Volume 37, Issue 3, March 1996
Index of content:

Null surfaces, initial values, and evolution operators for spinor fields
View Description Hide DescriptionWe analyze the initial value problem for spinor fields obeying the Dirac equation, with particular attention to the characteristic surfaces. The standard Cauchy initial value problem for first‐order differential equations is to construct a solution function in a neighborhood of space and time from the values of the function on a selected initial value surface. On the characteristic surfaces the solution function may be discontinuous, so the standard Cauchy construction breaks down. For the Dirac equation the characteristic surfaces are null surfaces. An alternative version of the initial value problem may be formulated using null surfaces; the initial value data needed differs from that of the standard Cauchy problem. We study, in particular, the intersecting pair of characteristics t=x and t=−x (and supress the y and z dependence). In this case the values of separate components of the spinor function on the intersecting pair of null surfaces comprise the necessary initial value data. We present an expression for the construction of a solution from the null surface data; two analogs of the quantum mechanical Hamiltonian operator determine the evolution of the system.

The monopole equations in topological Yang–Mills
View Description Hide DescriptionWe twist the monopole equations of Seiberg and Witten and show how these equations are realized in topological Yang–Mills theory. A Floer derivative and a Morse functional are found and are used to construct a unitary transformation between the usual Floer cohomologies and those of the monopole equations. Furthermore, these equations are seen to reside in the vanishing self‐dual curvature condition of an OSp(1‖2)‐bundle. Alternatively, they may be seen arising directly from a vanishing self‐dual curvature condition on an SU(2)‐bundle in which the fermions are realized as spanning the tangent space for a specific background.

Generalization of the Calogero–Cohn bound on the number of bound states
View Description Hide DescriptionIt is shown that for the Calogero–Cohn‐type upper bounds on the number of bound states of a negative spherically symmetric potential V(r), in each angular momentum state, that is, bounds containing only the integral ∫^{∞} _{0}‖V(r)‖^{1/2} dr, the condition V′(r)≥0 is not necessary, and can be replaced by the less stringent condition (d/dr)[r ^{1−2p }(−V)^{1−p }]≤0, 1/2≤p<1, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on p and l, and tend to the standard value for p=1/2.

Group theoretical foundations of fractional supersymmetry
View Description Hide DescriptionFractional supersymmetry denotes a generalization of supersymmetry which may be constructed using a single real generalized Grassmann variable, θ=θ̄,θ^{ n }=0, for arbitrary integer n=2,3,.... An explicit formula is given in the case of general n for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalized derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as θ.

Zeta‐regularization of the O(N) nonlinear sigma model in D dimensions
View Description Hide DescriptionThe O(N) nonlinear sigma model in a D‐dimensional space of the form R ^{ D−M }×T ^{ M }, R ^{ D−M }×S ^{ M }, or T ^{ M }×S ^{ P } is studied, where R ^{ M }, T ^{ M }, and S ^{ M } correspond to flat space, a torus, and a sphere, respectively. Using zeta‐regularization and the 1/N expansion, the corresponding partition functions—for deriving the free energy—and the gap equations are obtained. In particular, the free energy at the critical point on R ^{2q+1}×S ^{2p+2} vanishes in accordance with the conformal equivalence to the flat space R ^{ D }. Numerical solutions of the gap equations at the critical coupling constants are given for several values of D. The properties of the partition function and its asymptotic behavior for large D are discussed. In a similar way, a higher‐derivative nonlinear sigma model is investigated, too. The physical relevance of our results is discussed.

Complete sets of non‐self‐adjoint observables: An unbounded approach
View Description Hide DescriptionThe notion of completeness of a set S of compatible observables represented by maximal symmetric operators is discussed directly in terms of unbounded operators. In contrast with what happens for self‐adjoint observables, the present framework forces us to involve some partial algebraic structures such as the partial GW*‐algebra generated by S. In this way the previous approaches based on von Neumann algebras and on O*‐algebras are generalized.

