A novel method to solve functional differential equations application to quantum field theory

Abstract

We present a method to solve differential equations containing the variational operator as the derivation operation. We call them variational differential equations (VDE). The solution to a VDE should be a function containing the derivatives, with respect to the base space coordinates, of the fields up to a generic orders: as-th-order function. The variational operator doubles the order of the function on which it acts. Therefore, in order to make compatible the orders of the different terms appearing in a VDE, the solution should be a function containing the derivatives of the fields at all orders. But this takes us again back to functional methods. In order to avoid this, one must restrict the considerations, in the case of second-order VDEs, to the space ofs-th-order functions on which the variational operator acts transitively. We characterize these functions for a one-dimensional base space for the first-and second-order cases. These functions turn out to be polynomial in the highest-order derivatives of the fields with functions of the lower-order derivatives as coefficients. The VDEs reduce to a system of coupled partial differential equations for the coefficients we mentioned above. The importance of the method lies on the fact that the solution to VDEs are in a one-to-one correspondence with the solutions of functional differential equations. The previous method finds direct applications in quantum field theory, where the Schrodinger equation plays a central role. Since the Schrodinger equation is reduced to a system of coupled partial differential equations, this provides a nonperturbative scheme for quantum field theory. As an example we consider the massless scalar field.