Unified Field Theory
Theory of everything, unifying all forces, matter & spacetime.

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1: Required Reading
The scientific notation I will be using throughout the rest of the document, where we will be lower indices for covectors (rows) and upper indices for contravariant vectors (columns) as per Einstein index notation. You create the covector by taking the complex conjugate transpose of the vector. For in depth explanation, please expand the various subjects below:
Einstein Notation:
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.
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.
Bra-Ket Notation:
In quantum mechanics, bra–ket notation is a common notation for quantum states, i.e. vectors in a \CC 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 \CC 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 (anti-linear) 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 \CC^n, kets can be identified with column vectors, and bras with row vectors.
Wikipedia Articles:
Example Math:
\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}
Inner, Dot & Scalar Product:
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.
The dot and inner product are commutative, meaning:
A \cdot B = \overline{B \cdot A}
For all vectors A and B.
Figure 1(b): Dot Product.
Wikipedia Articles:
Example Math:
\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}
Cross Product:
The cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (\RR^3) and is denoted by the symbol \times. 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 right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
Exterior & Wedge Product:
The exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional 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.
Both the cross product and wedge product are anti-commutative, 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.
Figure 1(c): Cross & Wedge Product.
Wikipedia Articles:
Example Math:
\begin{aligned} A^{\mu} \times B^{\nu} &= \begin{bmatrix} a \\ b \end{bmatrix} e_i \times \begin{bmatrix} c \\ d \end{bmatrix} e_j \\ &= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} e_k \\ &= |ad - bc| e_k \end{aligned}
\begin{aligned} A^{\mu} \wedge B^{\nu} &= \begin{bmatrix} a \\ b \end{bmatrix} e_i \wedge \begin{bmatrix} c \\ d \end{bmatrix} e_j \\ &= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\ &= |ad - bc| \end{aligned}
Outer & Tensor Product:
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.
Kronecker Product:
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.
Like the cross product, the outer product is anti-commutative, meaning that.
A \otimes B = - B \otimes A
For all vectors A and B.
Wikipedia Articles:
Example Math:
\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}
Hadamard Product:
The Hadamard product (also known as the element-wise, 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.
The Hadamard product is associative and distributive. Unlike the matrix product, it is also commutative.
Wikipedia Articles:
Example Math:
\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} &= \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} &= \langle A | \odot \langle B | \\ &= B_{\nu} \odot A_{\mu} \\ &= {B}{A} \end{aligned}
Pauli Matrices:
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 \CC 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
Gamma Matrices:
In mathematical physics, the gamma matrices, also known as the Dirac matrices, are a set of conventional matrices with specific anti-commutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(\RR). It is also possible to define higher-dimensional 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 spin-½ 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
\gamma^0 = \gamma^t is the time-like, hermitian matrix. \gamma^1 = \gamma^x, \gamma^2 = \gamma^y, \gamma^3 = \gamma^z, are space-like, anti-hermitian matrices.
Wikipedia Articles:
Example Math:
\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}
Quaternion Numbers:
In mathematics, the \HH are a number system that extends the \CC numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of \HH is that multiplication of two \HH is none-commutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
We get the following when we calculate \HH^2 or the inner product between covariant conjugate vector and its contravariant form shown to the right:
\HH are generally represented in the form: a + \hat{i}b + \hat{j}c + \hat{k}d where a, b, c, and d are \RR numbers, and \hat{i}, \hat{j}, \hat{k} are the fundamental quaternion unit vectors.
Wikipedia Articles:
Example Math:
  • \hat{i} = \hat{j}\hat{k} = -\hat{k}\hat{j}
  • \hat{j} = \hat{k}\hat{i} = -\hat{i}\hat{k}
  • \hat{k} = \hat{i}\hat{j} = -\hat{j}\hat{i}
  • \hat{i}^2 = \hat{j}^2 = \hat{k}^2 = -1
  • \hat{i}\hat{j}\hat{k} = -1
\HH^{\mu} \HH_{\mu} = \begin{bmatrix} +1 & +\hat{i} & +\hat{j} & +\hat{k} \\ -\hat{i} & +1 & -\hat{k} & +\hat{j} \\ -\hat{j} & +\hat{k} & +1 & -\hat{i} \\ -\hat{k} & -\hat{j} & +\hat{i} & +1 \end{bmatrix}
Penrose Diagram:
In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an extension of a Minkowski diagram where the vertical dimension represents time, and the horizontal dimension represents a space dimension, and slanted lines at an angle of 45° correspond to light rays. The biggest difference is that locally, the metric on a Penrose diagram is conformally equivalent to the actual metric in spacetime. The conformal factor is chosen such that the entire infinite spacetime is transformed into a Penrose diagram of finite size, with infinity on the boundary of the diagram. For spherically symmetric spacetime, every point in the Penrose diagram corresponds to a 2-dimensional sphere.
