Classical Field Theory with Compact Time How does one find solutions to a classical field theory (say, $\phi^4$ theory) when the time dimension is periodic (so in $\mathbb{R}^n \times T^1$ or other compact topology), especially if the initial conditions have some randomness (so we're dealing with statistical physics)? It seems to me that the solutions should be very unstable to small perturbations and that there should be a cascade to smaller length scales, but I'm not sure how to show this. Additionally, solutions may be periodic because the time dimension is compact and the nature of the solution may depend on the choice of time topology, so there might be some 'crystalline' structures that would be absent in more general spacetimes with CTCs.
 A: Let time be periodic with period $1$. If you fix $\phi(0)$ (perhaps randomly) and your time evolution is deterministic, then $\phi(1)$ is already fully determined. A periodic time dimension implies $\phi(0) = \phi(1)$. This will usually be true for some choices of $\phi(0)$ and false for other choices; in the latter cases, no consistent solution exists. In other words, the effect of a periodic time dimension is purely to restrict the set of "allowed" initial conditions.
There's still an interesting problem left, which is that of finding the set of allowed initial conditions. Since $T^1$ and $\mathbb{R}$ are the only options for time topology, this problem is equivalent to that of finding periodic solutions of the analogous field theory on $\mathbb{R}^{n} \times \mathbb{R}$ (with a specified period).
Cascades to arbitrarily small length scales are generally irreversible, which means they would be impossible in this system. A transfer of energy from large length scales to small (but not arbitrarily small) length scales, followed by an inverse cascade transferring the energy back up, is possible in principle. However, if it happens in a system with $T^1$ time, it will also happen in the analogous system with $\mathbb{R}$ time, so the behavior shouldn't really be thought of as a result of the time topology.
