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%\shorttitle{Unification}
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\begin{document}
\title{Unification of Fundamental Physics Using Lagrangian-Euclidian Space, Where GR = QFT.}
\iflatexml\else
\author[0000-0003-1510-7786]{Hontas Farmer}
\email{hfarmer@ccc.edu}
\affil{College of DuPage\\
Divison of Mathematics and Natural Sciences \\
425 Fawell Blvd\\
Glen Ellyn, Illinois, United States of America}
\affil{City Colleges of Chicago-Wilbur Wright College\\
Department of Natural Science\\
226 W. Jackson St\\
Chicago, Illinois, United States of America}
\affil{Triton College\\
Department of Science\\
2000 N Fifth Ave\\
River Grove, Illinois, United States of America}
\fi
\vspace{-1em}
\date{\today}
\begingroup
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\maketitle
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\selectlanguage{english}
\begin{abstract}
Suppose we were to treat the Lagrangians of the theories we wish to unite as coordinates in a space of Lagrangians. In this space the Lagrangians
for general relativity (Einstein-Hilbert), the standard model, and
the dark sector act as coordinate axes. From here a functional derivative
equation is set up for a functional, T, which gives as a result another
functional, U, which denotes the Lagrangian for our universe. From
there the action $(\mathscr{S})$ due to U is computed and shown to
converge even at infinite energy. Finally with $\mathscr{S}$ a generating
functional for all possible interactions is computed and a one inch
equation of everything is given$(Z=e^{i\mathscr{S}})$. It is the assertion
of this paper that with the generating functional detailed herein
one may compute any physical quantity of interest including gravitational
and dark sector corrections and arrive at a finite result. Specifically graviton-graviton and graviton photon interactions are shown to result in finite observable quantities. It is noted that this is a fully worked out model where GR and QFT are treated on an equal footing by relativizing QFT, or put simply GR=QFT. %%
\end{abstract}%
\sloppy
\section{Introduction\label{introduction}}
The fundamental question to ask is why our universe is so well described
by at least three Lagrangians, which may not be simply unified, and
which derive from very different looking physical foundations? These
Lagrangians are the standard model of particle physics, the Einstein-Hilbert
Lagrangian and the Lagrangian for the dark sector. There is one widely
used and well studied effort at unification M-Theory. However, M-Theory
is based on SUSY and can't work without it. The Large Hadron Colliders
two big collaborations have not found signs of any particles predicted
by the modified super symmetric standard models which are the low
energy states of string theory and hence M-Theory \cite{2017arXiv170408493A,2017APS..APR.H3005F,CMS-PAS-SUS-16-050}.
New ideas are here needed, so here is a new idea.
Rather than looking for an answer that has a certain anticipated form,
that of a SUSY string or M theory, I propose a change of focus to answering
three fundamental questions. What is the underlying reason that our
universe has the Lagrangian that it does? Can we find a simpler more
fundamental representation of known physics? Can we at least quantify
the effect that dark sector physics could have on standard model particles
and gravity? This work builds on the principles of relativization
previously published in articles such as \cite{Farmer_2014,Farmer_2015,Farmer_2014a,Farmera,Farmer,2017APS..APR.H3005F}
\section{A New View Towards Unification}
\label{a-new-view-towards-unification}
Unification in particle physics traditionally means that as we increase
the energy scale in our experiments different forces begin to look
the same. This worked for the electro-weak and strong force but does
not work for gravity. We don't know if this will work for dark matter
or dark energy which are most of the universe but we can safely assume
that it will. The form of unification proposed in this paper is more
akin to the unification of electricity with magnetism in the form
of Maxwell's equations.
There are three fundamental Lagrangians each of which corresponds
to a dimension in an otherwise standard Euclidean three-space,$\mathscr{\left(L\right)}$.
One dimension gives the standard model of particle physics~$(\mathcal{L}_{SM})$,
another the Einstein-Hilbert Lagrangian~$(L_{EH})$ and the as yet
unknown Lagrangian for the dark sector~$(\mathcal{O})$ (for other).