One important point to make is that the light-like infinities are just the same as space-like and time-like infinities, just rotated at an angle of 45°, so just as we see space and time as being unique, we to need to see \RR and \II light-like events as unique too. We do as \RR gives us U(1) for electromagnetism (photons) and \hat{i}, \hat{j}, \hat{k} give us SU(3) of strong force (gluons).
Figure 1(h): Penrose Spacetime.
We can now take this penrose block, and stack them end to end in a similar way to creating a jigsaw puzzle. Only making sure we rotate each block at 90° in either direction, ultimately changing what was originally a time-like cone to time-like sphere. With all past events being inside the sphere and all future events likewise being outside. In the same way, we will also create spherical bubbles of space. This will create something akin to the hypercube or tesseract, with all past time-like events in the center portion (aka the visible universe we can see), and all space-like events in the outer portions (aka what we call dark matter and energy), with light-like events being the vertexes.
Wikipedia Articles:
2: Introduction
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 using [+,-,-,-] Minkowski metric, with [t,x,y,z] representing \RR scalar values of length (using c=1 for clarity):
\begin{aligned} (\hat{1} s)^2 &= \hat{1}^2 \times \left( (\hat{1} t)^2 + (\hat{i} x)^2 + (\hat{j} y)^2 + (\hat{k} z)^2 \right) \\ &= t^2 - x^2 - y^2 - z^2 \end{aligned}
Using this metric and choice of unit vectors for time and space, spacetime event length s is now expressible in one dimensional \HH form, aka that of a single particle or wave described by one number. The event length being the radius of a 4D sphere, just like r^2 = x^2 + y^2 + z^2 for a standard 3D sphere, only difference is space and time are always imaginary orthogonal to each other. I would then like to add that both space and time can also be defined using \HH numbers, to further give us the view of spacetime from four unique particles, in our case one type of electron (the \RR part we see outside the nucleus) and three colours of quarks (the \II parts we can't see inside the nucleus). This can be seen by looking at the \hat{1}, \hat{i}, \hat{j}, \hat{k} rotations of the spacetime event interval s. One with metric [+,-,-,-] and three with metric [-,+,+,+]:
\begin{aligned} (\hat{i} s)^2 &= \hat{i}^2 \times \left( (\hat{1} t)^2 + (\hat{i} x)^2 + (\hat{j} y)^2 + (\hat{k} z)^2 \right) \\ &= x^2 + y^2 + z^2 - t^2 \end{aligned}
So while the electrons are inside their own spacetime event cone s and thus time-like causally connected events to each other, the quarks are always space-like causally disconected events to the electrons a fundamental reason we see them as seperate particles but quarks have time-like event cone \hat{i} s, \hat{j} s, \hat{k} s with each other which is why we can't seperate out quarks individually but the behave in a non-commutative way just as multiplication of \II numbers do. In short the electrons and quarks are not causally connected, except through rotations via the weak force, which is the light-like event that is part electromagnetism and part strong force. If we think of electromagnetism and strong force as straight lines, they can only have momentum and no mass, but weak force being two halves of a straight line at 90° must have both momentum and mass.
This is important as \RR numbers are commutative, but \II part of \CC or \HH numbers are anti-commutative. Commutative properties give the same result, regardless of the order of operations, non-commutative is when the order of operation changes the result. I shown above we just extend that to the special relativity spacetime intervals as well and this gives us the different properties of electrons and quarks as well as electromagnetism, strong and weak forces.
This links the fact we see spacetime as four dimensional because we are made up of matter that is inherently four dimensional, specifically three of one type of non-commutative particle (quarks) giving us our connected view of three dimensional space, and one further particle giving us one dimensional time (electron). Ergo the observed spacetime is intrinsically linked to the dimensionality and connectivity of the energy observing it. But not only that we have duality because we can only view spacetime from the perspective of the quarks or electrons (rotational viewpoints), leading to viewing real space with metric [-,+,+,+] or real time with metric [+,-,-,-]. For this reason, we can never view a single particle as both wave and point source as both these viewpoints are orthogonal to each other, and we have to use both Minkowski spacetime metrics simultaneously, \RR interval of electrons giving us [+,-,-,-], and \II interval of quarks giving us [-,+,+,+], my argument being we need to start using both and thinking of the spacetime interval as fully \HH to unite both quark and electron, strong, weak and electromagnetic force interactions together in a single unified event interval of the atom. Thereby creating a unified wavefunction of the atom encoding all particles and forces into one theory.
Electromagnetic Force U(1) being that of the \RR commutative part of time, SU(3) strong force being that of \II non-commutative parts of time and finally the weak force being the rotations between \RR and \II. This is a departure from normal standard model that teaches us that the weak force is independent of the strong and electromagnetic forces, rather I postulate it is instead the rotational link between the electromagnetic and strong forces. E.g. the real time-like event s^2 \gt 0 for electrons and electrical charge, s^2 \lt 0 for quarks and coloured charge, and {\pm}s^2 = 0 for gluons and photons, finally with {\pm}(\hat{i}, \hat{j}, \hat{k})s^2 = 0 for weak bosons.
