Volume 43, Issue 6, June 2002
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Phase space Feynman path integrals
View Description Hide DescriptionA rigorous mathematical formulation of the “phase space Feynman path integral” is given in a general setting. This is then applied to yield a representation of solutions of the Schrödinger equation with potential depending both on the position and momentum variables.

Multidimensional Schrödinger equations with Abelian potentials
View Description Hide DescriptionWe consider two and threedimensional complex Schrödinger equations with Abelian potentials and a fixed energy level. The potential, wave function, and the spectral Bloch variety are calculated in terms of the Kleinian hyperelliptic functions associated with a genus two hyperelliptic curve. In the special case in two dimensions when the curve covers two elliptic curves, exactly solvable Schrödinger equations are constructed in terms of the elliptic functions of these curves. The solutions obtained are illustrated by a number of plots.

Geometric quantization of mechanical systems with timedependent parameters
View Description Hide DescriptionQuantum systems with adiabatic classical parameters are widely studied, e.g., in the modern holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with timedependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena.

Quantum threebody system in dimensions
View Description Hide DescriptionThe independent eigenstates of the total orbital angular momentum operators for a threebody system in an arbitrary dimensional space are presented by the method of group theory. The Schrödinger equation is reduced to the generalized radial equations satisfied by the generalized radial functions with a given total orbital angular momentum denoted by a Young diagram [μ, ν, 0,…, 0] for the group. Only three internal variables are involved in the functions and equations. The number of both the functions and the equations for the given angular momentum is finite and equal to

The reduction of a quantum system of three identical particles on a plane
View Description Hide DescriptionQuantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a threedimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.

Reduction of quantum systems with symmetry, continuous and discrete
View Description Hide DescriptionReduction of dynamical systems is closely related with symmetry. The purpose of this article is to show that Fourier analysis both on compact Lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing. The reduction procedure is applied to systems of many identical particles lying in which admit the action of a rotation group SO(3) and of a symmetric or permutation group.

Internal Lifshitz tails for Schrödinger operators with random potentials
View Description Hide DescriptionIn this short note, we prove that, for periodic Schrödinger operators perturbed by an Andersontype random potential with longrange single site potential, the density of states always show a Lifshitz tail at a band edge. Thus, we correct a mistake made in an earlier paper [F. Klopp, Duke Math. J. 98, 335–396 (1999)].

Noncommutative tori and universal sets of nonbinary quantum gates
View Description Hide DescriptionWe address the problem of universality in simulation of evolution of quantum system and in theory of quantum computations related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific unitary transformations (quantum gates) from given set. In an earlier paper application of Clifford algebras to constructions of universal sets of binary quantum gates was shown. For application of a similar approach to nonbinary quantum gates in present work we used rational noncommutative torus A set of universal nonbinary twogates is presented here as one example.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


A note on the improvement ambiguity of the stress tensor and the critical limits of correlation functions
View Description Hide DescriptionI study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies and Compatible results follow from the analysis of the RG equations. I perform a number of selfconsistency checks and discuss the issues in a large set of theories.

Hierarchy of Dirac, Pauli, and Klein–Gordon conserved operators in TaubNUT background
View Description Hide DescriptionThe algebra of conserved observables of the SO(4,1) gaugeinvariant theory of the Dirac fermions in the external field of the Kaluza–Klein monopole is investigated. It is shown that the Dirac conserved operators have physical parts associated with Pauli operators that are also conserved in the sense of the Klein–Gordon theory. In this way one gets simpler methods of analyzing the properties of the conserved Dirac operators and their main algebraic structures including the representations of dynamical algebras governing the Dirac quantum modes.

SuperGrassmannian and large limit of quantum field theory with bosons and fermions
View Description Hide DescriptionWe study a large limit of a twodimensional Yang–Mills theory coupled to bosons and fermions in the fundamental representation. Extending an approach due to Rajeev we show that the limiting theory can be described as a classical Hamiltonian system whose phase space is an infinitedimensional superGrassmannian. The linear approximation to the equations of motion and the constraint yields the ’t Hooft equations for the mesonic spectrum. Two other approximation schemes to the exact equations are discussed.

Structure conserving parametrization of Feynman diagrams
View Description Hide DescriptionUsing xspace parameters instead of Feynman parameters, dimensionally regularized Feynman diagrams are expanded with respect to the external momenta, the internal masses, the logarithms of these masses and the regularization parameter A general formula for an arbitrary Feynman diagram is obtained for any dimension All ultraviolet divergences appear in a direct and transparent way as poles. Their residues in the limit are subseries that are recognized as subdiagrams. Relations between diagrams are transparent in all orders of perturbation theory.

