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Lecture notes in mathematics

Quantum Hamiltonians associated with finite-dimensional Lie algebras and factorized s-matrices

David V. Chudnovsky, Gregory V. Chudnovsky

Published Jan 1, 1982• 4 citations
Publisher
DOI
10.1007/bfb0093506

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Quantum Hamiltonians Associated with Finite-Dimensional Lie Algebras and Factorized S Matrices

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Quantum Hamiltonian systems of lattice type are considered which are associated with classical finite-dimensional Lie algebras [like MH(2), SO(3), etc.], and possess complete integrability properties. The canonical Heisenberg ${p}_{n}$, ${q}_{n}$ variables are found for these Hamiltonian systems and it is shown how to construct their $S$ matrices. These new Hamiltonians are generalizations of the quantum Toda lattice Hamiltonian as well as $\mathrm{XXX}$ models of statistical mechanics.

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Abstract We are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.

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Drinfel'd Twists and Algebraic Bethe Ansatz

Jean Michel Maillet, J.M. Sánchez de Santos

We study representation theory of Drinfel'd twists, in terms of what we call F matrices, associated to finite dimensional irreducible modules of quantum affine algebras, and which factorize the corresponding (unitary) R matrices. We construct explicitly such factorizing F matrices for irreducible tensor products of the fundamental representations of the quantum affine algebra sl2 and its associated Yangian. We then apply these constructions to the XXX and XXZ quantum spins chains of finite length in the framework of the Algebraic Bethe Ansatz.

arXiv:q-alg/9612012•Dec 10, 1996•2 citations
Quantum Hamiltonian systems of lattice type are considered which are associated with classical finite-dimensional Lie algebras [like MH(2), SO(3), etc.], and possess complete integrability properties. The canonical Heisenberg pn{p}_{n}pn​, qn{q}_{n}qn​ variables are found for these Hamiltonian systems and it is shown how to construct their SSS matrices. These new Hamiltonians are generalizations of the quantum Toda lattice Hamiltonian as well as XXX\mathrm{XXX}XXX models of statistical mechanics.
Abstract We are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.
We study representation theory of Drinfel'd twists, in terms of what we call F matrices, associated to finite dimensional irreducible modules of quantum affine algebras, and which factorize the corresponding (unitary) R matrices. We construct explicitly such factorizing F matrices for irreducible tensor products of the fundamental representations of the quantum affine algebra sl2 and its associated Yangian. We then apply these constructions to the XXX and XXZ quantum spins chains of finite length in the framework of the Algebraic Bethe Ansatz.