An introduction to Lax pairs and the zero curvature representation

Krishnaswami, Govind S.; Vishnu, T. R.

doi:10.48550/arxiv.2004.05791

Cited by 3 publications

(2 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where k = 1, 2, ..., N , can be treated as the covariant derivative [45] and the system of equations (2.6) is overdetermined. Thus, X H k , where k = 1, 2, ..., N , must satisfy a compatible condition leading to the the zero-curvature condition (2.9).…”

Section: Zero-curvature Condition Of the Hamiltonian Vector Fieldsmentioning

confidence: 99%

Quantum integrability: Lagrangian 1-form case

Kongkoom¹,

Yoo-Kong²

2022

Preprint

View full text Add to dashboard Cite

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the 1-form structure is captured in the geometrical language both in classical and quantum realms. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation in various contexts, e.g. Hamiltonian vector fields, Lagrangian vector fields, temporal Lax matrices and Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature in various types of mappings (which will be expressed in terms of the Wilson line), e.g. a family of maps in N -parameter groups on contangent and tangent bundles, time evolution maps in the Lax pair level and unitary multi-time evolution operators in the Schrödinger picture. The main highlights of this work are the following. The new mathematical objects, called Hamilton vector field and Lagrange vector field defined in the space of time variables, are introduced to alternatively capture integrability of the systems. To ensure integrability, these vector fields must be conservative and irrotational. Another important result is the formulation of the continuous multi-time propagator. This new type of the propagator exhibits the path-independent feature on the space of time variables. Consequently, a new perspective on summing all possible paths unavoidably arises as not only all possible paths on the space of dependent variables but also on the space of independent variables must be taken into account.

show abstract

Section: Zero-curvature Condition Of the Hamiltonian Vector Fieldsmentioning

confidence: 99%

Quantum integrability: Lagrangian 1-form case

Kongkoom¹,

Yoo-Kong²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…A word on terminology-formally, a Lax pair is a pair of scalar operators and not their matrix analogs, but we will not make this distinction here and will refer to any L, M pair as a Lax pair and the L operator as a Lax operator. The reader is assumed to have some familiarity with the history and basic theory of integrable systems-for an introduction, see [19].…”

Section: Introductionmentioning

confidence: 99%