New connection between spinors and geometry

We study those nonlinear infinitesimal realizations ofSL(2,C) that leave invariant the quadratic function of the four-velocity components of a particle. These transformations are defined as maps of a larger manifold, which includes the four-velocity space, into itself in such a way that transformations of the depend upon other functions in the manifold. The requirement that remain invariant limits the types of other functions that can contribute in the transformation of the . However, among those allowed are the spinors and a three-dimensional space that transforms nonlinearly and recently associated with electric charge. We point out and explore two interesting aspects of these nonlinear realizations. First, they generally necessitate interactions since is not a covariant equation. Second, with superposition of solutions, exact measurement of the four-velocity or space-time position, is impossible. This and related features of nondeterministic measurement inherent to these realizations are discussed.     

On the support of the Ashtekar-Lewandowski measure

We show that the Ashtekar-Isham extension of the configuration space of Yang-Mills theories is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.These results are then used to prove that is contained in a zero measure subset of with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on . Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the extended configuration space .     

Quantum action principle in curved space

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL _c( , x)=(M/2)g_ij(x) ^{i j}–v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian.     

Relativistic electronic structure of the Pb (001) surface

The surface electronic band structure of the Pb (001) was calculated using the self-consistent, first-principles linear-augmented-plane-wave method and the norm-conserving pseudopotentiai method. In the nonrelativistic case, forbidden gaps appear above and below the Fermi levelin the bulk projected band structure of lead. An occupied surface state at the point and two surface states in a wide forbidden gap above E_F are found. A characteristic feature of the electronic structure of the Pb (001) surface is the absence of a surface state within the forbidden S-P gap in the vicinity of the point. The inclusion of scalar-relativistic effects leads to the merger of several S-P gaps into one wide gap extending throughout the entire Brillouin zone. At the same time, the occupied state at point extends to point and its energy decreases by 2 eV. New, relatively weak surface states in the direction and unoccupied states in the vicinity of the point appear. An unoccupied surface state is found at the bottom of the forbidden gap at point . Including the contribution of the spinorbit pseudopotentiai leads to the appearance of two-spin orbit gas; however, the surface level structure is practically unchanged (except for the disappearance of the unoccupied surface state of P_z-symmetry at point ).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 47–53, October, 1991.     

Gauge choice in Witten's energy expression

Witten's equation can be interpreted as a gauge fixing condition for classical supergravity. We rigorously prove the existence of asymptotically constant solutions of the more general gauge condition for almost all endomorphismsA of the spin bundle. Each gives an expression for the gravitational energy similar to Witten's. These include the choice , which yields the particularly elegant energy expression first noticed by Deser.Partially supported by N. S. F. Grant MCS-82-02018 at Harvard University     

Symplectic approach to the theory of quantized fields. I

Our aim in this paper the first one of a series concerned with the problem of field quantization starting from the symplectic structure underlying the classical theory, is to build up the variational theory necessary to all further constructions. The basic notions are the vertical bundle and thestructure 1-form used to define thegeneralized infinitesimal contact transformation which allows us to state and solve the variational problem related to field physics.Giving a system of modulevalued differential forms of different degree on the vertical bundle which solutions are the stationary cross sections is the main result in the paper. In this scheme the Euler-Lagrange classical equations are the expressions induced by such a system of differential forms on any cross section of the vertical bundle. This gives us a complete linearization of the Euler-Lagrange equations and, starting from it, a natural globalization of these equations. Finally, the notion of variational problem invariant by a Lie group is defined in this scheme, Noether's theorem related to such invariant problem is formulated and an intrinsic version of the so-called Noether invariants of classical variational calculus is obtained.This work has been realized in the Seminar of Mathematical Physics, directed by ProfessorJ. Sancho, in the Faculty of Science at the University of Barcelona (Spain).     

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

Strange properties of early time evolution of the projection of a state vector

Using the equation for a projection of a state vector, the exponent of a damping factor for a given state, corresponding to the effective decay width appearing in the case of multiple measured unstable states, is obtained numerically in some models for early time period t 0. is found to be nonzero (even for stable states) at this time period; thus, for early times, this damping factor is always smaller than unity. It is shown that at finitet's may be smaller or greater than the standard decay width in Weisskopf-Wigner theory depending on the parameters of a model considered.     

