Matter–Geometry Unification Overview

• Matter–geometry unification is a framework where matter fields and spacetime geometry emerge from a single geometric and algebraic structure.
• It employs methods like hypercomplex structures, symmetry breaking of high-dimensional groups, and non-Riemannian measures to derive coupled field equations.
• The models predict novel phenomena including dark matter candidates, quantized black hole horizons, and regular cosmic evolution without singularities.

Matter–geometry unification refers to a range of frameworks in which matter fields and geometric structures of spacetime are not distinct, independently postulated ingredients, but instead emerge from a single underlying geometric or algebraic description. These unification programs typically aim to subsume gravity, gauge fields, and fermions within generalized geometric actions, often exploiting nontrivial group structures, measure theory, or higher-dimensional setups. The following exposition synthesizes several principal approaches from the literature, including hypercomplex geometry, degenerate metric models, measure-based actions, teleparallelism, and geometric scalar field quantization.

1. Hypercomplex Structures and Metric Compatibility

Unified Field Theoretical (UFT) models have been developed where the spacetime manifold is equipped with an underlying hypercomplex structure—i.e., an internal space at each point with hypercomplex or Hermitian algebraic character (Cirilo-Lombardo, 2012). The defining technical requirement is zero non-metricity: the covariant derivative of the metric vanishes,

$Vg = 0$

ensuring metric compatibility. This invariant condition dictates that all parallel transports preserve lengths and angles, thus maintaining the affine straightness of geodesics. Within this geometric setup, torsion components (particularly the totally antisymmetric part) do not represent ad hoc matter sources but emerge from the geometry itself, encoding matter-like degrees of freedom. In effect, fields corresponding to gravity and matter are inherently part of the same geometric structure rather than independently imposed.

2. Geometrical Actions via Group Symmetry Breaking

The dynamical content in these frameworks is often governed by geometric Lagrangians constructed from curvature tensors originating from a symmetry breaking of higher-dimensional group manifolds (Cirilo-Lombardo, 2012). Constructions following the Cartan–MacDowell–Mansouri mechanism start with a large gauge group, such as SO(1,4) (de Sitter group), which is broken down to SO(1,3) (Lorentz group) and associated fields. The connection splits into Lorentz-valued and vierbein parts, producing extended curvature: $R_{AB} = O(w_{AB}) + (w_{AC} \wedge w^{C}_{B})$ The core geometric actions take determinantal form reminiscent of Eddington–Born–Infeld or non-topological Lagrangians: $L_g = \det R^a_\mu$ or

$L_g = \det[X]$

where $X$ often encapsulates both metric and antisymmetric tensor (e.g., electromagnetic field). Variation of this action yields coupled field equations for gravity and matter within one geometric entity.

3. Fermionic Fields and Antisymmetric Torsion

These frameworks establish a connection between the torsion of the affine connection and the spinor structure of matter fields. When the torsion tensor is totally antisymmetric,

$$T_{\mu\nu\lambda} = \epsilon_{\mu\nu\lambda\sigma} h^\sigma$$

it is the dual of an axial vector, naturally identifying the Dirac operator structure as emerging from the geometry. The torsion-induced terms in the fermionic Lagrangian produce corrections to the gyromagnetic factor of fermionic particles. For example, the modified Dirac equation takes the form: $$[(P - eA)^2 - m^2 - C(F + \text{torsion terms})]\psi = 0$$ implying that the interaction vertex gets torsion corrections, generically leading to anomalous contributions to the magnetic moment.

4. Measure-Based Unification and Non-Riemannian Actions

An alternative route uses non-Riemannian volume forms for spacetime integration (Guendelman et al., 2015). Here, the action includes metric-independent measures constructed from auxiliary maximal rank antisymmetric tensor fields: $$(B) = \frac{1}{3!}\epsilon^{\mu\nu\kappa\lambda}\partial_\mu B_{\nu\kappa\lambda}$$ The scalar field couples to both the standard $\sqrt{-g}$ measure and the non-Riemannian $(B)$, leading to a dynamical constraint on the Lagrangian: $L(\varphi, X) = -2M$ where $M$ is an integration constant that acts as a dynamically generated cosmological constant. The energy-momentum tensor decomposes into a geometric cosmological constant and a pressureless dust fluid (dark matter), the latter governed by a hidden nonlinear Noether symmetry. This symmetry ensures geodesic flow for the dust component. Perturbations that break the symmetry retain geodesic motion and permit evolutionary scenarios such as growing dark energy.

