Unified Field Theory Theory of everything, unifying all forces, matter & spacetime.
Preface
I would like to start by clarifying how the dimensions of space and time are related
to one another in order to show how they link together in a four dimensional construct
most commonly seen in special relativity:
s^2 = x^2 + y^2 + z^2 + (ict)^2
I would like to now add that both space and time can be defined using ℍ
numbers. This is important as ℝ numbers are commutative, but ℑ
part of ℂ numbers are anticommutative. Commutative properties give the
same result, regardless of the order of operations, anticommutative is when the order of
operation changes the result.
Now we get to the notation I will be using throughout the rest of the document, which will
be lower indices for covectors (rows) and upper indices for contravariant vectors (columns)
as per Einstein index notation. You get the covector by taking the complex conjugate transpose
of the vector. For in depth explanation, please expand the various subjects below:
\begin{aligned}
A_{\mu} B^{\nu}
&= \langle A  B \rangle \\
&= A \cdot B
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&=  A \rangle \langle B  \\
&= A \otimes B
\end{aligned}
\begin{aligned}
A_{\mu} B^{\nu}
&= \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} c \\ d \end{bmatrix} \\
&= ac + bd
\end{aligned}
\begin{aligned}
A_{\mu} B^{\nu}
&= \langle A  B \rangle \\
&= A \cdot B \\
&= {A}^\dagger{B}
\end{aligned}
\begin{aligned}
A_{\mu} B^{\nu}
&= B_{\nu} A^{\mu} \\
&= \overline{B \cdot A} \\
&= {B}^\dagger{A}
\end{aligned}
\begin{aligned}
A^{\mu} \times B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \hat{i} \times \begin{bmatrix} c \\ d \end{bmatrix} \hat{j} \\
&= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \hat{k} \\
&= ad  bc \hat{k}
\end{aligned}
\begin{aligned}
A^{\mu} \wedge B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \hat{i} \wedge \begin{bmatrix} c \\ d \end{bmatrix} \hat{j} \\
&= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\
&= ad  bc
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \begin{bmatrix} c & d \end{bmatrix} \\
&= \begin{bmatrix} ac & ad \\ bc & bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&=  A \rangle \langle B  \\
&= A \otimes B \\
&= {A}{B}^\dagger
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&=  B^{\nu} A_{\mu} \\
&=  B \otimes A \\
&=  {B}{A}^\dagger
\end{aligned}
\begin{aligned}
A^{\mu} \odot B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \odot \begin{bmatrix} c \\ d \end{bmatrix} \\
&= \begin{bmatrix} ac \\ bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A_{\mu} \odot B_{\nu}
&= \begin{bmatrix} a & b \end{bmatrix} \odot \begin{bmatrix} c & d \end{bmatrix} \\
&= \begin{bmatrix} ac & bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A^{\mu} \odot B^{\nu}
&=  A \rangle \odot  B \rangle \\
&= B^{\nu} \odot A^{\mu} \\
&= {A}{B}
\end{aligned}
\begin{aligned}
A_{\mu} \odot B_{\nu}
&= \langle A  \odot \langle B  \\
&= B_{\nu} \odot A_{\mu} \\
&= {B}{A}
\end{aligned}
\sigma_0 = \begin{bmatrix} +1 & 0 \\ 0 & +1 \end{bmatrix}
\sigma_1 = \begin{bmatrix} 0 & +1 \\ +1 & 0 \end{bmatrix}
\sigma_2 = \begin{bmatrix} 0 & i \\ +i & 0 \end{bmatrix}
\sigma_3 = \begin{bmatrix} +1 & 0 \\ 0 & 1 \end{bmatrix}
\gamma_0 =
\begin{bmatrix}
+1 & 0 & 0 & 0 \\
0 & +1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\gamma_1 =
\begin{bmatrix}
0 & 0 & 0 & +1 \\
0 & 0 & +1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix}
\gamma_2 =
\begin{bmatrix}
0 & 0 & 0 & i \\
0 & 0 & +i & 0 \\
0 & +i & 0 & 0 \\
i & 0 & 0 & 0
\end{bmatrix}
\gamma_3 =
\begin{bmatrix}
0 & 0 & +1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & +1 & 0 & 0
\end{bmatrix}
\gamma_5 =
\begin{bmatrix}
0 & +1 & 0 & 0 \\
+1 & 0 & 0 & 0 \\
0 & 0 & 0 & +1 \\
0 & 0 & +1 & 0
\end{bmatrix}
 i = jk = kj
 j = ki = ik
 k = ij = ji
 i^2 = j^2 = k^2 = 1
 ijk = 1
\mathbb{H}_{\mu} \mathbb{H}^{\mu} =
\begin{bmatrix}
+1 & i & j & k \\
+i & +1 & k & +j \\
+j & +k & +1 & i \\
+k & j & +i & +1
\end{bmatrix}
Spacetime Topology
Previously we mentioned the special relativity interval:
s^2 = x^2 + y^2 + z^2 + (ict)^2
However we must also now show all four rotational
copies relating to ± ct, ± ict,
± jct and ± kct as we will define all
dimensions as ℍ of which only the
± ict and ± ct component is shown
in Figure 1.
This will give us a central single time dimension as being complete
and four dimensional with one ℜ and three ℑ
parts. The reason to do this is the ℜ time part will represent
spacetime from the view of electromagnetism, our observable domain and
the ℑ time parts will represent the three colours of the
strong force. Thus not only showing the connectivity between electrons
and quarks, but also between matter and antimatter.
This is where can see that for each quark or electron, spcetime looks
like that described with flat Minkowski spacetime however when we add
everything together it starts to look more like it has negative curvature
(saddle shaped) which is a antide Sitter spacetime. Most crucially we can
see that because electromagnetism is ℜ part of spacetime
it must be commutative however interactions between the various strong
force parts being ℑ must be anticommutative as the order
of operation now matters.
