Accession Number : AD0713998

Title :   (Phi sup 4)sub 4 - Wavepackets.

Descriptive Note : Technical rept.,

Corporate Author : BROWN UNIV PROVIDENCE R I DIV OF APPLIED MATHEMATICS

Personal Author(s) : Bisshopp,Frederic

Report Date : SEP 1970

Pagination or Media Count : 37

Abstract : It is shown here that the nonlinear wave equation Phi sub TT - del squared(Phi) + lambda (Phi cubed) = 0 which defines the unquantized, unrenormalized (Phi sup 4)sub 4-theory has spherically symmetric solutions that are localized in space. When lambda is negative the solutions can be either steady or oscillatory (in time), and when lambda is positive only the oscillatory solutions exist. Two measures of the total energy of a wavepacket are discussed. One, the integral of Phi squared over all space, is divergent in all cases; the other, the integral of Phi sub 4 over all space, is divergent in general, but has a finite value when the class of solutions is limited to those for which the time average of Phi over a period of the oscillation is zero. For these solutions mass renormalization is unnecessary if we choose to identify mass of a wavepacket with the second measure of total energy. Perhaps quantization is also unnecessary, for, in an approximation, the oscillatory wavepackets are governed by a nonlinear Schrodinger equation. (Author)

Descriptors :   (*QUANTUM THEORY, WAVE FUNCTIONS), DIFFERENTIAL EQUATIONS, INTEGRALS, DISPERSION RELATIONS, ASYMPTOTIC SERIES, DISTRIBUTION FUNCTIONS, PROBABILITY DENSITY FUNCTIONS, TRANSFORMATIONS(MATHEMATICS), NONLINEAR DIFFERENTIAL EQUATIONS

Subject Categories : Statistics and Probability
      Quantum Theory and Relativity

Distribution Statement : APPROVED FOR PUBLIC RELEASE