Otherwise your games with the “definition” of initial and final states and with the sign of t are completely immaterial. “Initial” and “final” states are, according to logic, qualitatively different things, and the usual convention for the sign of t is that t_{initial} < t_{final}. But I have never even used this convention.

Even if I had, it wouldn’t matter. One can easily rewrite all proofs to the opposite convention by replacing t with −t; all those things are physically vacuous. The non-vacuous claim is that the future and past don’t play symmetric roles in logic.

— Physics Stack Exchange

— Jan 25 ’12 at 9:49

— Lubos Motl

2013.09.26 Thursday ACHK

That’s why the retrodicted probabilities of initial states pi=P(Hi) always depend on some subjective choices. What we think about the past inevitably depends on other things we have learned about the past. This is a totally new property of retrodictions that doesn’t exist for predictions. Predictions may be probabilistic (and in quantum mechanics and statistical physics, they are inevitably “just” probabilistic) but the predicted probabilities are objectively calculable for certain input data. The formulae that objectively determine these probabilities are known as the laws of physics. But the retrodicted probabilities of the past are not only probabilistic; their values inevitably depend on the subjective knowledge, too!

— Prediction isn’t the right method to learn about the past

— Lubos Motl

The future is not fixed, in the sense that the present chooses among the potential futures to evolve to. Since each higher entropy macrostate by definition is corresponding to more microstates, it has a higher probability to occur. 

However, the past is fixed. The probability is subjective probability. The present cannot “choose” among the “potential” pasts from which the present is evolved. The fact that there are more one possible pasts is due to your subjective ignorance about the past. If someone else has more data about the past, his number of possible pasts will be much smaller. If that person has a record of the past, there will be only one possible past.

The probability for predicting the future is objective, because by the definition of the word “future”, no one can have any data about the future. No one can have a record of the future now.

— Me@2013-07-26 6:01 PM

The difference between the future and the past is that logically, no one can have any data of future, but someone may have some data of the past. Also, different people can have different sets of data about the past.

— Me@2013-08-08 8:43 AM

2013.08.08 Thursday (c) All rights reserved by ACHK

And as we have explained many times, the results of this inference – the retrodictions – always depend on our priors. So the knowledge of the present is enough to calculate the future (classically) or to predict the unique probabilities of various states in the future (quantum mechanically). But it is simply never enough to calculate the unique state or unique probabilities of various states in the past.

The reason has been explained many times. But we can say that at least in the macroscopic context (when some microscopic detailed information is being omitted, e.g. because it’s unmeasurable), different initial states “A,B” in the past may evolve into the same final state “C” in the future.

— Logical arrow of time and terminology

— Lubos Motl

2013.07.20 Saturday ACHK

However, the right definition of the past and the future is independent of these sign conventions for spacetime coordinates. The right definition says that

    The future is evolving from the past (and the present).

Correspondingly, the calculations that are designed to theoretically mimic this evolution have the same arrow:

    The future is calculated from the past (and the present) as long as we use the usual calculations that resemble the evolution.

It’s important that you can’t exchange the words “future” and “past” in the sentence above.

That doesn’t mean that science can never say anything about the past, by manipulating with the present data or the data from a closer past. But this type of calculation is different from predictions of the future. It follows different formulae, too. They’re the formulae of logical inference, e.g. Bayesian inference.

And as we have explained many times, the results of this inference – the retrodictions – always depend on our priors. So the knowledge of the present is enough to calculate the future (classically) or to predict the unique probabilities of various states in the future (quantum mechanically). But it is simply never enough to calculate the unique state or unique probabilities of various states in the past.

— Logical arrow of time and terminology

— Lubos Motl

2012.02.28 Tuesday ACHK

1.    Time is just another, imaginary dimension of space
2.    The question about the existence of the past depends on our definition of existence
3.    The future doesn’t exist at the present regardless of any definitions
4.    Some units of time are better than others
5.    Among conserved quantities, time has a special relationship with energy
6.    All physical systems with many degrees of freedom inevitably possess a future-time asymmetry, the so-called logical arrow of time
7.    The logical arrow of time implies the asymmetry of all macroscopic processes, i.e. the thermodynamic arrow of time and other arrows of time
8.    An exact time-reversing symmetry which holds in Nature is called CPT and only applies to the microscopic laws
9.    To properly understand the psychological perception of time, one needs to analyze the functions of the brain
10.   When times become as short as the Planck time, \(10^{-43}\) seconds, time develops some wholly unfamiliar properties

— Ten new things modern physics has learned about time

— Lubos Motl

2011.09.06 Tuesday ACHK