Canonical quantization

The dynamical variables of a classical system form a Lie algebra , where the Lie multiplication is given by the Poisson bracket. Following the ideas ofSouriau andSegal, but with some modifications, we show that it is possible to realize as a concrete algebra of smooth transformations of the functionals on the manifold of smooth solutions to the classical equations of motion. It is even possible to do this in such a way that the action of a chosen dynamical variable, say the Hamiltonian, is given by the classical motion on the manifold, so that the quantum and classical motions coincide. In this realization, constant functionals are realized by multiples of the identity operator. For a finite number of degrees of freedom,n, the space of functionals can be made into a Hilbert space using the invariant Liouville volume element; the dynamical variablesF become operators in this space. We prove that for any hamiltonianH quadratic in the canonical variablesq ₁...q _n ,p ₁...p _n there exists a subspace ₁ which is invariant under the action of and , and such that the restriction of to ₁ form an irreducible set of operators. Therefore,Souriau's quantization rule agrees with the usual one for quadratic hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realization. For non-linear problems in general, however, the operators form a reducible set on any subspace of invariant under the action of the Hamiltonian. In particular this happens for . Therefore,Souriau's rule cannot agree with the usual quantization procedure for general non-linear systems.The method can be applied to the quantization of a non-linear wave equation and differs from the usual attempts in that (1) at any fixed time the field and its conjugate momentum may form a reducible set (2) the theory is less singular than usual.For a particular wave equation , we show heuristically that the interacting field may be defined as a first order differential operator acting onc -functions on the manifold of solutions. In order to make this space into a Hilbert space, one must define a suitable method of functional integration on the manifold; this problem is discussed, without, however, arriving at a satisfactory conclusion.On leave from Physics Department, Imperial College, London SW7.Work partly supported by the Office of Scientific Research, U.S. Air Force.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when is a generalized Robertson-Walker spacetime.     

Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Two functionals and are introduced forC ^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals and . Our functionals and are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.     

Mass and charge release by the evaporation of particles from compound nuclei around mass 60

Reaction product cross sections have been measured for reactions of⁴He projectile with^54,56Fe,⁵⁸Ni and⁵⁹Co targets in the bombarding energy range 14–27 MeV. The cross-section-weighted average charge and mass removed from the compound system were deduced and compared with results reported for heavy-ion induced reactions. Variation of with E _CN ^* is independent of the initial angular momentum of compound nucleus in theA=60 mass region in contrary to the rare-earth region. This would suggest a strong angular momentum removal from the compound nucleus by evaporation of particles forA60 nuclei. The average evaporated charge depends on both the CN excitation energy E _CN ^* and the compound nucleusN andZ numbers. The and dependence on excitation energy and the projectile-target combination can be understood in terms of the statistical model calculations.We would like to acknowledge the interest and support of this work by Prof. A. Budzanowski. We wish to express our gratitude to Dr. E. Rurarz for his help with preparing the scattering chamber and Faraday cup used in this work. This work was supported by the Polish Government Central Research Program Grant CPBP 01.09 and the Polish Ministry of Education Grant G-MEN-207/90.     

Interaction of a neutrino with a circularly polarized plane wave accompanied by (n¯n) pair production

We consider the processes (n )and n , induced by the interaction of the anomalous magnetic moment of the neutron with the field of a circularly polarized plane wave. Using the invariant solution of the modified Dirac equation, we obtain expressions for the total probabilities of these processes, which consist of several different contributions. The asymptotic behavior of the results at high energy is analyzed and the possibility of observing the effects is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 46–53, July, 1990.The authors thank V. G. Bagrov for useful discussions.     

Bäcklund transformations for the equationz_{uv} = 2z_u z_v /z + \bar z

The equation for the complex variablez is equivalent to the field equation for a SO(2, 1)-invariant nonlinear -model in two dimensions. A connection on a SU(2) bundle is given, such that the requirement of the vanishing curvature of the connection implies that equation. Bäcklund transformations, considered as gauge transformations of the connection, are explicity obtained.     

On generalising Abraham-Lorentz equation

It is proved that runaway solutions persist if Abraham's force –m is generalised by adding to it afinite number of terms which are linear in higher derivatives of . The implication of this result to Eliezer's relativistic generalisation of the Lorentz-Dirac equation is discussed.Worked supported by the Minerva Foundation.     

Superconformal current algebras and their unitary representations

A natural supersymmetric extension is defined of the current (= affine Kac-Moody Lie) algebra ; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of are constructed. They extend to unitary representations of the semidirect sumS (G) of with the superconformal algebra of Neveu-Schwarz, for , or of Ramond, for =0.On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria     

Isospectral Hamiltonian flows in finite and infinite dimensions

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad^*-invariant Poisson submanifolds of the dual of the positive part of the loop algebra is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleTP ₁(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles over a suitably desingularized curve may be used to determine both the flow of matricial polynomialsL() and the Hamiltonian flow in the spaceM _N,r×M_N,r in terms of -functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrödinger) equation.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army grant DAA L03-87-K-0110     

Quantum cohomology of partial flag manifoldsF_{n_1 } \ldots _{n_k }

We compute the quantum cohomology rings of the partial flag manifolds . The inductive computation uses the idea of Givental and Kim [1]. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle (E) associated with the vector bundleE this ring is found.     

Instabilities of certain empty Robertson-Walker space-times

Results are established concerning perturbations of each empty Robertson-Walker space-time (M, g) with a nonvanishing cosmological constant. The perturbed space-times have the general form ( ) with an extension ofM, and lying in an open neighborhood of g in a type ofW ^m topology. These results indicate that large classes of such perturbations give rise to space-times which suffer from one of two types of incompleteness.