Non-Geometrical Gauge Theory Construction

• Non-geometrical gauge theory is an approach that formulates gauge interactions using algebraic and combinatorial methods instead of traditional differential geometry.
• It employs recursive quantum assembly with finite-dimensional modules to generate emergent gauge groups, interaction structures, and metric properties.
• The framework recovers classical continuous gauge theories as singular limits, addressing divergences by relying on inherently regular, discrete constructions.

A non-geometrical construction of gauge theories refers to frameworks that formulate gauge interactions without presupposing the conventional differential-geometric infrastructure of smooth manifolds, bundles, and connections. Instead, such constructions proceed via algebraic, combinatorial, categorical, operator-algebraic, or recursive quantum–combinatorial approaches. These frameworks often produce gauge symmetry, interaction structures, and even emergent metric or space-time concepts from fundamentally finite, discrete, or algebraic ingredients, recovering conventional geometric gauge theories only in singular or continuum limits.

1. Recursive Quantum Assembly and Non-Geometric Foundations

In Finkelstein’s recursive quantum gauge theory (Finkelstein, 2010), gauge structures are constructed from the ground up via iterative algebraic assembly rather than geometric postulation. The starting point is a seed Hilbert space $R$, which is recursively assembled by the repeated application of a Grassmann quantification functor (“Grass”), mirroring a recursive power set operation but in a fermionic context. Each new “rank” in this recursion is a finite-dimensional Grassmann algebra:

• $2^0 := R$
• $2^{r+1} := \text{Grass}(2^r)$

Each assembly step produces finite-dimensional modules (“cells”) built from lower-rank modules (“monadics”), and these combinatorially generated modules form the fundamental objects (“quantum plexus”) of the theory. Notably, at every step, Fermi–Dirac statistics are imposed, meaning all building blocks are “unitized” as antisymmetric fermions, and the exclusion principle is enforced at each level.

This recursive combinatorial approach is non-geometric: there is no presupposed continuous manifold or smooth space–time. Instead, all structure is emergent, with gauge group actions, symmetry properties, and even the metric arising at higher recursive levels as collective properties of iterated algebraic assemblies.

2. Finite-Dimensional Representation Spaces and Combinatorial Gauge Structure

A central aspect of the non-geometric approach is the concrete, finite-dimensionality at every (finite) stage of recursion (Finkelstein, 2010). Each module or “cell” is a combinatorial aggregate of spinors or semivector representations. Unlike canonical quantum field theory, which employs infinite-dimensional Hilbert spaces and fibrations over smooth manifolds, recursive quantum gauge theory strictly builds all physical objects out of finite vector spaces—though the dimensions increase hyperexponentially with iteration.

Gauge groups in this framework are not associated with diffeomorphism groups of bundles over manifolds. Instead:

• At the cell level (“gauge of the first kind”), the group $SL(n)$ acts on the vertices of each cell.
• Event-level gauge structure (“gauge of the second kind”) emerges from the group actions on assemblies of cells, with the collective gauge group approximating complex gauge groups such as those of the Standard Model or gravitation only as a singular limit (when the number of recursions becomes very large).

This intrinsic algebraic (rather than geometric) definition means gauge invariance is understood as invariance under transformations acting directly on the quantum modules, not as invariance under coordinate transformations of a smooth space.

3. Canonical Gauge Theories as Singular Emergent Limits

Traditional geometric gauge theories—such as Yang–Mills theory, general relativity, or the electroweak sector—arise not as fundamental inputs but as organized singular limits of the recursive, finite-dimensional theory (Finkelstein, 2010). As the number of recursive quantification steps increases (particularly in the laminar, or large-scale limit), the combinatorial algebra approximates:

• Canonical commutator structures: e.g., $[x, x] \sim W$, $[x, p] \sim \hbar$, $[p, p] \sim \hbar W$
• Emergent continuum properties: infinite-dimensional representation spaces, continuous spectrum operators, and smooth manifold approximations

In this limit, gauge groups acting on the combinatorial structures become indistinguishable from the continuous Lie groups known from geometric gauge theory, and divergences familiar in conventional field theory (e.g., ultraviolet singularities) appear as artifacts of this limiting process.

