1. Physics
  2. Theory
  3. Unified Field Theory

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 anti-commutative. 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:
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) gives 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 ℂn, kets can be identified with column vectors, and bras with row vectors.
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} &= \langle A | B \rangle \\ &= A \cdot B \end{aligned}
\begin{aligned} A^{\mu} B_{\nu} &= | A \rangle \langle B | \\ &= A \otimes B \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.
The dot and inner product are commutative, meaning that A \cdot B = \bar{B} \cdot \bar{A} for all vectors A and B.
\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}
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 (\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 right-hand 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 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 anticommutative, meaning that A \times B = - B \times A for all vectors A and B. Unlike the cross product, the wedge product is associative.
\begin{aligned} A^{\mu} \times B^{\nu} &= e_i \cdot \begin{bmatrix} a \\ b \end{bmatrix} \times e_j \cdot \begin{bmatrix} c \\ d \end{bmatrix} \\ &= e_k \cdot det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\ &= e_k \cdot |ad - bc| \end{aligned}
\begin{aligned} A^{\mu} \wedge B^{\nu} &= e_i \cdot \begin{bmatrix} a \\ b \end{bmatrix} \wedge e_j \cdot \begin{bmatrix} c \\ d \end{bmatrix} \\ &= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\ &= |ad - bc| \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.
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} 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 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.
\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}
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 (σ), they are occasionally denoted by tau (τ) 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
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 Cℓ1,3(R). 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 I_2 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3
\gamma^0 is the time-like, hermitian matrix. The other three \gamma^1, \gamma^2, \gamma^3, are space-like, antihermitian matrices.
\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}
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 three-dimensional 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 three-dimensional 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:
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.
  • 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. 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 anti-de 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 T0 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 T0 and peaks of Sα and Sβ 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 T0 is created we must also have the seperation of matter and antimatter space Sα and Sβ. 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 Sα and Sβ do not wrap around but instead connect to a different universe (as defined by its own central T0), 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±, Z0) 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 Klein-Gordon 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
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
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.