"Unification of Fundamental Physics Using Lagrangian-Euclidian Space, Where GR = QFT." Hontas Farmer,

Authorea preprint,10/18/2017 DOI: 10.22541/au.150832792.28888379

**First a question, what is a Lagrangian?**At the most basic a Lagrangian is a function which is the

*difference between the kinetic and potential energies*of the system under consideration. Kinetic energy is energy of motion, potential energy is stored energy or in field theory terms the energy of an interaction. The Lagrangian satisfies Hamilton's principle of least action, physical systems follow the path which will result in the least action. Where the Lagrangian appears and minimizes the action of a physical system. A homespun example of this principle at work is the old adage that a river will find the path of least resistance to the ocean.

All that aside, the Lagrangian is ENERGY. QFT and GM both can be expressed in the form of a relativized Lagrangian, That is QFT can be recast in a form where it obeys the principles of general relativity in any local patch of space time where interactions actually occur. In simpler terms, at the end of the day, energy is energy is energy.

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 (script S) due to U is computed and shown to converge even at infinite energy. Finally with (script S) a generating functional for all possible interactions is computed and a one inch equation of everything is given . 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.

**This is the abstract of the paper.**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 (script S) due to U is computed and shown to converge even at infinite energy. Finally with (script S) a generating functional for all possible interactions is computed and a one inch equation of everything is given . 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.

The result of much functional calculus in a space formed by the Lagrangians of three theories treated as coordinates I get a single equation short equation in terms of the action of the universal Lagrangian (script S) and the Lagrangians of dark matter, the standard model, and general relativity.

This is the "One Inch Equation" as Michio Kaku once quipped the final theory would look like.

Z is what is known as a generating functional from which field theory results can be computed by taking functional derivatives.

Z is what is known as a generating functional from which field theory results can be computed by taking functional derivatives.

**On the latest LIGO observation.**

In the paper I give an equation for the correlation function between fermions and anti-fermions interacting via both electromagnetic and gravitational forces. The short version is the theory predicts nothing which one would not expect from general relativity and qft in that situation. LIGO observed two neutron stars, a collection of fermions and some anti-fermions in the mix colliding with a huge release of high energy photons and gravitational waves.

I have not done detailed numerical calculations yet. Intuition based on my model tells me that when and if we are able to examine such events closely the result will be a black hole. That may sound obvious but it isn't obvious that it needs to be a black hole. If enough mass is annihilated to photons and escapes the result could be a neutron star or a more massive, dense, and exotic object called a quark star.

I have not done detailed numerical calculations yet. Intuition based on my model tells me that when and if we are able to examine such events closely the result will be a black hole. That may sound obvious but it isn't obvious that it needs to be a black hole. If enough mass is annihilated to photons and escapes the result could be a neutron star or a more massive, dense, and exotic object called a quark star.

**On future finds in particle physics.**The lower case omega in the above equation gives relies on there being a simple proportionality between how strongly dark matter particles and particles which we may find at the LHC interact with standard model particles. This model predicts we will find new particles both fundamental and composite at accelerators in the future but does not require any super symmetry.

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