# Unified Field Theory Gauge Field Theory - Unifying All Forces

## Special Unitary Groups

Here we have the standard notation for special unitary group SU(3) used a lot in Q.C.D. to explain the gluon field:

$\mathrm{SU\left(3\right)}=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]×\left[\begin{array}{ccc}\stackrel{-}{x}& \stackrel{-}{y}& \stackrel{-}{z}\end{array}\right]$

$\mathrm{SU\left(3\right)}=\left[\begin{array}{ccc}x\stackrel{-}{x}& x\stackrel{-}{y}& x\stackrel{-}{z}\\ y\stackrel{-}{x}& y\stackrel{-}{y}& y\stackrel{-}{z}\\ z\stackrel{-}{x}& z\stackrel{-}{y}& z\stackrel{-}{z}\end{array}\right]$

With diagonal determinant:

$\mathrm{DET =}\left[x\stackrel{-}{x}+y\stackrel{-}{y}+z\stackrel{-}{z}\right]$

$\mathrm{DET =}\left[{x}^{2}+{y}^{2}+{z}^{2}\right]$

However we need to allow for multiple dimensions of time (negative curvature, unfolding) and space (positive curvature, folding), the only difference is the time dimensions will have the complex conjugate and standard vectors swapped due to multiplication of $\mathrm{± i}$.

The following example while arbitrary slects three dimensions of time, and six dimensions of space:

$\mathrm{SU\left(6 ± 3i\right)}=\left[\begin{array}{c}\stackrel{{-}}{{r}}\\ \stackrel{{-}}{{s}}\\ \stackrel{{-}}{{t}}\\ {x}\\ {y}\\ {z}\\ {u}\\ {v}\\ {w}\end{array}\right]×\left[\begin{array}{ccccccccc}{r}& {s}& {t}& \stackrel{{-}}{{x}}& \stackrel{{-}}{{y}}& \stackrel{{-}}{{z}}& \stackrel{{-}}{{u}}& \stackrel{{-}}{{v}}& \stackrel{{-}}{{w}}\end{array}\right]$

