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From what I've heard about the "arrow of time" problem, I've begun to wonder how particles would behave if "unobserved" systems could go backwards in time as well as forward. I label this notion the "pendulum of time." In essence, a system of entangled entities would undergo some sort of "random walk" in time and space, with no more preference for a path "forward" in time than for a preference to go "eastward" in space.

I imagine that such a system would "oscillate" in time and space until something caused it to decohere. After decoherence, the system would begin to oscillate again, moving forward in time and then back to the moment of decoherence, like a pendulum that swings back and forth and side to side, but always goes past the bottom of the swing, which, in this model, is the previous moment of decoherence. The "pendulum" could not revert to a point in time BEFORE the decoherence, but it could "explore" all physically possible paths AFTER it. (Sorry to use English to describe something that would be unambiguous as a mathematical formula, but that's why I'm here asking for help.)

My core question is whether such a "pendulum of time" would be consistent with double-slit experiments. As I understand the findings of modern physics, a particle passing through two slits exhibits a wave-like interference pattern. I am guessing that an unobserved particle in a "pendulum of time" model MIGHT produce exactly the same interference pattern. An observed particle, by contrast, would decohere after a single "swing" and would not produce an interference pattern.

It seems to me that I've heard the term "Poincare distribution" used in context of quantum wavefunctions. If I knew what a "Poincare distribution" was I'd try to figure out whether it might apply to this "pendulum of time" behavior.

Is there a mathematician in the house?