Relativity in Clifford's Geometric Algebras of Space and Spacetime

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cl{1,3} of Minkowski space, and in the algebra of physical space (APS)Cl{3}. STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA+ of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA+ are all that is needed to calculate physically measurable quantities because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA+ and hermitian conjugation are observer dependent, and the proper basis vectors are persistent vectors that sweep out timelike planes. In the relative version, the mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors. We relate the two versions of APS as consistent models within the same algebra.Comment: 22 pages, no figure

Electron scattering on 4^4He from coupled-cluster theory

We present a coupled-cluster calculation for the electron-$^4$He scattering in the region of the quasi-elastic peak. We show the longitudinal and transverse responses separately, and discuss results within two distinct theoretical methods: the Lorentz integral transform and spectral functions. The comparison between them allows to investigate the role of final state interactions, two-body currents and relativistic effects.Comment: 10 pages, 2 figure

16^{16}O spectral function from coupled-cluster theory: applications to lepton-nucleus scattering

We calculate the $^{16}$O spectral function by combining coupled-cluster theory with a Gaussian integral transform and by expanding the integral kernel in terms of Chebyshev polynomials to allow for a quantification of the theoretical uncertainties. We perform an analysis of the spectral function and employ it to predict lepton-nucleus scattering. Our results well describe the $^{16}$O electron scattering data in the quasi-elastic peak for momentum transfers $|\mathbf{q}|\gtrapprox500$ MeV and electron energies up to 1.2 GeV, extending therefore the so-called first principles approach to lepton-nucleus cross sections well into the relativistic regime. To prove the applicability of this method to neutrino-nucleus cross sections, we implement our $^{16}$O spectral functions in the NuWro Monte Carlo event generator and provide a comparison with recently published T2K neutrino data.Comment: 12 pages, 9 figure

Classification of Low Dimensional Lie Super-Bialgebras

A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1|2) and super-e(2) is presented. Combined technique of computer algebraic computations and a subsequent identification of equivalent structures is applied. In all the cases Poisson-Lie brackets on supergroups are found. Possibility of quantizing them in order to obtain quantum groups is discussed. It turns out to be straightforward for all but one structures for super-E(2) group.Comment: 15 pages, LaTe

Kappa-contraction from SUq(2)SU_q(2) to Eκ(2)E_{\kappa}(2)

We present contraction prescription of the quantum groups: from $SU_q(2)$ to $E_{\kappa}(2)$. Our strategy is different then one chosen in ref. [P. Zaugg, J. Phys. A {\bf 28} (1995) 2589]. We provide explicite prescription for contraction of $a, b, c$ and $d$ generators of $SL_q(2)$ and arrive at $^*$ Hopf algebra $E_{\kappa}(2)$.Comment: 3 pages, plain TEX, harvmac, to be published in the Proceedings of the 4-th Colloqium Quantum Groups and Integrable Systems, Prague, June 1995, Czech. J. Phys. {\bf 46} 265 (1996

Spectral function for 4^4He using the Chebyshev expansion in coupled-cluster theory

We compute spectral function for $^4$He by combining coupled-cluster theory with an expansion of integral transforms into Chebyshev polynomials. Our method allows to estimate the uncertainty of spectral reconstruction. The properties of the Chebyshev polynomials make the procedure numerically stable and considerably lower in memory usage than the typically employed Lanczos algorithm. We benchmark our predictions with other calculations in the literature and with electron scattering data in the quasi-elastic peak. The spectral function formalism allows one to extend ab-initio lepton-nucleus cross sections into the relativistic regime. This makes it a promising tool for modeling this process at higher energy transfers. The results we present open the door for studies of heavier nuclei, important for the neutrino oscillation programs.Comment: 12 pages, 5 figure