5. Degenerate Metrics and Higher-Dimensional Geometrization

Models using degenerate metrics, notably in five-dimensional settings, unify gravity and electromagnetism by extending the covariant metric $Y_{ab}$ to satisfy

$\det(Y_{ab}) = 0$

(Searight, 2018). The structure decomposes into tensor, vector (two photon-like fields), and scalar (Brans–Dicke type) components. Gauge-like (eigengauge) invariance and reflection symmetry bind these fields, ensuring one vector interacts with matter (ordinary photon) and the other remains electromagnetically hidden (dark photon). Admitting wave variation along the fifth dimension introduces mass terms for photons via: $A_\mu = a_\mu \exp[i(k_\nu x^\nu + k_5 w)]$ where $k_5$ is the fifth-dimensional wave number, producing massive, noninteracting photons that source gravity—proposed as dark matter candidates.

6. Minimal Coupling and Torsion Constraints in Generalized Geometry

The minimal coupling principle (MCP), essential for promoting flat-space field theories to curved spacetime, faces ambiguities in the presence of torsion within general affine geometries. When the affine connection $\Gamma$ contains torsion $T^\alpha_{\mu\nu}$, standard gauge fields acquire non-minimal couplings, which threaten gauge invariance: $$F = \nabla_{[\mu} \Lambda^N_{\nu]} - T^\alpha_{\mu\nu} \Lambda^N_\alpha + \ldots$$ (Jimenez et al., 2020). The resolution is symmetric teleparallelism, where the geometry is integrable and torsion-free. There, MCP is consistent for bosonic and fermionic fields, and generalized Bianchi identities constrain connection dynamics without introducing pathologies or second clock effects.

7. Geometric Scalar Field Quantization and Singularity Resolution

Interpretations of the Wheeler–DeWitt equation, along with ADM-based approaches, recast scalar matter fields as geometric fields on superspace (the space of 3-metrics) (Purohit, 2019, Purohit, 2020). The quantum theory employs ladder operators defined on this space, with couplings tied to the intrinsic curvature $R^{(3)}$: $$\mu^2(\zeta, \zeta^A) = -\frac{3\zeta^2}{32} R^{(3)}$$ Quantum states are peaked on definite 3-metrics with sharply defined energy but non-sharp momentum, reflecting the non-linear geometric structure. Importantly, the quantum theory dynamically prevents classical singularities: cosmological evolution initiates from a finite volume rather than a zero-volume big bang, and black hole horizons acquire quantum area quantization, eliminating central singularities.

8. Cosmological Solutions and Matter–Geometry Phenomenology

Unified frameworks predict new cosmological behaviors. In torsion-rich models (Cirilo-Lombardo, 2012), solutions with totally antisymmetric torsion may generate wormhole-like spacetimes; tratorial patterns (torsion from scalar gradients) preclude wormholes but yield expanding universes. Measure-based theories (Guendelman et al., 2015) enable simultaneous evolution of dark energy and matter from a single scalar field. Degenerate metric constructions (Searight, 2018) produce candidates for dark matter via massive, "dark" photons. Geometric quantization schemes (Purohit, 2019, Purohit, 2020) ensure regular cosmic evolution and provide alternative black hole structure.

9. Summary and Outlook

Matter–geometry unification encompasses models in which all fundamental fields—including gravitational, gauge, fermionic, and scalar sectors—arise as facets of generalized geometric structures, often exploiting group-theoretic symmetry breaking, non-Riemannian measures, higher-dimensional metrics, or intrinsic quantization on the space of geometries. Signatures include:

• Decisive role of torsion and hypercomplex structures in encoding matter degrees of freedom,
• Determinantal curvature actions generating non-topological dynamics,
• Dynamical emergence of cosmological constants and dust fluids from auxiliary measures,
• Prediction and interpretation of dark matter candidates (massive vector fields) from extended geometric settings,
• Quantum regulation mechanisms that prevent classical singularities.

Open directions involve rigorous exploration of phenomenological predictions (e.g., deviation from standard Einstein–Maxwell theory, dark sector signatures), further development of generalized symmetry breaking mechanisms, and quantization schemes that extend the unification paradigm to cover the full Standard Model field content, potentially within nontrivial microstructure or topological classes of spacetime.