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

Abolished adherence alters signaling pathways in phorbol ester-induced human U937 cells

Phorbol ester (TPA) treatment of human U937 myeloid leukemia cells is associated with increasing adherence and monocyte-like maturation whereby the role of β2 integrin-mediated attachment for subsequent growth properties and the differentiation program remains unclear. Here, stably-transfected U937 cells with a pMTH1 vector containing the β2 integrin gene of CD11b in antisense orientation (asCD11b-U937) demonstrated a significantly reduced proliferative capacity in contrast to control vector transfectants (pMTH1-U937) or wild-type U937 cells. Phorbol ester exposure induced adherence and growth arrest in more than 90% of pMTH1-U937 and wild-type U937 cells after 72 h. In contrast, TPA-treated asCD11b-U937 failed to attach and the proliferation continued in more than 30% of the cells. Moreover, increased apoptosis appeared in asCD11b-U937 after TPA induction in contrast to pMTH1-U937 cells. In addition, non-specific inhibition of adherence on an agarose surface demonstrated internucleosomal DNA fragmentation in both, pMTH1-U937 and asCD11b-U937 after TPA treatment indicating a functional relationship between abolished adherence, regulation of proliferation and induction of apoptosis. Western blot analysis revealed differences in the expression levels and altered phosphorylation patterns of Pyk-2, pp60src and p42/p44 MAP kinases between pMTH1-U937 and asCD11b-U937 following TPA exposure which was also substantiated by Pyk-2 immunoprecipitation. These findings suggested that induced adherence predominantly mediated by a functional CD11b/CD18 integrin in U937 cells is involved in the activation of downstream signaling kinases and contributes to cell cycle regulation and apoptosis during monocytic maturation

Involvement of CD11b integrin in the alteration of metabolic factors after phorbol ester stimulation of human myeloid leukemia cells

<p>Abstract</p> <p>Previous work has demonstrated that phorbol ester (TPA)-induced adherence of human U937 myeloid leukemia cells can be blocked upon down-modulation of the β2-integrin CD11b after stable transfection of U937 cells with a pMTH1 vector-containing the CD11b gene in antisense orientation (asCD11b-U937) <it>[Otte</it> et al.<it>, (2011)]</it>. In the present study, alterations in metabolism-associated factors, particularly intra- and extracellular proteases were investigated. A measurement of telomerase activity in the leukemic cells revealed continuously decreasing telomere adducts within 72 h of TPA treatment in pMTH1-U937 cells. In contrast, telomerase activity sustained in asCD11b-U937 upon TPA-induced differentiation. Flow cytometric analysis confirmed unchanged CD11b levels in TPA-induced asCD11b-U937 in contrast to elevated levels in pMTH1-U937 whereby the expression of other β2-integrins including CD11a, CD11c and CD18 was increased in both populations after TPA treatment. Moreover, adherent pMTH1-U937 demonstrated the expression of monocytic differentiation markers including F4-80 and CD14 and an increased MIP-1α production which remained at low or undetectable in TPA-induced asCD11b-U937. These effects indicated an altered response of the different cell populations to the TPA-induced differentiation process. Indeed, Western blot analysis revealed differences in the expression levels of intracellular metabolic factors including MnSOD and p97/VCP and after measurement of 20 S proteasomal proteolytic activity. In addition, increased levels of extracellular metabolic factors including the matrix metalloproteinases MMP-1, MMP-7 and MMP-9 were observed in pMTH1-U937 cells in contrast to unaltered levels in asCD11b-U937 cells.</p

Time resolved coherence imaging spectrometer on WEGA stellarator

Imaging sensor technologies such as charge coupled devices and complementary metal-oxide-semiconductor have made remarkable progress in recent years. Fast imaging systems based on these new technologies are now being routinely employed for advanced fusion diagnostics. Since two-dimensional imaging considerably improves the investigation of three-dimensional structural physics, we have installed and operated the first high-speed two-dimensional coherence imaging camera system for the study of ion temperatures and flow velocities in the WEGA stellarator based on the Doppler broadening of 468.6 nm He II line emission. The coherence imaging camera was able to image the complete plasma poloidal cross-section over a toroidal region spanning 10°. The camera was used for basic plasma studies, including electron cyclotron resonance heating (ECRH) power step experiments. The ion temperature of helium plasmas in WEGA is found to be 1.5-2.0 eV at maximum (26 kW) ECRH power. The plasma rotates in the E × B direction with speeds between 500 and 1000 m s-1, increasing at higher ECRH power. It was confirmed that the flow direction reverses with the direction of the magnetic field. The observed ion temperatures and flows were cross checked against a multi-channel Echelle spectrometer and satisfactory agreement obtained