Draft:Geometric Unity

Geometric Unity is a speculative work in progress program of research that is evaluating multiple instantiations of an idea which aspires to define a Unified Field Theory which replaces General Relativity with something less restrictive which allows for a more elaborate and symmetrical Topological Quantum Field Theory.

It builds upon the work done by its author Eric Weinstein when he obtained his PhD in Mathematical Physics from Harvard University in 1992 for his dissertation entitled:

Extension of Self-Dual Yang-Mills Equations Across the Eighth Dimension

The ABSTRACT mentions that an extension to all higher even numbered dimensions is sketched. Geometric Unity proposes that there exists a Chimeric Fiber Bundle $C$ which exists as a tangent to a 14 dimensional space $Y^{14}$ upon which there exists a pervasive single unified field named $\omega$ (omega) that is characterised by the elaborate Lie group of $U(64,\,64)$ metric Weyl spinors.

This Lie group underwent a Decomposition step from another Lie group of $U(128,\mathbb {C} )$ topological Dirac spinors, which arose from the Lie group of $U(128)$ undergoing a necessary Complexification step, so that both sides of this isomorphism had the same form:

$ad(P_{U(128)})\otimes \mathbb {C} \,\,\,{\overset {vect}{=}}\,\,\,Cl_{\mathbb {C} }^{*}\,=\,\Lambda ^{*}(C)\otimes \mathbb {C} \,=\,\Lambda ^{*}(TY^{14})\otimes \mathbb {C}$

This Complexification to $U(128,\mathbb {C} )$ would lead to pathological physics were it not for the Decomposition step to $U(64,\,64)$ as evidenced by the slide of the Fermionic Field content having a pair of 64s either side of the horizontal dashed line, which Eric Weinstein presented at the conclusion of the speculative Simonyi lecture at Oxford University he had been invited to give on his work in progress program for Geometric Unity by Professor Marcus du Sautoy on the 23rd May 2013. This important detail was subsequently clarified on 1st April 2021 when Eric Weinstein published the first version of his draft paper on Geometric Unity where page 51 shows a diagram of the same Fermionic Field content also based on the Lie group of $U(64,\,64)$ metric Weyl spinors.

He has said that a quantum field theorist would be needed to calculate the energy scales at which his predicted fields with right handed spin properties would manifest at within the Large Hadron Collider or its successor.

This 14 dimensional space $Y^{14}$ is grown endogenously from a purely mathematical, non Euclidean, pseudo Riemannian differential manifold, with 4 spatial dimensions and no concept of time, where this base space $X^{4}$ has no inherent Metric. This intentionally removes the assumptions of General Relativity where Albert Einstein selected the Lorentzian split-signature Metric of (1, 3) for his space; which he denoted $M^{1,3}$

Eric Weinstein adopted the following generalised formula:

${\begin{array}{lcl}\\{\textrm {\$}}_{C}^{U(u,u)}&{\textrm {where}}&u=\lfloor w/2\rfloor &{\textrm {and}}\quad w=2^{n}\\\uparrow \\Y^{n,n}&{\textrm {where}}&n=\lfloor m/2\rfloor &{\textrm {and}}\quad m=d+e\\\uparrow \\X^{d}&{\textrm {where}}&d\in \mathbb {N} >2&{\textrm {and}}\quad e=((d^{2}-d)\div 2)+d\end{array}}$

When $d=\Sigma (1,3)$ then $e$ provides the formula for the number of degrees of freedom in Einstein's Metric tensor $g_{\mu \nu }$

Because $g_{\mu \nu }$ has subscripts $\mu$ and $\nu$ that each independently range over 0-3 they then reference every entry within their 4x4 Rank 2 tensor. This yields 16 entries. However, due to this being symmetric about its descending diagonal which conveys "length" information 4 entries can be temporarily subtracted to yield 12 entries which are "mirrored". Only half of these are needed and so the division yields 6 entries which convey "angles". Together you have 4 + 6 = 10 degrees of freedom and consequently only need to calculate 10 of the Einstein Field Equations which are summarised by Einstein Summation Notation into this compact form:

$R_{\mu \nu }-{\tfrac {1}{2}}R_{\mu \nu }+\Lambda _{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$

