An approach to a relativistic strong coupling theory

Abstract

As is well known, the vacuum expectation value of the S-matrix provides us with the knowledge of all Feynman amplitudes and is equivalent to the S-matrix itself, if we introduce appropriate source functions. As the vacuum expectation value, which we call S VAC, is a c-number quantity, it is often more convenient to treat S vac than the S-matrix itself. Here we treat three cases: (i) two neutral scalar fields, ψ and φ, interacting through the interaction gψφ; (ii) the charged scalar field interacting with an external neutral scalar field φ through gψ ∗ψφ ; (iii) interacting charged and neutral scalar fields ψ and φ through the interaction gψ ∗ψφ . We derive several functional differential equations which S vac should satisfy and in which the coupling constant g and the source functions are variables. Then we obtain the solution in the form of an inverse power expansion of the coupling constant by solving the functional differential equations. In case (i), we can sum up the inverse power series and the result agrees with the exact solution. In case (ii), the only relevant Feynman amplitude is the modified propagator for the ψ-field, which is also expanded into an inverse power series of the coupling constant. In case (iii), S vac is expressed in a form involving only functional differentiations, but no more path integrals. If we carry out the functional differentiations, the result will also be an inverse power series of the coupling constant. Extension to other cases is also discussed.