Majoranized Feynman rules

We point out that the compact Feynman rules for Majorana fermions proposed by Denner et al. are in fact a convention for the complex phases of (anti)spinors, valid for both Majorana and Dirac fermions. We establish the relation of this phase convention with that common in the use of spinor techniques.     

SERS scaling rules

We demonstrate the qualitative analysis of surface-enhanced Raman scattering (SERS) intensity and optical extinction by experimentally and numerically. This analytical methods are well matched not only the simple square lattice array of nanostructures, but also the rectangular lattices. We also demonstrate SERS selectivity of modes controlling the optical extinction of excitation and scattering wavelength. Both square lattice and rectangular lattice have similar tendency, but the rectangular lattice structures have much higher selectivity of SERS modes.     

Electronuclear sum rules

Generalized sum rules are derived by integrating the electromagnetic structure functions along lines of constant ratio of momentum and energy transfer. For non-relativistic systems these sum rules are related to the conventional photonuclear sum rules by a scaling transformation. The generalized sum rules are connected with the absorptive part of the forward scattering amplitude of virtual photons. The analytic structure of the scattering amplitudes and the possible existence of dispersion relations have been investigated in schematic relativistic and non-relativistic models. While for the non-relativistic case analyticity does not hold, the relativistic scattering amplitude is analytical for time-like (but not for space-like) photons and relations similar to the Gell-Mann-Goldberger-Thirring sum rule exist.     

Continuous-parameter sum rules

Using Laplace transform and Regge high-energy behaviour, continuous-parameter sum rules are found for strong interaction amplitudes. The finite-energy version of these sum rules is discussed and equations for the Regge parameters obtained. Forward πN scattering is considered as an example and numerical values are found for the parameters of and ′ trajectories. A mathematical appendix on the incomplete gamma function is added.     

Bosonization rules in dimensions

We derive the bosonization rules for free fermions on a half-line with physically sensible boundary conditions for Luttinger fermions. We use path-integral methods to calculate the bosonized fermionic currents on the half-line and derive their commutation relations for a system with a boundary. We compute the fermion determinant of the fermionic fluctuations for a system with a boundary using Forman's approach. We find that the degrees of freedom induced at the boundary do not to modify the commutation relations of the bulk. We give an explicit derivation of the bosonization rules for the fermion operators for a system with boundaries. We derive a set of bosonization rules for the Fermi operators which include the explicit effect of the boundaries and of boundary degrees of freedom. As a byproduct, we calculate the one-particle Green function and determine the effects of the boundaries on its analytic structure.     

Local rules for quasicrystals

The relationship of local ordering and long-range order is studied for quasicrystalline tilings of plane and space. Two versions of the concept of local rules are introduced: strong and weak. Necessary conditions of the existence of strong local rules are found. They are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals. For planar quasicrystals the quadratic irrationalities play an important role. For three-dimensional quasicrystals not only quadratic but also cubic irrationalities are allowed. The existence of weak local rules is established for almost all two-dimensional quasicrystals based on quadratic irrationalities and for the three-dimensional quasicrystal having icosahedral symmetry.     

Selection rules for hybridization

Selection rules for the hybridization of the crystal-field levels of a magnetic ion are investigated by group theoretical methods. They appear only in a few cases. In absence of strict selection rules the possibility for drastic quantitative differences in the hybridization of different levels is discussed.This work was supported by the Deutsche Forschungsgemeinschaft through SFB 125 (Aachen-Jülich-Köln)     

Meson baryon sum rules

The PCAC consistency condition, the Adler-Weisberger sum rule and the dispersion superconvergent sum rule have been saturated with resonance contributions only. Strangeness-changing currents and strangeness-conserving currents have been considered and compared. The results obtained also support PCAC for the strangeness conserving current. While in theπN case PCAC and the Adler-Weisberger sum rule are almost saturated with the famous N ₃₃ ^* resonance only, in the KN caseY ₁ ^* (1385),Y ₀ ^* (1405) andY ₀ ^* (1520) seem to be sufficient. The superconvergent sum rule works very well in the case ofπN scattering, including the contribution of twelve resonances. For KN scattering it can be neither established nor disapproved.     

Feynman rules for superfields

The notion of functional differentiation with respect to superfields is defined and used to set up formal rules for computing Green functions and scattering amplitudes in supersymmetric models.     

Branes and wrapping rules

We show that the branes of ten-dimensional IIA/IIB string theory must satisfy, upon toroidal compactification, specific wrapping rules in order to reproduce the number of supersymmetric branes that follows from a supergravity analysis. The realization of these wrapping rules suggests that IIA/IIB string theory contains a whole class of generalized Kaluza-Klein monopoles.     

Some useful sum rules

Some useful sum rule expressions are derived and applied to give bounds on 〈x^N〉 values for the ground state wavefunction associated with an x⁴ potential.     

Optical Thomas-Reiche-Kuhn sum rules

The Thomas-Reiche-Kuhn sum rule is a fundamental consequence of the position-momentum commutation relation for an atomic electron and it provides an important constraint on the transition matrix elements for an atom. Analogously, the commutation relations for the electromagnetic field operators in a magnetodielectric medium constrain the properties of the dispersion relations for the medium through four sum rules for the allowed phase and group velocities for polaritons propagating through the medium. These rules apply to all bulk media including the metamaterials designed to provide negative refractive indices. An immediate consequence of this is that it is not possible to construct a medium in which all the polariton modes for a given wavelength lie in the negative-index region.     

Intersection theory rules symbology

We propose a novel method to determine the structure of symbols for any family of polylogarithmic Feynman integrals. Using the d log-bases and simple formulas for the leading order and next-to-leading contributions to the intersection numbers, we give a streamlined procedure to compute the entries in the coefficient matrices of canonical differential equations, including the symbol letters and the rational coefficients. We also provide a selection rule to decide whether a given matrix element mu...     

Polarizability sum rules in QED

The well founded total photoproduction and the, assumed substraction free, longitudinal photoproduction polarizability sum rules are checked in QED at the lowest non-trivial order. The first one is shown to hold, whereas the second one turns out to need a subtraction, which makes its usefulness for determining the electromagnetic polarizabilities of the nucleons quite doubtful.     

Feynman rules for probability amplitudes

Starting with very simple assumptions, Feynman rules for the quantum mechanical amplitudes and the associated probabilities are derived. These rules emerge as the only consistent rules for manipulating complex amplitudes assigned to processes. The probability of a process to which an amplitudex has been assigned is determined asp(x)=|x| , 0<2. If virtual processes are allowed,=2.