# Arrow and/or Pendulum of Time

I'm an interested amateur who DEFINITELY lacks the math and physics background I need to even set this problem up properly, much less solve it--but I'd appreciate some help in turning a "hunch" into an answerable question.

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?

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No takers yet? Let me add a little more definition to the problem.

Start with an extremely simplified “unobserved system” that consists of a single particle moving at some initial velocity towards two slits, with some kind of screen on the other side that can detect the particle. Instead of treating time as the independent variable and three spatial dimensions as dependent variables, specify a new parametric variable “p.” The particle’s position and velocity are defined at p=0. Let the particle’s position and velocity at p=1 be randomly selected from all possible states within the limits of Heisenberg’s uncertainty principle, and then repeat in a “random walk” pattern without any preference for increasing t. Continue this until something causes the system to “decohere.”

If we were to plot the path of this paramaterized particle in x, y, and t we would see a “fractal” shape–I think it would look a bit like an old-fashioned shaving brush.

As far as I can tell, this method would generate the interference patterns one sees in a double-slit experiment. In a system with a detector on one of the slits, the system would decohere the first time the p parameter randomly got the particle out to the detector. The moment of decoherence would reset the system with the particle now located near one slit, and all possible paths of that particle would now wind up on the screen at the far end looking just like a single particle going through a single slit.

By contrast, a system without a detector at one of the slits would yield an infinite number of different paths from the original starting point to the screen at the far end–but those paths would show just the kind of interference patters that make double-slit experiments so interesting.

What I can't do on my own is define the mathematics of this parameterized particle with enough precision to determine whether it yields results that are identical to the double-slit experiments. It may be that it is true by definition--the math that defines my parameterized particle should be essentially the same as quantum mechanics produces. The only difference will be that my parameterized path looks like a fractal while the standard model of a quantum wavefunction is a probability distribution. These two ways of representing the particle might well be indistinguishable.

DrChinese