Discrete Semantic States and Hamiltonian Dynamics in LLM Embedding Spaces

We investigate the structure of Large Language Model (LLM) embedding spaces using mathematical concepts, particularly linear algebra and the Hamiltonian formalism, drawing inspiration from analogies with quantum mechanical systems. Motivated by the observation that LLM embeddings exhibit distinct states, suggesting discrete semantic representations, we explore the application of these mathematical tools to analyze semantic relationships. We demonstrate that the L2 normalization constraint, a characteristic of many LLM architectures, results in a structured embedding space suitable for analysis using a Hamiltonian formalism. We derive relationships between cosine similarity and perturbations of embedding vectors, and explore direct and indirect semantic transitions. Furthermore, we explore a quantum-inspired perspective, deriving an analogue of zero-point energy and discussing potential connections to Koopman-von Neumann mechanics. While the interpretation warrants careful consideration, our results suggest that this approach offers a promising avenue for gaining deeper insights into LLMs and potentially informing new methods for mitigating hallucinations.