So just as we have found a link between electromagnetism and weak force to give electroweak theory U(1) x SU(2), we should also find a link between strong and weak forces as SU(2) x SU(3). So rather than thinking of the nucleus as having electrical charge, we instead need to think of how it is manipulating the weak force, and that then manipulating the electromagnetic force.
Hence the concept of all dimensions being \HH, including the spacetime interval s. Then the interval is not only in relation to any one particle, both up spin \HH_0 and down spin \HH^0, but also how each particle combines to create a compound event for either mesons with \CC^2 complexity, or atoms with \HH^2 complexity. Massless bosons being events that stay within \RR or \II spacetime so energy and momentum are equal (think straight line), massive bosons being those that cross from \RR to \II or vice-versa so energy and momentum are not equal, resulting in them having mass (think line with right angle in centre). Aka momentum of strong force is converted to mass of electrons, or mass of quarks is converted to momentum of electromagnetism.
3: Spacetime Topology
Previously we mentioned the special relativity interval. The main reason for doing this is so we can now show all four rotational copies relating to \pm t_0, \mp \hat{i} t^1, \mp \hat{j} t^2 and \mp \hat{k} t^3 as we will define all dimensions as \HH of which only the \pm t_0 and \mp \hat{i} t^1 component is shown in Figure 3(a).
We can see in Figure 3(a) the light-light infinite singularity event T^{0}, and the ever increasing event horizon (shown as the first black square from the centre), followed by the infinite future light like event horizon. The concept of now is redefined from a point (as would be seen by a single particle with standard special relativity), to a surface with everything inside being a previous event which was causally connected (time-like) to the event horizon in some way. In our case the big bang event which is the start of the causal connection for all matter. Different particles now represent a fraction of the total surface area with light-like events having two possible start and end points, fist is the \pm \infin edges \gamma, g as they stay within a block, second is the W^{\pm}, Z^0 that directly connect the electron and quarks event position.
Assuming we start with \HH_{0} = t_0 +\hat{i}x^0 +\hat{j}y^0 +\hat{k}z^0, and \HH_{1} = -\hat{i}\HH^{0}, \HH_{2} = -\hat{j}\HH^{0}, \HH_{3} = -\hat{k}\HH^{0}, aka looking at spacetime as one dimensional number where t, x, y, z are all \RR numbers defining scale, but the \RR and \II unit vector parts defining uniqueness and orthogonality.
This will give us a central single \HH complexity time dimension as being complete and still one dimensional with one \RR and three \II parts. The reason to do this is the \RR time part will represent spacetime from the view of electromagnetism and electrons, our observable domain and the \II time parts will represent the three colours of the strong force and quarks. Thus not only showing the connectivity between electrons and quarks, electromagnetism, weak and strong forces, but also between matter and antimatter.
Figure 3(a): Penrose Spacetime (2D)
One other thing to note is all \RR dimensions are parallel to each other as are all \II dimensions (regardless if they are related to space or time). Giving us S^{\alpha}, S^{\gamma} as \RR light-like infinities, and S^{\beta}, S^{\delta} as \II light-like infinities. We can further say all electromagnetic force light-like events \gamma are with \RR time \II space and strong force light-like events g with \II time and \RR space. Weak force being rotations being half \RR and \II light-like events.
Next I will switch from t_0 +\hat{i}t^1 +\hat{j}t^2 +\hat{k}t^3 to \color{#E619E6}{\bar{w}} +\color{#FF0000}{r} +\color{#00B050}{g} +\color{#0080FF}{b} and from t^0 -\hat{i}t_1 -\hat{j}t_2 -\hat{k}t_3 to \color{#E619E6}{w} +\color{#FF0000}{\bar{r}} +\color{#00B050}{\bar{g}} +\color{#0080FF}{\bar{b}}, aka we are just using the time components of \HH_{\mu}(t) above, obviously we will also have another three matrixes representing the spatial connectivity, giving us a total 4x4x4 matrix. The time based quantities of electromagnetic and colour charges now give us quantum chromodynamics, quantum electroweak and quantum electrodynamics connectivity.
