Hidden symmetries in integrable models David Osten IMPRS Particle Physics Colloquium MPP M¨ unchen, 14.12.2017
Hidden symmetries motivation in integrable models David Osten classical • conceptual challenges in quantum field theory integrability - symmetries and non-perturbative behaviour (spectrum, asymptotic freedom, geometry solitons),... quantum integrability - symmetries and the S -matrix applications 1 / 13
Hidden symmetries motivation in integrable models David Osten classical • conceptual challenges in quantum field theory integrability - symmetries and non-perturbative behaviour (spectrum, asymptotic freedom, geometry solitons),... quantum integrability - • standard approaches do not really help symmetries and the S -matrix • perturbation theory - expansion around free theory applications technical complications, resumming issues, ... • lattice calculations 1 / 13
Hidden symmetries motivation in integrable models David Osten classical • conceptual challenges in quantum field theory integrability - symmetries and non-perturbative behaviour (spectrum, asymptotic freedom, geometry solitons),... quantum integrability - • standard approaches do not really help symmetries and the S -matrix • perturbation theory - expansion around free theory applications technical complications, resumming issues, ... • lattice calculations • here: exactly solvable (or integrable ) toy models simple but non-trivial interacting theories 1 / 13
Hidden symmetries motivation in integrable models David Osten classical • conceptual challenges in quantum field theory integrability - symmetries and non-perturbative behaviour (spectrum, asymptotic freedom, geometry solitons),... quantum integrability - • standard approaches do not really help symmetries and the S -matrix • perturbation theory - expansion around free theory applications technical complications, resumming issues, ... • lattice calculations • here: exactly solvable (or integrable ) toy models simple but non-trivial interacting theories • What is a ’complete’ or ’exact’ solution? • simplicity ↔ (hidden) symmetries? • applications? 1 / 13
Hidden symmetries overview in integrable models David Osten classical integrability - symmetries and geometry 1 classical integrability - symmetries and geometry quantum integrability - symmetries and the S -matrix applications 2 quantum integrability - symmetries and the S -matrix 3 applications 2 / 13
Hidden symmetries the (unexpected) beauty of in integrable models David Osten the Kepler problem classical integrability - symmetries and p m geometry • effective one-body quantum x integrability - problem in central A symmetries and the S -matrix potential V = − α r applications 3 / 13
Hidden symmetries the (unexpected) beauty of in integrable models David Osten the Kepler problem classical integrability - symmetries and p m geometry • effective one-body quantum x integrability - problem in central A symmetries and the S -matrix potential V = − α r applications • conserved charges: • standard - energy E , angular momentum � L 3 / 13
Hidden symmetries the (unexpected) beauty of in integrable models David Osten the Kepler problem classical integrability - symmetries and p m geometry • effective one-body quantum x integrability - problem in central A symmetries and the S -matrix potential V = − α r applications • conserved charges: • standard - energy E , angular momentum � L • accidental/’hidden’ - perihel, Runge-Lenz vector � p × � L − α m � r A = � r • � L and � A : Noether charges of ’hidden’ SO ( 4 ) • algebraic solution via hidden symmetries 3 / 13
Hidden symmetries symmetries and geometry in integrable models David Osten Consider a system with n degrees of freedom classical → 2 n -dimensional phase space M with H ( q , p ) and { , } . integrability - symmetries and geometry • When is this system called integrable? quantum integrability - symmetries and the S -matrix applications 4 / 13
Hidden symmetries symmetries and geometry in integrable models David Osten Consider a system with n degrees of freedom classical → 2 n -dimensional phase space M with H ( q , p ) and { , } . integrability - symmetries and geometry • When is this system called integrable? quantum Definition (Liouville integrability): integrability - symmetries and • m functions, independent (on almost all M ), the S -matrix applications { f i , f j } = 0, { f k , H } = 0 f k ( q , p ) with • n ≤ m ≤ 2 n − 1: (super) integrable ⇒ Kepler problem: ’maximally’ integrable ( n =3, m =5) 4 / 13
