# Semantic Wave Functions: Exploring Meaning in Large Language Models Through Quantum Formalism

arXiv: https://arxiv.org/abs/2503.10664 · HTML: https://mlsanity.com/paper/2503.10664/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 33 references that are in the map.
- First posted on arXiv 9 Mar 2025.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [What do you learn from context? Probing for sentence structure in contextualized word representations](https://mlsanity.com/paper/1905.06316/) (2019)
- [Linear Algebraic Structure of Word Senses, with Applications to Polysemy](https://mlsanity.com/paper/1601.03764/) (2016)
- [Introduction to the Special Issue on Quantum Cognition](https://mlsanity.com/paper/1309.5673/) (2013)
- [Should Semantic Vector Composition be Explicit? Can it be Linear?](https://mlsanity.com/paper/2104.06555/) (2021)

## Built on by (most cited first, 2 of 2)

- [The Quantum LLM: Modeling Semantic Spaces with Quantum Principles](https://mlsanity.com/paper/2504.13202/) (2025)
- [Discrete Semantic States and Hamiltonian Dynamics in LLM Embedding Spaces](https://mlsanity.com/paper/2601.11572/) (2025)

## Abstract (arXiv)

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 real-valued 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 Schr"odinger 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.