Volume 36, Issue 12, December 1995
Index of content:

New spin‐one gauge theory in three dimensions
View Description Hide DescriptionThis paper presents a new type of nonlinear gauge theory for spin‐one fields in three dimensions. The theory is constructed by a systematic process of generalization applied to the linear gauge theory of spin‐one fields. The process takes advantage of vector cross product and curl operations available only in three dimensions, leading to a highly novel form for the generalization of the field equations and gauge symmetry. Features of the new theory are discussed, and some possible generalizations are outlined.

Vortex solutions in a class of 2‐d symmetry breaking Skyrme‐like models
View Description Hide DescriptionWe find vortex solutions in 2+1 dimensions belonging to a class of Skyrme‐like models, defined in terms of a global O(2) invariant unconstrained field and featuring a symmetry breaking potential. In the static limit all these models support finite‐energy topologically stable vortices characterized by a nonvanishing value of the field at infinity, but they do not support nontopological vortices characterized by vanishing value of the field at infinity. We have introduced a modified version of one of our models, which in a stationary field configuration supports a nontopological solution.

Jaynes–Cummings model governed by the Milburn equation without diffusion approximation
View Description Hide DescriptionIn this paper, we study the Jaynes–Cummings model governed by the Milburn equation for quantum mechanics. We find the exact solution of the Milburn equation without diffusion approximation and apply this solution to studying nonclassical properties of Jaynes–Cummings model (JCM) in the full range of decoherence parameter γ. The influence of the decoherence on nonclassical properties of JCM for different ranges of γ is investigated in detail.

Semiclassical sum rules and generalized coherent states
View Description Hide DescriptionVarious semiclassical sum rules are obtained for matrix elements of smooth observables using semiclassical estimates of time evolved coherent states together with stationary phase theorems.

Berry phase and supersymmetric topological index
View Description Hide DescriptionThe sequences of supersymmetry for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian are revised. The condition (so‐called supersymmetric adiabatic evolution condition) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek–Zee (degenerated case) phases of superpartners are taking place is pointed out. The analog of the Witten index (supersymmetric Berry index) is determined. The final expression for the new index has compact form of ind_{ BH }=sDet U≡Det U ^{τ}, where U is the cyclic evolution operator generated by supersymmetric Hamiltonian H and τ is a supersymmetric involution. As the examples of the suggested concept of the supersymmetric adiabatic evolution the holomorphic quantum mechanics on a complex plane and meromorphic quantum mechanics on Riemann surfaces are considered. The supersymmetric Berry indices for the models are calculated.

Statistical interpretation of p‐adic quantum theories with p‐adic valued wave functions
View Description Hide DescriptionThe development of p‐adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events are p‐adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also in p‐adic topologies.

Modified symmetry generators related to solvable scattering problems
View Description Hide DescriptionBy employing the noncompact groups SO(n,1) we show how matrix valued differential operators for the realization of the so(n,1) algebra can be used to obtain multichannel scattering via the occurrence of LS‐type interaction terms. These realizations are in terms of coordinates on the hyperboloids H ^{ n } regarded as cosets SO(n,1)/SO(n). The matrix‐valued nature of such realizations is connected with a finite dimensional unitary irreducible representation of the compact subgroup SO(n). The associated scattering problems are solvable one dimensional ones. The interaction terms are of LS‐type multiplied by Pöschl–Teller potentials, with S playing the role of the SO(n) spin. We also show that scattering Hamiltonians based on such realizations can be related to some effective Hamiltonian coming from a coupled system of slow and fast degrees of freedom after decoupling adiabatically the fast variables. The SO(n) symmetry in this picture can be identified as the residual symmetry group of the fast subsystem surviving the adiabatic decoupling. The SO(n) spin, manifesting itself through the appearance of the aforementioned irreducible representation, also originates from the fast dynamics.

Removal of the resolvent‐like energy dependence from interactions and invariant subspaces of a total Hamiltonian
View Description Hide DescriptionThe spectral problem (A+V(z))ψ=zψ is considered where the main Hamiltonian A is a self‐adjoint operator of sufficiently arbitrary nature. The perturbation V(z)=−B(A′−z)^{−1} B* depends on the energy z as resolvent of another self‐adjoint operator A′. The latter is usually interpreted as a Hamiltonian describing an internal structure of physical system. The operator B is assumed to have a finite Hilbert–Schmidt norm. The conditions are formulated when one can replace the perturbation V(z) with an energy‐independent ‘‘potential’’ W such that the Hamiltonian H=A+W has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The Hamiltonian H is constructed as a solution of the non‐linear operator equationH=A+V(H). It is established that this equation is closely connected with the problem of searching for invariant subspaces of the Hamiltonian H=[_{ B*} ^{ A } _{ A′} ^{ B }]. The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian H=A+W. Scattering theory is developed for this Hamiltonian in the case where the operator A has continuous spectrum.

Comments on Good’s proposal for new rules of quantization
View Description Hide DescriptionIn a recent paper [J. Math. Phys. 35, 3333 (1994)] Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyze the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the ‘‘mass spectrum’’ for the Klein–Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative ‘‘masses.’’ Finally, two major reasons are presented against the viability of this alternative proposal: (a) It does not lead to the correct energyspectrum for the hydrogen atom. (b) For field models, the standard quantum theory cannot be recovered from this alternative description.