Classical history theory of vector fields
View Description Hide DescriptionWe consider the extension of classical history theory to the massive vector field and electromagnetism. It is argued that the action of the two Poincaré groups introduced by Savvidou suggests that the history fields should have five components. The extra degrees of freedom introduced to make the fields five dimensional result in an extra pair of second class constraints in the case of the massive vector field, and in an extended gauge group in the case of electromagnetism. The total gauge transformations depend on two arbitrary parameters, and contain “internal” and “external” U(1) gauge transformations as subgroups.

Poincaré invariance for continuoustime histories
View Description Hide DescriptionWe show that the relativistic analog of the two types of time translation in a nonrelativistic history theory is the existence of two distinct Poincaré groups. The “internal” Poincaré group is analogous to the one that arises in the standard canonical quantization scheme; the “external” Poincaré group is similar to the group that arises in a Lagrangian description of the standard theory. In particular, it performs explicit changes of the space–time foliation that is implicitly assumed in standard canonical field theory.

Large limit of gauge theory of fermions and Bosons
View Description Hide DescriptionIn this paper we study the large limit of gauge theory coupled to a Majorana field and a real scalar field in 1+1 dimensions extending ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We show that the phase space of the resulting classical theory of bilinears, which are the mesonic operators of this theory, is where HH refers to the underlying complex graded space of combined oneparticle states of fermions and bosons and corresponds to the positive frequency subspace. In the begining to simplify our presentation we discuss in detail the case with Majorana fermions only [the purely bosonic case is treated in Toprak and Turgut, J. Math. Phys. 43, 1340 (2002)]. In the Majorana fermion case the phase space is given by where H refers to the complex oneparticle states and to its positive frequency subspace. The meson spectrum in the linear approximation again obeys a variant of the ’t Hooft equation. The linear approximation to the boson/fermion coupled case brings an additonal bound stateequation for mesons, which consists of one fermion and one boson, again of the same form as the wellknown ’t Hooft equation.

 GENERAL RELATIVITY AND GRAVITATION


The embedding of space–times in five dimensions with nondegenerate Ricci tensor
View Description Hide DescriptionWe discuss and prove a theorem which asserts that any ndimensional semiRiemannian manifold can be locally embedded in an dimensional space with a nondegenerate Ricci tensor which is equal, up to a local analytic diffeomorphism, to the Ricci tensor of an arbitrary specified space. This may be regarded as a further extension of the Campbell–Magaard theorem. We highlight the significance of embedding theorems of increasing degrees of generality in the context of higher dimensional space–times theories and illustrate the new theorem by establishing the embedding of a general class of Ricciflat space–times.

 DYNAMICAL SYSTEMS


New Liouville integrable noncanonical Hamiltonian systems from the AKNS spectral problem
View Description Hide DescriptionLiouville integrable noncanonical Hamiltonian systems with variable coefficient symplectic forms are generated from the AKNS matrix Lax pairs by binary symmetry constraints of the AKNS hierarchy. These integrable systems provide new integrable factorization for every AKNS system in the hierarchy.

Bruhat Poisson structure on and integrable systems
View Description Hide DescriptionIn this note we present a simple formula for the Bruhat Poisson structure on complex projective spaces in terms of the momentum coordinates. We also give a simple description of a family of functions in involution on compact Hermitian symmetric spaces obtained via the biHamiltonian approach using the Bruhat Poisson structure and an invariant symplectic structure. We compute these functions explicitly on and relate them to the Gelfand–Tsetlin coordinates. We also show how the Lenard scheme can be applied.

The coupled modified Korteweg–de Vries equations: Similarity reduction, Lie–Bäcklund symmetries and integrability
View Description Hide DescriptionThe Lie point symmetries of coupled modified Korteweg–de Vries equations is derived and shown that the similarity reduction associated with symmetries passes the Painlevé property indicating its integrability. The nonclassical symmetry analysis of Bluman and Cole and the direct method of Clarkson and Kruskal to equations show that there exists no new similarity reductions. A sequence of Lie–Bäcklund symmetries for equations is derived explicitly, establishing its complete integrability. The question of constructing the recursion operator of a characteristic of integrable systems, is also discussed.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Generalized symmetries in mechanics and field theories
View Description Hide DescriptionGeneralized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are characterized in a Lagrangian context by means of the transformation rules of the Poincaré–Cartan form and the (generalized) Nöther theorem is applied to obtain conserved quantities (first integrals in mechanics). In the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of Hamilton–Jacobi equations recovering standard results by means of this new method. Supersymmetries (Wess–Zumino model) are considered as an intriguing example in field theory.
