Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved spacetime. This canonical form is needed to apply the Whitham Lagrangian method. The latter method, unlike the WKB method, places no restriction on the magnitude of Planck's constant to obtain wave packets, and furthermore preserves the symmetries of the Dirac Lagrangian. We show using canonical Dirac fields in a curved spacetime, that the probability current has a Gordon decomposition into a convection current and a spin current, and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck's constant. We further discuss the classical-quantum correspondence in a curved spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved spacetime are a direct consequence of Whitham's Lagrangian method, and not just a physical hypothesis as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.Comment: PDF, 32 pages in referee format. Added significant material on canonical forms of Dirac equations. Simplified Theorem 1 for normal Dirac equations. Added section on Gordon decomposition of the probability current. Encapsulated main results in the statement of Theorem

Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields

Spatially referenced data often have autocovariance functions with elliptical isolevel contours, a property known as geometric anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) with respect to a reference axis and the aspect ratio of the principal correlation lengths. Since these parameters are unknown a priori, sample estimates are needed to define suitable spatial models for the interpolation of incomplete data. The distribution of the anisotropy statistics is determined by a non-Gaussian sampling joint probability density. By means of analytical calculations, we derive an explicit expression for the joint probability density function of the anisotropy statistics for Gaussian, stationary and differentiable random fields. Based on this expression, we obtain an approximate joint density which we use to formulate a statistical test for isotropy. The approximate joint density is independent of the autocovariance function and provides conservative probability and confidence regions for the anisotropy parameters. We validate the theoretical analysis by means of simulations using synthetic data, and we illustrate the detection of anisotropy changes with a case study involving background radiation exposure data. The approximate joint density provides (i) a stand-alone approximate estimate of the anisotropy statistics distribution (ii) informed initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford algebras'. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1} \otimes Cl_{1,1}. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q-deformed structures through nontrivial vacuum states in quantum theories is outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa, Mexico, June 27 - July 4, 199

Kitaev's quantum double model from a local quantum physics point of view

A prominent example of a topologically ordered system is Kitaev's quantum double model $\mathcal{D}(G)$ for finite groups $G$ (which in particular includes $G = \mathbb{Z}_2$, the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of $\mathcal{D}(G)$, and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups $G$ is more complicated. We outline how one could use amplimorphisms, that is, morphisms $A \to M_n(A)$ to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015). Mainly revie

Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject

Two-Fermion Bound States within the Bethe-Salpeter Approach

To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results given by light front dynamics.Comment: 32 pages, XIII Tables, 8 figure

The dependence of dijet production on photon virtuality in ep collisions at HERA

The dependence of dijet production on the virtuality of the exchanged photon, Q^2, has been studied by measuring dijet cross sections in the range 0 < Q^2 < 2000 GeV^2 with the ZEUS detector at HERA using an integrated luminosity of 38.6 pb^-1. Dijet cross sections were measured for jets with transverse energy E_T^jet > 7.5 and 6.5 GeV and pseudorapidities in the photon-proton centre-of-mass frame in the range -3 < eta^jet <0. The variable xg^obs, a measure of the photon momentum entering the hard process, was used to enhance the sensitivity of the measurement to the photon structure. The Q^2 dependence of the ratio of low- to high-xg^obs events was measured. Next-to-leading-order QCD predictions were found to generally underestimate the low-xg^obs contribution relative to that at high xg^obs. Monte Carlo models based on leading-logarithmic parton-showers, using a partonic structure for the photon which falls smoothly with increasing Q^2, provide a qualitative description of the data.Comment: 35 pages, 6 eps figures, submitted to Eur.Phys.J.

Experimental Assessment of the Water Quality Influence on the Phosphorus Uptake of an Invasive Aquatic Plant: Biological Responses throughout Its Phenological Stage

International audienceUnderstanding how an invasive plant can colonize a large range of environments is still a great challenge in freshwater ecology. For the first time, we assessed the relative importance of four factors on the phosphorus uptake and growth of an invasive macrophyte Elodea nuttallii (Planch.) St. John. This study provided data on its phenotypic plasticity, which is frequently suggested as an important mechanism but remains poorly investigated. The phosphorus uptake of two Elodea nuttallii subpopulations was experimentally studied under contrasting environmental conditions. Plants were sampled in the Rhine floodplain and in the Northern Vosges mountains, and then maintained in aquaria in hard (Rhine) or soft (Vosges) water. Under these conditions, we tested the influence of two trophic states (eutrophic state, 100 mu g.l(-1) P-PO43- and hypertrophic state, 300 mu g.l(-1) P-PO43-) on the P metabolism of plant subpopulations collected at three seasons (winter, spring and summer). Elodea nuttallii was able to absorb high levels of phosphorus through its shoots and enhance its phosphorus uptake, continually, after an increase of the resource availability (hypertrophic > eutrophic). The lowest efficiency in nutrient use was observed in winter, whereas the highest was recorded in spring, what revealed thus a storage strategy which can be beneficial to new shoots. This experiment provided evidence that generally, the water trophic state is the main factor governing P uptake, and the mineral status (softwater > hardwater) of the stream water is the second main factor. The phenological stage appeared to be a confounding factor to P level in water. Nonetheless, phenology played a role in P turnover in the plant. Finally, phenotypic plasticity allows both subpopulations to adapt to a changing environment

Angular and Current-Target Correlations in Deep Inelastic Scattering at HERA

Correlations between charged particles in deep inelastic ep scattering have been studied in the Breit frame with the ZEUS detector at HERA using an integrated luminosity of 6.4 pb-1. Short-range correlations are analysed in terms of the angular separation between current-region particles within a cone centred around the virtual photon axis. Long-range correlations between the current and target regions have also been measured. The data support predictions for the scaling behaviour of the angular correlations at high Q2 and for anti-correlations between the current and target regions over a large range in Q2 and in the Bjorken scaling variable x. Analytic QCD calculations and Monte Carlo models correctly describe the trends of the data at high Q2, but show quantitative discrepancies. The data show differences between the correlations in deep inelastic scattering and e+e- annihilation.Comment: 26 pages including 10 figures (submitted to Eur. J. Phys. C

Beauty photoproduction measured using decays into muons in dijet events in ep collisions at s\sqrt{s}=318 GeV

The photoproduction of beauty quarks in events with two jets and a muon has been measured with the ZEUS detector at HERA using an integrated luminosity of 110 pb$^{- 1}$. The fraction of jets containing b quarks was extracted from the transverse momentum distribution of the muon relative to the closest jet. Differential cross sections for beauty production as a function of the transverse momentum and pseudorapidity of the muon, of the associated jet and of $x_{\gamma}^{jets}$, the fraction of the photon's momentum participating in the hard process, are compared with MC models and QCD predictions made at next-to-leading order. The latter give a good description of the data.Comment: 32 pages, 6 tables, 7 figures Table 6 and Figure 7 revised September 200