Topological sectors and measures on moduli space in quantum Yang–Mills on a Riemann surface
View Description Hide DescriptionPrevious path integral treatments of Yang–Mills on a Riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations. This paper extends the path integral formulation to treat separately each topological sector. The formulation is sufficiently explicit to calculate Wilson line expectations exactly. Further, it suggests two new measures on the moduli space of flat connections, one of which proves to agree with the small‐volume limit of the Yang–Mills measure.

On the time decay of a wave packet in a one‐dimensional finite band periodic lattice
View Description Hide DescriptionNonstationary Schrödinger equation with a periodic finite band potential p(x) is considered. The Green’s functionG(x,x′,t) of this equation is investigated when t→∞. Asymptotics for G(x,x′,t) are specified. It is shown that for large ‘‘velocities’’ v=(x−x′)/t the principal term in asymptotics of G(x,x′,t), t→∞ coincides with the Green’s function for p=0. The principal term in the asymptotics of G(x,x′,t) in the case v→∞ is equal to a sum of Green’s functions of unperturbed problems for particles whose masses are equal to effective masses of the Hill operator under investigation.

Variational formulation of a moment problem quantization method
View Description Hide DescriptionThe eigenvalue moment method (EMM) has proven to be an effective technique for generating converging lower and upper bounds to the bosonic ground state energy of singular, strongly coupled, quantum systems. Application of EMM theory requires an appropriate linearization of the highly nonlinear Hankel–Hadamard (HH) moment determinant constraints for the (n+1)×(n+1) Hankel matrices M^{ n }[u]≡M̂^{ n } _{0}+∑_{ i=1} ^{ m s } M̂^{ n } _{ iu } i), dependent on the missing moment variables {u(i)}≡u. We propose an alternate variational formulation utilizing the functions Det(M^{ n+1}[u])/Det(M^{ n }[u]), which we prove to be locally convex over the missing moment subset satisfying the HH positivity conditions Det(M^{ν}[u])≳0, for ν≤n. Additional features of this variational formulation facilitate its application to important problems such as the octic, sextic, and quartic anharmonic oscillators.

Geometric phase for isotopic spin coherent states
View Description Hide DescriptionThe concept of geometric phase for a closed circuit in the ray space is applied to the manifold of generalized coherent states defined as the eigenstates of isotopic spin charges. The geometry of the state manifold is elucidated through a calculation of the Gaussian curvature.

Quantum chaos in the group‐theoretical picture
View Description Hide DescriptionThe dynamical‐group approach is developed and applied to investigate the problems of controllability and quantum chaos in two fundamental models of the matter–radiation interaction. It provides a new insight into the dynamics of nonstationary quantum process of the interaction between two‐level atoms and a single‐mode radiation field without and with the feedback. A sequence of transitions from the quasiperiodicity to chaos has been numerically observed for two‐level atoms interacting with a self‐consistently generated radiation field. The unitary irreducible representations of the SU(2) group of dynamical symmetry in a noncanonical parametrization is constructed, allowing one to use the results for describing the time evolution of any driven quantum system with the underlying SU(2) symmetry.

Geometric phase, bundle classification, and group representation
View Description Hide DescriptionThe line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel–Weil–Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel–Weil–Bott theorem to Berry’s adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry’s connection. The results about the fiber bundles and group theory are used to introduce a procedure to reduce the problem of the nonadiabatic(geometric) phase to Berry’s adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non‐Abelian monopoles is pointed out.

Essential spectrum of the Dirac Hamiltonian for a spin 1/2 neutral particle with an anomalous magnetic moment in an asymptotically constant magnetic field
View Description Hide DescriptionThe both lower and upper estimates of the lower bound λ^{2} _{0} of the essential spectrum σ_{ ess }(H ^{2}) of the square of the Dirac Hamiltonian H for a spin 1/2 neutral particle with an anomalous magnetic moment in an asymptotically constant magnetic field are obtained. It is found that in a restricted case, λ^{2} _{0}≤m ^{2}, where m is the mass of the particle. Moreover, it is proven that σ_{ ess }(H ^{2})=[λ^{2} _{0},∞). In particular, in the case where the space dimension d is odd and d≥3, σ_{ ess }(H)=(−∞,−λ_{0}]∪[λ_{0},∞). In the case where d=2 and 3, σ_{ ess }(H) is exactly identified.

