More on complexity of operators in quantum field theory

• Published in: The journal of high energy physics Vol. 2019; no. 3; pp. 1 - 41
• Main Authors: Yang, Run-Qiu, An, Yu-Sen, Niu, Chao, Zhang, Cheng-Yong, Kim, Keun-Young
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2019; Springer Nature B.V; SpringerOpen
• Subjects: Axioms; Classical and Quantum Gravitation; Complexity; Curvature; Elementary Particles; Field theory; Gauge-gravity correspondence; Geometry; High energy physics; Holography and condensed matter physics (AdS/CMT); Invariants; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; Saturation; String Theory; Topology; Gauge-gravity correspondence; Holography and condensed matter physics (AdS/CMT)
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP03(2019)161

A bstract Recently it has been shown that the complexity of SU( n ) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p -norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k -local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU( n ) groups.