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An Investigation of the Low Equation and the Chew-Mandelstam Equations


Wilson, Kenneth Geddes (1961) An Investigation of the Low Equation and the Chew-Mandelstam Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VCDA-4Q31.


We show that for scalar theories without a cutoff the asymptotic form for large energies ω of the perturbation expansion of the Low equation in the one-meson approximation is a double power series in the coupling constant g² and lnω. The method applied by Gell-Mann and Low to the photon propagator in electrodynamics is used to show that if the crossing matrix has only one negative eigenvalue this power series reduces to a series in a single variable
g = g₁ / 1 - g₁θ/π lnω
where g₁ = g₁(g²), and θ is a constant. The series in y is evaluated for several interactions and for some crossing matrices with no physical interpretation; for the former the series is a simple algebraic function, while for the latter it usually diverges for all values of y ≠ o. We obtain the exact solution of the one-meson approximation for the symmetric scalar pion-nucleon interaction; it is a multiple-valued function of g². We compare perturbation approximation to the determinantal function of Baker and the cotangent of the phase shift, with the numerical solution of Salzman, for the symmetric pseudoscalar theory with a cutoff; they are found to be often accurate to a few percent. We show that Chew and Mandelstam's approximate equations for pion-pion scattering have no solution for positive coupling λ, and that the perturbation expansion of the solution of their equations for isotopic spin o pions diverges for λ ≠ o. For pions with I = 1 we present calculations of the perturbation expansion to sixth order in λ.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Physics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Awards:The Nobel Prize in Physics, 1982.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Gell-Mann, Murray
Group:Nobel Prize
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1961
Additional Information:The Nobel Prize in Physics 1982 was awarded to Kenneth G. Wilson "for his theory for critical phenomena in connection with phase transitions."
Record Number:CaltechETD:etd-10222002-104500
Persistent URL:
Related URLs:
URLURL TypeDescription Prize announcement (Physics, 1982)
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4205
Deposited By: Imported from ETD-db
Deposited On:22 Oct 2002
Last Modified:06 Oct 2020 01:13

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