Unified Field Theory
Theory of everything, unifying all forces, matter & spacetime.
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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}
In mathematics, especially in applications of linear algebra to physics, the Einstein notation
or Einstein summation convention is a notational convention that implies summation over a set of
indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a
notational subset of Ricci calculus; however, it is often used in applications in physics that
do not distinguish between tangent and cotangent spaces. It was introduced to physics by
Albert Einstein in 1916.
In quantum mechanics, bra–ket notation is a common notation for quantum states, i.e. vectors
in a complex Hilbert space on which an algebra of observables acts. More generally the notation
uses the angle brackets (the \rangle and \langle symbols) and
a vertical bar (the  symbol), for a ket (for example,  A \rangle) to denote
a vector in an abstract usually complex vector space A and a bra, (for example,
\langle f ) to denote a linear functional f on A.
The natural pairing of a linear function f = \langle f  with a vector
v =  v \rangle is then written as \langle f  v \rangle. On
Hilbert spaces, the scalar product (\ ,\ ) (with anti linear first argument)
given an (antilinear) identification of a vector ket \phi =  \phi \rangle with
a linear functional bra (\phi,\ ) = \langle \phi . Using this notation, the
scalar product (\phi,\psi) = \langle \phi  \psi \rangle. For the vector space
ℂ^{n}, kets can be identified with column vectors, and bras with row
vectors.
 Wikipedia: Einstein Notation
 Wikipedia: BraKet Notation
Einstein notation will be what I will use throughout the rest of the document, which will
employ the standard of 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.
\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}
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and
the cosine of the angle between them. In the case of vector spaces, the dot product
is used for defining lengths (the length of a vector is the square root of the dot
product of the vector by itself) and angles (the cosine of the angle of two vectors
is the quotient of their dot product by the product of their lengths).
An inner product space is a vector space with an additional structure called an inner
product. This additional structure associates each pair of vectors in the space with a
scalar quantity known as the inner product of the vectors. Inner products allow the
rigorous introduction of intuitive geometrical notions such as the length of a vector
or the angle between two vectors. They also provide the means of defining orthogonality
between vectors (zero inner product). Inner product spaces generalize Euclidean spaces
(in which the inner product is the dot product, also known as the scalar product) to
vector spaces of any (possibly infinite) dimension, and are studied in functional
analysis.
More precisely, for a real vector space, an inner product A_{\mu} B^{\nu}
satisfies the following properties shown.
 Wikipedia: Dot Product
 Wikipedia: Inner Product
The dot and inner product are commutative, meaning:
A \cdot B = \bar{B} \cdot \bar{A}
For all vectors A and B.
\begin{aligned}
A^{\mu} \times B^{\nu}
&= e_i \begin{bmatrix} a \\ b \end{bmatrix} \times e_j \begin{bmatrix} c \\ d \end{bmatrix} \\
&= e_k det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\
&= e_k ad  bc
\end{aligned}
\begin{aligned}
A^{\mu} \wedge B^{\nu}
&= e_i \begin{bmatrix} a \\ b \end{bmatrix} \wedge e_j \begin{bmatrix} c \\ d \end{bmatrix} \\
&= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\
&= ad  bc
\end{aligned}
The cross product or vector product (occasionally directed area product to emphasize the
geometric significance) is a binary operation on two vectors in threedimensional space
(\mathbb{R}^3) and is denoted by the symbol ×. Given two
linearly independent vectors A and B, the cross product A \times B is
defined as a vector C that is perpendicular (orthogonal) to both A and B, with a direction
given by the righthand rule and a magnitude equal to the area of the parallelogram that
the vectors span.
The exterior product or wedge product of vectors is an algebraic construction used in
geometry to study areas, volumes, and their higherdimensional analogues. The exterior
product of two vectors A and B, denoted by A \wedge B, is called a bivector
and lives in a space called the exterior square, A vector space that is distinct from the
original space of vectors. The magnitude of A \wedge B can be interpreted
as the area of the parallelogram with sides A and B, which in three dimensions can also
be computed using the cross product of the two vectors.
 Wikipedia: Cross Product
 Wikipedia: Exterior, Wedge Product
Both the cross product and wedge product are anticommutative, meaning:
A \times B =  B \times A
A \wedge B =  B \wedge A
For all vectors A and B. Unlike the cross product, the wedge product is associative.
\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}
The outer product of two coordinate vectors is a matrix. If the two vectors have dimensions n
and m, then their outer product is an n × m matrix. More generally, given two tensors
(multidimensional arrays of numbers), their outer product is a tensor. The outer product of
tensors is also referred to as their tensor product and can be used to define the tensor algebra.
The Kronecker product, sometimes denoted by \otimes is an operation on two
matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer
product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix
of the tensor product with respect to a standard choice of basis. The Kronecker product should
not be confused with the usual matrix multiplication, which is an entirely different operation.
