Investigations on pathologies in higher spin field theories

The present thesis deals with the studies on certain aspects of pathological higher field theories .It brings to light some new abnormalities and new samples of abnormal theories and also puts forward a novel approach towards the construction of trouble free theories

Studies in quantum field theory at finite temperature

The thesis deals with certain quantum field systems exhibiting spontaneous symmetry breaking and their response to temperature. These models find application in diverse branches such as particle physics, solid state physics and non~linear optics. The nature of phase transition that these systems may undergo is also investigated. The thesis contains seven chapters. The first chapter is introductory and gives a brief account of the various phenomena associated with spontaneous symmetry breaking. The chapter closes with anote on the effect of temperature on quantum field systems. In chapter 2, the spontaneous symmetry breaking phenomena are reviewed in more detail. Chapter 3, deals with the formulation of ordinary and generalised sine-Gordon field theories on a lattice and the study of the nature of phase transition occurring in these systems. In chapter 4, the effect of temperature on these models is studied, using the effective potential method. Chapter 5 is a continuation of this study for another model, viz, the m6 model. The nature of phase transition is also studied. Chapters 5 and 6 constitute a report of the investigations on the behaviour of coupling constants under thermal excitation D1 $4 theory, scalar electrodynamics, abelian and non-abelian gauge theories

Characterization of Probability Distributions Using the Residual Entropy Function

The present study on the characterization of probability distributions using the residual entropy function. The concept of entropy is extensively used in literature as a quantitative measure of uncertainty associated with a random phenomenon. The commonly used life time models in reliability Theory are exponential distribution, Pareto distribution, Beta distribution, Weibull distribution and gamma distribution. Several characterization theorems are obtained for the above models using reliability concepts such as failure rate, mean residual life function, vitality function, variance residual life function etc. Most of the works on characterization of distributions in the reliability context centers around the failure rate or the residual life function. The important aspect of interest in the study of entropy is that of locating distributions for which the shannon’s entropy is maximum subject to certain restrictions on the underlying random variable. The geometric vitality function and examine its properties. It is established that the geometric vitality function determines the distribution uniquely. The problem of averaging the residual entropy function is examined, and also the truncated form version of entropies of higher order are defined. In this study it is established that the residual entropy function determines the distribution uniquely and that the constancy of the same is characteristics to the geometric distribution

Studies on Non-Resonant Multiphoton Processes in Atomic Hydrogen

In the present thesis we have formulated the Dalgarno-Lewis procedure for two-and three-photon processes and an elegant alternate expressions are derived. Starting from a brief review on various multiphoton processes we have discussed the difficulties coming in the perturbative treatment of multiphoton processes. A small discussion on various available methods for studying multiphoton processes are presented in chapter 2. These theoretical treatments mainly concentrate on the evaluation of the higher order matrix elements coming in the perturbation theory. In chapter 3 we have described the use of Dalgarno-Lewis procedure and its implimentation on second order matrix elements. The analytical expressions for twophoton transition amplitude, two-photon ionization cross section, dipole dynamic polarizability and Kramers-Heiseberg are obtained in a unified manner. Fourth chapter is an extension of the implicit summation technique presented in chapter 3. We have clearly mentioned the advantage of our method, especially the analytical continuation of the relevant expressions suited for various values of radiation frequency which is also used for efficient numerical analysis. A possible extension of the work is to study various multiphoton processcs from the stark shifted first excited states of hydrogen atom. We can also extend this procedure for studying multiphoton processes in alkali atoms as well as Rydberg atoms. Also, instead of going for analytical expressions, one can try a complete numerical evaluation of the higher order matrix elements using this procedure.

Acoustic radiation and scattering from cylindrical bodies and analysis of transducer arrays

