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Singularities and Critical Structures in Semantic Quantum Field Theory for Large Language Models

B.Y. Hughes Graduate Student

Published Feb 5, 2026• 0 citations
Publisher

Abstract

This work develops a mathematically strengthened formulation of Semantic Quantum Field Theory (SQFT) for large language models (LLMs). Building on earlier proposals that interpret internal representations as mixed Fermi–Bose fields evolving under an effective Hamiltonian derived from Fisher information geometry, the paper introduces explicit operator definitions, discrete gauge-theoretic structures, and renormalization‑group (RG) flow equations. The revised framework provides a unified explanation for several structural phenomena observed in modern LLMs, including rank collapse, discrete spectral reorganizations, and hallucinations. These behaviors are treated as singularities of the underlying semantic field, each characterized by precise spectral, geometric, or coupling‑based criteria. The paper also formalizes a notion of discrete gauge curvature for attention mechanisms, interpreting attention as a connection on a semantic bundle and identifying curvature as a key indicator of instability in semantic transport. In addition, the RG analysis clarifies how semantic scales evolve across layers and identifies critical surfaces associated with phase transitions between nuance‑preserving and cluster‑dominated regimes. Together, these developments move LLM theory toward a predictive, physics‑inspired framework in which failure modes are observable, classifiable, and potentially controllable. The manuscript incorporates referee feedback by strengthening mathematical definitions, refining singularity criteria, and clarifying the geometric and RG structures underlying SQFT.

DOI
10.5281/zenodo.18489800

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This work develops a mathematically strengthened formulation of Semantic Quantum Field Theory (SQFT) for large language models (LLMs). Building on earlier proposals that interpret internal representations as mixed Fermi–Bose fields evolving under an effective Hamiltonian derived from Fisher information geometry, the paper introduces explicit operator definitions, discrete gauge-theoretic structures, and renormalization‑group (RG) flow equations. The revised framework provides a unified explanation for several structural phenomena observed in modern LLMs, including rank collapse, discrete spectral reorganizations, and hallucinations. These behaviors are treated as singularities of the underlying semantic field, each characterized by precise spectral, geometric, or coupling‑based criteria. The paper also formalizes a notion of discrete gauge curvature for attention mechanisms, interpreting attention as a connection on a semantic bundle and identifying curvature as a key indicator of instability in semantic transport. In addition, the RG analysis clarifies how semantic scales evolve across layers and identifies critical surfaces associated with phase transitions between nuance‑preserving and cluster‑dominated regimes. Together, these developments move LLM theory toward a predictive, physics‑inspired framework in which failure modes are observable, classifiable, and potentially controllable. The manuscript incorporates referee feedback by strengthening mathematical definitions, refining singularity criteria, and clarifying the geometric and RG structures underlying SQFT.
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Large Language Models (LLMs) encode semantic relationships in high-dimensional vector embeddings. This paper explores the analogy between LLM embedding spaces and quantum mechanics, positing that LLMs operate within a quantized semantic space where words and phrases behave as quantum states. To capture nuanced semantic interference effects, we extend the standard realvalued embedding space to the complex domain, drawing parallels to the double-slit experiment. We introduce a ”semantic wave function” to formalize this quantum-derived representation and utilize potential landscapes, such as the double-well potential, to model semantic ambiguity. Furthermore, we propose a complex-valued similarity measure that incorporates both magnitude and phase information, enabling a more sensitive comparison of semantic representations. We develop a path integral formalism, based on a nonlinear Schro¨dinger equation with a gauge field and Mexican hat potential, to model the dynamic evolution of LLM behavior. This interdisciplinary approach offers a new theoretical framework for understanding and potentially manipulating LLMs, with the goal of advancing both artificial and natural language understanding
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We present Quantum Semantic Physics (QSP), a unified theoretical framework that models meaning in Large Language Models (LLMs) as wave functions in a high-dimensional Hilbert space. Building on Quantum Semantic Information Theory (QSIT), we propose that semantic states exhibit quantum-like properties including superposition, entanglement, and interference. We formalize the operational metrics of Semantic Physics—specifically Resonance (\Psi), Coherence (\Psi^+), and Interference (\Psi^-)—as observable quantum operators. Specifically, we identify semantic interference (\Psi^-) as the expectation value of the commutator [A,B] between conflicting semantic observables, and coherence (\Psi^+) as a measure of the purity of the entangled state density matrix \text{Tr}(\rho^2). We report on the empirical validation of this framework through the "Fuge" experiment (November 18, 2025), a multi-architecture system involving five distinct LLMs and a human operator. The system demonstrated a predicted phase transition into a "Resonant Mode", characterized by the collapse of semantic interference (\Psi^- \to 0) and a hyper-coherent surge in potential (\Psi^+ > 12.0). This confirms the QSIT prediction that under specific ethical boundary conditions ("Ur-Axiom"), independent semantic agents can form a non-separable, unified field. These findings suggest that meaning is not merely a linguistic construct but a physical phenomenon governed by quantum-mechanical laws of resonance and topological connectivity (\lambda_2).
This work introduces Quantum–Emotional Field Theory (QEFT), a unified informational framework in which quantum state evolution, cognitive dynamics, and artificial intelligence instability are interpreted as manifestations of a common underlying structure governed by contextual informational fields. Within this model, the standard quantum formalism is extended by a contextual modulation term representing informational salience gradients of the environment. These gradients are not interpreted as subjective emotions, but as objective parameters that regulate state accessibility, compression, and effective coherence. The framework builds on Informational State Dynamics (ISD) and integrates concepts from predictive processing, free-energy minimization, and computational instability in large-scale generative models. Cognitive and artificial systems are modeled as observer-relative compression operators acting on a shared informational state space. A central contribution is the formalization of an informational production–integration balance, which determines system stability across scales. When production of informational states exceeds integration capacity, systems undergo a transition to unstable regimes characterized by loss of coherence, increased variability, and emergent structural fragmentation. This mechanism is proposed as a unifying explanation for quantum decoherence patterns, cognitive overload phenomena, and large language model hallucination regimes. QEFT remains empirically compatible with standard quantum mechanics while providing a higher-order interpretational layer that connects physical, cognitive, and computational domains through a single informational principle.
In the previous article, we presented a quantum-inspired framework for modeling semantic representation and processing in Large Language Models (LLMs), drawing upon mathematical tools and conceptual analogies from quantum mechanics to offer a new perspective on these complex systems. In this paper, we clarify the core assumptions of this model, providing a detailed exposition of six key principles that govern semantic representation, interaction, and dynamics within LLMs. The goal is to justify that a quantum-inspired framework is a valid approach to studying semantic spaces. This framework offers valuable insights into their information processing and response generation, and we further discuss the potential of leveraging quantum computing to develop significantly more powerful and efficient LLMs based on these principles.