Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because another formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators that may cause divergence of the classical perturbation series. The results that are established here link the concept of quantum-mechanical integrability to a technical question, namely, the behavior of specific perturbation series. \textcopyright{} 1996 The American Physical Society.
Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because another formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators that may cause divergence of the classical perturbation series. The results that are established here link the concept of quantum-mechanical integrability to a technical question, namely, the behavior of specific perturbation series. \textcopyright{} 1996 The American Physical Society.
Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.
For a (classically) integrable quantum-mechanical system with two degrees of freedom, the functional dependence H=HQ(J1,J2) of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship H(p1,q1;p2,q2)=HC(J1,J2) in the classical limit of that system. The former converges toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, nonuniform. The existence of the function H=HQ(J1,J2) in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.
In the present thesis we perform a step towards understanding the effec-
tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small
excitations: small number of reversed spins or, equivalently, small number of
fermions per site. The calculation of the first order normal form Hamiltonian
amounts to erasing all the nonresonant interaction terms, and allows to draw
interesting conclusions on the dynamics of the system, e.g. to determine ap-
proximately conserved quantities. Moreover, we get a good correspondence
between our perturbative energy spectrum and the exact one computed via
the Bethe ansatz, the approximation being of course the more accurate the
smaller is the number of fermions per site. The present method, here tested
in an exactly solvable case, can be applied to any nonintegrable system of
weakly interacting fermions (and/or bosons).
Considering physics as it is done today one notices that physics separates into two formally and conceptually different parts. There is on the one hand classical physics that deals with macroscopic phenomena and there is on the other hand quantum physics that deals with microscopic phenomena. Penrose [96] described this situation as a disturbing analogy to ancient Greece, where two different sets of laws for earth and heaven were applied. So immediately the question arises if one really has to use different formalisms on different scales or if it is possible to describe physics in a unified way. In order to investigate this question one first has to consider how classical and quantum physics are related. Quantum mechanics results from classical physics by a procedure called quantization and classical physics is reobtained by taking the classical limit. Both procedures are heavily plagued by problems [61, 100]. In the first chapter the mathematical problems of quantization will be addressed and it will be described how the star product formalism circumvents these problems. The star product formalism has the advantage that there is no formal break if one goes from classical physics to quantum physics. This formal advantage and the resulting beauty is then used as a guiding principle for the further development. The first question that arises is if the spin can be described in the star product formalism. That this is indeed possible is shown in the second chapter. As a starting point the spin description with pseudoclassical mechanics as it was developed by Berezin is used. One can then construct a fermionic star product and apply it for deformation quantization of pseudoclassical mechanics. In analogy to the bosonic star product formalism one obtains spin Wigner functions that act as spin projectors. Besides the nonrelativistic case it is also possible to formulate Dirac theory with star products. The Clifford algebra of the gamma matrices is hereby described as a deformed version of a four dimensional Grassmann algebra. The fermionic star product in combination with the bosonic Moyal product leads to a supersymmetric star product formalism that can be used to describe supersymmetric quantum mechanics and in the relativistic case to describe the supersymmetric structure of Dirac theory. The other direction of generalization of the star product formalism is the application of star products in quantum field theory that is described in chapter three. After constructing a suitable normal product it is shown that the algebraic structures of perturbative quantum field theory appear also in the star product formalism, which is an expression of the algebra morphism of the operator and the star product formalism. But moreover the quantum group structure that was recently found in perturbative quantum field theory is shown to be a natural algebraic structure of the star product. The essential advantage of deformation quantization is that the classical limit has a well defined meaning. In the context of the spin description with star products this leads to the question of the classical limit of spin, or equivalently to the question of the physical status of pseudoclassical mechanics. In chapter four it is shown that this question is solved if one realizes that the fermionic sector together with the fermionic star product describe the underlying geometric structure. The deformation of a Grassmann algebra leads to a Clifford calculus that is equivalent to geometric algebra. The formulation of geometric algebra in the star product formalism is given in chapter four. One sees there that geometric algebra as the most fundamental geometric formalism that unifies all geometric structures that appear in physics can be described in a supersymmetric manner that parallels the bosonic star product structures. Having obtained a formulation of geometry with fermionic star products this formalism is then applied in the fifth chapter to physical problems. As examples for the application of geometric algebra in classical physics the rigid body and the Kepler problem are considered. In both cases the formalism of geometric algebra gives the most elegant formulation of the problem. In the quantum case one can then combine the fermionic star product formalism that describes the underlying geometric structure with the bosonic star product that describes the noncommuative structure of quantum mechanics. The result is a noncommutative version of geometric algebra that leads to a natural appearance of spin terms. The same idea applied on the phase space leads to the split in supersymmetric partner systems. Geometric algebra gives in this way a natural geometric foundation of supersymmetric quantum mechanics. Similarly one can interpret the hidden BRST-structure of classical mechanics that was found by E. Gozzi and M. Reuter in the path integral formalism from a star product point of view.