Narrow Results

In this paper, we study left and right $n$-regular functions that originally were introduced in [I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics II, accepted in Adv. Theor. Math. Phys]. When $n=1$, these functions are the usual quaternionic left and right regular functions. We show that $n$-regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy–Fueter type reproducing formulas. Arguably, these Cauchy–Fueter type reproducing formulas for $n$-regular functions are quaternionic analogues of Cauchy’s integral formula for the $n$th-order pole

We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due to the partial isomorphism between CA(3, 1) and the centrally extended Sp( 8), we perform a nonlinear realization of the centrally extended (CE)Sp( 8) in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie algebra and then, by the nonlinear realization, we define the coset field transformations, the Cartan forms and the inverse Higgs constraints. Finally, we discuss the geometrical Lagrangians where all the information on matter fields and their interactions can be contained.

I give a geometrical description of conformal gauge gravitational theory (CGTG) from the viewpoint of symmetries and affine structure. In the frames of the CGTG incorporating gravitation with torsion space-time into Standard Model of electro-weak interaction (EWI) the multi-muon events produced at the Fermilab Tevatron collider were studied. The CGTG gives the value of the torsion pseudotrace–spinor (muon) universal coupling f_T = 4.388 ⋅ 10^-17 G_F, and with limits from known experiments — torsion mass m_T = 0.4700 ⋅ 10^-7 eV or m_T = 0.445 ⋅ 10^-15 muon mass. So the value of the constant of effective four-fermions interaction f_T/m_T = 0.988, indeed may lead to multi-muon events production. The model of interaction of quantum oscillator with the tensor potential W_μν of traceless part of the torsion lead to 2 cm displacement of quark–lepton system as a whole in the magnetic field of collider in accordance with a significant sample of events related to production and decay in which at least one of the muon candidates is produced outside of the beam pipe of radius 1.5 cm. A traceless part of the torsion in CGTG does not vanish in the Newtonian limit of nonzero mass. Torsion gravity potential W_μν gives conservation of a special conformal current and may be produced in the condition of a spontaneous breaking of gauge symmetry where the gravitation mass M_X defect is 1–3 Tev ⋅ c^-2 or 10^-13M_X. This effect may be possible at known effects on top pair asymmetries at the Tevatron and LHC and takes place as the known energy dissipation above 3 TeV of the Galaxy gamma-ray and neutrino spectrum from two bubbles outside the Galaxy plane.

The minimal representation π of the indefinite orthogonal group O(m+1, 2) is realized on the Hilbert space of square integrable functions on ℝ^m with respect to the measure |x|⁻¹dx₁ ⋯ dx_m. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m+3, ℂ) by means of the Bessel function. Taking its ‘boundary value’, we also find the integral kernel of the ‘inversion operator’ corresponding to the inversion element on the Minkowski space ℝ ^{m, 1}.