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Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems First Edition Edition
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This notion of integrability will not work for finite-dimensional quantum systems, as the author shows by using a system that hypothetically has a set L of mutually commuting operators, this set also including the Hamiltonian. He shows that no two commuting operators are algebraically independent, and at most D commuting operators are linearly independent, where D is the dimension of the eigenspace of one of the operators. The author then presents a notion of integrability that is less trivial, in that it will give information on the dynamics of the quantum system.
Since quantum systems are typically systems of particles that are interacting with each other, the dynamical events of interest are the scattering events. Indeed, the scattering theory of quantum systems is highly developed, and has inspired an enormous amount of research in both physics and mathematics.Read more ›