Minkowski Representation Hypothesis

• MRH is a framework asserting that core features in physics, geometry, and AI are represented using Minkowskian geometry, invariant measures, and convex structures.
• Its methodology integrates algebraic reconstruction and spectral representations to derive classical metrics from quantum observables, supporting non-perturbative field theories.
• Applications span cosmology, discretized geometry, and neural network interpretability by employing Minkowski tensors and convex sum structures for robust representations.

The Minkowski Representation Hypothesis (MRH) refers to a broad class of frameworks in mathematics, physics, and machine learning that hypothesize or rigorously demonstrate how certain objects, observables, or internal representations can and should be represented in terms of Minkowskian geometry, structure, or algebraic properties. The MRH encompasses diverse domains, including quantum physics, cosmology, field theory, geometry, and the interpretability of neural network representations, all unified by the central idea that Minkowski-type representations capture essential structural, dynamical, or conceptual content.

1. Core Definitions and Conceptual Foundations

At its essence, the MRH posits that the key structural or functional features of a system—be it physical space(-time), quantum observables, representation spaces in AI, or geometric functionals—can be effectively (and sometimes uniquely) cast in terms of representations aligned with Minkowski space, generalized Minkowski algebra, or convex/Minkowski combinatorial structures. This hypothesis often entails the identification of:

• Underlying symmetry groups (Lorentz/Poincaré or, more generally, pseudo-Riemannian automorphism groups)
• Convexity or summation structures, sometimes formalized as Minkowski sums
• Invariant (or covariant) measures, intervals, or metric tensors of Lorentzian or pseudo-Euclidean type
• Spectral or functional representations whose properties mirror those of Minkowski geometry or algebra

The specific form and implications of the MRH depend on the context, but the unifying feature is the emergence or explicit encoding of Minkowskian structure as the organizing principle for the representation of data, states, observables, or geometric objects.

2. Physical Origins: Minkowski Space from Symmetry and Quantum Observables

In mathematical physics, the MRH arises primarily from efforts to rigorously justify, reconstruct, or generalize the spacetime framework of special relativity. Foundational results demonstrate that Minkowski space is not a mere postulate but an emergent concept tied to the symmetries of physical systems:

• Quantum observables and the Poincaré group: In variants of Penrose's Spin Geometry Theorem extended to the Poincaré group E(1,3), the classical metric structure of Minkowski space is recovered directly from the algebra of quantum mechanical observables (notably the four-momentum $p_a$ and angular momentum $J_{ab}$) without postulating spacetime a priori. Empirical distances between timelike world-lines can be constructed as functions of these observables, with the classical Minkowskian interval emerging in the appropriate limit (Szabados, 2023).
• Operator algebra and emergent geometry: In approaches based on extended Hilbert spaces, the MRH is realized by identifying Minkowski space as the sector of normalized eigenvectors of generalized "position" operators with zero eigenvalue. The metric signature and spacetime dimension arise from regularization procedures, and the holographic perspective identifies Minkowski spacetime as the conformal boundary of higher-dimensional spaces (e.g., AdS₅) (Plewa, 2019).
• Coordinate-free derivations: The existence of Minkowski space as a physical object is established independently of coordinate systems by showing that equivalence relations among event data (Einstein time and real positions) reconstruct the spacetime manifold and transformation group (Wagner, 2016).

These perspectives support the claim that Minkowski representations are operationally and conceptually intrinsic to the structure of physically meaningful theories.

3. Algebraic and Geometric Generalizations

The MRH extends far beyond the standard four-dimensional setting, encompassing:

• Generalized Clifford algebra representations: In geometric algebra $C\ell(\mathbb{R}^3)$, the MRH expresses four-dimensional spacetime as a slice of an eight-dimensional algebraic structure in which spin, helicity, and time receive a natural, intrinsically multidimensional and geometric interpretation (Chappell et al., 2012, Chappell et al., 2015).
• Algebraic reconstruction of physical laws: Maxwell's equations and conservation laws emerge algebraically from requirements of Minkowski-type invariance and Clifford conjugation, without the need to impose "external" physical arguments (Chappell et al., 2012, Chappell et al., 2015).
• Whitney forms and discrete geometry: The MRH also applies to geometric discretization, where Whitney forms (central in finite element exterior calculus) and their Hodge duals admit representation identities that are fully compatible with Minkowski and more general flat pseudo-Riemannian manifolds. Covector and vector formulations exhibit duality and the preservation of key geometric and physical properties under Lorentz transformations (Salamon et al., 2014).