From Figure 3(a) all the contravariant light-like vectors denoted by \color{#E619E6}{w}, \color{#FF0000}{r}, \color{#00B050}{g}, \color{#0080FF}{b} all flow from the light-like \RR \pm \infin (S^{\alpha}, S^{\gamma}) and the covariant light-like vectors denoted by \color{#E619E6}{\bar{w}}, \color{#FF0000}{\bar{r}}, \color{#00B050}{\bar{g}}, \color{#0080FF}{\bar{b}} all flow from the light-like \II \pm \infin (S^{\beta}, S^{\delta}). So \color{#E619E6}{w} \color{#E619E6}{\bar{w}} represents U(1) of the electromagnetic force \gamma with time-like vectors being the electrons, rotation (\color{#FF0000}{r} +\color{#00B050}{g} +\color{#0080FF}{b}) \color{#E619E6}{w} would be the W^{+} weak boson, rotation \color{#E619E6}{\bar{w}} (\color{#FF0000}{\bar{r}} +\color{#00B050}{\bar{g}} +\color{#0080FF}{\bar{b}}) would be the W^{-} weak boson, and diagonal translation \color{#FF0000}{r} \color{#FF0000}{\bar{r}} + \color{#00B050}{g} \color{#00B050}{\bar{g}} + \color{#0080FF}{b} \color{#0080FF}{\bar{b}} \leftrightarrow \color{#E619E6}{w} \color{#E619E6}{\bar{w}} being the Z^{0} weak boson. The remainder is SU(3) of the strong nuclear force with time-like vectors \color{#FF0000}{r} \color{#FF0000}{\bar{r}} \color{#00B050}{g} \color{#00B050}{\bar{g}} and \color{#0080FF}{b} \color{#0080FF}{\bar{b}} representing the three coloured quarks. Strong and electromagnetic force carriers \gamma, g being light-like events that pass through particle events, weak force carrier bosons pass between particle events.
Figure 3(b): Light Rays
As you can see, the outer T^1 := \mp \{ 1,i \} \infin square is best seen as the event horizon of a black hole, and the inner T^0 := \pm \{ 1,i \} \infin as the singularity, both are \infin far away from the moment of here and now, being the surface area defined by the middle square. The points \{ T^\alpha, T^\gamma \} = iT^1 and \{ T_\beta, T_\delta \} = T_1, and as both points T^0, T^1 are singularities, defining the structure of the universe between them. One singularity T^0 being 2D, defining the structure of all fields withing the universe, and the other T^1 being 4D defining how those fields are folded around each other to give us the different particle properties and spatial structure. E.g the inner singualrity T^0 \in \CC^2, and outer singularity T^1 \in \HH^2.
So overall the progression of the universe, start from a singular field and gradually gets more and more complex as it folds. E.g. \RR^2 \mapsto \CC^2, \CC^2 \mapsto \HH^2, \HH^2 \mapsto \OO^2, then \OO^2 \mapsto \SS^2 and so on. Each visible universe mapped between two singularities, in our case \CC^2 \mapsto \HH^2.
Combining the above, we now get:
\left[ \HH^{\mu} \HH_{\mu} \right] (t) = \left[ \def \arraystretch{1.2} \begin{array}{cccc} \color{#E619E6}{\bar{w}} \\ \hdashline \color{#FF0000}{r} \\ \color{#00B050}{g} \\ \color{#0080FF}{b} \end{array} \right] \left[ \def \arraystretch{1.2} \begin{array}{c:ccc} \color{#E619E6}{w} & \color{#FF0000}{\bar{r}} & \color{#00B050}{\bar{g}} & \color{#0080FF}{\bar{b}} \end{array} \right]
\left[ \HH^{\mu} \HH_{\mu} \right] (t) = \left[ \def \arraystretch{1.2} \begin{array}{c:ccc} \color{#E619E6}{\bar{w}} \color{#E619E6}{w} & \color{#E619E6}{\bar{w}} \color{#FF0000}{\bar{r}} & \color{#E619E6}{\bar{w}} \color{#00B050}{\bar{g}} & \color{#E619E6}{\bar{w}} \color{#0080FF}{\bar{b}} \\ \hdashline \color{#FF0000}{r} \color{#E619E6}{w} & \color{#FF0000}{r} \color{#FF0000}{\bar{r}} & \color{#FF0000}{r} \color{#00B050}{\bar{g}} & \color{#FF0000}{r} \color{#0080FF}{\bar{b}} \\ \color{#00B050}{g} \color{#E619E6}{w} & \color{#00B050}{g} \color{#FF0000}{\bar{r}} & \color{#00B050}{g} \color{#00B050}{\bar{g}} & \color{#00B050}{g} \color{#0080FF}{\bar{b}} \\ \color{#0080FF}{b} \color{#E619E6}{w} & \color{#0080FF}{b} \color{#FF0000}{\bar{r}} & \color{#0080FF}{b} \color{#00B050}{\bar{g}} & \color{#0080FF}{b} \color{#0080FF}{\bar{b}} \end{array} \right]
The \RR time gives us the commutative properties of electromagnetism, and \II time gives us the non-commutative properties of the strong force. The weak force now being described as rotations between electromagnetism and strong force, which means it to must be commutative in nature as combination of \RR and \II is always commutative.
The concept of the big bang is now obvious to see as it represents a moment when T^{0} had zero radius but had \HH complexity, a point when electrons and quarks were unified just like a ripple in a pond doesn't exist before a stone is thrown in, there is no concept of before time or negative radius. Expanding spacetime is when the trough of T^{0} and peaks of S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta} are increasing in separation which would have the effect of driving matter and antimatter apart, as well as forcing electromagnetism and strong forces apart and giving us the concept of Higgs mechanism. This greater this separation relating to the diameter of T^0, the greater the Higgs field coupling and greater the mass of the particles. This should have the effect of coalescing quarks into mesons then baryons and atoms as the universe cools down, shown as reducing frequency. The interaction between positive and negative \RR parts gives us time-like \RR matter and antimatter events as seen by the interaction of electrons and electromagnetism, the interaction between \II gives us time-like \II matter and antimatter of quarks and strong force, which is simultaneously space-like event to electromagnetism and electrons.
