May3-03, 12:38 AM
In classical physics, we have the naive notion that, given the necessary parametres, and given the correct mathematical framework, one might deduce (if one had nothing better to do and several eons to perform the calculations) the events of the future. Put differently, given one set of initial conditions, the outcomes of the universe and the course of history are uniquely determined. This is, to some extent, philosophically satisfying.
Of course, qm soon put away such foolish notions to death. So at any point in time, we may only answer "what is the probability the universe will turn out this way", not "will the universe turn out this way". Similarly, suppose that excessive weed-smoking has killed off all the hippocampal cells of our otherwise reasonably intelligent brains. Then by running the rules of mathematics in a world where the time axis has been reversed, we can similarly only answer questions such as "what is the probability that I had frosted corn-flakes for breakfast this morning?", but not "what did I actually have for breakfast?"
On the other hand, suppose we do possess memory. The mathematics has not changed, but we do know, with some small uncertainty, the events which we are not able to predict (postdict?)
Let me make this more precise. Let us fix ourselves at time 0. The further ahead we look in time, the less certain we are about whether an event will occur. This should be obvious; the longer a time interval we allow ourselves, the greater the number of outcomes the system (the universe) can take at the end of that time, and so the less probable each particular outcome is. For example, let our simplified universe be a single electron confined in a sphere of radius one (the electron will not leak out of the confinement as in usual confinement situations; we do not allow space to exist outside the sphere). Suppose we have just performed an information which tells us, with uncertainty dx, the position of the electron at time t=0. At t=1, the electron has some wavefunction psi. There is now a non0 probability that it can be found in any region of finite volume inside the sphere, whereas at t=0, we had that the probability of finding it outside of (x+/-dx, y+/-dx, z+/-dx) was 0. The wavefunction is like gas with no container; it spills everywhere and makes a great big mess. As time goes on, the wavefunction becomes more and more even. That is, say, at t=10^10^400, p(A)/p(B) is roughly v(A)/v(B), where A, B are regions in the sphere, p(X) is the probability of finding the electron in region X, and v(X) is the volume of X. So we see that as time goes on, uncertainty increases.
Not so if time goes backwards. Since the math is the same, using qm we conclude that at t=-10^10^400, the wavefunc exhibits the same behaviour as that described above. However, suppose that we had been doing experiments every 10 seconds to determine with the same uncertainty dx the position of the electron. Our experiments tell us with an uncertainty whose magnitude is bounded the position of the e, while our calculations tell us with increasingly less precision where the electron is. (The actual wavefunction will not be as we have described, since we must take into account the presence of an experimental physicist and his equipment; but the gist is that uncertainty as postdicted by theory increases without bound, while the uncertainty as remembered by memory remains bounded by the largest uncertainty over all experiments.)
So... uh... I think I had a point when I started rambling... I no longer have something shocking to say... Instead I will just say that this seems strange to me.
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