Volume 41, Issue 5, May 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Structure and properties of Hughston’s stochastic extension of the Schrödinger equation
View Description Hide DescriptionHughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston’s equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston’s equation to earlier work on normpreserving stochastic equations, and show that Hughston’s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energybased approach, such as Hughston’s, suffices to give an objective interpretation of the measurement process in quantum mechanics.

Extensions of convexity models
View Description Hide DescriptionThe notion of convexity model is introduced to provide a general frame for statistical theories of physical interest: this frame encompasses, in particular, the classical and the quantum cases. In a convexity model the states of the physical system, and the convex structure they form, play a basic role; observables and related quantities are then naturally defined. The notion of extensions of a convexity model is studied: it appears physically relevant to cope with several needs, paradigmatically with the one of viewing the physical system as a part of a compound system. We focus attention on quantumlike extensions of both the usual classical and quantum convexity models, as well as on classicallike extensions of the quantum model. The behavior of state overlapping and state superposition under model extension is briefly examined.

expansions in nonrelativistic quantum mechanics
View Description Hide DescriptionAn extensive number of numerical computations of energy series using a recursive Taylor series method are presented in this paper. The series are computed to a high order of approximation and their behavior on increasing the order of approximation is examined.

Group theoretical quantization and the example of a phase space
View Description Hide DescriptionThe group theoretical quantization scheme is reconsidered by means of elementary systems. Already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions. A systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed, twodimensional prototypes of which are and (interpreted as restrictions of and to positive coordinate and momentum, respectively). In this framework (and under an additional, natural condition) an action on results as the unique admissible group action. Furthermore, for symplectic manifolds which are (specific) parts of phase spaces with known quantum theory a simple “projection method” of quantization is formulated. For and equivalent results to those of more established (but more involved) quantization schemes are obtained. The approach may be of interest, e.g., in attempts to quantize gravity theories where demanding nondegenerate metrics of a fixed signature imposes similar constraints.

Symmetry requirement for a deformation operator related to density functional theory
View Description Hide DescriptionWe study symmetry properties of an operator that has been introduced in Quantum Chemistry under the name of “Local Scaling Method,” or “Local Scaling Transformation.” This operator is defined using deformations of the space It has previously been used in order to obtain densityfunctional approximations of the electron problem, and new representability results. In order that the operator satisfies a natural symmetry requirement associated with the symmetry group of a molecule, we show that only the deformations that commute with all operations of the symmetry group may be used. These deformations are listed and practical consequences explained.

Information content for quantum states
View Description Hide DescriptionA method for representing probabilistic aspects of quantum systems by means of a density function on the space of pure quantum states is introduced. In particular, a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix. This result is applied to derive the Shannon entropy of a quantum state. The information theoretic quantum entropy thereby obtained is shown to have the desired concavity property, and to differ from the conventional von Neumann entropy. This is illustrated explicitly for a twostate system.

Renormalized Hamiltonian for a peptide chain: Digitalizing the protein folding problem
View Description Hide DescriptionA renormalized Hamiltonian for a flexible peptide chain is derived to generate the longtime limit dynamics compatible with a coarsening of torsional conformation space. The renormalization procedure is tailored taking into account the coarse graining imposed by the backbone torsional constraints due to the local steric hindrance and the local backbonesidegroup interactions. Thus, the torsional degrees of freedom for each residue are resolved modulo basins of attraction in its socalled Ramachandran map. This Ramachandran renormalization (RR) procedure is implemented so that the chain is energetically driven to form contact patterns as their respective collective topological constraints are fulfilled within the coarse description. In this way, the torsional dynamics are digitalized and become codified as an evolving pattern in a binary matrix. Each accepted Monte Carlo step in a canonical ensemble simulation is correlated with the real mean first passage time it takes to reach the destination coarse topological state. This realtime correlation enables us to test the RR dynamics by comparison with experimentally probed kinetic bottlenecks along the dominant folding pathway. Such intermediates are scarcely populated at any given time, but they determine the kinetic funnel leading to the active structure. This landscape region is reached through kinetically controlled steps needed to overcome the conformationalentropy of the random coil. The results are specialized for the bovine pancreatic trypsin inhibitor, corroborating the validity of our method.

Dressing symmetries of holomorphic BF theories
View Description Hide DescriptionWe consider holomorphic BF theories, their solutions and symmetries. The equivalence of Čech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described.

The branch process of cosmic strings
View Description Hide DescriptionIn light of the φmapping method and the topological tensor current theory, the topological structure and the topological quantization of topological defects are obtained under the condition that Jacobian When it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the generation, annihilation, and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of φmapping, respectively.

