Quantum Physical Systems Research Articles

DRINFEL'D TWIST AND q-DEFORMING MAPS FOR LIE GROUP COVARIANT HEISENBERG ALGEBRAE

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are characterized by their being covariant under the action of the quantum group Uhg, q:=eh. We present a systematic procedure for determining all possible corresponding changes of generators, together with the corresponding realizations of the Uhg-action. The intriguing relation between g-invariants and Uhg-invariants suggests that these changes of generators might be employed to simplify the dynamics of some g-covariant quantum physical systems.

Entropy and information of open systems

Of the two definitions of 'information' given by Shannon and employed in the communication theory, one is identical to that of Boltzmann's entropy and gives in fact a measure of statistical uncertainty. The other involves the difference of unconditional and conditional entropies and, if properly specified, allows the introduction of a measure of information for an open system depending on the values of the system's control parameters. Two classes of systems are identified. For those in the first class, an equilibrium state is possible and the law of conversation of information and entropy holds. When at equilibrium, such systems have zero information and maximum entropy. In self-organization processes, information increases away from the equilibrium state. For the systems of the other class, the equilibrium state is impossible. For these, the so-called 'chaoticity norm' is introduced and also two kinds of self-organization processes are considered and the concept of information is appropriately defined. Common information definitions are applied to classical and quantum physical systems as well as to medical and biological systems.

Hilbert Space or Gel'fand Triplet. Time-Symmetric or Time-Asymmetric Quantum Mechanics

Intrinsic microphysical irreversibility is thetime asymmetry observed in exponentially decayingstates. It is described by the semigroup generated bythe Hamiltonian H of the quantum physical system, not by the semigroup generated by a Liouvillian Lwhich describes the irreversibility due to the influenceof an external reservoir or measurement apparatus. Thesemigroup time evolution generated by H is impossible in the Hilbert space (HS) theory, which allowsonly time-symmetric boundary conditions and a unitarygroup time evolution. This leads to problems with decayprobabilities in the HS theory. To overcome these and other problems (nonexistence of Dirac kets)caused by the Lebesgue integrals of the HS, one extendsthe HS to a Gel'fand triplet, which contains not onlyDirac kets, but also generalized eigenvectors of the self-adjoint H with complex eigenvalues(ER – iΓ/2) and a Breit-Wignerenergy distribution. These Gamow statesψG have a time-asymmetric exponentialevolution. One can derive the decay probability of the Gamow state into the decay productsdescribed by Λ from the basic formula of quantummechanics ℘(t) = Tr(|ψG ›‹ψG|Λ), which in HS quantum mechanicsis identically zero. From this result one derives the decay rate ℘(t) and all the standard relations between℘(0), Γ, and the lifetimeτR used in the phenomenology of resonancescattering and decay. In the Born approximation oneobtains Dirac's Golden Rule.

Quantizing constrained systems

We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order hbar^2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order hbar^2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics, Feynman path integrals, and semiclassically in terms of adiabatic actions.

Non-Abelian Berry phase in a quantum mechanical environment

We investigate a quantum physical system that can be naturally separated into fast and slow moving components. A modification of the conventional molecular Born-Oppenheimer approximation is considered by taking the intermolecular position vector to be a slowlyvarying quantum mechanical parameter. It is found that the fast motion (electronic degrees of freedom) induces a non-Abelian vector potential (Berry connection) into the dynamics of the slow system (nucleus), thereby modifying the commutation relations of the slow variables.

Jauch-Piron logics with finiteness conditions

We show that there are no non-Boolean block-finite orthomodular posets possessing a unital set of Jauch-Piron states. Thus, an orthomodular poset representing a quantum physical system must have infinitely many blocks.

Coherent states: Theory and some applications

In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.

Quantum Logics and Relative Lindenbaum Property

AbstractThe relative Lindenbaum property is considered in orthomodular quantum logic and in partial class ical logic. The properties of these models are connected with the possibility of hidden‐variable reconstruction of particular (1/2‐spin particle e.g.) quantum physical systems.

A modification of the axiom system of quantum mechanics

A general axiom system, including both classical and quantum mechanics as special cases, is proposed. On the basis of the axioms assumed it is shown that the logic of experimentally verifiable propositions concerning any (classical or quantum) physical system may be embedded into an atomistic complete lattice. Moreover, in the quantum case (characterized in the paper by validity of the superposition principle) a generalization of the Piron's representation theorem for the logic is stated.