## Exact solutions of classical scalar field equations

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Frasca, Marco

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We give a class of exact solutions of quartic scalar field theories. These
solutions prove to be interesting as are characterized by the production of
mass contributions arising from the nonlinear terms while maintaining a
wave-like behavior. So, a quartic massless equation has a nonlinear wave
solution with a dispersion relation of a massive wave and a quartic scalar
theory gets its mass term renormalized in the dispersion relation through a
term depending on the coupling and an integration constant. When spontaneous
breaking of symmetry is considered, such wave-like solutions show how a mass
term with the wrong sign and the nonlinearity give rise to a proper dispersion
relation. These latter solutions do not change the sign maintaining the
property of the selected value of the equilibrium state. Then, we use these
solutions to obtain a quantum field theory for the case of a quartic massless
field. We get the propagator from a first order correction showing that is
consistent in the limit of a very large coupling. The spectrum of a massless
quartic scalar field theory is then provided. From this we can conclude that,
for an infinite countable number of exact classical solutions, there exist an
infinite number of equivalent quantum field theories that are trivial in the
limit of the coupling going to infinity.

Comment: 7 pages, no figures. Added proof of existence of a zero mode and two more references. Accepted for publication in Journal of Nonlinear Mathematical Physics

Comment: 7 pages, no figures. Added proof of existence of a zero mode and two more references. Accepted for publication in Journal of Nonlinear Mathematical Physics

##### Keywords

Mathematical Physics, High Energy Physics - Phenomenology, High Energy Physics - Theory, 35C05, 35C20