We also have the concept of spacetime rotation, where space and time swap roles. If you can see the wave-like nature of spacetime in Figure 3(a) you can imagine this rotation as the wave moving in the various combinations of S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta} seen as phase shifting. If there is no movement then we could classify the universe as being a standing wave, and just as mass can have both rest mass and momentum, so too can the universe. This possibly could explain how mass lost to black holes contributes to the rotation of the universe, and our universe is but one of an infinite series. The universe we see is only the standing wave portion, energy phase shifted out of our universe and into another adjacent universe, see as energy being rotated a full wavelength away from us. So mass and energy are never lost in a black hole, but rotated 90° to another universe, likewise matter entering our universe from a black hole or vacuum is rotated in from another universe. The event horizon of the black hole being where events change from time-like to space-like through light-like. So we should think of the interior of all black holes to be space-like \II matter that connects universes (aka outside of universe), and \RR matter what connects black holes (inside of universe), with light mapping the spacetime framework between the two. As shown in Figure 3(a) as the expanding circumference of T^0, or cosmic microwave background (CMB) where differences in red shift will indicate greater quantities of either time-like \RR or space-like \II matter.
  • e^{-}_{L}, e^{+}_{R} :\rightarrow \color{#E619E6}{w} \color{#E619E6}{\bar{w}} .
  • e^{-}_{R}, e^{+}_{L} :\rightarrow \color{#E619E6}{\bar{w}} \color{#E619E6}{w} .
  • q_{L}, \bar{q}_{R} :\rightarrow \color{#FF0000}{r} \color{#FF0000}{\bar{r}}, \color{#00B050}{g} \color{#00B050}{\bar{g}}, \color{#0080FF}{b} \color{#0080FF}{\bar{b}} .
  • q_{R}, \bar{q}_{L} :\rightarrow \color{#FF0000}{\bar{r}} \color{#FF0000}{r}, \color{#00B050}{\bar{g}} \color{#00B050}{g}, \color{#0080FF}{\bar{b}} \color{#0080FF}{b} .
  • H^0 :\rightarrow light-like \pm \infin (with \HH complexity).
You may also see that when time T^{0} is created we must also have the separation of matter and antimatter spaces S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta}. We can also say this is the point where the Higgs field H^0 centred around T^{0} became none zero with \HH complexity and is increasing, while H^0 centred around T^{\alpha}, T^{\gamma} and T^{\beta}, T^{\delta} must be decreasing. Think of this as ever decaying number of possible future events, but ever expanding number of past events. Very similar to how we would see the flow of time in an hour glass, with each particle of sand representing a unique event in spacetime.
So in our example, the Higgs field H^0 that defines the seperation of all the matter and light we can see in our universe, must be itself based on \HH complexity number with \RR portion giving us electrons, electromagnetism and the light we see, and \II parts the quarks, strong force we see within the nucleus. It is both source of all light, and seperation of mass (aka Higgs effect). The secondary decaying negative Higgs field as described above along with the incresing T^0 volume is most likely what we call dark energy, and related to the collection of all past events (T^0 volume) no matter if they were light-like, space-like or time-like as well as the reduction in volume of T^{\alpha}, T^{\gamma} and T^{\beta}, T^{\delta} representing the total life left and all possible future events. So as T^0 volume is increasing to the positive, we see entropy and flow of time as positive, eventually it will get to a point where it either stops or reverses and entropy and flow of time will similarly either stop pr turn negative. Giving us not only the possability of heat death but also cyclical universe, it also gives us a connection to a continuous wave like multiverse of universes as well, where each universe relates to 90° for its expansion phase and 90° for its collapsing phase, making expansion, entropy and the overal flow of time cyclical in nature as well.
We can now also define four categories of matter:
  • Matter: (0°) Common positive spacetime (positive time and space).
  • Dark Matter: (90°) Partially rotated spacetime (negative time, positive space).
  • Antimatter: (180°) Common negative spacetime (negative time and space).
  • Dark Energy: (270°) Partially rotated spacetime (positive time, negative space).
Mesons being the only solution using \CC^2 (2D) complexity, atoms of one colour electron and three coloured quarks the only solution using \HH^2 (4D) complexity. The idea of negative space and time may be hard to grasp, but if you think of them as hidden within the spacetime framework as we do for vacuum energy, and spacetime framework as being the walls we can't see through (null vectors). This way spacetime is both moulded on the positive side with matter, and negative side with dark matter, dark energy and vacuum energy. Particle creation from vacuum energy or dark matter annihilation is based on a rotation between negative and positive spacetime. As you can see, we are separated from antimatter by both dark matter and energy.
