Volume 43, Issue 5, May 2002
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Controllability of quantum mechanical systems by root space decomposition of su(N)
View Description Hide DescriptionThe controllability property of the unitary propagator of an level quantum mechanical system subject to a single control field is described using the structuretheory of semisimple Lie algebras. Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases.

Generalized coherent and squeezed states based on the algebra
View Description Hide DescriptionStates which minimize the Schrödinger–Robertson uncertainty relation are constructed as eigenstates of an operator which is an element of the algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute general Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes–Cummings Hamiltonian.

Reversing quantum dynamics with nearoptimal quantum and classical fidelity
View Description Hide DescriptionWe consider the problem of reversing quantum dynamics, with the goal of preserving an initial state’s quantum entanglement or classical correlation with a reference system. We exhibit an approximate reversal operation, adapted to the initial density operator and the “noise” dynamics to be reversed. We show that its error in preserving either quantum or classical information is no more than twice that of the optimal reversal operation. Applications to quantum algorithms and information transmission are discussed.

Optimal control in laserinduced population transfer for two and threelevel quantum systems
View Description Hide DescriptionWe apply the techniques of control theory and of subRiemannian geometry to laserinduced population transfer in two and threelevel quantum systems. The aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences. SubRiemannian geometry and singularRiemannian geometry provide a natural framework for this minimization, where the optimal control is expressed in terms of geodesics. We first show that in twolevel systems the wellknown technique of “πpulse transfer” in the rotating wave approximation emerges naturally from this minimization. In threelevel systems driven by two resonant fields, we also find the counterpart of the “πpulse transfer.” This geometrical picture also allows one to analyze the population transfer by adiabatic passage.

Two families of superintegrable and isospectral potentials in two dimensions
View Description Hide DescriptionAs an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky–Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators, and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Generalized boundary conditions for the Aharonov–Bohm effect combined with a homogeneous magnetic field
View Description Hide DescriptionThe most general admissible boundary conditions are derived for an idealized Aharonov–Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on selfadjoint extensions yields a fourparameter family of boundary conditions; the other two parameters of the model are the Aharonov–Bohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the Aharonov–Bohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.

Phaseamplitude method for numerically exact solution of the differential equations of the twocenter Coulomb problem
View Description Hide DescriptionA numerically exact phaseamplitude method that has been presented earlier by P. O. Fröman, Larsson, and Hökback [J. Math. Phys. 40, 1764–1779 (1999)] is particularized and modified in order to make it adapted to the solution of the differential equations of the twocenter Coulomb problem.

On gauge transformations of Bäcklund type and higher order nonlinear Schrödinger equations
View Description Hide DescriptionWe introduce a new, more general type of nonlinear gauge transformation in nonrelativistic quantum mechanics that involves derivatives of the wave function and belongs to the class of Bäcklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schrödinger equation results in higher order partial differential equations. As an example, we derive a general family of sixthorder nonlinear Schrödinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current σ=ρ∇Δ ln ρ, where We derive gauge invariant quantities, and characterize the subclass of the sixthorder equations that is gauge equivalent to the free Schrödinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.

Quantum information geometry and standard purification
View Description Hide DescriptionWe investigate relations between Uhlmann’s parallelism, monotone Riemannian metrics and dual affine connections on the space of density matrices.

Exact evolution equations for SU(2) quasidistribution functions
View Description Hide DescriptionWe derive an exact (differential) evolution equation for a class of SU(2) quasiprobability distribution functions. Linear and quadratic cases are considered as well as the quasiclassical limit of the large dimension of representation,

DetDet correlations for quantum maps: Dual pair and saddlepoint analyses
View Description Hide DescriptionAn attempt is made to clarify the ballistic nonlinear sigma model formalism recently proposed for quantum chaotic systems, by looking at the spectral determinant for quantized maps and studying the correlator By identifying as one member of a dual pair acting in the spinor representation of the expansion of in powers of is shown to be a decomposition into irreducible characters of In close analogy with the ballistic nonlinear sigma model, a coherentstate integral representation of is developed. For generic this integral has saddle points and the leadingorder saddlepoint approximation turns out to reproduce exactly, up to a constant factor. This miracle is explained by interpreting as a character of and arguing that the leadingorder saddlepoint result corresponds to the Weyl character formula. Unfortunately, the Weyl decomposition behaves nonsmoothly in the semiclassical limit and to make further progress some additional averaging needs to be introduced. Several schemes are investigated, including averaging over basis states and an “isotropic” average. The saddlepoint approximation applied in conjunction with these schemes is demonstrated to give incorrect results in general, one notable exception being a semiclassical averaging scheme, for which all loop corrections vanish identically. As a side product of the dual pair decomposition with isotropic averaging, the crossover between the Poisson and CUE limits is obtained.