The concept of the big bang is now obvious to see as it represents
a moment when T^{0} had zero radius, just like a ripple
in a pond doesn't exist before a stone is thrown in, as there is no
concept of before or negative radius. Expanding spacetime is when the
trough of T^{0} and peaks of S^{α}
and S_{β} are increasing in seperation which would
have the effect of driving matter and antimatter apart as well as forcing
electromagnetism and strong forces apart (greater mass seperation).
Electrons (e^{−}) would sit in the + ℍ
part, positrons (e^{+}) in the − ℍ part and
r,
g,
b
quarks in the +i ℍ, +j ℍ and
+k ℍ parts respectively. The
r,
g,
b
antiquarks in the −i ℍ, −j ℍ
and −k ℍ parts respectively.
You may also see that when time T^{0} is created we must
also have the seperation of matter and antimatter space S^{α}
and S^{β}. We can now also define three categories of
matter:
 Matter: Share both common space and time dimensions.
 Dark Matter: Share only common space dimension.
 Dark Energy: Share only common time dimension.
As you can see in Figure 1, the electrons and positrons only
share the same common time, so positrons created at the big bang
would count as some of the total dark energy. But from the perspective
of quarks or antiquarks, positrons and electrons share common space
and time so would be seen as matter. Mesons being the only solution
using ℂ spacetime, atoms of one colour electron and
three coloured quarks the only solution using ℍ
spacetime.
It is also important to note that S^{α} and
S^{β} do not wrap around but instead connect to
a different universe (as defined by its own central T^{0}),
as we must define the overall multiverse as continuous with no hard edges.
Last point to note is that with penrose diagrams the origin or destination
for light must always be the diagonal edges, and matter moves from the
central time area outward in each direction. For spacelike ℜ
mass must stay within the red areas, and for timelike ℑ mass
within the blue areas. Electromagnetic force (γ) must stay within
the electron area, and the strong force (g) in the quark areas. Only the
weak bosons (W^{±}, Z^{0}) can cross between quark
and electron areas and will be made of part ℜ momentum and
ℑ momentum, which is why they must have mass and be
commutative in nature.
Figure 1: Penrose Spacetime.
Energy Equivalence
Now while using the top most penrose spacetime block, I propose to now look
at the rates of change of time and space instead, by doing the following
two tranforms:
 Temporal: t :\rightarrow \hbar \gamma^{0} \partial_{0}
 Spatial: x :\rightarrow \hbar \gamma^{j} \partial_{j}
We must also add add the factor of c to the spatial axis to change what
would be units of momentum to units of energy, so it matches the units used
in the temporal axis.
This then gives us the image as seen in Figure 2.
We have now highlighted the four quadrants for spacelike and timelike matter,
and the black lines represents lightlike matter. The blue and red curved
survaces now represent the particles energy and momentum levels respectively.
One thing that is now shown is that for each energy level, the gap between
each level must decrease as they bunch up towards the corners of infinite
energy seperation and zero energy and momentum represented by the cross of
the black axis in the center of the diagram. We are using the Minkowski metric
[,+,+,+] here to show the timelike energy is always negative (usually
depicted by the colour blue) and spacelike energy is always positive (usually
depicted by the colour red).
We can now calculate mass as we would the radius of a 2D circle, using:
 x^2 + y^2 = r^2
Transposing:
 x :\rightarrow i\hat{p}c = \hbar c \gamma^j \partial_j
 y :\rightarrow \hat{E} = i \hbar c \gamma^0 \partial_0
 r :\rightarrow mc^2
Which gives us the following two formula:
(i\hat{p}c)^2 + (\hat{E})^2 = (mc^2)^2
(\hbar c \gamma^j \partial_j)^2 
(\hbar c \gamma^0 \partial_0)^2
= (mc^2)^2
Normalisation of the second formula becomes:
\nabla^2  \frac{1}{c^2} \frac{\partial^2}{\partial{t^2}}
= (\frac{mc}{\hbar})^2
Which is the KleinGordon equation as:
\partial_{0}^{2} =
\frac{1}{c^2} \frac{\partial^2}{\partial t^2}
and
\partial_{j}^{2} =
\frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} +
\frac{\partial^2}{\partial z^2} =
\nabla^2
(\gamma^{\mu})^2 = \gamma_{\mu} \gamma^{\mu} = I_4
Figure 2: Energy Equivalence.
Special Relativity
This explains the need for two Minkowski metrics which must be used together,
in our case as we have the following:
 Electron, Positron: [ct, ix, iy, iz], Metric: [+,−,−,−]
 Red, Antired Quark: [ict, x, y, z], Metric: [−,+,+,+]
 Green, Antigreen Quark: [jct, x, y, z], Metric: [−,+,+,+]
 Blue, Antiblue Quark: [kct, x, y, z], Metric: [−,+,+,+]
In terms of momentum and energy, electrons and quarks must all be
moving at right angles to each other (defining that they are unique
particles). The event horizon between the electrons and quarks is the
gap between the electron cloud and quark lowest energy level.
The higher the energy level of the electron, the further out it goes.
The higher energy level of the quark, the more the electron is pulled
closer. This stretching of the negative part of time between the electron
and quarks, gives us the effects of charge seperation and gravity.

