Classical Solutions to Double Oscillator Field Theory

4 Abstract The present paper aims at studying the physics of the double oscillator field theory from a classical, non-perturbative perspective. In introducing the topic a short but to the point treatment of the physics of the quantum mechanics double oscillator is presented. A subsequent section of the paper summarizes the current state of research in the self-interacting scalar fields theory (with an emphasis on the specific physical systems generating spontaneous symmetry breaking). We further discuss in a non-perturbative framework a few classes of solutions to the double oscillator field theory. The focus is on analyzing the classes of upshots for the equation )()( 2222 rJaSignmm t =−+∇−∂ φφφφ , where J(r) is a source of the form )(0 rQ & δ . We particularize this problem, looking at solutions of the form ax ±= )(ηφ . The classes of solutions are separately discussed in function of their complexity. Several results in this or in related domains are also acknowledged and further applied where possible. The paper introduces and leaves open the issue concerning similarity between the double oscillator field theory and the 4λφ theory. 1. Introduction We will start this paper by recalling that when we quantize the harmonic oscillator the creation operator evolves in a way that completely mimics the evolution of the classical solutions [6]. This is a settled result in physics and does not need detailed argumentation. The essence of the proof lies in the fact that since coherent states are built from the vacuum by hitting it with exponentiated creation operators, it’s also true that coherent states evolve in a way which completely mimics the evolution of the corresponding classical solutions. We contend that there is no reason to think that such an argument would not apply as well for the quantum field theory. Consequently it is worth trying to find classical solutions to the double oscillator field theory, for instance. Our target is to find classical solutions to the double well oscillator field theory, leaving a discussion open on the similarity between this theory and the theory of the self-interacting scalar fields ( 4λφ ). To illustrate this analogy, we will draw your attention to the explicit expression and behavior of the potentials in the case of the quantum mechanics double oscillator on the one hand and the 4λφ field theory on the other hand. While in the first situation the potential will be of the form