Geometric Unity on Superphysics

Abstract

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Abstract and Introduction

An attempt is made to address a stylized question posed to Ernst Strauss by Albert Einstein regarding the amount of freedom present in the construction of our field theoretic universe.

From Unified Field Theory to Quantum Gravity and Back

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Starting in 1984 it became common to hear from leaders of the theoretical physics community that theoretical or fundamental physics was not about traditional unification of the kind sought for by the like of Albert Einstein.

From Unified Field Theory to Quantum Gravity and Back

Thu, 01 Apr 2021 00:00:00 +0000

1.4.1 Ehresmannian Geometry Advantages

Ehresmannian geometry gives the freedom to choose the internal symmetries of a physical theory without being confined to symmetries tied to the tangent bundle of Space-time.

From Unified Field Theory to Quantum Gravity and Back

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2 Incompatibility and Incompleteness Blocking Geometric Unification

Einstein’s theory of gravity cannot be unified with the Standard Model of quantum field theory because there is no known way to renormalize a quantum theory of metrics.

Geometric Unity

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2.3 Higgs Sector Remains Geometrically Unmotivated

For most of the 20th century, fundamental physics was split into two halves, only one of which was geometric. Then, in the mid 1970s, the quantum sector was discovered to have a basis in differential geometry with the advent of the Wu-Yang dictionary of Simons, Wu and Yang. Gauge potentials corresponded to the geometer’s notion of a connection and Fermi fields fit with Atiyah and Singer’s rediscovery of the Dirac operator in a bundle theoretic context.

The Observerse Recovers Space-Time

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In order to make progress beyond modern General Relativity and Quantum Field Theory, Space-Time itself should be sacrificed from the outset as being fundamental.

As such, in Geometric Unity we will proceed without loss of generality to consider not a single space, but pairs of spaces linked by maps.

The Chimeric Bundle

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We now define two metric bundles which, with the above assignments, are canonically isomorphic:

where each of these Chimeric bundles may be thought of as ‘semi-canonically’ equivalent to the Tangent and Co-Tangent bundles according to:

Chimeric Spinors and Heterogeneous Spin Bundles

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We have abstractly defined spinor bundles on Y , it behoves us to understand what kind of spinors we may have coaxed into existence.

We should begin by noting that the choice of (1, 3) metric signature (which can be mirrored by choice of a (3, 1) and thus is not meaningfully distinguished from it) is treated by us as anthropic data. If it were otherwise, there would likely be no life to evolve to observe these structures.

The Main Principal Bundle

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We have always found the byzantine intricacies of Clifford Algebras confusing and an attempt to recollect the various containments just discussed is offered here:

where we are privileging one particular path up the diagram that contains the metric and unitary representations:

Topological Spinors and Their Observation

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The appearance of topological spinors may then finally be interrogated under observation byג

. Let x ∈ U ⊂ X be a point in the neighborhood of a local observation גU .

Pati-Salam

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One way of looking at all of this is as a geometric setting for Grand Unified theories.

The Georgi-Glashow model of SU(5) and its associated Spin(10) enlargement is more popular.

Unified Field Content: The Inhomogeneous Gauge Group and Fermionic Extension

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Up until now we have been dealing with finite-dimensional constructions that superficially appear to be geometric (e.g. spinorial), but are actually mostly topological in nature. In this section we focus on the field content of GU which brings us to infinite dimensional algebraic constructions.

Infinite Dimensional Function Spaces: A, H, N

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The so-called Gauge Group of automorphisms of PH is defined to be:

H = Γ∞(PH ×Ad H) (5.3)

where the space of connections for PH

A = Conn(PH) (5.4)

is an affine space modeled on the right H-module

The Distinguished connection A0 and its consequences

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All of the preceding is general and did not depend on the choice of any particular connection. However, besides the ability to contract and project within the Einstein-Riemann paradigm, the secondary benefit we have discussed is the existence of a distinguished Levi-Civita connection.

Stabilizer Subgroup

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If we act on the space of connections using the natural right action of the inhomogeneous gauge group G we may ask what the stabilizer subgroup is for the Levi-Civita spin connection A0. To this end, solving for g ∈ G stabilizing A0 we have:

The Augmented or Displaced Torsion

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The Torsion Tensor has always presented a puzzle. It is traditionally introduced very early on in the study of Riemannian geometry and is almost never heard from thereafter except in niche explorations. One modern interpretation of this stylized fact could be that the torsion of a connection is afflicted with a disease that keeps it from being ‘gauge covariant’ and thus useful to the mainstream of modern theory.

The Family of Shiab Operators

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With this machinery notated and established, we can turn to the second major advantage of working in the Meta-Riemannian paradigm.

Figure 5: Ship In a Bottle Construction.

In essence, gauge theory and relativity have been disconnected because of the incompatibility of contraction and gauge covariance of terms within the action.

Lagrangians

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There are multiple considerations in putting forward Lagrangians in this context. In particular there are issues of redundant equations, Bianchi Identities, cohomological considerations for deformation complexes, so-called Supersymmetry, agreement with prior physical equations, and the issue of DiracPairs where one set of more restrictive (usually first order) equations guarantees the solution of the equations of a different related (usually second order) Lagrangians.

Lagrangians

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9.2 Second Order Euler-Lagrange Equations

One of the Claims of Geometric Unity is that we have been unsuccessful in Unifying the four basic equations for Gravity, Non-Gravitational force, Matter and Higgs phenomena because they belong to a Dirac Pair.

Deformation Complex

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The expected way to have our field equations arise naturally as the obstruction to a cohomology theory, is to first ask about the moduli space of solutions to the equations. That is, if ω

Observed Field Content

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One of the features that arises when doing away with the primary nature of space-time and replacing a single metric space with a tension between two separate but related spaces linked by metrics, is that we find ourselves in the novel situation where must relate fields that are native to different spaces. The principal means of doing this is via the pull back operation.

Observed Field Content

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12.5 The Failure of Unification May Be Solved by Dirac Square Roots

If we accept the colloquial description of the Dirac equation as the square root of the Klein-Gordon equation, we see that solutions of a first order operator can guarantee solutions of a more general second order equation.

Summary

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The approach of Geometric Unity as a candidate physical theory of our world is to work with a variety of different bundles in both finite and infinite dimensions which are all generated from a single space X4.

Chirality Is Merely Effective

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12.9 Chirality Is Merely Effective and Results From Decoupling a Fundamentally Non-Chiral Theory

Consider a stylized system of equations for a world Y with metric g, having scalar curvature R(y), and endowed with a non-chiral Dirac operator operating on full Dirac Spinors,

Final Thoughts

Fri, 30 May 2025 00:00:00 +0000

To sum up, let us revisit the Witten synopsis to see what GU has to say about it:

Figure 7: Edward Witten Synopsis.

Geometric Unity may be considered an alternative narrative that tweaks familiar concepts in various ways. As the author sees it, it is really a collection of interconnected ideas about shifting our various perspectives. Given the apparent stagnation in the major programs, GU has sought an alternate interpretation of either or both of the two incompatible models for fundamental physics of the Standard Model or General Relativity.