for their mentor-ship. Finally I thank my
parents, Marianne Fairey and Joseph O’Dwyer. I dedicate this paper to the
memory of Anthony O’Dwyer.

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