The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.

Retrocausal models of quantum mechanics add further weight to the conflict between causality and the possible existence of free will. We analyze a simple closed causal loop ensuing from the interaction between two systems with opposing thermodynamic time arrows, such that each system can forecast future events for the other. The loop is avoided by the fact that the choice to abort an event thus forecasted leads to the destruction of the forecaster's past. Physical law therefore enables prophecy of future events only as long as this prophecy is not revealed to a free agent who can otherwise render it false. This resolution is demonstrated on an earlier finding derived from the two-state vector formalism, where a weak measurement's outcome anticipates a future choice, yet this anticipation becomes apparent only after the choice has been actually made. To quantify this assertion, weak information is described in terms of Fisher information. We conclude that an already existing future does not exclude free will nor invoke causal paradoxes. On the quantum level, particles can be thought of as weakly interacting according to their past and future states, but causality remains intact as long as the future is masked by quantum indeterminism. Quanta 2016; 5: 53–60.