Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory

Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functions in general requires complex numbers to be formulated. However, it soon turned out to even require some hypercomplex algebra. Incorporating Special Relativity leads to an equation (Dirac 1928) requiring pairwise anti-commuting coefficients, usually $4\times 4$ matrices. A unitary ring of square matrices is an associative hypercomplex algebra by definition. Since only the algebraic properties and relations of the elements matter, we replace the matrices by biquaternions. In this paper, we first consider the basics of non-relativistic and relativistic QT. Then we introduce general hypercomplex algebras and also show how a relativistic quantum equation like Dirac's one can be formulated using biquaternions. Subsequently, some algebraic preconditions for operations within hypercomplex algebras and their subalgebras will be examined. For our purpose equations akin to Schr\"odinger's should be able to be set up and solved. Functions of complementary variables should be Fourier transforms of each other. This should hold within a purely non-real subspace which must hence be a subalgebra. Furthermore, it is an ideal denoted by $\mathcal{J}$. It must be isomorphic to $\mathbb{C}$, hence containing an internal identity element. The bicomplex numbers will turn out to fulfil these preconditions, and therefore, the formalism of QT can be developed within its subalgebras. We also show that bicomplex numbers encourage the definition of several different kinds of conjugates. One of these treats the elements of $\mathcal{J}$ like the usual conjugate treats complex numbers. This defines a quantity what we call a modulus which, in contrast to the complex absolute square, remains non-real (but may be called `pseudo-real'). However, we do not conduct an explicit physical interpretation here but we leave this to future examinations.Comment: 21 pages (without titlepage), 14 without titlepage and appendi

Colloquium: Atomic spin chains on surfaces

In the present Colloquium, we focus on the properties of 1-D magnetic systems on solid surfaces. From the emulation of 1-D quantum phases to the potential realization of Majorana edge states, spin chains are unique systems to study. The advent of scanning tunnelling microscope (STM) based techniques has permitted us to engineer spin chains in an atom-by-atom fashion via atom manipulation and to access their spin states on the ultimate atomic scale. Here, we present the current state of research on spin correlations and dynamics of atomic spin chains as studied by the STM. After a brief review of the main properties of spin chains on solid surfaces, we classify spin chains according to the coupling of their magnetic moments with the holding substrate. This classification scheme takes into account that the nature and lifetimes of the spin-chain excitation intrinsically depend on the holding substrate. We first show the interest of using insulating layers on metals, which generally results in an increase in the spin state's lifetimes such that their quantized nature gets evident and they are individually accessible. Next, we show that the use of semiconductor substrates promises additional control through the tunable electron density via doping. When the coupling to the substrate is increased for spin chains on metals, the substrate conduction electron mediated interactions can lead to emergent exotic phases of the coupled spin chain-substrate conduction electron system. A particularly interesting example is furnished by superconductors. Magnetic impurities induce states in the superconducting gap. Due to the extended nature of the spin chain, the in-gap states develop into bands that can lead to the emergence of 1-D topological superconductivity and, consequently to the appearance of Majorana edge states

Text Entry Performance and Situation Awareness of a Joint Optical See-Through Head-Mounted Display and Smartphone System

Optical see-through head-mounted displays (OST HMDs) are a popular output medium for mobile Augmented Reality (AR) applications. To date, they lack efficient text entry techniques. Smartphones are a major text entry medium in mobile contexts but attentional demands can contribute to accidents while typing on the go. Mobile multi-display ecologies, such as combined OST HMD-smartphone systems, promise performance and situation awareness benefits over single-device use. We study the joint performance of text entry on mobile phones with text output on optical see-through head-mounted displays. A series of five experiments with a total of 86 participants indicate that, as of today, the challenges in such a joint interactive system outweigh the potential benefits.Comment: To appear in IEEE Transactions on Visualization and Computer Graphics On page(s): 1-17 Print ISSN: 1077-2626 Online ISSN: 1077-262

Kinetics of Martensite Decomposition and Microstructure Stability of Ti-6246 during Rapid Heating to Service Temperatures

The aerospace alloy Ti-6246 was subjected to inductive heat treatments with high heating and quenching rates (up to 1500 K/s) while being applied to an in situ diffraction study at the HEMS beamline P07B at DESY. Thereby, the characterization of the emerging phases was possible at any point in the process. The heat treatment schedules include the preparation of Ti-6246 samples by means of a homogenization treatment and subsequent quenching to trigger α″-martensite formation. In order to simulate fast reheating within the scope of application, the samples were reheated to the upper range of possible service temperatures (550–650 °C) with a heating rate of 100 K/s. In a second heat treatment design, the homogenized and quenched sample state was exposed to high-temperature tempering at 840 °C, which aims for the elimination of α″. Again, fast reheating to the same service temperatures was executed. With the aim of this approach, the stability of the microstructure consisting of α-Ti, β-Ti and α″-martensite was characterized. Further, the martensite decomposition path was analyzed. It shows a two-tier nature, firstly approaching the bcc β-unit cell in the low-temperature range (<400 °C) but subsequently transforming into an hcp-like unit cell and later on into equilibrium α-Ti

The Effect of Shock Control Bumps on the Transonic Flutter and Buffeting Characteristics of a Typical Wing Section

We investigate the effects of small-scale airfoil geometry modifications, in particular so-called shock control bumps (SCB), on the transonic flutter and buffeting behavior of a classical two-degree-of-freedom model employing linearized stability analysis. The unsteady aerodynamic forces are modelled by solving the discretized RANS equations with respect to small-perturbation mesh deformation input. Special emphasis is put on the coupling of fluid modes and structural modes. SCBs that are designed initially towards purely aerodynamic buffet suppression perform also well in the aeroelastic analysis with respect to fluid-mode flutter. Classical flutter and buffeting are found to be separated by an apparent wall of stability. It is further noted that SCBs can potentially serve to improve the classical flutter behavior in the transonic dip region

Three-tiered approach for standard information requirements for polymers requiring registration under REACH

Polymers are a very large class of chemicals comprising often complex molecules with multiple functions used in everyday products. The EU Commission is seeking to develop environmental and human health standard information requirements (SIRs) for man-made polymers requiring registration (PRR) under a revised Registration, Evaluation, Authorisation and Restriction of Chemicals (REACH) Regulation. Conventional risk assessment approaches currently used for small molecules may not apply to most polymers. Therefore, we propose a conceptual three-tiered regulatory approach for data generation to assess individual and groups of polymers requiring registration (PRR). A key element is the grouping of polymers according to chemistry, physico-chemical properties and hazard similarity. The limited bioavailability of many polymers is a prominent difference to many small molecules and is a key consideration of the proposed approach. Methods assessing potential for systemic bioavailability are integral to Tier 1. Decisions for further studies are based on considerations of properties and effects, combined with systemic bioavailability and use and exposure considerations. For many PRRs, Tier 1 data on hazard, use and exposure will likely be sufficient for achieving the protection goals of REACH. Vertebrate animal studies in Tiers 2 and 3 can be limited to targeted testing. The outlined approach aims to make use of current best scientific evidence and to reduce animal testing whilst providing data for an adequate level of protection