Bivariate averaging functions,translation and scale autocorrelations,Fourier and Mellin transforms,the Wiener-Khinchine theorem and their inter-relationships

We first draw attention to the fact that the position operator, , its translation generator, , and its scale generator, , form an important group of triplet of operators that appear in the Heisenberg uncertainty relation stated in its most general form. The pair forms the phase-space and they have led to Fourier transform pair, the autocorrelation function, the Wiener-Khinchine theorem, and the Wigner function with many different applications to wave phenomena. The importance of the pairs and has been pointed out by Moses and Quesada (1972, 1973, 1974) who showed that we must then consider a Mellin transform pair, a scale autocorrelation function, and a corresponding Wiener-Khinchine theorem. In the present paper, we define and explore properties of a bivariate averaging function defined in a new “phase-space” involving the Mellin transform variable and its partner which can either be the position or momentum, analogous to the Wigner function. The not-necessarily positive feature of the bivariate averaging functions is traced to the general Heisenberg uncertainty mentioned above. The properties and their inter-relationships among the averaging functions are given. We hope this will be of use in discussing physical phenomena involving fractals, turbulence, and near phase transitions where the scaling properties are of importance.     

A convolution integral for Fourier transforms on the groupSL(2,C)

The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.     

Causal correlation functions and Fourier transforms: Application in calculating pressure induced shifts

By adopting a concept from signal processing, instead of starting from the correlation functions which are even, one considers the causal correlation functions whose Fourier transforms become complex. Their real and imaginary parts multiplied by 2 are the Fourier transforms of the original correlations and the subsequent Hilbert transforms, respectively. Thus, by taking this step one can complete the two previously needed transforms. However, to obviate performing the Cauchy principal integrations required in the Hilbert transforms is the greatest advantage. Meanwhile, because the causal correlations are well-bounded within the time domain and band limited in the frequency domain, one can replace their Fourier transforms by the discrete Fourier transforms and the latter can be carried out with the FFT algorithm. This replacement is justified by sampling theory because the Fourier transforms can be derived from the discrete Fourier transforms with the Nyquis rate without any distortions. We apply this method in calculating pressure induced shifts of H₂O lines and obtain more reliable values. By comparing the calculated shifts with those in HITRAN 2008 and by screening both of them with the pair identity and the smooth variation rules, one can conclude many of shift values in HITRAN are not correct.     

A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms

We analyze the security of a recently proposed asymmetric cryptosystem that based on the phase-truncated Fourier transforms (PTFTs), and describe a specific attack method to break the cryptosystem. This specific attack, which is based on a two-step iterative amplitude retrieval approach and works by using the public keys and ciphertexts, would allow an attacker to reveal the encrypted information and the decryption keys that generated in the encryption procedure. The numerical simulation results, which are given to show the effectiveness of the proposed specific attack, imply that some appropriate measurements should be made to enhance the resistance of the PTFT-based cryptosystem against the specific attack when it is used as a public-key cryptosystem.     

Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.     

Lightest scalar in the SU(L)2 x SU(R)2 linear sigma model

We consider the lightest scalar meson in the frame of the SU(L)2 x SU(R)2 linear sigma model, keeping in mind that this model could be the low energy realization of two-flavor QCD. We show that the sigma field is described by its four-quark component at least in the sigma resonance energy (virtuality) region and the sigma-->gammagamma decay is the four-quark transition. We emphasize that residues of the sigma pole in the pipi-->pipi and gammagamma-->pipi amplitudes do not give an idea about the sigma meson nature, and the progress in studying the sigma meson production mechanisms in different processes could essentially further us in understanding its nature.     

Axion depending on the Higgs sector inSU(2)L×SU(2)R×U(1)

Using Higgses with quantum numbers of fermion bilinears we discuss the axion in four different Higgs sectors inSU(2)_L×SU(2)_R×U(1). Three of the cases are similar to the “standard axion” in the Salam-Weinberg model and in one case the axion can be made invisible.     

SU(2)L×SU(2)R×U(1) gauge model

We determine here the most general electroweak interaction based on the groupSU(2)_L×SU(2)_R×U(1). When we rotate theZ ₁,Z ₂ basis to theZ,D basis such that the total interaction ofZ with the right-handed current is zero, we obtain an interaction that is free of triangle anomalies. This condition enables us to know the angle through whichZ ₁,Z ₂ basis is to be rotated. We show that the triangle anomaly free interaction obtained by others is contained here as a special case. We also determine the triangle anomaly free weak interaction whenever the neutral (Z,D) bosons are mass eigenstates and show that it reduces to the neutral sector of the standard model whenever g _R ² goes to infinity. The charged sector is also developed here. The most general elements of the masssquared matrix of theZ,D bosons are evaluated. The masses of the left- and right-handed charged bosons are also determined.     

Complex order fractional Fourier transforms and their use in diffraction theory. Application to optical resonators

The electromagnetic field transfer by diffraction from a spherical emitter to a spherical receiver is expressed by a fractional Fourier transform whose order depends on the distance between the emitter and the receiver and their curvature radii. Special attention is paid to complex order transforms: scaled variables and scaled field amplitudes are defined by complying with mathematical consistency. For a given distance of observation, a simple geometrical condition is set up to determine whether the corresponding transform order is a real or a complex number. The result is applied to check the stability of optical resonators.     

A note on energy-momentum conservation in Palatini formulation of L(R) gravity

By establishing that Palatini formulation of L(R) gravity is equivalent to ω=−3/2 Brans-Dicke theory, we show that energy-momentum tensor is covariantly conserved in this type of modified gravity theory.     

Proton decay in theSU(4) c ×SU(2) L ×SU(2) R model

Possible proton decays in theSU(4) _C ×SU(2) _L ×SU(2) _R unification model are discussed. There are some characteristics in the decay products, which are different from those in the standardSU(5) orSO (10) model, in certain cases.     

Optical entangled fractional Fourier transform derived via non-unitary SU(2) bosonic operator realization and its convolution theorem

Based on the two mutually conjugate tripartite entangled state representations (TESR), we introduce the 3-mode optical entangled fractional Fourier transform (EFrFT) through a new non-unitary bosonic operator realization of the SU(2) generators and the tripartite squeezing operator. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, is not the direct product of three single-variable FrFTs. We show that the EFrFT adapts to the transform between two mutually conjugate quantum mechanical TESR, i.e., we can recast the EFrFT into the matrix element of in the TESR. In addition, we define the two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.     

Lowering the dimension of polynomial vector fields in R(2) and R(3)

We classify the normal forms associated to polynomial vector fields with dimensions two and three whose principal part is linear. Then we reduce by one the dimension of the associated differential systems. This is achieved by means of the extension of a symmetry of the unperturbed part to the whole system, up to a certain order of approximation. The corresponding transformation is formal in the sense that we have no estimation on its radius of convergence. We calculate the invariants, reduced phase spaces and differential systems associated to each reduction. This is the general setting for the study of the dynamics of the reduced system in order to obtain information about the original one. (c) 2001 American Institute of Physics.     

Supersymmetric SU C (4) × SU L (2) × SU R (2) and spontaneous breakdown of symmetries

A supersymmetric version of the left right symmetric partial unification group SU _C (4) × SU _L (2) × SU _R (2) is presented. The spontaneous breakdown of gauge symmetry in a favourable chain of descent has been studied in detail. The mass spectra have been calculated. The method of O’Raifeartaigh has been used to break supersymmetry. The lifting of degeneracy of mass levels between physical multiplets has been shown to occur due to radiative corrections.