Class Of Quantum Systems Research Articles

GEOMETRY IN QUANTUM NON-INTEGRABILITY

We present a framework within which the way classical behavior emerges from arbitrary quantum dynamics can be studied systematically. This behavior is characterized by a quenching (topological) index Ξ, which controls the quenching of quantum correlations and signals the existence or the lack of classical limit. The compactness of geometry distinguishes two classes of quantum systems. The kick examples given in this letter suggest that quantum correlation suppresses chaos for compact geometry, while the opposite is found in the non-compact case.

Constrained dynamics and exterior differential systems

The Dirac analysis of constrained Hamiltonian mechanics is one of the conventional precursors to the quantization of classical systems. In this paper the analysis is reformulated in the language of exterior differential systems, starting from the Lagrangian, moving through the generation of primary and secondary constraints and leading to the construction of symmetry generators for gauge symmetries. This reformulation extends the procedure to non-coordinate systems. A computer algebra implementation of the procedure in REDUCE is also described.

New class of frustrated quantum spin systems with an exactly known ground state.

We present a new class of quantum systems whose ground state can be found exactly. This construction involves an assembly of finite frustrated clusters connected by fixed spins. By contrast to solvable dimerized models, our systems exhibit long-range magnetic order, providing a striking example of ordering by quantum fluctuations in the presence of frustration.

Integral and fractional quantization of a class of quantum systems.

Integral and fractional quantization of a class of quantum systems.

The quantization of classical systems

§1 lists axioms desirable in any quantization rule for the functions of the q's. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, we obtain in §7−§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule Λ assigning a unique self-adjoint operator to each such function f; a unitary operator W L corresponding to L and an equation Λ(f∘L)=W -1 LΛƒW L . Contents 1. 1. Prequantization schemes 2. 2. Momentum observables 3. 3. The set L( M) 4. 4. Quantization schemes 5. 5. An example 6. 6. Systems of imprimitivity 7. 7. Quantization schemes on vector spaces 8. 8. Standard quantization schemes 9. 9. Admissible quantization schemes Appendix References

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory

Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so-called eigenoperators γ(t) belonging to an A-eigen-integral decomposition of Φ with respect to a self-adjoint operator A on G. We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.

On the quantization of classical systems

It is proved that no quantization rule can be invariant under all canonical transformations of the classical variablesp andq. The maximum invariance of a quantization rule is determined.