[Paper Review] Gravity as a Higgs Field. I.the Geometric Equivalence Principle

G. Sardanashvily|ArXiv.org|May 6, 1994

Relativity and Gravitational Theory6 references3 citations

TL;DR

This paper proposes that gravity can be understood as a Higgs field arising from spontaneous symmetry breaking, with the tetrad field acting as the Higgs mechanism's order parameter. By reformulating the Equivalence Principle via Klein-Chern geometries, it shows that Dirac fermions couple to gravity through a composite spinor bundle over a tetrad parameter space, leading to a total Dirac operator incorporating connections along both spacetime and tetrad coordinates.

ABSTRACT

{\it If gravity is a metric field by Einstein, it is a Higgs field.} Gravitation theory meets spontaneous symmetry breaking in accordance with the Equivalence Principle reformulated in the spirit of Klein-Chern geometries of invariants. In gravitation theory, the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the connected Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is Dirac fermion matter possessing only exact Lorentz symmetries. The associated Higgs field is a tetrad gravitational field $h$ represented by a global section of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs field issues from the fact that, in the presence of different tetrad fields, Dirac fermion fields are described by spinor bundles associated with different reduced Lorentz subbundles of $LX$, and we have nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that a fermion field must be regarded only in a pair with a certain tetrad field. These pairs fail to be represented by sections of any product bundle $S imes\Si$, but sections of the composite spinor bundle $S o\Si o X^4$. They constitute the so-called fermion-gravitation complex where values of tetrad gravitational fields play the role of coordinate parameters, besides the familiar world coordinates. In Part 1 of the article, geometry of the fermion-gravitation complex is investigated. The goal is the total Dirac operator into which components of a connection on $S o\Si$ along tetrad coordinate directions make contribution. The Part II will be devoted to dynamics of fermion-gravitation complex. It is a constraint system to describe which we use the covariant multisymplectic generalization of the Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only the time.

Motivation & Objective

• To reformulate the Equivalence Principle using Klein-Chern geometries of invariants to unify gravity and fermionic matter.
• To establish a geometric framework where gravity arises as a Higgs field due to spontaneous Lorentz symmetry breaking.
• To describe the coupling of Dirac fermions to gravity not via a product bundle, but via a composite spinor bundle over the space of tetrad fields.
• To lay the geometric foundation for a constrained Hamiltonian dynamics of the fermion-gravitation complex in a multisymplectic formalism.
• To show that different tetrad fields lead to inequivalent representations of cotangent vectors via Dirac matrices, necessitating a fibered structure over the tetrad space.

Proposed method

• Reduces the structure group of the frame bundle $LX$ over a 4D world manifold $X^4$ to the Lorentz group $SO(3,1)$, motivated by exact Lorentz symmetry of Dirac fermions.
• Identifies the tetrad gravitational field $h$ as a global section of the quotient bundle $\Sigma = LX / SO(3,1)$, defining the Higgs field in this context.
• Constructs the fermion-gravitation complex as a composite spinor bundle $S \to \Sigma \to X^4$, where the tetrad field acts as a coordinate parameter.
• Derives a total Dirac operator that includes connections along both spacetime and tetrad coordinate directions, unifying spacetime and internal symmetries.
• Applies a covariant multisymplectic generalization of Hamiltonian formalism, assigning canonical momenta to derivatives with respect to all world coordinates.
• Uses the geometry of principal bundles and associated bundles to model the interaction between fermions and gravity as a gauge-like structure with Higgs-like symmetry breaking.

Experimental results

Research questions

• RQ1 How can the Equivalence Principle be reformulated in terms of Klein-Chern geometries to unify gravity and fermionic matter?
• RQ2 What is the geometric role of the tetrad field in realizing gravity as a Higgs field via spontaneous Lorentz symmetry breaking?
• RQ3 Why is the fermion-gravitation system not representable by a product bundle $S \times \Sigma$, and what structure replaces it?
• RQ4 How does the total Dirac operator emerge from connections along both spacetime and tetrad parameter directions?
• RQ5 What is the role of the multisymplectic formalism in describing the dynamics of the fermion-gravitation complex as a constrained system?

Key findings

• The tetrad field $h$ serves as a Higgs field for gravity, with its global section in $\Sigma = LX / SO(3,1)$ encoding the spontaneous breaking of the full linear frame symmetry to Lorentz symmetry.
• Dirac fermions are coupled to gravity only in pairs with specific tetrad fields, and such pairs are described by sections of the composite spinor bundle $S \to \Sigma \to X^4$, not by product bundles.
• Different tetrad fields lead to inequivalent representations of cotangent vectors via Dirac matrices, invalidating a universal spinor bundle structure.
• The total Dirac operator is constructed as a sum of components from connections along both spacetime and tetrad coordinate directions, unifying geometric and dynamical aspects.
• The fermion-gravitation complex forms a constraint system, with dynamics governed by a covariant multisymplectic Hamiltonian formalism where momenta are assigned to all world coordinate derivatives.
• The geometric framework establishes a natural setting for a future dynamical theory of gravity and fermions, with the Higgs-like mechanism rooted in the reduction of the frame bundle structure group.