\Delta \approx \frac{\partial}{\partial t}Linear (first order) shear, relates to momentum and charge like attraction.

\Delta \approx \frac{\partial^2}{\partial {t^2}}Curved (second order) shear, relates to acceleration and gravitational attraction.
This shows charge between electrons and quarks acts as a special relativistic
boost between the two, and the rate of change of these boosts act like
curvature and gravity. This is how quarks and electrons alone do not hold
gravity and why it has been so difficult to unify special and general
relativity or strong and electromagnetic forces using only a single real
valued metric of spacetime.
However the combination of quarks and electrons that make mesons, baryons
or atoms as well as the use of a fully connected ℍ model
of spacetime, it is possible to unify both special, general relativity and
all forces together in one unifying model.
Treating the seperate particles as standing waves, would only give the
rest mass and special relativistic and charge connectivity. Adding in
the further rate of vibrational fluctuation of each particle will now
also show the general relativistic connection. In the case of the atom,
the quarks are changing energy far more often than the electrons as they
not only change in energy level the same as electrons, but are also constantly
swapping colour charge as well. This net difference in rates of change in
energy levels will create a gravitational attraction towards the center of
mass defined by the net movement of quarks in the nucleus.
General Relativity
We now have defined what dark matter and energy are, as well as how
they both must transfer energy and momentum to normal matter by further
defining how they are connected to matter through space and time.
Causality is now the net direction of all of the various spacetime
elements combined. So we see time moving forward, and entropy as
always being positive because there is more dark energy pulling on
the outer timelike connections (four corners of Figure 1).
The has the effect of increasing charge seperation of electrons and
quarks, decaying free neutrons. While the smaller amount of dark matter
helps pull matter together, increasing the chances of nuclear fusion.
Now putting this all together we create a wave front moving in one
direction as shown in Figure 3. However from a stationary point,
it will look as if the universe is expanding to a maximum point, then
contracting back to nothing. Matter now has three states being
minuimums when energy or momentum are timelike (shown as blue), maximums
when they are spacelike (shown in red), connected by lightlike type
(horizontal plane along z=50, center of the green areas).
Depending what part you are in, spacetime is being continuously created
and destroyed, think of this as the big bang and crunch happening at
the same time, multiple times, but constantly moving location with
the overall energy staying constant. If stationary you would see
energy changing from spacelike to timelike through lightlike and back
again continuously, that change would show as changes to entropy, and
causality, e.g. time flowing backwards, heat going from hot to cold,
etc. This resolves the paradoxical answer to how we can both have a
big bang creating spacetime, but energy can neither be created or
destroyed. Nor is the idea the point of creation as being a static point,
or singular; like everything else it is a wave and continuous.
Figure 3: General Relativity.
Calculated Tensor
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
Summary
This framework shows that for any spacetime domain, for example that of the electrons must connect to both normal matter and dark matter through spatial
connectivity and similarly normal matter and dark energy through temporal connectivity. Matter is then based on the complexity of the temporal and spatial connectivity as shown:

\mathbb{C}1D+1T (Meson, Quark + Antiquark Pair, Electron + Positron Pair).

\mathbb{H}3D+1T (Atom, Baryon)

\mathbb{O}7D+1T

\mathbb{S}15D+1T
As there are no definable numbers between ℂ and
ℍ, we will not see any type of compound particles
That are purely 3D, say a meson with an electron as it is not allowed.
Also below ℂ is also not possible as you must create
spacelike dimension as well as timelike dimensions in order to satify
the wavelike nature and continuous topology overall.
While we have good proof to determine our own universe as being 4D, the
universes that connect externally to ours don't necessarilly have to be
4D as well. We also can't assume that the amount of fermions and quarks
are equal too, as dark matter is connected spatially it may have greater
pull to electrons, or one of the colours of quarks in different areas of space.
Likewise for dark energy, only it will vary over time; which has pretty much
been determined already in cosmology shows a growth spurt earlier in the
evolution of our universe.