For the sake of simplicity the Vacuum Field Equations are preferable where the value of the Stress-Energy-Momentum tensor $T_{\mu \nu }$ is zero due to it being a vacuum. Eric Weinstein replaces this Cosmological constant with the Vacuum Expectation Value of some of his newly predicted bosonic fields, where a non uniform distribution of this Dark Energy would yield local variations in the accelerating expansion of our Universe as $\Lambda _{\mu \nu }$ would only have a constant value throughout an initially empty cosmos, so that would characterise mysterious temperature fluctuations within the Cosmic Microwave Background as gravitation pulled back a partial sample of omega into the void. Which implies that without this initially constant VEV there would be nothing to bootstrap our tangible reality into existence:

$R_{\mu \nu }-{\tfrac {1}{2}}R_{\mu \nu }\,+\Lambda _{\mu \nu }=\,0$

However, Eric Weinstein intends to recover our $X^{1,3}$ space-time from his Chimeric Fiber Bundle $C$ as an arbitrary selection amongst all that are possible within $Y^{7,7}$

${\begin{array}{lcl}\\{\textrm {\$}}_{C}^{U(64,64)}&{\textrm {where}}&64=\lfloor 128/2\rfloor &{\textrm {and}}\quad w=2^{7}=128&{\textrm {The}}\,\,{\textrm {Rules}}\,\,{\textrm {Of}}\,\,{\textrm {The}}\,\,{\textrm {Game}}\\\uparrow \\Y^{7,7}&{\textrm {where}}&7=\lfloor 14/2\rfloor &{\textrm {and}}\quad 14=4+10&{\textrm {The}}\,\,{\textrm {Observed}}\\\uparrow \\X^{1,3}&{\textrm {where}}&d\in \mathbb {N} =4&{\textrm {and}}\quad e=((4^{2}-4)\div 2)+4&{\textrm {The}}\,\,{\textrm {Observer}}\end{array}}$

This generalisation of General Relativity essentially adopts an unrestricted set of dimensional measures for which to chart its base space $X^{4}$ the cardinality of which is 14:

• 4 axes: w x y z
• 6 angles between each pair of those axes: x&y y&z z&x w&x w&y w&z
• 4 component parts of a vector needed to measure length within that space: [ W X Y Z ]

= 14 dimensional measures

So, General Relativity is more restrictive that Geometric Unity as it only deals with the last two which measure the deformation of non Euclidean space-time in terms of how its angles and length changes within $g_{\mu \nu }$ whilst Eric Weinstein also incorporates location information. This extra "headroom" allows him to divide those 14 dimensions into a symmetrical split-signature $(7,\,7)$ of seven temporal dimensions and seven spatial dimensions. Although $(9,\,5)$ and $(5,\ 9)$ are also viable choices they imply there exists a more fundamental Unified Field Theory which would explain why the split-signature $(t,\,s)\neq (s,\,t)$ when this is automatically the case when $t=s$ as it is in $(7,\,7)$

The Observerse can then be thought of as our Universe contemplating itself into existence through the consideration of all choices of The Observer $X^{t,s}$ observing $Y^{7,7}$which is The Observed. These two spaces are coupled in what Eric Weinstein calls "The Observerse":

${\begin{array}{lcl}\\{\textrm {\$}}_{C}^{U(64,64)}&&{\textrm {The}}\,\,{\textrm {Rules}}\,\,{\textrm {Of}}\,\,{\textrm {The}}\,\,{\textrm {Game}}\\\uparrow \\Y^{7,7}&&{\textrm {The}}\,\,{\textrm {Observed}}\\\uparrow &{\textrm {where}}\,{\textrm {the}}\,{\textrm {field}}\,{\textrm {content}}\,{\textrm {is:}}\quad \\X^{t,s}&Spin(7-t,7-s)&{\textrm {The}}\,\,{\textrm {Observer}}\end{array}}$

During his 2nd April 2020 upload of his 23rd May 2013 Oxford University lecture he discussed in the Supplementary Slide Explainer he appended to it and its Introduction to regular visitors to his podcast The Portal that it was in the tradition of a Unified Field Theory, but that he was mindful of what a hopeful candidate for a "Theory of Everything" might look like and that it was worth contemplating how to go about defining a final fundamental theory of everything in physics which would not be replaced by another that was more fundamental. This led to him admitting that he did not see how something could be made out of nothing, which is why he begins with $X^{4}$ as his initial assumption.