$\mathrm{SU\left(6 ± 3i\right)}=\left[\begin{array}{ccccccccc}\stackrel{{-}}{{r}}{r}& \stackrel{{-}}{{r}}{s}& \stackrel{{-}}{{r}}{t}& \stackrel{{-}}{{r}}\stackrel{{-}}{{x}}& \stackrel{{-}}{{r}}\stackrel{{-}}{{y}}& \stackrel{{-}}{{r}}\stackrel{{-}}{{z}}& \stackrel{{-}}{{r}}\stackrel{{-}}{{u}}& \stackrel{{-}}{{r}}\stackrel{{-}}{{v}}& \stackrel{{-}}{{r}}\stackrel{{-}}{{w}}\\ \stackrel{{-}}{{s}}{r}& \stackrel{{-}}{{s}}{s}& \stackrel{{-}}{{s}}{t}& \stackrel{{-}}{{s}}\stackrel{{-}}{{x}}& \stackrel{{-}}{{s}}\stackrel{{-}}{{y}}& \stackrel{{-}}{{s}}\stackrel{{-}}{{z}}& \stackrel{{-}}{{s}}\stackrel{{-}}{{u}}& \stackrel{{-}}{{s}}\stackrel{{-}}{{v}}& \stackrel{{-}}{{s}}\stackrel{{-}}{{w}}\\ \stackrel{{-}}{{t}}{r}& \stackrel{{-}}{{t}}{s}& \stackrel{{-}}{{t}}{t}& \stackrel{{-}}{{t}}\stackrel{{-}}{{x}}& \stackrel{{-}}{{t}}\stackrel{{-}}{{y}}& \stackrel{{-}}{{t}}\stackrel{{-}}{{z}}& \stackrel{{-}}{{t}}\stackrel{{-}}{{u}}& \stackrel{{-}}{{t}}\stackrel{{-}}{{v}}& \stackrel{{-}}{{t}}\stackrel{{-}}{{w}}\\ {x}{r}& {x}{s}& {x}{t}& {x}\stackrel{{-}}{{x}}& {x}\stackrel{{-}}{{y}}& {x}\stackrel{{-}}{{z}}& {x}\stackrel{{-}}{{u}}& {x}\stackrel{{-}}{{v}}& {x}\stackrel{{-}}{{w}}\\ {y}{r}& {y}{s}& {y}{t}& {y}\stackrel{{-}}{{x}}& {y}\stackrel{{-}}{{y}}& {y}\stackrel{{-}}{{z}}& {y}\stackrel{{-}}{{u}}& {y}\stackrel{{-}}{{v}}& {y}\stackrel{{-}}{{w}}\\ {z}{r}& {z}{s}& {z}{t}& {z}\stackrel{{-}}{{x}}& {z}\stackrel{{-}}{{y}}& {z}\stackrel{{-}}{{z}}& {z}\stackrel{{-}}{{u}}& {z}\stackrel{{-}}{{v}}& {z}\stackrel{{-}}{{w}}\\ {u}{r}& {u}{s}& {u}{t}& {u}\stackrel{{-}}{{x}}& {u}\stackrel{{-}}{{y}}& {u}\stackrel{{-}}{{z}}& {u}\stackrel{{-}}{{u}}& {u}\stackrel{{-}}{{v}}& {u}\stackrel{{-}}{{w}}\\ {v}{r}& {v}{s}& {v}{t}& {v}\stackrel{{-}}{{x}}& {v}\stackrel{{-}}{{y}}& {v}\stackrel{{-}}{{z}}& {v}\stackrel{{-}}{{u}}& {v}\stackrel{{-}}{{v}}& {v}\stackrel{{-}}{{w}}\\ {w}{r}& {w}{s}& {w}{t}& {w}\stackrel{{-}}{{x}}& {w}\stackrel{{-}}{{y}}& {w}\stackrel{{-}}{{z}}& {w}\stackrel{{-}}{{u}}& {w}\stackrel{{-}}{{v}}& {w}\stackrel{{-}}{{w}}\end{array}\right]$

The determinant, event horizon and Higgs field ${H}^{0}$ is now:

$\mathrm{DET =}\left[x\stackrel{-}{x}+y\stackrel{-}{y}+z\stackrel{-}{z}+u\stackrel{-}{u}+v\stackrel{-}{v}+w\stackrel{-}{w}+\stackrel{-}{r}r+\stackrel{-}{s}s+\stackrel{-}{t}t\right]$

$\mathrm{DET =}\left[{x}^{2}+{y}^{2}+{z}^{2}+{u}^{2}+{v}^{2}+{w}^{2}-{r}^{2}-{s}^{2}-{t}^{2}\right]$

That may look unweildy, but you can see this is not SU(9) as the time dimensions are now reversed to the space dimensions and the model for spacetime with three space dimensions and one time dimension is a subset of this. So quarks will see this as 6D+3iT (inwards spin) and leptons will see it as 6D-3iT (outward spin).

The red area is what will lead to space like magnetic and strong fields ${\psi }_{B}$, green areas are the spacetime charges ${\psi }_{E}$ and blue areas are time like magnetic and strong fields ${\psi }_{B}$.

Dark energy is the effect of time like fields and charges outside of the visible spacetime $\mathrm{\left[x, y, z, t\right]}$, and dark matter is the effect of space like fields.