4. Spectral Representations and Non-perturbative Field Theory

In quantum field theory, the MRH asserts that non-perturbative propagators of fundamental particles can be represented in generalized Källén-Lehmann (KL) spectral form directly in Minkowski space:

• Nakanishi integral representation (NIR): The infinite tower of Dyson-Schwinger equations for propagators is encoded as a coupled system of integral equations for unknown Nakanishi weight functions (NWFS), yielding a complete description of dressed propagators via spectral densities consistent with the analytic and symmetry structure required by Minkowski space (Mezrag et al., 2020).
• Analyticity, renormalizability, and gauge invariance: The MRH is strongly supported by the explicit appearance of Minkowskian structure in the denominator of spectral integrals, and by the ability to maintain physical (e.g., causality, locality) and gauge-theoretic requirements in this framework, including anomalies and non-perturbative chiral symmetry breaking (Belyea, 2010, Mezrag et al., 2020).

5. Minkowski Representation in Geometry, Integral Geometry, and Statistics

The MRH generalizes the representation of valuations, tensors, and geometric invariants through Minkowski-type formulae:

• Even Minkowski valuations: Translation-invariant, SO(n)-equivariant valuations are shown to admit unique integral representations involving area measures and generating (kernel) functions, directly paralleling the classical theory of the Moore–Penrose inverse in inner-product spaces. Operators such as Alesker’s Hard Lefschetz operators preserve Minkowski representation structures under derivative and integration operations (Schuster et al., 2014, Gao et al., 2023).
• Anisotropy quantification via Minkowski tensors: The MRH underlies the use of interfacial Minkowski tensors to characterize the anisotropy and shape of level sets in Gaussian random fields. The tensor hierarchy is governed by the covariance structure of the gradient, and higher-order signatures collapse to predictions from lower-rank tensors in Gaussian ensembles, providing model-independent hypothesis testing (Klatt et al., 2021).

6. Applications and Implications in Machine Learning

Contemporary research applies the MRH to the analysis and interpretation of neural network representations, particularly in vision transformers:

• Minkowski sum structure of token embeddings: MRH posits that internal token embeddings are not mere sparse linear combinations of axis-aligned features but are assembled from Minkowski sums of convex polytopes—each corresponding to a set of archetypal "concepts" or "landmarks." Multi-head attention naturally yields convex combinations per head, and their additive sum forms the token structure (Fel et al., 8 Oct 2025).
• Interpretability and steering constraints: Unlike traditional latent direction paradigms, MRH implies that meaningful semantic steering or intervention operates within bounded convex sets, and exceeding these manifolds leads to saturation or semantic inversion, with direct implications for concept editing, explainability, and robustness (Fel et al., 8 Oct 2025).

7. Cosmological and Foundational Reinterpretations

The MRH offers alternative cosmological frameworks:

• Cosmology in Minkowski representation: By recasting Friedmann-Lemaître-Robertson-Walker metrics via conformal transformations into Minkowski form, all physical observables become encoded as evolutions of fundamental scales (masses, times, lengths), not in expanding geometry. This reframing addresses problems such as the cosmological constant and observational tensions, and provides a geometric origin for dark matter, dark energy, inflation, and baryogenesis (Lombriser, 2023).

Summary Table: Variants of the Minkowski Representation Hypothesis

Domain	Structural Core of MRH	Key Implication/Result
Quantum/Relativistic Physics	Representation via symmetry group observables (Poincaré or conformal invariance)	Emergence/reconstruction of Minkowski spacetime, metric, and intervals (Szabados, 2023, Plewa, 2019)
Quantum Field Theory	Spectral-integral (KL/Nakanishi) representation of propagators	Viable non-perturbative methods in Minkowski space (Mezrag et al., 2020)
Clifford/Geometric Algebra	Multivector amplitudes, higher-dimensional extensions	Unified framework for spin, helicity, multidimensional time (Chappell et al., 2012, Chappell et al., 2015)
Discretized Geometry/FEEC	Dual covector/vector Whitney forms, mesh invariance	Covariant numerics on non-Euclidean meshes (Salamon et al., 2014)
Integral Geometry/Statistics	Kernel representations, Minkowski tensors/valuations	Quantification and prediction of anisotropy; block matrix inverses (Schuster et al., 2014, Klatt et al., 2021, Gao et al., 2023)
Representation Learning (AI)	Tokens as Minkowski sums of convex archetypal tiles	Interpretable, bounded, and task-relevant neural embeddings (Fel et al., 8 Oct 2025)
Cosmology	Conformal reparameterization to Minkowski frame	Reinterpretation of expansion, masses, and constants (Lombriser, 2023)

Concluding Remarks

The Minkowski Representation Hypothesis formalizes a recurring mathematical and physical insight: that many fundamental objects—geometric, analytic, statistical, or algorithmic—exhibit a deep compatibility with Minkowskian structure, whether through symmetries, spectral decompositions, convexity properties, or algebraic frameworks. The MRH both unifies diverse methodologies under a geometric paradigm and provides concrete operational and computational advantages, especially in contexts requiring manifest covariance, non-perturbative analysis, or interpretable representations. It continues to inspire new directions in foundational physics, numerical analysis, statistical geometry, and machine learning.