On the purification of factor states

Let be aC*-algebra and be an opposite algebra. Notions of exact andj-positive states of are introduced. It is shown, that any factor state of can be extended to a pure exactj-positive state of . The correspondence generalizes the notion of the purifications map introduced by Powers and Størmer. The factor states ₁ and ₂ are quasi-equivalent if and only if their purifications and are equivalent.     

Effective Potential for \mathcal{P}\mathcal{T}-Symmetric Quantum Field Theories

Recently, a class of -invariant scalar quantum field theories described by the non-Hermitian Lagrangian = () ² +g ² (i) was studied. It was found that there are two regions of . For <0 the -invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the -invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for -symmetric quantum field theories.     

Inequivalent quantizations for non-linear σ model

We compute the homotopy groups ₀ and ₁ of the classical configuration space of anO(3) invariant field theory on ×, where is a compact two dimensional manifold for arbitrary genusg and- denotes the time coordinate. We also present the finite dimensional, unitary, irreducible, inequivalent representations of the appropriate fundamental groups and comment on some of their implications.     

Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

ГАЛЬВАНОМАГНИТНЫИ Э ФФЕКТ В МАРГАНЕЦ-ЦИНК ОВЫХ ФЕРРИТАХ В ОКРЕСТНОС ТИ ТОЧКИ КЮРИ

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The magnetoresistance in zinc-manganese ferrites in the vicinity of the Curie point

The paper describes an exact method for measuring the adiabatic and isothermal magnetoresistant effects in ferrites. It gives the results of studying the temperature dependence of the magnetoresistant effect, which is negative near the Curie point in ferrites and the temperature dependence of which has a maximum of absolute values inT _c . It is also shown that the pronounced maximum of the adiabatic magnetoresistant effect is to a certain extent caused by the magnetocaloric effect. When measuring the dependence of the magnetoresistant effect on the field strength for a temperature equal toT _c , certain small deviations from the theoretically assumed dependence were found; the influence of different factors on these deviations is discussed and a proposal for their explanation is given in analogy to the results known for metals.

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Algebras of observables with continuous representations of symmetry groups

Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory.Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU _g of a symmetry group in Hilbert space and a von Neumann algebra on invariant with respect toU _g . The set of all operatorsX such thatU _g X U _g ^–1 , as a function ofg , is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra is shown to be the weak (or strong) closure of .Infield theory, a unitary representation up to a factorU(a, ) of the proper inhomogeneous Lorentz group and local von Neumann algebras _C for finite open space-time regionsC are assumed, with the usual transformation properties of underU(a, ). The collection of allX_C giving uniformly continuous functionsU (a, )X U ^–1 (a, ) on is then a localC*-algebra , called thealgebra of local observables. The algebra is again weakly (or strongly) dense in _c . The norm-closed union of the for allC is calledalgebra of quasilocal observables (or quasilocal algebra).In either case, the group is represented by automorphisms V _g resp. V(a, ) — with V _g X=U _g X U _g ^–1 — of theC*-algebra , and this is astrongly continuous representation of on the Banach space . Conditions for V (a, ) can then be formulated which correspond to the usualspectrum condition forU (a, ) in field theory.Work supported in part by the Deutsche Forschungsgemeinschaft.     

Central decomposition of invariant states applications to the groups of time translations and of euclidean transformations in algebraic field theory

With aC*-algebra with unit andgG _g a homomorphic map of a groupG into the automorphism group ofG, the central measure of a state of is invariant under the action ofG (in the state space of ) iff is -invariant. Furthermore if the pair { ,G} is asymptotically abelian, is ergodic iff is ergodic. Transitive ergodic states (corresponding to transitive central measures) are centrally decomposed into primary states whose isotropy groups form a conjugacy class of subgroups. IfG is locally compact and acts continuously on , the associated covariant representations of { , } are those induced by such subgroups. Transitive states under time-translations must be primary if required to be stable. The last section offers a complete classification of the isotropy groups of the primary states occurring in the central decomposition of euclidean transitive ergodic invariant states.     