Classical mechanics with lapse
View Description Hide DescriptionMechanics is developed over a differentiable manifold as space of possible positions. Time is considered to fill a one‐dimensional Riemannian manifold, so having the metric as lapse. Then the system is quantized with covariant instead of partial derivatives in the Schrödinger operator.

Solvable (nonrelativistic, classical) n‐body problems on the line. II
View Description Hide DescriptionA solvable n‐body problem is exhibited, which features equations of motion of Newtonian type, , j=1,...,n, with ‘‘forces’’ that are linear and quadratic in the particle velocities,, and depend highly nonlinearly on the positions , k=1,...,n, of the n ‘‘particles’’ on the line. Explicit expressions of the functions , in terms of elliptic functions, are given; they contain n+4 arbitrary constants, in addition to the n ‘‘masses’’ and to n arbitrary functions . Special cases in which the elliptic functions reduce to trigonometric or rational functions are of course included. The technique whereby this model has been arrived at entails that its initial‐value problem is solvable by quadratures [for any n and arbitrary initial data x(0) and ẋ(0)]. A discussion of the actual behavior of the solution, and of special cases, is postponed to future papers.

The stress tensor for nonlocal field equations
View Description Hide DescriptionConstruction of the stress tensor for a low differential order field theory on translation‐invariant space is routine. If the underlying space is rotation, or Lorentz, invariant, an equivalent symmetric tensor can be found as well. Extension to nonlocal field equations, common, for example, in the statistical mechanical theory of fluids, is not routine and is carried out in this paper.

Gentle perturbations of the free Bose gas. The critical regime
View Description Hide DescriptionThe modular structure of the free Bose gas in the critical regime of couplings is completely described by certain Gaussian Markovian (on a circle) nonergodic thermal process. Gentle perturbations of the arising Gaussian stochastic structure are controlled rigorously and the arising non‐quasi‐free W* KMS structure is shown to be nonergodic in certain range of couplings. The preserved nonergodicity seems to be connected to the presence of the Bose–Einstein condensate in the constructed models of interacting bosons.

Statistical mechanics of the deformable droplets on flat surfaces
View Description Hide DescriptionA comprehensive statistical mechanics treatment of (non)interacting deformable planar droplets of arbitrary rigidity is developed. Closed form exact analytic results are obtained for the area statistics of a single droplet and for the decay rates of metastable states characteristic of the first‐order phase transitions for an assembly of such droplets. To select the correct form of the interaction between the droplets, the reparametrization invariance is taken into account. Most of the known two‐dimensional lattice models are obtained as the limiting cases of the interacting dropletmodel discussed in the text.

Statistical mechanics of the deformable droplets on Riemannian surfaces: Applications to reptation and related problems
View Description Hide DescriptionThe statistical mechanics treatment of the Laplace–Young‐type problems developed for the flat surfaces is generalized to the case of surfaces of constant negative curvature and connected with them to Riemannian surfaces. Obtained results are mainly used to supply an additional support of the quantum Hall effect (QHE) analogy employed in recent work [J. Phys. 4, 843 (1994)], which provides theoretical justification of the tube concept used in polymerreptationmodels. As a byproduct, close links between QHE, quantum chaos, and the non‐Abelian Chern–Simons quantum mechanics are indicated.

On a nonlinear stationary problem arising in transport theory
View Description Hide DescriptionIn this article a nonlinear one‐dimensional stationary transport equation with general boundary conditions is considered where an abstract boundary operator relates the incoming and the outgoing fluxes. Existence results are proved in the case where the collision operator is of the Hammerstein type. In particular, it is shown that these results remain valid for multidimensional geometry with vacuum boundary conditions. Sufficient conditions are given in terms of collision frequency and scattering kernel assuring the existence and uniqueness of solutions. The article ends with the discussion of the case of multiplying boundary conditions.