This space of Lagrangians has all the usual features closure under
scalar multiplication dot product etc. Furthermore, this space supports
calculus with the functional derivative. A vector in this space gives
the Lagrangian to describe the physics of a particular universe, equation
\ref{eq:Uvec}.
\begin{equation}
U(\alpha,\beta,\omega)=\alpha\mathcal{L}_{SM}+\beta L_{EH}+\omega\mathcal{O}\label{eq:Uvec}
\end{equation}
Each one of $(L),$$(\mathcal{L})$, and~$(\mathcal{O})$ are as
independent as x y and z coordinates. Only$(\mathcal{L})$ for the
standard model of particle needs to be a quantum field theory. We
can treat Einstein-Hilbert Lagrangian~$(L)$ as such a theory. Will
the infinities cancel out? The answer is yes, and I show how below
(equation \ref{eq:gen}) but to understand the result read on.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig1/Fig1}
\caption{{{{The space of physically meaningful Lagrangians~$\mathscr{\left(L\right)}$
is three dimensional, Euclidean, and supports calculus.}%
}%
}}
\end{center}
\end{figure}
The algebra to show that any such combination of Lagrangians is also
a Lagrangian and that they form a vector space of Lagrangians is trivial.
\section{Deriving the Lagrangian for Our Universe}
\label{deriving-the-lagrangian-for-our-universe}
We know that all Lagrangian will have to obey the principle of least
action stated in the most general form possible.
\[
\delta\int U\left(\alpha,\beta,\omega\right)=0
\]
All Lagrangians must satisfy this principle in addition to matching
all the observed physical data. So one may consider a space of Lagrangians
$(\mathscr{L})$ in which all the Lagrangians differ from those in
our universe by a scalar parameter.
If we define a gradient on this space using functional differentiation
as follows.
\[
\nabla\equiv\frac{\delta}{\delta\mathcal{L}}+\frac{\delta}{\delta L}+\frac{\delta}{\delta\mathcal{O}}
\]
Using this derivative one may define a differential equation for a
functional on $(\mathscr{L})$ which can give various universes.
\begin{equation}
\nabla T=l(\alpha,\beta,\omega)T\label{eq:defT}
\end{equation}
is a generic Lagrangian T is a functional from which Lagrangians can
be derived.
To get a well defined first order partial differential equation we
need one boundary condition or one initial condition. For this differential
equation the derivative must converge uniformly to the Lagrangian
of a particular universe as the Lagrangian goes to zero and as T goes
to one. For our universe this means the solution must approach that
of our universe in the low energy limit. To give finite results it
should also be a mathematically well behaved functional on$(\mathscr{L})$.
The simple anzats that will satisfy this for any given universe is
equation {\ref{eq:anzats}}.
\begin{equation}
T=e^{\beta L\left(\alpha\mathcal{L}-\omega\mathcal{O}\right)}\label{eq:anzats}
\end{equation}
\[
\nabla T=\nabla e^{\beta L\left(\alpha\mathcal{L}-\omega\mathcal{O}\right)}=\nabla\left(\beta L\left(\alpha\mathcal{L}-\omega\mathcal{O}\right)\right)e^{\beta L\left(\alpha\mathcal{L}-\omega\mathcal{O}\right)}\rightarrow U(\alpha,\beta,\omega)
\]
For our universe let $(\beta=\alpha=1)$,$(\omega\ll1)$
\begin{equation}
U=\left(L+\mathcal{L}-\omega\left(\mathcal{O}+L\right)\right)e^{L\left(\mathcal{L}-\omega\mathcal{O}\right)}\label{eq:OurUniv}
\end{equation}
The value of the exponential functional approaches one as the value
of the Lagrangians, which are it's arguments, approach zero. Therefore,
at low energies the exact physics we have observed will be seen.
If one chooses to Taylor expand U in equation {\ref{eq:OurUniv}}
they will find an infinite progression of corrective terms. Taking
those steps does not make the math that follows any easier and so
showing it will be omitted from this paper.
\section{Generating Functional for Field Interactions.}
\begin{equation}
Z=e^{\imath\intop d^{4}xU}\label{eq:gen}
\end{equation}
Having the generating function for the theory completely defines how
its operators are related, in quantum field theory, but this is NOT
a quantum field theory in the truest sense of the word. Each Lagrangian
$L$, $\mathcal{L}$, and $\mathcal{O}$, is assumed to have been written in a form which has the following
features, it is invariant under all space-time diffeomorphisms and
the units have been canceled out perhaps by dividing by the Planck
energy. Likewise the units on the actions will have been canceled
out as well.