Emissivity of CO2-H2O mixtures at temperatures up to 1000° K

The emissivity in the 2.7 m range is examined with a spectrometer having 25 cm^–1 for 2.5, ,7.5 cm·atm,4 8cm·atm, 400T1000° K; 150P730 mm Hg. It is found that relation (1) is obeyed to within /0.1, though the calculated transmission is usually less than the measured value. It is shown that the relation is obeyed on account of the mutual position of the CO₂ and H₂O lines in the band, i.e., one gas may be considered as unselective relative to the other.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Phase separation of symmetrical polymer mixtures in thin-film geometry

Monte Carlo simulations of the bond fluctuation model of symmetrical polymer blends confined between two neutral repulsive walls are presented for chain lengthN _A=N _B=32 and a wide range of film thicknessD (fromD=8 toD=48 in units of the lattice spacing). The critical temperaturesT _c (D) of unmixing are located by finite-size scaling methods, and it is shown that , wherev ₃0.63 is the correlation length exponent of the three-dimensional Ising model universality class. Contrary to this result, it is argued that the critical behavior of the films is ruled by two-dimensional exponents, e.g., the coexistence curve (difference in volume fraction of A-rich and A-poor phases) scales as , where ₂ is the critical exponent of the two-dimensional Ising universality class ( ₂=1/8). Since for largeD this asymptotic critical behavior is confined to an extremely narrow vicinity ofT _c (D), one observes in practice effective exponents which gradually cross over from ₂ to ₃ with increasing film thickness. This anomalous flattening of the coexistence curve should be observable experimentally.     

The construction of a representation of loop algebras

By considering the cohomology of the loop algebraL , a representation ofL is constructed. the construction is based on a derivation ofL and a two-dimensional closed cochain ofl with coefficients in real numbersR ¹. In the case of =0, the differential of the energy representation of the corresponding loop groupLG is derived.This work was supported in part by the National Natural Science Foundation of China.     

Von neumann entropy as information rate

Recently it has been shown that quantum theory can be viewed as a classical probability theory by treating Hilbert space as a measure space (H, B(H)) of events or hidden states. Each density operator defines a set of probability measures such that(E _n )=w _n (alln). Coding elements H by subspacesE _n entails distortion. We show that the von Neumann entropyS() = -trInequals the effective rate at which the Hilbert space produces information with zero expected distortion, and comment on the meaning of this.     

Projectively Equivariant Symbol Calculus

The spaces of linear differential operators acting on -densities on and the space of functions on which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect (ⁿ) of vector fields of ⁿ. However, these modules are isomorphic as sl(n + 1,)-modules where is the Lie algebra of infinitesimal projective transformations. In addition, such an -equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the -equivariant symbol map to study the of kth-order linear differential operators acting on -densities, for an arbitrary manifold M and classify the quotient-modules .     

On the Leibniz Homology of Poisson Algebras

We study the Leibniz homology of the Poisson algebra of polynomial functions over (²ⁿ ,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some _2n ( )-modules and obtain some explicit result in low degree.     

Superderivations ofC *-algebras implemented by symmetric operators

The paper studies unbounded symmetric and dissipative implementations (S,G) of^*-superderivations ofC ^*-algebras . It associates with them representations _S of the domainsD() of on the deficiency spacesN(S) of the symmetric operatorsS. A link is obtained between the deficiency indicesn _±(S) ofS and the dimensions of irreducible representations of . For the case when (S,G) is a maximal implementation and max(n _±(S))<, some conditions are given for the representation _S to be semisimple and to extend to a bounded representation of .     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Isospectral hamiltonian flows in finite and infinite dimensions

A moment map is constructed from the Poisson manifold _A of rank-r perturbations of a fixedN×N matrixA to the dual of the positive part of the formal loop algebra =gl(r)[[, ^–1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in _A. The latter may be identified with flows on finite dimensional coadjoint orbits in and linearized on the Jacobi variety of an invariant spectral curveX _r which, generically, is anr-sheeted Riemann surface. Reductions of _A are derived, corresponding to subalgebras ofgl(r, ) andsl(r, ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.This research was partially supported by NSF grants MCS-8108814 (A03), DMS-8604189, and DMS-8601995     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

Distribution of the error term for the number of lattice points inside a shifted circle

We investigate the fluctuations inN (R), the number of lattice pointsnZ ² inside a circle of radiusR centered at a fixed point [0, 1)². Assuming thatR is smoothly (e.g., uniformly) distributed on a segment 0RT, we prove that the random variable has a limit distribution asT (independent of the distribution ofR), which is absolutely continuous with respect to Lebesgue measure. The densityp (x) is an entire function ofx which decays, for realx, faster than exp(–|x|^4–). We also obtain a lower bound on the distribution function which shows thatP (–x) and 1–P (x) decay whenx not faster than exp(–x ⁴⁺). Numerical studies show that the profile of the densityp (x) can be very different for different . For instance, it can be both unimodal and bimodal. We show that , and the variance depends continuously on . However, the partial derivatives ofD are infinite at every rational point Q ², soD is analytic nowhere.     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.