Some remarks concerning Nambu mechanics

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. We prove the free flux-across-surfaces theorem, which was conjectured by Combes, Newton and Shtokhamer (Phys. Rev. D. 11 (1975), 366), and which relates the integrated quantum flux to the usual quantum mechanical formula for the cross-section. The integrated quantum flux is equal to the probability of outward crossings of surfaces by Bohmian trajectories in the scattering regime.     

Asymptotic triviality of the Møller operators in Galilei invariant quantum field theories

Asymptotic triviality of the Møller operators is shown for fermions interacting via nonlocal pair-potentials with a momentum cut-off. Such systems have stable Hamiltonians, defining the time evolution in the Heisenberg representation, and strong mixing properties.     

C *-algebraic quantization and the origin of topological quantum effects

Concepts from the theory of abstract operator algebras are used to solve the problem of quantizing a particle moving on an arbitrary locally compact homogeneous space. Inequivalent quantizations are identified with inequivalent irreducible representations of the corresponding C ^*-algebra. Topological terms in the action (or Hamiltonian) are found to be representation-dependent, and are automatically induced by the quantization procedure. Known charge quantization conditions turn out to be identically satisfied. Several examples are considered, among them the Dirac monopole and the Aharonov-Bohm effect.     

Causal Construction of the Massless Vertex Diagram

The massless one-loop vertex diagram is constructed by exploiting the causal structure of the diagram in configuration space, which can be translated directly into dispersive relations in momentum space.     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941     

Stochastic quantisation of a gauge field in the spatial axial gauge

The stochastic quantisation method of Nelson can be applied to quantise an Abelian gauge field in the spatial axial gauge.     

Propagator of a particle characterized by an internal symmetry

We examine the propagators associated with particles characterized by an internal symmetry. Of particular interest is the case of spontaneously broken Kac-Moody algebra.     

q-transformation theory

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU _q (2) algebra. To obtain transition probabilities, one must take the Woronowicz square.     

Transformation theory ofq-quantum mechanics

Although there is no empirical motivation for replacing the commutators of dynamically conjugate operators in quantum mechanics byq-commutators, it appears possible to construct a consistent mathematical formulism based on this idea. To examine such a possibility further, we have studied the relation of this proposal to the Schwinger action principle, since the entire quantum mechanical formulism may be inferred from this principle. In particular, we have discussed the quantum transformation theory within this framework.To Julian Schwinger, 1918–1994, one of the creators of quantum field theory, and a giant of twentieth-century physics     

Trace identities from tau functions

An approach to trace identities is described based on the-function formalism for mKdV and affine Toda systems. The set-up is given for discrete and continuous spectrum solutions by exploiting the relation between Jost solutions, diagonal entries of the monodromy matrix, and-functions. Examples for solutions with discrete scattering data are worked out.     

Fock spaces with generalized statistics

Fock representations for quantum fields which obey generalized statistics are explicitly constructed. The main features of these representations are investigated.     

On the holomorphic factorization for superconformal fields

For a generic value of the central charge, we prove the holomorphic factorization of partition functions for free superconformal fields which are defined on a compact Riemann surface without boundary. The partition functions are viewed as functionals of the Beltrami coefficients and their fermionic partners, which variables parametrize superconformal classes of metrics.     

A new limit for the noncommutative space–time parameter

The spectrum of the hydrogen atom in the framework of noncommutative quantum mechanics is studied and the related phenomenology is presented. We find that the noncommutative effects are similar to those obtained by considering the extended charged nature of the proton in the atom. To the first order in the noncommutative parameter, it is equivalent to an electron in the fields of a Coulomb potential and an electric dipole and this allows us to get a bound for this parameter. In a second step, we compute noncommutative corrections of the energy levels and find that they are at the second order in the parameter of noncommutativity. By comparing our results to those obtained from experimental spectroscopy, we get another limit for this parameter.     

Fermion number creation via electromagnetism

Let an external current, whose support is confined to the space-like slab |x ⁰| < T in two-dimensional spacetime, build up a localized charge density which vanishes for times |x ⁰| > T. We show that the zero mass Dirac quantum field reacts to this current by a c-number shift of the fermion number, i.e. Q _out=Q _in+Q, with , where q(x ⁰) denotes the total external charge. For the shift of the axial charge we obtain an extension of existing results.     

On the schwarz problem for the [^(\textsu)]\text2 \widehat{{\text{su}}}_{\text{2}} Knizhnik-Zamolodchikov equation

We study the monodromy representations ^k,Iof the mapping class group ₄ acting on 4-point blocks satisfying the Knizhnik-Zamolodchikov equation for the levelk su₂ current algebra. We classify all irreducible ^k,Iwhich are realized by finite groups; we also display finite irreducible components for the reducible representations corresponding tok = 10.Supported by the Federal Ministry of Science and Research, Austria.     

Dispersive Calculation of the Massless Multi-loop Sunrise Diagram

The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and explained. The result displays the high energy behaviour of the massive sunrise diagrams, whose calculation is involved already for the two-loop case.     

Scattering theory for a model of an atom in a quantized field

We compute the scattering operator for a simplified model of an atom interacting a with a quantized field. The field is restricted to the vacuum and one-particle sectors, and the atom has only two states. We also solve the inverse scattering problem for the same model. The methods used rely on the particular form of the interaction, which is chosen to mimic the interaction between radiation and matter.This work was partially supported by NSF Grant DMS-8922941.     

Topological representations of U q (sl2(ℂ)) on the torus and the mapping class group

We compute the mapping class group action on cycles on the configuration space of the torus with one puncture, with coefficients in a local system arising in conformal field theory. This action commutes with the topological action of the quantum group U _q (sl₂()), and is given in vertex form.