## 1 Introduction

^{3N}. The product of initial and final probabilities characterizing the required low-entropy may then represent a phase space volume smaller than a Planck cell – thus indicating the absence of any solution.

## 2 Retarded and Advanced Fields

## 3 Cat Maps

## 4 (Anti-)Causality

_{0}(see Fig. 1). (Finding much lower entropy values numerically would be too time-consuming for this large number of particles.) Unfortunately, the results do not confirm Schulman’s claim that these solutions are “affected” by the perturbation only in the direction away from the relevant low entropy boundary (that is, towards the “physical future”) [3]. Evidently, this concept of causality, defined by means of perturbations, is insufficient. The very concept of a “perturbation” seems to be ill-defined for two-time boundary conditions.

**Figure 1.**Four random two-time boundary solutions (forming a narrow bundle in the diagram) are compared with two other ones, selected by trial and error for their slightly lower entropy values at t

_{0}= 200 or t

_{0}= −200. Values for t < 0 are identical with those at t

_{f}− t = 200.000 − t, although the final condition is actually irrelevant in the range shown. Entropy scattering around t = 1300 is accidental. (See Appendix of [7] for details of the model and an elementary Mathematica program for your convenience.)

_{0}). Both boundary conditions are then violated by the new solution arising from this perturbed state, used as a complete “initial” condition. The results (shown in Fig. 2) are now most dramatic towards the former “past”, demonstrating the relevance of fine-grained information (similar to Borel’s example) for correctly calculating “backwards in time”. Deviations from the original two-time boundary solution close to t

_{0}also on the right are due to the fact that the coarse graining assumed in this model does not define a very good master equation (as discussed in [7]).

**Figure 2.**Time-symmetric “effect” on a solution “caused” by a perturbation of the microscopic state at time t

_{0}= 200, defined by an accidental entropy minimum at this time. The perturbed solution drastically violates both boundary conditions that were valid for the unperturbed solution.

## 5 Cosmology and Gravitation

## 6 Quantum Aspects

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