A critical implication is that the divergences and infinities of canonical quantum field theory are not fundamental; they arise exclusively from the singular approximation of a more fundamental, regular, finite algebraic theory.

4. Emergence of Metric, Spin-Statistics, and Higgs Sectors

The recursive algebraic construction seamlessly encodes elements traditionally inserted by hand in geometric constructions:

• Spin–statistical correlation: Every fundamental object is a fermion by construction, and spinor structures are “grown” recursively; each level’s spinors become the semivectors for the next rank, leading to Clifford algebra relations:

$\{Y_\mu, Y_\nu\} = 2g_{\mu\nu}$

• Metric emergence: Within the algebraic duplex (Cartan) construction, the space (denoted $W$) becomes a quadratic space with an indefinite (neutral) inner product:

$$W : \text{Dup } Q,\quad \langle v, v' \rangle = v' \circ v$$

The standard Minkowski metric structure arises only in the continuum (singular) limit.

• Higgs field as emergent operator: The quantized imaginary unit ($\mathbb{L}i$), upon renormalization, generates not only the phase structure of quantum amplitudes but also electric charge and the Higgs field. In the singular limit, the operator satisfies $\hat{i}^2 = -1$ and functions as an order parameter, connecting quantum phase, symmetry breaking, and mass generation.

5. The Six–Iteration Structure and Finite Discreteness

A key quantitative result is the assertion that six iterations of recursive quantification are both necessary and sufficient to replicate all known physical structure (Finkelstein, 2010). For example:

• The “cell” (local module, rank $C \approx 4$) has just enough structure to encode spinor, gauge, and local interaction properties.
• The “event” (a more complete structure including space–time and field degrees of freedom, rank $E \approx 6$) encompasses all features observed in present physical models.

Both ranks are finite, and their algebraic “plexus” can, in singular approximation, reproduce the phenomenology of continuous gauge field theories. This finiteness eliminates the need for regularization techniques typically needed in infinite-dimensional quantum field theory.

6. Conceptual Advances and Unification

The recursive quantum, non-geometrical approach brings several conceptual shifts to gauge theory construction:

• Unified quantization and gauging: Both are rendered as recursive, algebraic quantification steps, not as distinct geometric or analytic processes. There is no separate stage where “geometry” is quantized—the entire structure is quantum and algebraic from inception.
• Discrete underpinning of continuous symmetries: Continuous gauge groups (e.g., $SU(3)\times SU(2)\times U(1)$) are now understood as singular approximations to finite, discrete group actions defined on the quantum cells.
• Divergence and regularity: Pathologies of canonical field theory, such as infinities and non-renormalizability, are resolved by founding the theory on regular, finite-dimensional structures, only recovering familiar singularities upon taking geometric limits.

A plausible implication is that quantum gravity and Standard Model interactions may be reinterpreted as singular approximations to fundamentally finite, recursively assembled combinatorial gauge structures.

7. Connections to Broader Non-Geometric Paradigms

This recursive, finite, and combinatorial approach represents one major direction within the broader landscape of non-geometrical gauge theory. Other parallel efforts—such as noncommutative geometry (Masson, 2012, Suijlekom, 2014, Hamilton, 21 May 2025), operator-algebraic constructions (Brothier et al., 2019), algebraic web constructions for quiver gauge theories (Kimura et al., 2019), canonical and constraint-based approaches (Miskovic et al., 2019), and categorical and higher-algebraic (2-group, 3-group) methods (Radenkovic et al., 2020, Kim, 2023)—all share the guiding principle of eschewing the primacy of classical smooth geometry. Instead, they derive gauge structures from intrinsic algebraic, operator, or combinatorial data and recover standard geometric results as limiting cases where appropriate.

In summary, the non-geometrical construction of gauge theories in the recursive quantum sense provides a route to foundational regularity, intrinsic gauge and metric emergence, and the unification of quantum and gauge structure without recourse to a preexisting manifold. This program offers both technical tools for evading divergences and a conceptual realignment of the ontology underpinning gauge interactions (Finkelstein, 2010).