New mathematical methods to analytically investigate linear acoustic radiation and scattering from cylindrical bodies and transducer arrays are presented. Three problems of interest involving cylinders in an infinite fluid are studied. In all the three problems, the Helmholtz equation is used to model propagation through the fluid and the beam patterns of arrays of transducers are studied. In the first problem, a method is presented to determine the omni-directional and directional far-field pressures radiated by a cylindrical transducer array in an infinite rigid cylindrical baffle. The solution to the Helmholtz equation and the displacement continuity condition at the interface between the array and the surrounding water are used to determine the pressure. The displacement of the surface of each transducer is in the direction of the normal to the array and is assumed to be uniform. Expressions are derived for the pressure radiated by a sector of the array vibrating in-phase, the entire array vibrating in-phase, and a sector of the array phase-shaded to simulate radiation from a rectangular piston. It is shown that the uniform displacement required for generating a source level of 220 dB ref. μPa @ 1m that is omni directional in the azimuthal plane is in the order of 1 micron for typical arrays. Numerical results are presented to show that there is only a small difference between the on-axis pressures radiated by phased cylindrical arrays and planar arrays. The problem is of interest because cylindrical arrays of projectors are often used to search for underwater objects. In the second problem, the errors, when using data-independent, classical, energy and split beam correlation methods, in finding the direction of arrival (DOA) of a plane acoustic wave, caused by the presence of a solid circular elastic cylindrical stiffener near a linear array of hydrophones, are investigated. Scattering from the effectively infinite cylinder is modeled using the exact axisymmetric equations of motion and the total pressures at the hydrophone locations are computed. The effect of the radius of the cylinder, a, the distance between the cylinder and the array, b, the number of hydrophones in the array, 2H, and the angle of incidence of the wave, α, on the error in finding the DOA are illustrated using numerical results. For an array that is about 30 times the wavelength and for small angles of incidence (α<10), the error in finding the DOA using the energy method is less than that using the split beam correlation method with beam steered to α; and in some cases, the error increases when b increases; and the errors in finding the DOA using the energy method and the split beam correlation method with beam steered to α vary approximately as a7 / 4 . The problem is of interest because elastic stiffeners – in nearly acoustically transparent sonar domes that are used to protect arrays of transducers – scatter waves that are incident on it and cause an error in the estimated direction of arrival of the wave. In the third problem, a high-frequency ray-acoustics method is presented and used to determine the interior pressure field when a plane wave is normally incident on a fluid cylinder embedded in another infinite fluid. The pressure field is determined by using geometrical and physical acoustics. The interior pressure is expressed as the sum of the pressures due to all rays that pass through a point. Numerical results are presented for ka = 20 to 100 where k is the acoustic wavenumber of the exterior fluid and a is the radius of the cylinder. The results are in good agreement with those obtained using field theory. The directional responses, to the plane wave, of sectors of a circular array of uniformly distributed hydrophones in the embedded cylinder are then computed. The sectors are used to simulate linear arrays with uniformly distributed normals by using delays. The directional responses are compared with the output from an array in an infinite homogenous fluid. These outputs are of interest as they are used to determine the direction of arrival of the plane wave. Numerical results are presented for a circular array with 32 hydrophones and 12 hydrophones in each sector. The problem is of interest because arrays of hydrophones are housed inside sonar domes and acoustic plane waves from distant sources are scattered by the dome filled with fresh water and cause deterioration in the performance of the array.

Properties of Equilibrium Distributions of Order n

The present work is intended to discuss various properties and reliability aspects of higher order equilibrium distributions in continuous, discrete and multivariate cases, which contribute to the study on equilibrium distributions. At first, we have to study and consolidate the existing literature on equilibrium distributions. For this we need some basic concepts in reliability. These are being discussed in the 2nd chapter, In Chapter 3, some identities connecting the failure rate functions and moments of residual life of the univariate, non-negative continuous equilibrium distributions of higher order and that of the baseline distribution are derived. These identities are then used to characterize the generalized Pareto model, mixture of exponentials and gamma distribution. An approach using the characteristic functions is also discussed with illustrations. Moreover, characterizations of ageing classes using stochastic orders has been discussed. Part of the results of this chapter has been reported in Nair and Preeth (2009). Various properties of equilibrium distributions of non-negative discrete univariate random variables are discussed in Chapter 4. Then some characterizations of the geo- metric, Waring and negative hyper-geometric distributions are presented. Moreover, the ageing properties of the original distribution and nth order equilibrium distribu- tions are compared. Part of the results of this chapter have been reported in Nair, Sankaran and Preeth (2012). Chapter 5 is a continuation of Chapter 4. Here, several conditions, in terms of stochastic orders connecting the baseline and its equilibrium distributions are derived. These conditions can be used to rede_ne certain ageing notions. Then equilibrium distributions of two random variables are compared in terms of various stochastic orders that have implications in reliability applications. In Chapter 6, we make two approaches to de_ne multivariate equilibrium distribu- tions of order n. Then various properties including characterizations of higher order equilibrium distributions are presented. Part of the results of this chapter have been reported in Nair and Preeth (2008). The Thesis is concluded in Chapter 7. A discussion on further studies on equilib- rium distributions is also made in this chapter.