Particle-wave duality: a dichotomy between symmetry and asymmetry

Symmetry plays a central role in many areas of modern physics. Here we show that it also underpins the dual particle and wave nature of quantum systems. We begin by noting that a classical point particle breaks translational symmetry whereas a wave with uniform amplitude does not. This provides a basis for associating particle nature with asymmetry and wave nature with symmetry. We derive expressions for the maximum amount of classical information we can have about the symmetry and asymmetry of a quantum system with respect to an arbitrary group. We find that the sum of the information about the symmetry (wave nature) and the asymmetry (particle nature) is bounded by log(D) where D is the dimension of the Hilbert space. The combination of multiple systems is shown to exhibit greater symmetry and thus more wavelike character. In particular, a class of entangled systems is shown to be capable of exhibiting wave-like symmetry as a whole while exhibiting particle-like asymmetry internally. We also show that superdense coding can be viewed as being essentially an interference phenomenon involving wave-like symmetry with respect to the group of Pauli operators.Comment: 20 pages, 3 figure

Integral Grothendieck-Riemann-Roch theorem

We show that, in characteristic zero, the obvious integral version of the Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck's strategy: we use resolution of singularities and the weak factorization theorem for birational maps.Comment: 24 page

Hyperinsulinism-hyperammonaemia syndrome: novel mutations in the GLUD1 gene and genotype-phenotype correlations

Background: Activating mutations in the GLUD1 gene (which encodes for the intra-mitochondrial enzyme glutamate dehydrogenase, GDH) cause the hyperinsulinism–hyperammonaemia (HI/HA) syndrome. Patients present with HA and leucine-sensitive hypoglycaemia. GDH is regulated by another intra-mitochondrial enzyme sirtuin 4 (SIRT4). Sirt4 knockout mice demonstrate activation of GDH with increased amino acid-stimulated insulin secretion. Objectives: To study the genotype–phenotype correlations in patients with GLUD1 mutations. To report the phenotype and functional analysis of a novel mutation (P436L) in the GLUD1 gene associated with the absence of HA. Patients and methods: Twenty patients with HI from 16 families had mutational analysis of the GLUD1 gene in view of HA (n=19) or leucine sensitivity (n=1). Patients negative for a GLUD1 mutation had sequence analysis of the SIRT4 gene. Functional analysis of the novel P436L GLUD1 mutation was performed. Results: Heterozygous missense mutations were detected in 15 patients with HI/HA, 2 of which are novel (N410D and D451V). In addition, a patient with a normal serum ammonia concentration (21 µmol/l) was heterozygous for a novel missense mutation P436L. Functional analysis of this mutation confirms that it is associated with a loss of GTP inhibition. Seizure disorder was common (43%) in our cohort of patients with a GLUD1 mutation. No mutations in the SIRT4 gene were identified. Conclusion: Patients with HI due to mutations in the GLUD1 gene may have normal serum ammonia concentrations. Hence, GLUD1 mutational analysis may be indicated in patients with leucine sensitivity; even in the absence of HA. A high frequency of epilepsy (43%) was observed in our patients with GLUD1 mutations

Moduli Spaces of Lumps on Real Projective Space

Harmonic maps that minimize the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge three lumps is a D2-symmetric 7-dimensional manifold of cohomogeneity one. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formula for various geometric quantities. We also discuss the implications for lump decay

Grothendieck groups and a categorification of additive invariants

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e., a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.Comment: 27 pages, to appear in International J. Mathematic

SRB Environment Evaluation and Analysis. Volume 3: ASRB Plume Induced Environments

Contract NAS8-37891 was expanded in late 1989 to initiate analysis of Shuttle plume induced environments as a result of the substitution of the Advanced Solid Rocket Booster (ASRB) for the Redesigned Solid Rocket Booster (RSRB). To support this analysis, REMTECH became involved in subscale and full-scale solid rocket motor test programs which further expanded the scope of work. Later contract modifications included additional tasks to produce initial design cycle environments and to specify development flight instrumentation. Volume 3 of the final report describes these analyses and contains a summary of reports resulting from various studies

Heisenberg antiferromagnet on the square lattice for S>=1

Theoretical predictions of a semiclassical method - the pure-quantum self-consistent harmonic approximation - for the correlation length and staggered susceptibility of the Heisenberg antiferromagnet on the square lattice (2DQHAF) agree very well with recent quantum Monte Carlo data for S=1, as well as with experimental data for the S=5/2 compounds Rb2MnF4 and KFeF4. The theory is parameter-free and can be used to estimate the exchange coupling: for KFeF4 we find J=2.33 +- 0.33 meV, matching with previous determinations. On this basis, the adequacy of the quantum nonlinear sigma model approach in describing the 2DQHAF when S>=1 is discussed.Comment: 4 pages RevTeX file with 5 figures included by psfi

Connection Conditions and the Spectral Family under Singular Potentials

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wave functions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well-defined even if the wave functions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 / | x |$ and the harmonic oscillator with square inverse potential $V(x) = (m \omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined solely by the eigenvalues of the characteristic matrix $U \in U(2)$.Comment: TeX, 18 page