Dynamical breakdown of time reversal invariance and causality

Irreversibility and acausality of a subsystem are established in exactly soluble harmonic models with reversible and causal dynamics. It is shown that initial conditions, imposed on some dynamical degrees of freedom may break time reversal invariance for other degrees of freedom. This happens if observations carried out in any large but finite amount of time can not resolve the spectrum of the eliminated degrees of freedom, namely, when the spectrum has a condensation point at the ground state. Acausality follows due to the dominance of the dynamics by almost time-independent modes.

Figure 1

The apparent spectral function ${\rho }_m^{(\sigma )}$ for $m=1$, ${\omega }_j={\omega }_0/j$, $N=20$, (a) $T=2000$, (b) $T=700$, (c) $T=100$ as functions of $\omega /{\omega }_0$.