 Wikipedia: Outer Product
 Wikipedia: Kronecker Product
 Wikipedia: Tensor Product
Like the cross product, the outer product is anticommutative, meaning that.
A \otimes B =  B \otimes A
For all vectors A and B.
\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}
The Hadamard product (also known as the elementwise, entrywise or Schur product) is
a binary operation that takes two matrices of the same dimensions and produces another
matrix of the same dimension as the operands where each element i, j is the product of
elements i, j of the original two matrices. It should not be confused with the more
common matrix product.
 Wikipedia: Hadamard Product
The Hadamard product is associative and distributive. Unlike the matrix product, it
is also commutative.
\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} +\sigma_0 & 0 \\ 0 & \sigma_0 \end{bmatrix}
\gamma^1 = \begin{bmatrix} 0 & +\sigma_1 \\ \sigma_1 & 0 \end{bmatrix}
\gamma^2 = \begin{bmatrix} 0 & +\sigma_2 \\ \sigma_2 & 0 \end{bmatrix}
\gamma^3 = \begin{bmatrix} 0 & +\sigma_3 \\ \sigma_3 & 0 \end{bmatrix}
\gamma^5 = \begin{bmatrix} 0 & +\sigma_0 \\ +\sigma_0 & 0 \end{bmatrix}
I_4 = \begin{bmatrix} +\sigma_0 & 0 \\ 0 & +\sigma_0 \end{bmatrix}
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2
complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter
sigma (\sigma), they are occasionally denoted by tau (\tau)
when used in connection with isospin symmetries.
 \sigma_0 = \sigma_t = I_2
 \sigma_1 = \sigma_x
 \sigma_2 = \sigma_y
 \sigma_3 = \sigma_z
 Wikipedia: Pauli Maatrices
In mathematical physics, the gamma matrices, also known as the Dirac matrices, are a set
of conventional matrices with specific anticommutation relations that ensure they generate
a matrix representation of the Clifford algebra Cℓ_{1,3}(R). It is also possible
to define higherdimensional gamma matrices. When interpreted as the matrices of the action
of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column
vectors on which the matrices act become a space of spinors, on which the Clifford algebra of
spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations
and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular
are fundamental to the Dirac equation for relativistic spin½ particles.
 \gamma^0 = \sigma_3 \otimes \sigma_0
 \gamma^j = i\sigma_2 \otimes \sigma_j
 \gamma^5 = \sigma_1 \otimes \sigma_0 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3
 Wikipedia: Gamma Matrices
\gamma^0 is the timelike, hermitian matrix. The other three
\gamma^1, \gamma^2, \gamma^3, are spacelike,
antihermitian matrices.
 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}
In mathematics, the quaternions are a number system that extends the complex numbers. They
were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to
mechanics in threedimensional space. A feature of quaternions is that multiplication of
two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two
directed lines in a threedimensional space or equivalently as the quotient of two vectors.
We get the following when we calculate \mathbb{H}^2 or the inner product
between covariant conjugate vector and its contravariant form shown to the right:
 Wikipedia: Complex
 Wikipedia: Quaternion
 Wikipedia: Octonion
 Wikipedia: Sedenion
Quaternions are generally represented in the form: a + ib + jc + kd where
a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.
Spacetime Topology
Figure 1: Penrose Spacetime.
Previously we mentioned the special relativity interval. 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.
Energy Equivalence
Figure 2: 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 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).
Using Pythagoean Theorem:
 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
Giving Us:
(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
Normalising:
(\hat{E})^2 = (mc^2)^2 + (\hat{p}c)^2
\nabla^2  \frac{1}{c^2} \frac{\partial^2}{\partial{t^2}} = (\frac{mc}{\hbar})^2
Resulting In Einstein & KleinGordon Equations.
Dirac Equation By Summing Vectors:
\hat{E}  \hat{p}c  mc^2 = 0
Normalising:
i \hbar \gamma^{\mu} \partial_{\mu}  mc = 0
As:
\left[
\partial_{0}^{2} =
\frac{1}{c^2} \frac{\partial^2}{\partial t^2}
\right]
\left[
\partial_{j}^{2} = \nabla^2 =
\frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} +
\frac{\partial^2}{\partial z^2}
\right]
\left[
\gamma_{\mu} \gamma^{\mu} =
I_4
\right]
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
Figure 3: 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.
Calculated Tensor
NOTE: This is currently a work in progress to show how the ℍ
math can be shown in n x n format for 3 x ℍ for space dimensions and 1 x
ℍ for time dimension. While this normally would be shown as a 4 x 4 tensor
array, this is designed for ℂ mathematics instead.
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
The X/Y rotation will rotate space to time, given the Minkowski relationship which requires one
to be imaginary when the other is real. The Z rotation does the same, only it is in and out of the
page. This shows how each time and space component consists of bubbles (waves) which repeat
infinitely in both X/Y and Z planes.
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, Particle Pairs).

\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. 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.