Direct sum decompositions and indecomposable TQFTs
View Description Hide DescriptionThe decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFTs in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one‐dimensional, and indecomposable two‐dimensional theories are classified.

Quantum canonical transformations revisited
View Description Hide DescriptionA preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory.

The inverse boundary problem for the Rayleigh system
View Description Hide DescriptionThe inverse matrix Sturm–Liouville problem associated to the classical Rayleigh system is studied. This problem differs from previously investigated ones in several ways: the matrix potential is not Hermitian, the spectral parameter occurs also in the boundary condition, and both the potential and boundary condition depend also on a frequency parameter. We characterize all symmetries and identities satisfied by the corresponding scattering data and deduce an integral equation of the Gelfand–Levitan type that solves the inverse Rayleigh boundary problem. On the basis of these results we propose to study the interesting asymptotic properties of the Rayleigh boundary problem for large values of the frequency ω.

‘‘Falling cat’’ connections and the momentum map
View Description Hide DescriptionWe consider a standard symplectic dynamics on TM generated by a natural LagrangianL. The Lagrangian is assumed to be invariant with respect to the action TR _{ g } of a Lie groupG lifted from the free and proper action R _{ g } of G on M. It is shown that under these conditions a connection on principal bundle π:M → M/G can be constructed based on the momentum map corresponding to the action TR _{ g }. A simple explicit formula for the connection form is given. For the special case of the standard action of G=SO(3) on M=R ^{3}×...×R ^{3} corresponding to a rigid rotation of an N‐particle system the formula obtained earlier by Guichardet and Shapere and Wilczek is reproduced.

Lax–Phillips theory for scattering by thin elastic bodies
View Description Hide DescriptionThe scheme of construction of linear self‐adjoint operators for acoustic systems with an interaction of sound and vibration is proposed. Kinetic energy of a system induces the basic Hilbert spaceL _{2}(Ω), where Ω denotes the domain filled with fluid. Potential energy of the system induces the quadratic form on this Hilbert space and, consequently, the positive operator that governs the evolution. Unlike earlier theories, our operators are perturbations of the Laplace operator in L _{2} spaces. The operators may also be obtained as extensions of symmetric operator (−Δ) defined on C ^{∞} _{0}(Ω) into a wider Hilbert space. Functions from their domains have an additional smoothness of their derivatives on the boundary; the smoothness follows from the elasticity of the boundary. The scheme allows one to describe inhomogeneities of vibrating objects concentrated at a point or on a line (such as cracks and stiffeners). It is demonstrated that Lax–Phillips scattering theory may be generalized for scattering by thin elastic objects. The necessary condition for the existence of the incoming and the outgoing subspaces is derived in terms of spectral properties of the operator of the isolated elastic object.

Homogeneity in integral representations for effective properties of composite materials
View Description Hide DescriptionThe effective complex conductivity σ* of an n‐component composite material is considered. Recently, an integral representation that treats the component conductivities σ_{1},...,σ_{ n } symmetrically was developed. This representation has the advantage that the moments of the positive measure in the integral are directly related to the coefficients in a perturbation expansion of σ* around a homogeneous medium. However, the admissible class of measures has been difficult to characterize, due to the condition that σ* is a homogeneous function of the component conductivities. Here, this admissible class is characterized in terms of linear relations among the moments associated with tetrahedra in Fourier space Z ^{ n }. The homogeneity is used to derive a new formula for σ* in the two‐component case, in which σ* for general media is expressed in terms of the conductivities of laminates of second rank.

Lie algebra of anomalously scaled fluctuations
View Description Hide DescriptionFor quantum lattice systems, it is proven that anomalously scaled fluctuations have a natural Lie algebrastructure. The harmonic lattice in the ground state is given as an illustration of the general theorem.

Diffusion through a slab
View Description Hide DescriptionBoltzmann’s equation is solved for the diffusion of nonabsorbing particles through a thick slab. The dimensionless thickness D=dμ_{ s }, where d is the thickness and μ_{ s } is the rate of scattering. Our solution neglects terms of order O(e ^{−D }) and is exact in the limit that D≫1. Simple expressions are obtained for the reflection and transmission coefficients, as well as for the angular distributions on the front and back sides of the slab.

Hamilton’s principle in stochastic mechanics
View Description Hide DescriptionIn this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi‐like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero‐sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle‐point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time‐independent equation.
It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle‐point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle‐point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex‐valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real‐valued diffusions with complex‐valued drift and driving noise processes. From the variational principle we derive a Newton‐type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator.

Hodograph equations in relativistic gas dynamics admitting an infinite number of symmetries
View Description Hide DescriptionA straightforward application of the Lie‐group theory to the linear equation, which describes the relativistic one‐dimensional flow in the hodograph plane, is carried out. It is shown that, for particular equations of state, one has an infinite number of symmetries. In this case, transformations can be found that reduce the hodograph equation to a form directly integrable. The remarkable aspect of the analysis is that one of these equations of state is identified as that characterizing a one‐dimensional fully degenerate Fermi gas. Astrophysicists might find this result useful in practical applications.

Boundary value problems for integrable equations compatible with the symmetry algebra
View Description Hide DescriptionBoundary value problems for integrable nonlinear partial differential equations are considered from the symmetry point of view. Families of boundary conditions compatible with the Harry‐Dym, KdV, and mKdV equations and the Volterra chain are discussed. We also discuss the uniqueness of some of these boundary conditions.