The claim of equation \ref{eq:gen} needs to be verified experimentally.
In this section I will show that the functional integrals in equation
\ref{eq:gen} will converge and that for low energies they will give
the same action, and interactions that we observe at the energies
we have been able to prove so far. Furthermore, I will show some of
the higher energy predictions made and which may soon be testable.
\[
\intop d^{4}xU=\intop d^{4}x\left(L+\mathcal{L}-\omega\left(\mathcal{O}+L\right)\right)e^{L\left(\mathcal{L}-\omega\mathcal{O}\right)}
\]
To simplify this expression I will integrate by parts and make use
of the fact that the action of the standard model, $\mathcal{S}$,
and the Einstein-Hilbert action, s, are well known. The action of
the dark sector is not known but never the less the integral of it's
Lagrangian will by definition be a Lagrangian, S. Carrying out this
integration is not complicated given we know parts of the integral
already. We will call the value of this integral $\mathscr{S}$
for the action due to the unifiedLagrangian U.
\[
\mathscr{S}=\intop d^{4}xU=\left(s+\mathcal{S}-\omega\left(\mathbb{S}+s\right)\right)e^{L\left(\mathcal{L}-\omega\mathcal{O}\right)}
\]
Clearly if the Lagrangians go to infinity the actions also go to infinity.
The relative negative sign between them ensures those infinities cancel
out. The value of the exponential will approach one due to the infinities
canceling to zero. Now the partition which describes the whole universe
with all interactions and corrections and gives finite results can
be written down as equation \ref{eq:genfin} .
\begin{equation}
Z=e^{\imath\left(s+\mathcal{S}-\omega\left(\mathbb{S}+s\right)\right)e^{L\left(\mathcal{L}-\omega\mathcal{O}\right)}}=e^{\imath\mathscr{S}}\label{eq:genfin}
\end{equation}
With equation \ref{eq:genfin} in hand one may compute the correlation
functions for any desired interactions to any desired degree of precision.
It is even possible to constrain the unknown effects of dark sector
interactions on standard model particles in the confines of this framework.
As a demonstration of this I will derive the Graviton Graviton crossection
~and show that the corrections converge to a finite quantity.
\subsection{Graviton-Graviton Correlation Function}
The major challenge in quantum gravity is to come up with a formulation
which does not contain ultraviolet divergences. ~To demonstrate that
this model does not have the UV divergence problem we now derive the
correlation function from the generating functional formalism for
graviton graviton interaction. ~Many steps will be skipped, and detailed
calculations are in the supplemental material for this paper.
\begin{equation}
G\left(RR'\right)=Z\left[J\right]^{-1}\left(-i\frac{\delta}{\delta J}\right)\left(-i\frac{\delta}{\delta J'}\right)Z\left[J\right]\rvert_{_{J=0}}
\end{equation}
\begin{equation}
=Z\left[J\right]^{-1}\left(-iiRR'\right)Z\left[J\right]\rvert_{_{J=0}}=e^{-\imath\mathscr{S}}RR'e^{\imath\mathscr{S}}
\end{equation}
Therefore the correlation fuction
\begin{equation}
G\left(RR'\right)=RR'
\end{equation}
Therefore when one graviton interacts with another graviton finite
input gives finite output. ~Every term in the taylor series will
have a corresponding counter term in this model. ~ Therefore, the
UV divergence problem is solved.
\subsection{Graviton- Electromagnetic interactions.}
Another interesting cross section from a practical application standpoint
would be the interaction of gravitons with electromagnetic waves and
charged particles. ~All of our technology is based on our understanding
of quantum electrodynamics and its classical limits, Maxwell's equations.