However, this isn't a "magic number" needed to make the mathematical physics create The Observerse whose geometric construction which can potentially yield an 'Arena':

${\begin{array}{lcl}\\{\textrm {\$}}_{C}^{U(64,64)}&&{\textrm {The}}\,\,{\textrm {Rules}}\,\,{\textrm {Of}}\,\,{\textrm {The}}\,\,{\textrm {Game}}\\\uparrow \\Y^{7,7}&&{\textrm {The}}\,\,{\textrm {Observed}}\\\uparrow &{\textrm {where}}\,{\textrm {the}}\,{\textrm {field}}\,{\textrm {content}}\,{\textrm {is:}}\quad \\X^{1,3}&Spin(6,4)&{\textrm {The}}\,\,{\textrm {Observer}}\end{array}}$

Where $(t=1,s=3)$ and consequently that choice of Lorentzian split-signature $(1,3)$ Metric allocates 4D from the 14D that are available and leaves a remainder pair of dimensions which have the split-signature of $(7-1,7-3)=(6,4)$ and this characterises the symmetries of the Quantum Field Theory as implicit dimensions $g^{6,4}$ that both resemble the 6 angles and 4 component parts of a vector needed to measure length within our space-time:

$X^{1,3}$

But it is also the exact number of dimensions needed to define the Lie group:

$Spin(6,4)$

This promises both a Grand Unification of all the known force mediating fields (through the geometry of Charles Ehresmann), as well as incorporating the classical manifold geometry used by Albert Einstein to construct deformable space-time for gravitation (which was based on the geometry of Bernhard Riemann). Albert Einstein's General Relativity makes use of algebraic equations which are incompatible with the analytic Gauge Theory of the Yang-Mills curvature equations which describe Spin-1 Bosonic fields that are "dancing" on $Y^{7,7}$ according to the Supplementary Slide Explainer appended to the lecture. Consequently, Eric Weinstein is using a modified version of Edward Witten's Topological Quantum Field Theory known as Chern-Simons Theory which harmonises with the Yang-Mills curvature equations as both are analytic. Eric Weinstein intends to use the same strategy that Paul Dirac employed in order to derive his 1st order Dirac equation for the momentum of an electron in flat space-time from the established 2nd order Klein-Gordon equation, only rather than recover Spin 1/2 electrons and positrons from a Spin 0 Klein-Gordon equation Eric Weinstein intends to recover Spin 3/2 Rarita-Schwinger matter as well from a 14 dimensional Self-Dual Yang-Mills curvature equations for Spin 1 bosonic fields and incorporate the Spin 0 bosonic field for the Higgs field as a quartic in this 2nd order equation operating on $Y^{14}$ which will look like a quadratic in the 1st order equation operating on $X^{1,3}$

As a result Eric Weinstein intends to define his 2nd order equation and then derive his 1st order equation using Chern-Simons Theory rather than General Relativity.

Eric Weinstein explores the use of Gauge Theory with a view to recover space-time as a $(1,3)$ sector within a 4D section of a 14D space which lacks a Metric, but which has a Chimeric Fiber Bundle that has a Lie group of $U(64,\,64)$ metric Weyl spinors from which he obtains one. Einstein went from his assumption of a $(1,3)$ space-time up the Levi-Civita connection into a flat Horizontal Vector Space where the mathematics was simpler. Weinstein does the reverse as he starts with $(7,\,7)$ and then has a Chimeric Fiber Bundle $C$ which is constructed from the direct sum of the horizontal bundle that is the pullback from the base space $X^{4}$ with the vertical tangent bundle along the fibers, which yields a Horizontal cotangent space of 4 dimensions which is at right angles to $Y^{14}$ direct sum the Vertical tangent space of 10 dimensions to $Y^{14}$