We now have to define the DEL operator representing momentum for both space and time:

${\nabla }_{\mathit{Space}}=\left(\frac{\partial }{\partial x},+,\frac{\partial }{\partial y},+,\frac{\partial }{\partial z},+,\frac{\partial }{\partial u},+,\frac{\partial }{\partial v},+,\frac{\partial }{\partial w}\right)$

${\nabla }_{\mathit{Time}}=\left(\frac{\partial }{\partial r},+,\frac{\partial }{\partial s},+,\frac{\partial }{\partial t}\right)$

And the Laplace operator representing acceleration for both space and time:

${{\nabla }^{2}}_{\mathit{Space}}=\left(\frac{{\partial }^{2}}{{\partial x}^{2}}+\frac{{\partial }^{2}}{{\partial y}^{2}}+\frac{{\partial }^{2}}{{\partial z}^{2}}+\frac{{\partial }^{2}}{{\partial u}^{2}}+\frac{{\partial }^{2}}{{\partial v}^{2}}+\frac{{\partial }^{2}}{{\partial w}^{2}}\right)$

${{\nabla }^{2}}_{\mathit{Time}}=\left(\frac{{\partial }^{2}}{{\partial r}^{2}},+,\frac{{\partial }^{2}}{{\partial s}^{2}},+,\frac{{\partial }^{2}}{{\partial t}^{2}}\right)$

We now have defined the dimensions as the determinent of thr array, but we also have two forms of mass the quarks with respect to $t$ which will spin inwards from the event horizon, and leptons with resect to $\stackrel{-}{t}$ which whill spin outwards from the even horizon.

With all stable atoms like nobel gasses, the charge and spin of the quarks must balance that of the leptons. Aka the overall spin and charge is zero, so there needs to be an electron to balance every proton and neutrino to balance every neutron. This doesn't mean it will have no mass, but rather it will be unreactive.

Shown below is the full diagram, now showing quarks trying to spin inward into the page, and leptons spinning outward out of the page. Here you can clearly see the inner box of 3D+1T spacetime as a subset of the 6D+3T outer box.

Comparison of Lepton and Quark Fields

Figure 1.

Now the electron and neutrino are compound particles with spatial charges giving rise to spin orbitals and photons and gluons are the same thing except for the direction they spin with respect to the Higgs field ${H}^{0}$. The ${W}^{±}$ and ${Z}^{0}$ are now just rotations from outside perspective to inside perspective or vice versa. As the Higgs field ${H}^{0}$ and event horizon are spin and charge neutral, anything passing though it with respect to space only must also be charged neutral (${Z}^{0}$), anything rotating with repsect to space and time can also carry charge (${W}^{±}$).

Ultimately though, we don't want to restrict the topology in either way. But we must now take into account the number of dimensions of each type and they are oriented with repsect to itself and its surrounding dimensions. So the 3D+1T visible universe is bubble inside the 6D+3T universe and that could be itself a bubble within a larger universe.

## Maxwell Equations

The Maxwell-Gauß equation describes electric charge density $\rho$, as the source of the electric field, ${\psi }_{E}$.

${\nabla }_{\mathit{Space}}.{\psi }_{E}=\frac{\rho }{{\epsilon }_{0}}$

The Maxwell-Flux equation asserts the conservative nature of the magnetic field, ${\psi }_{B}$.

${\nabla }_{\mathit{Space}}.{\psi }_{B}=0$

The Maxwell-Faraday equation asserts the induction law that creates an induction potential by relating the curl of the electric field, ${\psi }_{E}$, to the time variation of the magnetic field, ${\psi }_{B}$.

${\nabla }_{\mathit{Space}}×{\psi }_{E}=-{\nabla }_{\mathit{Time}}×{\psi }_{B}$

The Maxwell-Ampère equation describes how electric current density $J$, relates to time variations of the electric field, ${\psi }_{E}$, as the sources of magnetic fields, ${\psi }_{B}$.

${\nabla }_{\mathit{Space}}×{\psi }_{B}=\frac{1}{{\epsilon }_{0}{c}^{2}}J+\frac{1}{{c}^{2}}{\nabla }_{\mathit{Time}}×{\psi }_{E}$

We can use these dot and cross products for both electromagnetism as well as the strong fields, as these are just generalising the momentum calculations created by finding the first deritive of the field portion ${\psi }_{B}$ of the wave function, and ${\psi }_{B}$ for the charge portion of the wave function.