It is also important to note that while S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta} are all internally connected they do not wrap around but instead connect to a different universe (as defined by its own central time T^{\alpha}, T^{\gamma} and T^{\beta}, T^{\delta}, are better viewed as universes that are seperated by unique H^0 fields so as we must define the overall multiverse as continuous with no hard edges. This is the wave-like nature mentioned previously. We must also not that while the inner H^0 has \HH complexity, the outer H^0 field does not have to be the same, it could have \OO or \SS complexity and we only see its 4D shadow interacting with ours, think of it as 8D or 16D universe, with one or more 4D bubbles of which we are in one.
4: 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 transforms:
  • Temporal: t :\rightarrow \hbar \gamma^{0} \partial_{0}.
  • Spatial: x :\rightarrow \hbar \gamma^{j} \partial_{j}.
We must also add the factor of c back to the spatial and temporal axis to change what would be units of momentum to units of energy. You should notice this is still one dimensional as one axis is always \RR while the other is \II, mass being defined as the radius or hypotenuse created by what is now energy and momentum axis.
This then gives us the image as seen in Figure 4(a).
We have now highlighted the four quadrants for space-like and time-like events, and the black lines represents light-like events. The blue and red curved surfaces now represent the particles equal 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 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 time-like energy is always positive (usually depicted by the colour blue) and space-like energy is always negative (usually depicted by the colour red).
\hat{E} = +i{\hbar}c \gamma^{0} \partial_{0}
i \hat{p} c = -{\hbar}c \gamma^{j} \partial_{j}
i \hat{p} c = +{\hbar}c \gamma^{j} \partial_{j}
\hat{E} = -i{\hbar}c \gamma^{0} \partial_{0}
Figure 4(a): Energy Equivalence
Using Pythagorean 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
Normalizing:
  • (\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
Dirac Equation By Summing Vectors:
  • \hat{E} - \hat{p}c - mc^2 = 0
Normalizing:
  • i \hbar \gamma^{\mu} \partial_{\mu} - mc = 0
5: Special Relativity
We now need to show the four rotational copies of the Minkowski metric which must be used together, in our case we have the following:
Particle: \color{#FF0000}{q}, \color{#FF0000}{\bar{q}}
Vector: [-\hat{i}t^1, -x_1, +\hat{k}y_1, -\hat{j}z_1]
Metric: [-,+,-,-], [+,-,+,+]
Particle: \color{#00B050}{q}, \color{#00B050}{\bar{q}}
Vector: [-\hat{j}t^2, -\hat{k}x_2, -y_2, +\hat{i}z_2]
Metric: [-,-,+,-], [+,+,-,+]
Particle: \color{#0080FF}{q}, \color{#0080FF}{\bar{q}}
Vector: [-\hat{k}t^3, +\hat{j}x_3, -\hat{i}y_3, -z_3]
Metric: [-,-,-,+], [+,+,+,-]
Particle: \color{#E619E6}{e}, \color{#E619E6}{\bar{e}}
Vector: [+t_0, +\hat{i}x^0, +\hat{j}y^0, +\hat{k}z^0]
Metric: [+,-,-,-], [-,+,+,+]
\left[ \def \arraystretch{1.2} \begin{array}{c:ccc} \color{#E619E6}{+t_0} & \color{#E619E6}{+\hat{i}x^0} & \color{#E619E6}{+\hat{j}y^0} & \color{#E619E6}{+\hat{k}z^0} \\ \hdashline \color{#FF0000}{-\hat{i}t^1} & \color{#FF0000}{-x_1} & \color{#FF0000}{+\hat{k}y_1} & \color{#FF0000}{-\hat{j}z_1} \\ \color{#00B050}{-\hat{j}t^2} & \color{#00B050}{-\hat{k}x_2} & \color{#00B050}{-y_2} & \color{#00B050}{+\hat{i}z_2} \\ \color{#0080FF}{-\hat{k}t^3} & \color{#0080FF}{+\hat{j}x_3} & \color{#0080FF}{-\hat{i}y_3} & \color{#0080FF}{-z_3} \end{array} \right] = \left[ \def \arraystretch{1.2} \begin{array}{cccc} \color{#E619E6}{\bar{e}} \\ \hdashline \color{#FF0000}{q} \\ \color{#00B050}{q} \\ \color{#0080FF}{q} \end{array} \right]
\left[ \def \arraystretch{1.2} \begin{array}{c:ccc} \color{#E619E6}{+t^0} & \color{#FF0000}{+\hat{i}t_1} & \color{#00B050}{+\hat{j}t_2} & \color{#0080FF}{+\hat{k}t_3} \\ \hdashline \color{#E619E6}{-\hat{i}x_0} & \color{#FF0000}{-x^1} & \color{#00B050}{+\hat{k}x^2} & \color{#0080FF}{-\hat{j}x^3} \\ \color{#E619E6}{-\hat{j}y_0} & \color{#FF0000}{-\hat{k}y^1} & \color{#00B050}{-y^2} & \color{#0080FF}{+\hat{i}y^3} \\ \color{#E619E6}{-\hat{k}z_0} & \color{#FF0000}{+\hat{j}z^1} & \color{#00B050}{-\hat{i}z^2} & \color{#0080FF}{-z^3} \end{array} \right] = \left[ \def \arraystretch{1.2} \begin{array}{c:ccc} \color{#E619E6}{e} & \color{#FF0000}{\bar{q}} & \color{#00B050}{\bar{q}} & \color{#0080FF}{\bar{q}} \end{array} \right]
The reasoning to show the two lines above is to highlight the two light-like barriers, in the first matrix, the vertical line denotes the \RR \pm \infin that gives us \gamma and the colours \color{#E619E6}{w}, \color{#FF0000}{r}, \color{#00B050}{g}, \color{#0080FF}{b}, acting as the barrier between time and space and therefor the light we see. While the horizontal like denotes the \II \pm \infin that gives us the colours \color{#E619E6}{\bar{w}}, \color{#FF0000}{\bar{r}}, \color{#00B050}{\bar{g}}, \color{#0080FF}{\bar{b}} acting as the barrier between particles (E.g. the gap between the electron cloud and nucleus), these are the virtual bosons we can't see. For the conjugate matrix form, these two barriers are reversed with \RR bosons being the horizontal line, and \II virtual bosons the vertical line.