Coulomboscillator duality in spaces of constant curvature
View Description Hide DescriptionIn this paper we construct generalizations to spheres of the wellknown LeviCivita, Kustaanheimo–Steifel, and Hurwitz regularizing transformations in Euclidean spaces of dimensions two, three, and five. The corresponding classical and quantum mechanical analogs of the Kepler–Coulomb problem on these spheres are discussed.

Combinatorial formulas for Vassiliev invariants from Chern–Simons gauge theory
View Description Hide DescriptionWe analyze the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern–Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant and we describe the reconstruction of the full primitive Vassiliev invariants at orders two, three, and four. At orders two and three we recover known combinatorial expressions for these invariants. At order four we present new combinatorial expressions for the two primitive Vassiliev invariants present at this order.

Padé interpolation: Methodology and application to quarkonium
View Description Hide DescriptionA novel application of the Padé approximation is proposed in which the Padé approximant is used as an interpolation for the small and large coupling behaviors of a physical system, resulting in a prediction of the behavior of the system at intermediate couplings. This method is applied to quarkonium systems, and reasonable values for the and quark masses are obtained.

Dynamical symmetry approach to periodic Hamiltonians
View Description Hide DescriptionWe show that dynamical symmetry methods can be applied to Hamiltonians with periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf potential and its extensions using representations of and Energy bands and gaps are readily understood in terms of representation theory. We compute the transfer matrices and dispersion relations for these systems, and find that the complementary series plays a central role as well as nonunitary representations.

Path integral solution by sum over perturbation series
View Description Hide DescriptionA method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov–Bohm–Coulomb system is obtained by the method. Different from the earlier treatment based on the space–time transformation and infinite multiplevalued trasformation of Kustaanheimo–Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green’s function and an explicit path integral is avoided.

Absence of singular spectrum for Schrödinger operators with anisotropic potentials and magnetic fields
View Description Hide DescriptionWe study magnetic Schrödinger operators of the form in with We get a limiting absorption principle and the absence of singular spectrum under rather mild and especially anisotropic hypothesis. The magnetic field and the potential will be connected by some conditions, but in the variable there will be almost no constraints. If and our results contrast with the known fact that always has bound states if is negative.

Timedependent Schrödinger equations having isomorphic symmetry algebras. I. Classes of interrelated equations
View Description Hide DescriptionIn this paper, we focus on a general class of Schrödinger equations that are time dependent and quadratic in X and P. We transform Schrödinger equations in this class, via a class of timedependent mass equations, to a class of solvable timedependent oscillator equations. This transformation consists of a unitary transformation and a change in the “time” variable. We derive mathematical constraints for the transformation and introduce two examples.

Timedependent Schrödinger equations having isomorphic symmetry algebras. II. Symmetry algebras, coherent and squeezed states
View Description Hide DescriptionUsing the transformations from paper I, we show that the Schrödinger equations for (1) systems described by quadratic Hamiltonians, (2) systems with timevarying mass, and (3) timedependent oscillators all have isomorphic Lie space–time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of numberoperator states are constructed. The algebras and the states are used to compute displacementoperator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is demonstrated.

Three flavor neutrino oscillations in matter
View Description Hide DescriptionWe derive analytic expressions for three flavor neutrino oscillations in the presence of matter in the plane wave approximation using the Cayley–Hamilton formalism. Especially, we calculate the time evolution operator in both flavor and mass bases. Furthermore, we find the transition probabilities, matter mass squared differences, and matter mixing angles all expressed in terms of the vacuum mass squared differences, the vacuum mixing angles, and the matter density. The conditions for resonance in the presence of matter are also studied in some examples.

The origin of chiral anomaly and the noncommutative geometry
View Description Hide DescriptionWe describe scalar and spinor fields on a noncommutative sphere starting from canonical realizations of the enveloping algebra The gauge extension of a free spinor model, the Schwinger model on a noncommutative sphere, is defined and the model is quantized. The noncommutative version of the model contains only a finite number of dynamical modes and is nonperturbatively UV regular. An exact expression for the chiral anomaly is found. In the commutative limit the standard formula is recovered.

The spectrum of a magnetic Schrödinger operator with randomly located delta impurities
View Description Hide DescriptionWe consider a single band approximation to the random Schrödinger operator in an external magnetic field. The spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice. In this paper we generalize these results by letting the delta impurities have random positions as well as strengths; they are located in squares of a lattice with a general bounded distribution. We characterize the entire spectrum of this operator when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level is infinitely degenerate, and that the eigenfunctions corresponding to other Landau band energies are exponentially localized.
