PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

Rational Functions with Prescribed Global and Local Minimizers

A family of multivariate rational functions is constructed. It has strong local minimizers with prescribed function values at prescribed positions. While there might be additional local minima, such minima cannot be global. A second family of multivariate rational functions is given, having prescribed global minimizers and prescribed interpolating data.     

Orbits of arcs in PG(N,K) under projectivities

The symmetric group on k symbols is made to operate on a certain set of matrices in such a way that its orbits are in one-to-one correspondence with the orbits of the k-arcs of an N-dimensional projective space under the group of projectivities. This leads to a formula for the number of such orbits.     

The dynamics of speculative behaviour

A number of recent empirical studies cast some doubt on the random walk theory of asset prices and suggest these display significant transitory components and complex chaotic motion. This paper analyses a model of fundamentalists and chartists which can generate a number of dynamic regimes which are compatible with the recent empirical evidence.     

Infinite products for power series

An algorithm is introduced and shown to lead to a unique infinite product representation for a given formal power series A(z) with A(0) = 1. The infinite product is , where all b_n ≠ 0, r_n , and r_n+1 > r_n. The degree of approximation by the polynomial (1 + b₁z^r₁) · · · (1 + b_nz^r_n) is also considered.     

On interpolation by radial polynomials

A lemma of Micchelli's, concerning radial polynomials and weighted sums of point evaluations, is shown to hold for arbitrary linear functionals, as is Schaback's more recent extension of this lemma and Schaback's result concerning interpolation by radial polynomials. Schaback's interpolant is explored. Happy 60th and beyond, Charlie! Mathematics subject classifications (2000) 41A05, 41A6.     

TEM investigations on PECVD-titanium carbide layers

TiC_x-PECVD-layers were characterized by TEM. EDX analysis and electron diffraction. TiC_x-layers deposited using benzene showed a columnar structure, at which the column size decreases with rising excess carbon content. TiC_x-layers deposited using n-heptane presented a lamellar structure, at which the lamellar thickness diminishes with an increasing excess carbon content. In dependence on the layer thickness a periodic progress of the element contents was observed, at which a maximum for Ti and Cl correlates with a minimum for C. It was found that the incorporated chlorine is bonded to titanium. The lattice parameter depends on the chlorine content. Using TiCl₄/H₂/Ar-gas mixtures without any hydrocarbon, layers containing TiH₂ are formed.     

Enantioselective separations by capillary GC on derivatized cyclodextrins. Part 2: Determination of enantiomeric excess of some racemic 2,3-isopropylidene-1,2,3-cyclohexanetriol derivatives on permethyl α-, β-, and γ-cyclodextrins

Capillary GC on permethyl α-, β-, and γ-cyclodextrins has been applied to separate and quantify the enantiomers of some 2,3-iso-propylidene-1,2,3-cyclohexanetriol derivatives. Quantitative CGC data are compared to those obtained with chiral shift ¹H NMR.