Right. You want to dissect Wikipedia. Fine. Just don't expect me to find this remotely interesting. It's just more data, another layer of the edifice. I'll give you the facts, the structure, the bare bones. But don't mistake it for a conversation.

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Attempt to extend Yang–Mills theory to gravity

In the realm of quantum field theory, the persistent, some might say obsessive, pursuit of gauge gravitation theory represents an ongoing effort to extend the principles of Yang–Mills theory. This latter theory, you see, has proven remarkably adept at providing a unified and universal description of the fundamental interactions that govern our rather chaotic universe. The ambition, then, is to bend its elegant framework to encompass gravity itself, a force that has stubbornly resisted easy quantification within the quantum fold.

It is crucial, however, to maintain a certain clarity, a dispassionate delineation between similar concepts. Gauge gravitation theory should not be mistaken for gauge theory gravity. The latter is a formulation of classical gravitation, a rather quaint endeavor, expressed through the sophisticated lens of geometric algebra. Nor should it be confused with Kaluza–Klein theory. While Kaluza–Klein theory does employ gauge fields, its purpose is to describe particle fields, not the gravitational force itself. It’s a subtle distinction, I understand, but precision is, regrettably, a necessity in these matters.

Overview

The genesis of the idea of a gauge model for gravity can be traced back to 1956, a mere two years after the very concept of gauge theory was born. It was Ryoyu Utiyama who first ventured this suggestion. Initially, these early attempts to construct a gauge theory of gravity, drawing parallels with existing gauge models of internal symmetries, ran headfirst into a significant obstacle. The challenge lay in reconciling general covariant transformations with the establishment of the gauge status of a pseudo-Riemannian metric, or more precisely, a tetrad field. It was a conceptual knot that resisted immediate untangling.

To circumvent this particular difficulty, a novel approach was explored: the representation of tetrad fields as gauge fields associated with the translation group. The infinitesimal generators of general covariant transformations were then considered as the generators of this translation gauge group. In this framework, a tetrad (or coframe) field was intrinsically identified with the translational component of an affine connection defined on a world manifold.

This connection, let us denote it by $K$, can be decomposed into two parts:

$K = \Gamma + \Theta$

Here, $\Gamma$ represents a linear world connection, a standard mathematical construct. The more intriguing component is $\Theta$, the soldering form. It is defined as:

$\Theta = \Theta_{\mu}^{a} dx^{\mu} \otimes \vartheta_{a}$

where $\vartheta_{a}$ is a non-holonomic frame, expressed as:

$\vartheta_{a} = \vartheta_{a}^{\lambda} \partial_{\lambda}$

Consider, for instance, the Cartan connection. In this specific case, the soldering form $\Theta$ becomes the canonical soldering form on $X$, denoted as $\theta$:

$\Theta = \theta = dx^{\mu} \otimes \partial_{\mu}$

The physical interpretation of this translational part, $\Theta$, of an affine connection has been a subject of varied perspectives. Within the context of gauge theory applied to dislocations, for example, the field $\Theta$ is understood to describe a form of distortion. Simultaneously, when a linear frame $\vartheta_{a}$ is given, its decomposition $\theta = \vartheta^{a} \otimes \vartheta_{a}$ provides a compelling motivation for numerous researchers to treat the coframe $\vartheta^{a}$ as a translation gauge field.

The inherent difficulties in constructing a gauge theory of gravitation by direct analogy with Yang–Mills theory stem from fundamental differences in the nature of the gauge transformations involved. In theories of internal symmetries, the gauge transformations are essentially vertical automorphisms of a principal bundle $P \to X$, acting upon the bundle itself while leaving its base space $X$ invariant. Gravitation theory, conversely, is built upon the principal bundle $FX$ of tangent frames to $X$. This bundle falls under the category of natural bundles $T \to X$, where diffeomorphisms of the base space $X$ induce automorphisms of $T$. These transformations are precisely what we term general covariant transformations. It is this class of transformations that proves sufficient for restating Einstein's general relativity and metric-affine gravitation theory within a gauge theoretical framework.

Within the framework of gauge theory on natural bundles, linear connections on a world manifold $X$ are conceptualized as gauge fields, specifically as principal connections defined on the linear frame bundle $FX$. The metric (or tetrad) gravitational field, in this context, assumes the role of a Higgs field. Its function is to orchestrate the spontaneous symmetry breaking of general covariant transformations.

Spontaneous symmetry breaking, as a phenomenon, is typically observed in quantum mechanics when the vacuum state of a system is not invariant under the action of the symmetry group. In the realm of classical gauge theory, spontaneous symmetry breaking occurs when the structure group $G$ of a principal bundle $P \to X$ can be reduced to a closed subgroup $H$. This reduction implies the existence of a principal subbundle of $P$ with $H$ as its structure group. A well-established theorem posits a one-to-one correspondence between such reduced principal subbundles (often referred to as G-structures) and the global sections of the quotient bundle $P/H \to X$. These global sections are then interpreted as classical Higgs fields.

The conceptual leap that identifies the pseudo-Riemannian metric as a Higgs field emerged during the construction of non-linear (induced) representations of the general linear group GL(4, R). The Lorentz group, a fundamental component of special relativity, is a Cartan subgroup of GL(4, R). The geometric equivalence principle, which postulates the existence of a reference frame wherein Lorentz invariants are defined across the entire world manifold, provides the theoretical underpinning for reducing the structure group GL(4, R) of the linear frame bundle $FX$ to the Lorentz group. Consequently, the very definition of a pseudo-Riemannian metric on a manifold $X$—as a global section of the quotient bundle $FX/O(1, 3) \to X$—naturally leads to its physical interpretation as a Higgs field. The underlying physical impetus for this symmetry breaking is often attributed to the presence of Dirac fermion matter, whose inherent symmetry group is the universal two-sheeted covering SL(2, C) of the restricted Lorentz group, SO+(1, 3).

See also

• Ashtekar variables
• Metric-affine gravitation theory
• Einstein–Cartan theory
• Spontaneous symmetry breaking
• Teleparallelism
• Reduction of the structure group
• Higgs field (classical)
• General covariant transformations
• Equivalence principle (geometric)
• Affine gauge theory
• Classical unified field theories