$C=T_{H}^{4*}(Y^{14})\,\oplus \,T_{V}^{10}(Y^{14})$

Weinstein then intends to recover space-time by going down the Levi-Civita connection from this flat Horizontal Vector Space analogous to an inversion of the phenomenon of shadows being cast on the bottom of a swimming pool by ripples refracting light on its surface, here General Relativity would have the deformations of the surface of the water create geodesics which would through the Levi-Civita connection transcribe themselves as vectors within multiple flat Horizontal tangent spaces by dint of parallel transport. Geometric Unity does not have space-time to begin with, so it has to recover it from this flat Horizontal cotangent space as part of the Chimeric Fiber Bundle $C$ and that means vectors reconstruct the geodesics on the base space $X^{4}$ so that it inherits a Metric and is therefore aware of its current deformational curvature and how it ought to make celestial masses move by warping the geodesics which are just a generalisation of a straight line into the higher non Euclidean dimensions of $X^{1,3}$

However, this is complicated by this all being gauge theoretic, so it also has to preserve Ehresmannian gauge covariance in coming from $Y^{7,7}$ to $X^{4}$ to recover $X^{1,3}$ and here is where it uses the Levi-Civita connection to induce the Riemannian curvature of the spinor bundle. Unfortunately, this is too much information so as with Einstein the Weyl curvature contribution has to be eliminated (this is why the foremast representing this Weyl curvature tensor "breaks off" when the Ship representing the Riemannian curvature tensor is put into the Bottle which represents the Space of Connections, after the mainmast representing the traceless Ricci curvature tensor and the mizzenmast representing the full Ricci Scalar tensor are temporarily folded down to allow the Ship to be pushed stern first towards the base of the Bottle through the added constraint imposed by the narrow neck which represents the requirement to preserve Ehresmannian gauge covariance, after which the two remaining masts are drawn back up by threads (as it is not a String Theory), and the mizzenmast shrunk by half to be consistent with the Einstein Field Equations). In terms of analytic Chern-Simons theory there are a family of custom Ship In A Bottle operators (aka. SHIABs), however the one Ship In A Bottle operator that is required to recover physical space-time from the pure mathematics of the geometric unity of the Chimeric Fiber Bundle that includes $X^{1,3}$ as one of many possible arbitrary choices of observation of a sector of a section within The Observerse that includes $Y^{7,7}$has the general form:

$\overbrace {\begin{array}{lcl}\circledcirc _{\epsilon }\xi &=&[(\boxminus )\wedge (*\xi )]-{\frac {\textrm {*}}{2}}[(\boxminus )]\wedge *[(\boxplus )\wedge (*\xi )]]\\\boxminus &=&\epsilon ^{-1}\Phi ^{1}\epsilon \\\boxplus &=&\epsilon ^{-1}\Phi ^{2}\epsilon \end{array}} ^{{\textrm {Ricci}}\,{\textrm {tensor}}\,{\textrm {like}}\quad \quad {\textrm {Ricci}}\,{\textrm {scalar}}\,{\textrm {like}}}$

This SHIAB differs from the one shown at the top of this page that is the observation made by $Y^{7,7}$ of ${\textrm {\$}}_{C}^{U(64,64)}$ which recovers the pervasive single unified field $\omega$ (omega) that appears as both Bosonic $\beta$ and Fermionic $\chi$ fields with a variety of Spin values that include Eric Weinstein's prediction of Spin 3/2 Rarita-Schwinger matter:

$\omega =(\beta ,\chi )=((\underbrace {\epsilon } _{0}\ ,\,\underbrace {\varpi } _{1}),(\underbrace {\nu } _{\tfrac {1}{2}}\ ,\,\underbrace {\zeta } _{{\frac {1}{2}}\,,\ {\frac {3}{2}}}))$

This all comes out of this central $Spin(6,4)$ Lie group:

$Spin(1,3)\times Spin(6,4)\longrightarrow Spin(7,7)$

Here we have the double cover of the Lorentz group which describes the symmetries of our $(1,3)$ split-signature space-time that form the basis of Noether's theorem which define fundamental physical conservation laws along with another Lie Group which includes The Pati-Salam Model:

$SU(4)\times (SU(2)_{L}\times SU(2)_{R})$

Which includes The Standard Model:

$SU(3)_{C}\times SU(2)_{L}\times U(1)_{\gamma }$

Where $U(1)_{\gamma }$ is Weak Hypercharge which includes all of Electromagnetism:

$U(1)_{em}$

Which includes all of the colours of the rainbow. Which were found through the use of prisms to re-unify back into pure white light by Sir Issac Newton when he was sent home from Cambridge University during the pandemic of 1666, during which he also realised that celestial bodies obeyed the same physical laws of gravitation as on Earth.