Theoretical Physics Philip B. Burt

1. Huygens' Principle in quantum theory: Huygens' Principle generalizes to quantum scattering theory using space time transition amplitudes.  An application of these ideas leads to non perturbative analysis of scattering problems.
2. Properly Normalized quantum field theories: These are Intrinsically nonlinear, non perturbative quantum field theories which describe interactions without the inconsistencies present in perturbative theories.
3. Non perturbative solution of nonlinear differential equations:  The field equations of interacting quantum field theories are intrinsically nonlinear, meaning that the nonlinearity is always present.  A new, non perturbative technique for solving these equations is being studied.

Selected Publications: Huygens' Principle and the Unification of Dynamics, Proceedings of the Conference on Theoretical Physics, University of Georgia September 12, 1997(George Strobel, editor).  In this paper Huygens' Principle in its most general form is used to express the quantum scattering dynamics of an interacting system with an infinite, changing number of degrees of freedom.  The results are non perturbative, embodying the principles of proper normalization.  A new concept, the relative probability amplitude is introduced to describe intrinsically nonlinear interactions.

Quantum Mechanics and Nonlinear Waves (Harwood Academic Press, N.Y.1981).This monograph describes the research on intrinsically nonlinear quantum field theories, beginning with a re-examination of the fundamental ideas of non interacting theories.  The central role of the superposition principle of quantum theory is emphasized in the subsequent analysis of non perturbative, self interacting quantum field theories.  The new concept of proper normalization of interacting quantum field theories eliminates the inconsistencies of perturbative renormalization, eliminating the infinities which are common to such theories.  An extensive discussion of non perturbative solution of nonlinear field equations includes some of the most important nonlinear field equations occurring in physics.  Applications to elementary particle physics, superconductivity and other topics of high current interest form an integral part of the book.

Non perturbative Solution of Nonlinear Field Equations, Nuovo Cimento 100B,43,1987.  In this paper a new technique for solving nonlinear ordinary and partial differential equations independent of traditional methods is explored.  This technique employs a transformation of second order differential equations to a set first order generalizations of the Riccati equation.  Approximations are introduced in these new equations.  Applications to the Lane-Emden, nonlinear Klein Gordon and other equations are given.

Boundary Conditions in Quantum Field  Theories, Foundations of Physics 23, 965,1993.  From its inception quantum electrodynamics and other quantum field theories have contained serious inconsistencies.  The source of these inconsistencies has been the assumption that interactions among systems can develop out of non interacting systems.  In this paper quantum electrodynamics and some other important quantum field theories are shown to be persistently interacting.  The interaction strength must always appear in the solutions to the field equations.
 

Some Graduates Of the Program:
Mesgun Sebhatu:Dissertation: "Quantized Solitary Waves in Nonlinear Field Theories and Their Application to Nucleon-Nucleon and Nucleon-Antinucleon Interactions."  Presently Professor of Physics, Winthrop University.  Internationally recognized for his work in low energy nuclear physics and, in particular, for the development of SWEPs, Solitary Wave Exchange Potentials, Dr. Sebhatu received a King-Parks-Chavez Fellowship for sabbatical research in nuclear physics in 1991-92.

Beverley A.P.Taylor: dissertation;"Solitary Waves in Relativistically Invariant Field Theories and Their Applications to the Electromagnetic Interactions of Hadrons."  Presently Professor of Physics, Miami University-Hamilton.  Internationally recognized for innovations in physics education, Dr. Taylor was recognized by the American Association of Physics Teachers with its  Distinguished Service Citation, 1996.

William Ditto:Dissertation: "A Study of the Properly Normalized Lamb Shift and a New Non perturbative Method of Solution to Nonlinear Field equations Using Continued fractions."  Presently Associate Professor of Physics, Georgia Institute of Technology, Dr. Ditto has received significant international recognition for his work in chaos.  An excerpt from his research is a feature of the 1998 American Physical Society Calendar.

Graduate Student
Julie Talbot: M.S. Thesis,"Variational Calculation of PSWEP parameters,"  Ph.D. research in progress on the nonperturbative calculation of radiative corrections to the magnetic moments of charged vector mesons.

Some Earlier Work:

• Unstable Cyclotron Instabilities in a Bounded Plasma Shell, Physics of Fluids 4,1412,1961(with E.G.Harris). The first detailed study of instabilities in finite plasma, an instability was predicted.  It was later observed in many laboratories  and named the "Burt-Harris instability."
• Perturbation Theory For Toroidal Plasma, Physics Letters 34A,47,1970.  An important question in the analysis of plasma in toroidal geometries concerns the appropriate parameter to use in approximations.  In this paper the parameter is identified as powers of the aspect ratio of the torus multiplied by the logarithm of the aspect ratio.
• Integral  Approximations in Potential Scattering, Journal of Mathematical Physics12,2275,1971.  This paper introduced a non perturbative approximation for the scattering amplitude which encompasses both the WKB and the Born approximation utilizing  force as an essential concept.
• Solution of Nonlinear Partial Differential Equations from Base Equations, Journal of Mathematical Analysis and Applications 47,520,1974(with James L. Reid).  In this paper a large class of nonlinear partial differential equations are solved exactly in terms of solutions of related linear partial differential equations.  This is the first in a series of papersi investigating, as it happens, a very general property of nonlinear partial differential equations.  Since the solutions are exact valuable insights are obtained into the behavior of these systems when relevant parameters are varied.
• Solitary Waves in Nonlinear Field Theories, Physical Review Letters 32,1080,1974.  This paper presents a very surprising result.  Nonlinear field equations similar to those of gauge field theories have exact solutions analogous to classical solitary waves.  The solutions are functions of solutions of the linear Klein Gordon equation.
• Exact, Multiple Soliton Solutions of the Double Sine Gordon Equation, Proceeding of the Royal Society of London A359,479,1978 and Multidimensional Solutions of the Sine Gordon Equation, Physics Letters 82B,423,1979.  In these papers multidimensional solutions of two specific nonlinear partial differential equations are presented.  These solutions represent several noncolinear plane fronted waves which collide with only a phase shift.  As the wave vectors of the solutions become parallel, the waves collapse into a single wave, giving rise to the name "collapson" for the systems.  Further discussion of the solutions can be found in the monograph.

Substantial evidence is accumulating for the existence of coherent, spin zero meson excitations. In this note some properties of these excitations are discussed and a comparison of theoretical predictions with recent experiments is given.

Coherent spin zero excitations are described by an intrinsically nonlinear quantum field theory described by the field equation (1,2)



This equation has exact solutions describing coherent quantum fields for (2,3). The interesting case for this note is p=3/2. The propagator for these solutions for this case is (1,2)



where has no poles for k real. The poles of this propagator give the mass formula



when we apply this theory to neutral pions. These are the masses of the excited, coherent states of the system for all positive integers n. In the table, a comparison of the theoretical predictions with recent experimental observations is made (4,5,6).

There are several important points about these excitations which should be emphasized. In the first place, they are non perturbative. The solutions of the field equations are non perturbative, hence the mass spectrum itself is. The spectrum depends only on the exponent in the interaction, not on the coupling constants. This seems to be in good agreement with the experimental results, with perhaps small corrections necessary to improve the fit. Finally, the fact that these are coherent excitations of a single system means that they will appear in virtual processes as a part of the system propagator. The significance of this property is currently being explored in the context of the standard model (7).

n
0	135	135	----
1	540	547	.017
2	945	958	.032
3	1350	1385*	.086
4	1755	1760*	.012
5	2160	2150^	.025
6	2565	----	----
7	2970	2980	.025
8	3375	3417	.103