Hidden symmetries symmetries and geometry in integrable models David Osten Consider a system with n degrees of freedom classical → 2 n -dimensional phase space M with H ( q , p ) and { , } . integrability - symmetries and geometry • When is this system called integrable? quantum Definition (Liouville integrability): integrability - symmetries and • m functions, independent (on almost all M ), the S -matrix applications { f i , f j } = 0, { f k , H } = 0 f k ( q , p ) with • n ≤ m ≤ 2 n − 1: (super) integrable ⇒ Kepler problem: ’maximally’ integrable ( n =3, m =5) • How do the ’solutions’ look? 4 / 13
Hidden symmetries symmetries and geometry in integrable models David Osten Consider a system with n degrees of freedom classical → 2 n -dimensional phase space M with H ( q , p ) and { , } . integrability - symmetries and geometry • When is this system called integrable? quantum Definition (Liouville integrability): integrability - symmetries and • m functions, independent (on almost all M ), the S -matrix applications { f i , f j } = 0, { f k , H } = 0 f k ( q , p ) with • n ≤ m ≤ 2 n − 1: (super) integrable ⇒ Kepler problem: ’maximally’ integrable ( n =3, m =5) • How do the ’solutions’ look? Theorem (Arnold): Assume we have an integrable Hamiltonian system, if M f = { ( q , p ) ∈ M | f k ( q , p ) = c k } is compact and connected: M f ∼ T n : S 1 × S 1 × ... × S 1 . see e.g. harmonic oscillator, Kepler problem 4 / 13
Hidden symmetries conceptual lessons in integrable models David Osten • # conserved charges ≥ # d.o.f. classical → purely algebraic construction of solutions integrability - symmetries and geometry quantum integrability - symmetries and the S -matrix applications 5 / 13
Hidden symmetries conceptual lessons in integrable models David Osten • # conserved charges ≥ # d.o.f. classical → purely algebraic construction of solutions integrability - • standard examples: symmetries and geometry • 1 d systems (energy conservation) quantum integrability - • harmonic oscillator symmetries and the S -matrix • Kepler problem applications 5 / 13
Hidden symmetries conceptual lessons in integrable models David Osten • # conserved charges ≥ # d.o.f. classical → purely algebraic construction of solutions integrability - • standard examples: symmetries and geometry • 1 d systems (energy conservation) quantum integrability - • harmonic oscillator symmetries and the S -matrix • Kepler problem applications • disturbed integrable models: → violated conservation laws e.g. ’disturbed’ Kepler problem: perihel rotation 5 / 13
Hidden symmetries Classical field theories in integrable models David Osten classical integrability - symmetries and What about field theories? geometry so far only systems with finitly many degrees of freedom, quantum integrability - symmetries and integrability rather trivial the S -matrix • field theories have an ∞ -dimensional phase space applications degrees of freedom for every point in space • ∞ − ∞ = ? how organise enough symmetries, what is a complete solution? - no universal definition of integrability 6 / 13
Hidden symmetries Lax integrability in integrable models David Osten classical integrability - description for a big class of integrable theories symmetries and geometry • existence of a pair of (differential) operators L , M quantum integrability - symmetries and the S -matrix d applications e.o.m. ⇔ d t L = [ L , M ] 7 / 13
Hidden symmetries Lax integrability in integrable models David Osten classical integrability - description for a big class of integrable theories symmetries and geometry • existence of a pair of (differential) operators L , M quantum integrability - symmetries and the S -matrix d applications e.o.m. ⇔ d t L = [ L , M ] • ⇒ eigenvalues of L are conserved! infinite tower of conserved charges, generating an ∞ -dim. (hidden) symmetry group • exact solution? 7 / 13
Hidden symmetries quantum integrability - naive in integrable models David Osten classical generalisation of classical integrability integrability - symmetries: symmetries and geometry functions on phase space M → operators on Hilbert space H quantum integrability - symmetries and ∃ n = dim ( H ) independent operators ˆ I 1 , ..., ˆ the S -matrix I n : applications [ ˆ I i , ˆ [ ˆ I i , ˆ I j ] = 0, H ] = 0. 8 / 13
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