However one thing to note, is the diagonal of both matricies are always positive and \RR, for this reason the electron energy (time component) is commutative allowing free electrons, but the momentum (space component) is non-commutative giving us the variation of orbitals, greater the momentum, the more complex the orbital paths and the more electrons that can fill that shell, no different than higher harmonics. However the opposite is true for quarks, as the energy is non-commutative which is why they can never be free, and just like the momentum of the electrons, the energy levels of quarks will show harmonic structure too. The momentum however is commutative and free to move about, so colour swapping of quarks is no similar to the photoelectric effect of electrons, only instead of changing energy levels as in the case of electrons, quarks move location by swappping colour charges. So everything we have learned for electromagnetism can be used for quarks, only we need to change energy and momentum terms around, so colour charge is more like magnetism and colour momentum is more like electric charge. Thus showing both the duality and universality of the particles and forces that make up atoms.
The spatial and temporal components (momentum and energy) of all particles are seperated by these two light-like inner barriers, regardless if they are matter or antimatter, and furthermore by two light-like \pm \infin barriers that define begining and end of causal connection and what ultimately seperates matter and antimatter. Thus showing the walled garden each particle must play within. With the energy and momentum of both electron and quarks being unique, but also interconnected in very specific ways giving us the properties we see. So we can either look at the universe in terms of spacetime as with walls of 0, \pm \infin space and time to denote here, now (0) and the \infin future, past and distance, or as \RR and \II light-like walls having space and time dimensions as the diagonal lines and light as the verical and horizontal axis. Thus concepts such as positive curvature can be imagined as greater \II barrier between electrons and quarks giving us the decay of free neutrons, and greater \RR as the production and stretching of light. Thus giving us the properties of fusion and fission as well as effects of gravity we see. However if the curvature was negative, the opposite happens. Light is blue shifted and absorbed, neutrons would gain stability as the barrier between electrons and nucleons would also decrease and fission of heavier elements would cease. Thus light emiting stars show areas of net positive curvature, neutron stars and black holes show net negative curvature. The extreme being black holes where the light-like lengths turn negative and time and space, \RR and \II light swap places, so spacetime becomes timespace and the visible light of black holes is trapped within - which if you were inside the black hole would look exactly the same as we see the cosmic microwave background. The event horizon of the black hole being the T^0 moment of causal connection for all matter and light within it. Oddly this also will change electrons to quarks and quarks to electrons as spacetime is literally turned inside out. This folding of spacetime will create higher complexity on the inside of black holes (8D, 16D, etc) and lower complexity outside. So our universe is inside a 4D black hole, with 8D black holes inside it, and 2D black hole outside, it is the 2D black hole that gives us the duality of energy and momentum and why all fields within our universe have 2D base structure.
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, shown above as when the electron is contravariant vector, the quarks must always be covariant vectors no matter if we are talking about matter or antimatter and this is the case for both the time-like energy of the particles and space-like momentum. The \II light-like 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 pushed away. This stretching of the energy and momentum between the electron and quarks, gives us the effects of charge separation and gravity. Charge seperation being the \II, \RR light-like linear gradient aka massless transfer, and gravity being where the gradient is curved, so accelerating it. Think of the graviton as \RR gravity, being where space is more compressed than time, where the tachyon is \II gravity where time is more compressed than space.
  • \Delta \approx \partial_{\mu}
    Linear (first order) shear, relates to momentum and charge like attraction.
  • \Delta \approx \partial_{\mu}^{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 \HH model of spacetime, it is possible to unify both special, general relativity and all forces together in one unifying model. It also shows the fundamental energy and motion of quarks and electrons as well as the electromagnetic and strong forces are all the same, it is only the differences in location with respect to each other that define the differences in properties.
