DUAL HAMILTONIAN STRUCTURES IN AN INTEGRABLE HIERARCHY Vincent - PowerPoint PPT Presentation

DUAL HAMILTONIAN STRUCTURES IN AN INTEGRABLE HIERARCHY Vincent Caudrelier RAQIS’16, University of Geneva Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

Somes references and collaborators Based on joint work with J. Avan, A. Doikou and A. Kundu Lagrangian and Hamiltonian structures in an integrable hierarchy and space-time duality , Nucl. Phys. B902 (2016), 415-439. Multisymplectic approach to integrable defects in the sine-Gordon model , J. Phys. A 48 (2015) 195203. A multisymplectic approach to defects in integrable classical field theory , JHEP 02 (2015), 088 and some ongoing work with A. Fordy. Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

Plan 1. The fundamentals: classical and quantum R matrix The general scheme Tracing the origin of the classical r -matrix 2. Some new observations on the classical r -matrix New input from covariant field theory Poisson brackets for the “time” Lax matrix 3. Why the dual picture? Motivation: integrable defects The bigger picture: initial-boundary value problems 4. Speculations, outlook, quantum case Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Quantum Classical YBE YBE R =1+ � r R 12 R 13 R 23 = R 23 R 13 R 12 − − − − − − → [ r 12 , r 13 ] + [ r 12 , r 23 ] +[ r 13 , r 23 ] = 0 Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Quantum Classical YBE YBE R =1+ � r R 12 R 13 R 23 = R 23 R 13 R 12 − − − − − − → [ r 12 , r 13 ] + [ r 12 , r 23 ] +[ r 13 , r 23 ] = 0 � �   � � Lax matrix Lax matrix [ , ] → � { , } R 12 L 1 L 2 = L 2 L 1 R 12 − − − − − − → {L 1 , L 2 } = [ r 12 , L 1 L 2 ] R =1+ � r Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Quantum Classical YBE YBE R =1+ � r R 12 R 13 R 23 = R 23 R 13 R 12 − − − − − − → [ r 12 , r 13 ] + [ r 12 , r 23 ] +[ r 13 , r 23 ] = 0 � �   � � Lax matrix Lax matrix [ , ] → � { , } R 12 L 1 L 2 = L 2 L 1 R 12 − − − − − − → {L 1 , L 2 } = [ r 12 , L 1 L 2 ] R =1+ � r   ultralocality   � � Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Quantum Classical YBE YBE R =1+ � r R 12 R 13 R 23 = R 23 R 13 R 12 − − − − − − → [ r 12 , r 13 ] + [ r 12 , r 23 ] +[ r 13 , r 23 ] = 0 � �   � � Lax matrix Lax matrix [ , ] → � { , } R 12 L 1 L 2 = L 2 L 1 R 12 − − − − − − → {L 1 , L 2 } = [ r 12 , L 1 L 2 ] R =1+ � r   ultralocality   � � Monodromy matrix Monodromy matrix [ , ] → � { , } R 12 T 1 T 2 = T 2 T 1 R 12 − − − − − − → {T 1 , T 2 } = [ r 12 , T 1 T 2 ] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Classical discrete Classical continuous Lax matrix Lax matrix L =1+∆ U {L 1 , L 2 } = [ r 12 , L 1 L 2 ] − − − − − − → { U 1 , U 2 } = δ [ r 12 , U 1 + U 2 ] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.1 Classical and quantum R matrix: general scheme Classical discrete Classical continuous Lax matrix Lax matrix L =1+∆ U {L 1 , L 2 } = [ r 12 , L 1 L 2 ] − − − − − − → { U 1 , U 2 } = δ [ r 12 , U 1 + U 2 ]     � � Monodromy matrix Monodromy matrix {T 1 , T 2 } = [ r 12 , T 1 T 2 ] {T 1 , T 2 } = [ r 12 , T 1 T 2 ] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Examine in detail the fundamental relation (continuous case) { U 1 , U 2 } = δ [ r 12 , U 1 + U 2 ] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Examine in detail the fundamental relation (continuous case) { U 1 , U 2 } = δ [ r 12 , U 1 + U 2 ] • With all explicit dependences, it reads { U 1 ( x , λ ) , U 2 ( y , µ ) } = δ ( x − y )[ r 12 ( λ − µ ) , U 1 ( x , λ )+ U 2 ( y , µ )] where U 1 = U ⊗ 1 I, U 2 = 1 I ⊗ U and (rational case) r 12 ( λ ) = g P 12 P 12 permutation , g constant λ , Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Examine in detail the fundamental relation (continuous case) { U 1 , U 2 } = δ [ r 12 , U 1 + U 2 ] • With all explicit dependences, it reads { U 1 ( x , λ ) , U 2 ( y , µ ) } = δ ( x − y )[ r 12 ( λ − µ ) , U 1 ( x , λ )+ U 2 ( y , µ )] where U 1 = U ⊗ 1 I, U 2 = 1 I ⊗ U and (rational case) r 12 ( λ ) = g P 12 P 12 permutation , g constant λ , • Ultralocal Poisson algebra for the entries of the matrix U . Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix What is the origin of this fundamental relation ? Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix What is the origin of this fundamental relation ? → The Poisson brackets of the fields contained in U . Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix What is the origin of this fundamental relation ? → The Poisson brackets of the fields contained in U . Example: nonlinear Schr¨ odinger (NLS) equation iq t + q xx − 2 g | q | 2 q = 0 one imposes equal time Poisson brackets (at t = 0) { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) so that one can write NLS as a Hamiltonian system � � | q x | 2 + g | q | 4 � q t = { H NLS , q } , H NLS = dx [Zakharov, Manakov ’74] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix On the other hand, NLS as a PDE is obtained as the compatibility condition of the auxiliary problem � Ψ x = U Ψ Ψ t = V Ψ i.e. the zero curvature condition U t − V x + [ U , V ] = 0 with Lax pair � − i λ � − 2 i λ 2 + i | q | 2 � � q ( x ) 2 λ q + iq x U ( x , λ ) = , V = 2 λ gq ∗ − igq ∗ 2 i λ 2 − i | q | 2 gq ∗ ( x ) i λ x [Zakharov, Shabat ’71] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix Then, the canonical Poisson brackets on the fields { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) are equivalent to the ultralocal Poisson algebra for U { U 1 ( x , λ ) , U 2 ( y , λ ) } = δ ( x − y )[ r 12 ( λ − µ ) , U 1 ( x , λ )+ U 2 ( y , µ )] [Sklyanin ’79] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix Continue the reasoning: but then, what is the origin of the canonical PB for the fields { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) , source of all the rest of the approach? Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Go back to the Classics: Lagrangian/Hamiltonian mechanics Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Go back to the Classics: Lagrangian/Hamiltonian mechanics • Canonical fields are prescribed from a Lagrangian description Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Go back to the Classics: Lagrangian/Hamiltonian mechanics • Canonical fields are prescribed from a Lagrangian description For NLS L NLS = i 2( q ∗ q t − qq ∗ t ) − q ∗ x q x − g ( q ∗ q ) 2 Then π = ∂ L NLS π ∗ = ∂ L NLS = i = − i 2 q ∗ , 2 q ∂ q ∗ ∂ q t t and one requires { π ( x ) , q ( y ) } = δ ( x − y ) Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix • Go back to the Classics: Lagrangian/Hamiltonian mechanics • Canonical fields are prescribed from a Lagrangian description For NLS L NLS = i 2( q ∗ q t − qq ∗ t ) − q ∗ x q x − g ( q ∗ q ) 2 Then π = ∂ L NLS π ∗ = ∂ L NLS = i = − i 2 q ∗ , 2 q ∂ q ∗ ∂ q t t and one requires { π ( x ) , q ( y ) } = δ ( x − y ) • This yields the known brackets { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) (NB: Dirac procedure must be used). Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix Summary: the textbook approach [Faddeev, Takhtajan, ’87] { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy

1.2 Tracing the origin of the classical r -matrix Summary: the textbook approach [Faddeev, Takhtajan, ’87] { q ( x ) , q ∗ ( y ) } = i δ ( x − y ) ⇓ { U 1 ( x , λ ) , U 2 ( y , µ ) } = δ ( x − y )[ r 12 ( λ − µ ) , U 1 ( x , λ ) + U 2 ( y , µ )] Vincent Caudrelier Dual Hamiltonian Structures in an Integrable Hierarchy