## Kein-Gordon Equation

The Kein-Gordon equation asserts the mass of a system relates to the difference between spatial acceleration and temporal acceleration.

${\left(\frac{\mathrm{mc}}{\hslash }\right)}^{2}\psi =\left({{\nabla }^{2}}_{\mathit{Space}}-\frac{1}{{c}^{2}}{{\nabla }^{2}}_{\mathit{Time}}\right)\psi$

However as we are dealing with a system of greater than three dimensions of space and one dimension of time, we have to assume the wavefunction has portions of visible matter within the 3D+1T boundry, dark matter and dark energy outside the 3D+1T boundry.

 ${\left(\frac{\mathrm{mc}}{\hslash }\right)}^{2}\psi =$ $\left({{\nabla }^{2}}_{\mathit{Space}}-\frac{1}{{c}^{2}}{{\nabla }^{2}}_{\mathit{Time}}\right){{\psi }}_{{\mathit{M}}}$ $+$ $\left({{\nabla }^{2}}_{\mathit{Space}}-\frac{1}{{c}^{2}}{{\nabla }^{2}}_{\mathit{Time}}\right){{\psi }}_{{\mathit{DM}}}$ $+$ $\left({{\nabla }^{2}}_{\mathit{Space}}-\frac{1}{{c}^{2}}{{\nabla }^{2}}_{\mathit{Time}}\right){{\psi }}_{{\mathit{DE}}}$

However as dark matter is a space like event, and dark energy a time like event with only matter having both space and time like events.

We can also view dark matter and dark energy as one form of mass with both fields and charges like matter, but acting outside the standard $\mathrm{\left[x, y, z, t\right]}$ frame of reference for visible matter.

So the previous example can be simplified to:

 ${\left(\frac{\mathrm{mc}}{\hslash }\right)}^{2}\psi =$ $\left(\frac{{{\partial }}^{{2}}}{{{\partial x}}^{{2}}}{+}\frac{{{\partial }}^{{2}}}{{{\partial y}}^{{2}}}{+}\frac{{{\partial }}^{{2}}}{{{\partial z}}^{{2}}}{-}\frac{{1}}{{{c}}^{{2}}}\frac{{{\partial }}^{{2}}}{{{\partial t}}^{{2}}}\right)\psi$ $+$ $\left(\frac{{{\partial }}^{{2}}}{{{\partial u}}^{{2}}}{+}\frac{{{\partial }}^{{2}}}{{{\partial v}}^{{2}}}{+}\frac{{{\partial }}^{{2}}}{{{\partial w}}^{{2}}}\right)\psi$ $-$ $\frac{{1}}{{{c}}^{{2}}}\left(\frac{{{\partial }}^{{2}}}{{{\partial r}}^{{2}}}{+}\frac{{{\partial }}^{{2}}}{{{\partial s}}^{{2}}}\right)\psi$

And finally:

${\left(\frac{\mathrm{mc}}{\hslash }\right)}^{2}\psi =\left({{{\nabla }}^{{2}}}_{{\mathit{uvw}}}{-}\frac{{1}}{{{c}}^{{2}}}{{{\nabla }}^{{2}}}_{{\mathit{rs}}}\right)\psi +\left({{{\nabla }}^{{2}}}_{{\mathit{xyz}}}{-}\frac{{1}}{{{c}}^{{2}}}{{{\nabla }}^{{2}}}_{{\mathit{t}}}\right)\psi$

This now shows that mass can be created either by visible matter (green, spacetime event), dark matter (red, spacelike event), and dark energy (blue, timelike event).

It also shows that any visible universe (green, spacetime event), can also have matter which is none interactive with charges within it. But it can affect the visible universe by altering momentum and acceleration of the dimensions. Giving us gravity from dark matter and expansion of spacetime from dark energy.

The ratio of dark matter to dark energy will also give the topology of the visible universe it creates, as this ration is approximately 3:1 there needs to be three space like dimensions to balance out the one temporal dimension.