Treating the separate 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.
Special relativity is where we define the axis as \hat{E} and \hat{p}c and general relativity is where the axis becomes \hat{P} = \partial_0 \hat{E} and \hat{F} = \partial_j \hat{p}c. So while special relativity has axis of energy and momentum (both in momentum units), general relativity has axis of power and force (both in acceleration units). So general relativity is just the gradient of special relativity, if we use the triangle of energy, mass and momentum of special relativity, general relativity defines how that triangle is deformed. With positive de-Sitter curvature increasing the angles to greater than 180°, negative anti de-Sitter curvature decreasing the angles to less than 180° and flat Minkowski spacetime being the special case where the angle comes to exactly 180°. So special relativity will be the relation between electromagnetism, weak and strong forces, electrons and quarks, specifically through charge and momentum and general relativity is how the momentum and energy changes over space and time. So just as special relativity has bosons (light-like events) of photon \gamma, gluon g, W^{\pm} and Z^0, So too general relativity should have tachyon \Tau being \delta{^2}_t \gt \delta{^2}_s, graviton G as \delta{^2}_s > \delta{^2}_t, the acceleration equivalents of \gamma, g and V^{\pm} and Y^0 as the acceleration equivalents of the weak force, which again is a rotation between tachyon and graviton acceleration. Now we also have the concept of space-like, light-like and time-like curvature, time-like de-Sitter curvature gives us entropy and dark energy, space-like anti de-Sitter curvature gives us gravity and dark matter and finally light-like curvature gives us special relativity and flat Minkowski spacetime.
6: General Relativity
We now have defined what dark matter and energy could be, 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, yet still being physically separated so as not to directly interact with matter. 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 time-like connections shown as four corners of Figure 3(a). This has the effect of increasing charge separation 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 6(a) to show a rotating universe, expansion would be shown as the wave reducing in frequency. However looking at the rotating view from a stationary point, it may also look as if the universe is expanding to a maximum point, then contracting back to nothing. Matter now has three states being minimum when energy or momentum are time-like (shown as blue), maximum when they are space-like (shown in red), connected by light-like type (horizontal plane along z=50, center of the green areas). Heat death senario would look like the wave form stretching until it reaches 0 Hz (aka longitudinal stretch), cyclical would show the overal phase shift as shown in Figure 6(a).
Figure 6(a): General Relativity
Depending what part you are in, spacetime is being continuously created and destroyed, think of this as the big bang and crunch happening continuously, but constantly moving location with the overall energy staying constant. If stationary you would see energy changing from space-like to time-like through light-like 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 wave-like and continuous.
7: Calculated Tensor
NOTE: This is currently a work in progress to show how the \HH math can be shown in n x n format for 3 x \HH for space dimensions and 1 x \HH for time dimension. While this normally would be shown as a 4 x 4 x 4 tensor array.
4D
7D
9D
17D
35D
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
4D
7D
9D
17D
35D
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
8: 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:
  • \CC :\rightarrow
    1D + 1T (Meson, Particle Pairs).
  • \HH :\rightarrow
    3D + 1T (Atom, Baryon)
  • \OO :\rightarrow
    7D + 1T (Beyond Standard Model)
  • \SS :\rightarrow
    15D + 1T (Beyond Standard Model)
As there are no definable numbers between \CC and \HH, 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 \CC is also not possible as you must create space-like dimension as well as time-like dimensions in order to satisfy the wave-like 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 necessarily have to be 4D as well. We also can't assume that the amount of fermions and quarks are equal, as dark matter is connected spatially it may have greater pull to either 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.
The graph shown in Figure 3(a) is how rotation of the wave-like nature of the universe could be perceived, as space is converted to time, and vice versa. From any stationary point, this would look like spacetime expanding then contracting as one half of the wave form, then inverting to become timespace expanding and contracting for the second half of the wave form. If we add to this the time-like matter, we would first experience spacetime as matter then timespace as imaginary time-like matter, then spacetime as antimatter, then timespace as imaginary antimatter, before repeating a new cycle. However as we can only see like matter, there really is no way to tell which quadrant we are in currently, they will always look like spacetime and matter locally.
One important conclusion is rather than matter being four dimensional because of spacetime, but rather the inverse is true, that we see the universe as four dimensional because of the complexity of matter we are made of. This both allows for matter that is not causally connected to the big bang event and that event is only relevant to the matter we are made of. This renders the idea the universe has one beginning or end as meaningless, as even at the universal scale, time has only local relevance. Concepts such as dark matter and dark energy do not need to be fully connected to all the same dimensions, but rather overlap partially. The fact we haven't been able to find any connectivity between dark matter and electromagnetism would suggest as much. If this theory is correct, we may also need to check connectivity of things like dark matter with the strong and weak forces as well as quarks, not just electrons and the electromagnetic force. Light is not the only yard stick we can use to probe the universe.