Coherent states and annihilation–creation operators associated with the irreducible unitary representations of su(1,1)
View Description Hide DescriptionWe construct a kind of annihilation–creation operators related to the affine coherent states. Next, we reinterpret them as annihilation–creation operators associated with the irreducible unitary representation of the algebrasu(1,1), by adding another generator to the two generators of the unitary representation of the onedimensional affine group.

The Bargmann transform and canonical transformations
View Description Hide DescriptionThis paper concerns a relationship between the kernel of the Bargmann transform and the corresponding canonical transformation. We study this fact for a Bargmann transform introduced by Thomas and Wassell [J. Math. Phys. 36, 5480–5505 (1995)]—when the configuration space is the twosphere and for a Bargmann transform that we introduce for the threesphere It is shown that the kernel of the Bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation (a classical analog of the Bargmann transform). We show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in or We also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics. Some of these quantizations have been studied by Bargmann and Todorov [J. Math. Phys. 18, 1141–1148 (1977)] and the other quantizations are related to the work of Guillemin [Integ. Eq. Operator Theory7, 145–205 (1984)]. Since suitable infinite linear combinations of powers of the generating functions are coherent states for or we show finally that the studied Bargmann transforms are actually coherent states transforms.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


New solutions of relativistic wave equations in magnetic fields and longitudinal fields
View Description Hide DescriptionWe demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which effectively reduces the number of variables in the initial equations. Then we use the corresponding representations to construct new sets of exact solutions, which may have a physical interest. Namely, we present new sets of stationary and nonstationary solutions in magnetic field and in some superpositions of electric and magnetic fields.

Finite dimensional Hamiltonian formalism for gauge and quantum field theories
View Description Hide DescriptionWe discuss in this article the canonical structure of classical field theory in finite dimensions within the pataplectic Hamiltonian formulation, where we put forward the role of Legendre correspondence. We define the generalized Poissonpbrackets which are the analogs of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of pbrackets. As illustration of our formalism we present three examples: the interactingscalar fields, conformal string theory and the electromagnetic field.

Quantizing Yang–Mills theory on a twopoint space
View Description Hide DescriptionWe perform the Batalin–Vilkovisky quantization of Yang–Mills theory on a twopoint space, discussing the formulation of Connes–Lott as well as Connes’ real spectral triple approach. Despite the model’s apparent simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the Gribov problem.

Selfdual Chern–Simons vortices on Riemann surfaces
View Description Hide DescriptionWe study selfdual multivortex solutions of Chern–Simons Higgs theory in a background curved space–time. The existence and decaying property of a solution are demonstrated.

Local existence proofs for the boundary value problem for static spherically symmetric Einstein–Yang–Mills fields with compact gauge groups
View Description Hide DescriptionWe prove local existence and uniqueness of static spherically symmetric solutions of the Einstein–Yang–Mills (EYM) equations for an arbitrary compact semisimple gauge group in the socalled regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang–Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity.

Lax–Phillips scattering theory of a relativistic quantum field theoretical Lee–Friedrichs model and Lee–Oehme–Yang–Wu phenomenology
View Description Hide DescriptionThe scattering theory of Lax and Phillips, originally developed for classical wave equations, has recently been extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a welldefined function in the Lax–Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum fieldtheoretical form of the two (or more) channel relativistic quantum fieldtheoretical form of the Lee model. We show that this theory provides a rigorous underlying basis for the Lee–Oehme–Yang–Wu construction.

On higherdimensional dynamics
View Description Hide DescriptionTechnical results are presented on motion in manifolds to clarify the physics of branetheory, Kaluza–Klein theory, inducedmatter theory, and string theory. The socalled canonical or warp metric in five dimensions (5D) effectively converts the manifold from a coordinate space to a momentum space, resulting in a new force (per unit mass) parallel to the fourdimensional (4D) velocity. The form of this extra force is actually independent of the form of the metric, but for an unbound particle is tiny because it is set by the energy density of the vacuum or cosmological constant. It can be related to a small change in the rest mass of a particle, and can be evaluated in two convenient gauges relevant to gravitational and quantum systems. In the quantum gauge, the extra force leads to Heisenberg’s relation between increments in the position and momenta. If the 4D action is quantized then so is the higherdimensional part, implying that particle mass is quantized, though only at a level of or less, which is unobservably small. It is noted that massive particles which move on timeline paths in 4D can move on null paths in 5D. This agrees with the view from inflationary quantum field theory, that particles acquire mass dynamically in 4D but are intrinsically massless. A general prescription for dynamics is outlined, wherein particles move on null paths in an manifold which may be flat, but have masses set by an embedded 4D manifold which is curved.
