Classicalλφ4theory in 3 + 1 dimensions

We present a numerical study of the classical $\lambda {\phi }^4$ theory in 3 + 1 dimensions, starting from a spherically symmetric configuration with energy concentrated in a few high-frequency modes (such a configuration might be produced in a high-energy collision). We find that while the energy spreads out into modes on either side of the initial modes, at a rate proportional to the coupling constant $\lambda$, there is no cascading of energy into lower-frequency modes. The number of "quanta" in the field is nearly constant (to about 0.1%) in time. This is explained in terms of classical perturbation theory, which shows this number to be an adiabatic invariant apart from small high-frequency fluctuations. Our results suggest that the decay of coherent field configurations proceeds much more slowly than the simplest estimate $\tau \sim \frac{1}{\Delta \omega }$.