Quantum integrability: Lagrangian 1-form case
arXiv:2203.16914 · doi:10.1016/j.nuclphysb.2023.116101
Abstract
A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language for quantum level. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation, e.g. Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature of the mapping, e.g. multi-time evolution in the Schrödinger picture. Another important result is the formulation of the continuous multi-time propagator. With this new type of the propagator, a new perspective on summing all possible paths unavoidably arises as not only all possible paths in the space of dependent variables but also in the space of independent variables must be taken into account. The semi-classical approximation is applied to the multi-time propagator expressing in terms of the classical action and the fluctuation around it. Therefore, the extremum propagator, resulting in path independent feature on the space of independent variables, would guarantee the integrability of the system.
References in corpus (23)
- Classification of integrable equations on quad-graphs. The consistency approach
- Algebraic entropy
- Quantum Inverse Scattering Method. Selected Topics
- Lagrangian multiforms and multidimensional consistency
- A time-discretized version of the Calogero-Moser model
- Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
- Pure quantum integrability
- Discrete-time Calogero-Moser system and Lagrangian 1-form structure
- An integrable multicomponent quad equation and its Lagrangian formulation
- Variational symmetries and pluri-Lagrangian systems in classical mechanics
- Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems
- Non-Abelian Stokes theorem in action
- Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure
- Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
- Continuum limits of pluri-Lagrangian systems
- Multiform description of the AKNS hierarchy and classical r-matrix
- Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
- Integrability and Fusion Algebra for Quantum Mappings
- Integrable Quantum Mappings and Quantization Aspects of Integrable Discrete-time Systems
- Exactly solved models on planar graphs with vertices in
- On the Lagrangian 1-Form Structure of the Hyperbolic Calogero-Moser System
- Time-sliced path integrals with stationary states
- Variational symmetries and pluri-Lagrangian systems