On maass wave forms and the imaginary quadratic Doi-Naganuma lifting

Mathematische Annalen -     

EPR of Co+2 in Mg(CH3COO)2·4H2O

EPR spectra of the ⁵⁹Co⁺² ion in oriented crystals of Mg(CH₃COO)₂·4H₂O have been measured at 9.4 GHz and a temperature of 4.2 K. The data for each of the two metal ion sites per unit cell are well-described by a spin Hamiltonian for $\text{S =}\text{1}\text{2}\text{, I =}\text{7}\text{2}$ with g_x = 6.13, g_y = 389, g_z = 2.49 and A_x = 0.0193, A_y' = 0.075, A_z' = 0.0032 cm^?1. Only the x-axes of the g- and A-tensors coincide. Orientation of the principal directions relative to the crystal axes has been established for the g- and A-tensors at both sites. They are consistent with the weakly ferromagnetic canted antiferromagnetism found in Co(CH₃COO)₂·4H₂O below T_N ?85mK.     

Some examples of clans

It is shown that every space obtained by identifying exactly two points of a two-cell supports a clan, including the so-called “French Army Hat”. Examples are given of clans on spaces obtained by sewing together ice-cream cones.     

Hidden symmetry of the CKM and neutrino-mapping matrices

We propose that the smallness of the light quark masses is related to the smallness of the T (i.e. CP) violation in hadronic weak interactions. Accordingly, for each of the two quark sectors (“upper” and “lower”) we construct a 3 × 3 mass matrix in a bases of unobserved quark states, such that the “upper” and “lower” basis states correspond exactly via the W^± transitions in the weak interaction. In the zeroth approximation of our formulation, we assume T conservation by making all matrix elements real. In addition, we impose a “hidden symmetry” (invariance under simultaneous translations of all three basis quark states in each sector), which ensures a zero mass eigenstate in each sector.Next, we simultaneously break the hidden symmetry and T invariance by introducing a phase factor e^iχ in the interaction for each sector. The Jarlskog invariant J_CKM, as well as the light quark masses are evaluated in terms of the parameters of the model. Comparing formulas, we find that most unknown factors drop out, resulting in a simple relation with , to leading order in χ and m_s/m_b, with A, λ the Wolfenstein parameters. (Because of the large top quark mass, the contribution from upper quark sector can be neglected.) Setting J_CKM = 3.08 × 10⁻⁵, m_b = 4.7 GeV (1s mass), m_s = 95 MeV, A = 0.818, and λ = 0.227, we find , consistent with the accepted value m_d = 3 − 7 MeV.We make a parallel proposal for the lepton sectors. With the hidden symmetry and in the approximation of T invariance, both the masses of e and ν₁ are zero. The neutrino-mapping matrix V_ν is shown to be of the same Harrison-Perkins-Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant J_ν.     

Reply to: “Comment on: ‘Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory’ [Phys. Lett. A 372 (2008) 2514]” [Phys. Lett. A 372 (2008) 5732]

We compute the emission amplitude for the collective emission from a sphere of identical atoms in the scalar photon theory for both the cases of the complex kernel (i.e. including virtual photons) and real kernel. We explicitly show that the single mode theory based on the real kernel neglects the effects of the different decay rates and frequency shifts associated with the eigenfunctions belonging to the same angular index but with different radial indices. We show that these effects modify, for k₀R?1, both the time dependence of the emission amplitude and its angular distribution, in clear contradiction to the assertions made by the Comment's authors.     

A timeon model of quark and lepton mass matrices

It is proposed that T violation in physics, as well as the masses of electron and quarks, arise from a pseudoscalar interaction with a new spin 0 field τ(x), odd in P and T, but even in C. This interaction contains a factor iγ₅ in the quark and lepton Dirac algebra, so that the full Hamiltonian is conserving; but by spontaneous symmetry breaking, the new field τ(x) has a nonzero expectation value τ≠0 that breaks P and T symmetry. Oscillations of τ(x) about its expectation value produce a new particle, the “timeon”. The mass of timeon is expected to be high because of its flavor-changing properties.The main body of the paper is on the low energy phenomenology of the timeon model. As we shall show, for the quark system the model gives a compact three-dimensional geometric picture consisting of two elliptic plates and one needle, which embodies the 10 observables: six quark masses, three Eulerian angles and the Jarlskog invariant of the CKM matrix.For leptons, we assume that the neutrinos do not have a direct timeon interaction; therefore, the lowest neutrino mass is zero. The timeon interaction with charged leptons yields the observed nonzero electron mass, analogous to the up and down quark masses. Furthermore, the timeon model for leptons contains two fewer theoretical parameters than observables. Thus, there are two testable relations between the three angles and the Jarlskog invariant of the neutrino mapping matrix.     

What is Quantum Mechanics? A Minimal Formulation

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called “microscopic theory”, applicable to any closed system S of arbitrary size N, using concepts referring to S alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen–Specker–Bell theorem and Gleason’s theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.     

Cubic twists of <Emphasis Type="Italic">GL</Emphasis>(2) automorphic L-functions

Let and let be a cuspidal automorphic representation of . Consider the family of twisted L-functions L(s,) where ranges over the cubic Hecke characters of K. In this paper the mean value of this family of L-functions is computed; the result is consistent with the generalized Lindelöf hypothesis. From this mean value result a nonvanishing theorem is established: for given s there are infinitely many cubic twists such that the L-value at s is nonzero. At the center of the critical strip the number of such characters of norm less than X is . These results are obtained by introducing and studying three different families of weighted double Dirichlet series. These series are related by functional equations, some of which are obtained through the study of higher metaplectic Eisenstein series and the Hasse-Davenport relation. The authors establish the continuation of such series and then obtain their main result by Tauberian methods. Mathematics Subject Classification (1991) 11F66, 11F70, 11M41, 11N75, 22E55     

Limited superradiant damping of small samples

We show that for samples small compared to a wavelength the maximum possible superradiant damping is severely limited by the shortness of the radiation-field-induced relaxation time T_Δ as compared to the superradiant lifetime T_S.