~If we can understand fully how gravity can interact with electromagnetism
then we could, in time, develop practical technologies.~
With the generating functional worked out, and the lack of UV divergence
proven, it is trivial to show that the graviton-electromagnetic correlation
function will be given as follows~
\begin{equation}
G(\overline{\psi},\psi,A,R)=Z\left[J\right]^{-1}\overline{\psi}\gamma\psi AR\ Z\left[J\right]\rvert_{J=0}\ =\overline{\psi}\gamma\psi AR\
\end{equation}
This can be interpreted quite simply, the correlation between electromagnetism
and gravitation is linear. ~ If this model is correct, then modifying
gravity to engineer the local space-time manifold is a matter of manipulating
the electro-magnetic field, and nothing dramatic will occur. ~R will
change as the E and M field does but there is not a run away divergence
as the previous calculation shows.~
\section{Discussion}
\label{discussion}
Does this paper answer the fundamental questions listed in the introduction
\ref{introduction}? Yes.
The underlying reason that our universe has the Lagrangian $U$ (equation
\ref{eq:OurUniv}) is because it is the only one that satisfies the
differential equation for the function $T$ (equation \ref{eq:defT})
on the space of Lagrangians $\mathscr{L}$ given the boundary conditions
for our universe. One can then ask why our universe has the boundary
conditions, why are the fundamental constants what they are, and so
forth. They are what they are fundamentally because they lead to an
equation U that satisfies the boundary conditions of our universe.
If they were any different then U would describe a different universe.
To talk of other universes is as of right now a very popular speculation,
but speculation none the less. That said, the space of Lagrangians
$\mathscr{L}$ contains as many points as a 3D euclidean space can
this theory does not rule out other possibilities. Instead it constrains
us to just one of those Lagrangians (equation \ref{eq:OurUniv}).
A simpler more fundamental representation of the physics embodied
in the Lagrangian U is arguably given by any one of three equations
in this paper. The equation giving the T function for our universe
(equation\ref{eq:anzats}) is a candidate but it is simply a tool
used to derive the Lagrangian U. A better answer is furnished by the
generating functional Z (equation \ref{eq:genfin}).
The generating functional facilitates easy computation of correlation
functions between fields by way of simple functional derivatives.
~ Using that formalism it was shown that UV divgerence is not an
issue in this model for graviton-graviton, and graviton-electromagnetic
field calculations.~
Is this paper an example of what past publications have called a relativized model or where GR=QFT? In relativized Quantum Field Theory QFT is modified to make it obey the principles of General Relativity. The logic being that since QFT is the result of modifying quantum mechanics so that it is compatible with Special Relativity, we simply modifiy it a bit more. GR=QFT states put simply that GR is equally fundamental to QFT and that without modification one can work with both of them. The literature on this is a bit lacking in details on how to do it, for instance \cite{2017arXiv170803040S}. This model could also be interpreted in that manner, with QFT techniques applied to calculating quantities in GR and treating Ricci curvature as a field operator.
\section{Conclusion}
\label{conclusion}
This paper posed the question why is our universe is so well described
by at least three Lagrangians, which may not be simply unified, and
which derive from very different looking physical foundations? I postulated
a euclidean space of Lagrangians $\mathscr{L}$ in which the Lagrangians
for the standard model $\mathcal{L}$ the Einstein-Hilbert Lagrangian
$L$ and the action of the dark sector $\mathcal{O}$ (even though
the details of this last one are unknown) are coordinates. From that
assumption a differential equation for a functional $T$ on the space
$\mathscr{L}$ was derived. The result of that equation with the right
boundary conditions is the Lagrangian for our universe $U$. With
$U$ I defined the partition function $Z$. The action integral $\mathscr{S}$
was computed and found to converge at all energies. Therefore the
generating functional $Z$ can be used to compute physically meaningful
quantities.
From the generating functional Z the results of any interaction between
fields may be computed by taking functional derivatives. Equation
\ref{eq:genfin} also makes manifestly obvious that there can be no
infinitely energetic interactions in this universe. Equation \ref{eq:genfin}
can be written as what as a simple elegant ``one inch equation''.
Equation embodies all known physics, and allows calculations to constrain
the effects of all as of yet unknown physics of the dark sector, and
does so without any divergences, singularities, or infinities.
\[
Z=e^{\imath\mathscr{S}}
\]
This theory is not simply a quantum field theory although it builds
on it. This is not general relativity although it complies with the
constraints of that theory. This is a new type of theory. Let us call
it \textbf{Lagrangian-Euclidean Space Theory} (LEST), because the
one postulate of this theory is that there exist a three dimensional
Euclidean space of ~Lagrangian's in which all of these functionals
exist.~
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