Krylov Complexity in Quantum Field Theory
Kiran Adhikari, Sayantan Choudhury, Abhishek Roy
TL;DR
This work analyzes Krylov complexity $K(t)$ in quantum field theory and links it to holographic volume via a particle-number interpretation when the Krylov basis aligns with the Fock basis. By discretizing free scalar QFT to a lattice of coupled harmonic oscillators, the authors derive a mode-resolved complexity $K(t)=\sum_{\mathbf k} \frac{\sin^2(\tilde{\omega}_{\mathbf k} t/2)}{\sinh^2(\beta \tilde{\omega}_{\mathbf k}/2)}$, showing qualitative alignment with the holographic complexity-Volume relation. The framework is extended to systems with inverted oscillators, where $K(t)$ exhibits exponential growth, signaling chaotic dynamics in the field-theoretic Krylov description. The results suggest Krylov complexity captures particle production and volume-like growth in QFT and offers a robust alternative to Nielsen-based measures, with clear directions for incorporating interactions and fermionic fields.
Abstract
In this paper, we study the Krylov complexity in quantum field theory and make a connection with the holographic "Complexity equals Volume" conjecture. When Krylov basis matches with Fock basis, for several interesting settings, we observe that the Krylov complexity equals the average particle number showing that complexity scales with volume. Using similar formalism, we compute the Krylov complexity for free scalar field theory and find surprising similarities with holography. We also extend this framework for field theory where an inverted oscillator appears naturally and explore its chaotic behavior.