Time Reversal Symmetry: An Intuitive Picture Okay so here is my confusion. When reading into TR as applied both to Classical and Quantum Physics, I have found two opinions and somehow I do not know which one is more correct than the other. The arguments are all based on a very classical picture.
First opinion: imagine playing a movie forwards in time. Reverse it by playing it backwards, TR should imply that there is no difference. Now here is one of the big confusions: when playing the movie backwards, one would definitely know that it is being played backwards and the reverse motion IS DIFFERENT from when it was played forwards in time.
Second opinion: TR just implies switching $v$ to $-v$. In this case, imagine a runner running in the positive $x$-direction. Now apply TR, it just gives what you would otherwise see in a mirror. Here's my other confusion: the second opinion differs from the first in that this time, you will not see the runner running backwards, rather the image has been flipped and instead he would be running in the opposite direction. This is different from say, you "rewind" his motion.
Any help would be much appreciated.
 A: Time reversal (TR) does not flip $x$, or else:
$$ v = \frac{dx}{dt} $$
would violate TR.
It works as follows:
$$ x \rightarrow x' $$
$$ t \rightarrow -t' $$
$$ v' = \frac{dx'}{dt'} = \frac{dx}{-dt} = -\frac{dx}{dt} = -v $$
Position is even, velocity is odd, acceleration is even, jerk is odd, and so on.
A: There is perhaps another way to look at this issue: If you’re dealing with equations that include time only in second order, or multiples thereof, the equations will be time reversal invariant. As it happens, the laws that govern the behavior of classical systems are invariant under time reversal. This means that, for example, the conservation of energy and momentum, Newton’s 2nd law, Lorentz invariance, etc. do not change when t is replaced by $-t$. In every case, the negative sign cancels, and your left with the same equation.
Let’s not confuse this with reversing the motion of a physical system. Time reversal applies to the equations that are the laws of physics. Starting with the same initial conditions, the system will evolve in the same way, regardless of whatever is assumed to be the direction of time. Take Newton’s 2nd law as an example. Replace $t$ with $-t$, and it still reads $F = ma$. The change in sign cancels. Call this a feature or a flaw, but it’s a property of the equations in classical physics.
Let’s do a Bob and Alice thought experiment: Bob and Alice are each in their own reference frames moving in negative time relative to each other. Bob sees Alice moving in negative time relative to his frame of reference, and Alice sees Bob moving in negative time relative to her frame of reference. If both start a physical process under the same initial conditions, they will both evolve in the same way. There is nothing in classical physics that would distinguish one system from the other.
The examples given in this question are normally used to demonstrate a violation of the 2nd law of thermodynamics. It isn’t clear that they have much to do with time reversal symmetry. They demonstrate the improbability of a decrease in entropy.
In the 2nd example, $v$ is replaced by $-v$, and we see the runner’s reflection in a mirror. No matter because the runner is still moving forward in time. Turning the runner around and having her run in the opposite direction doesn’t affect the direction of time. It isn’t clear how time reversal symmetry would apply to this example.
Quantum mechanics, in and of itself, is not a proper description of nature: Among other things, it doesn’t obey the Theory of Special Relativity. To describe the world of subatomic particles Quantum Field Theory is required. In QFT, elementary particles are identified by quantum numbers—i.e., electric charge, and  parity. Time reversal invariance holds universally in QFT, except in the case of the weak nuclear force. The discovery of a violation in CP invariance, that is a violation in the product of charge and parity conjugation also implied a violation in time reversal invariance.  That’s because causality requires that CPT be conserved, and therefore time reversal invariance would also be broken. (Causality includes the condition that an event can affect only future events that occur within the forward light cone.)
A: Except for the correction JEB pointed out, your second opinion is the right one.
Some system starts in an initial state, and proceeds under the laws of physics to a final state. TR says that you can start with everything in the position of the final state and all the velocities reversed. The system will then proceed under the laws of physics back to the original state, except that the initial velocities will be reversed.
The only difference is that many things are wildly improbable when run backwards.

*

*A runner normally runs forward. A runner running backwards is not something you see, but it violates no laws of physics.

*If you start with all the air on one side of the room, it will flow to spread out evenly. It would violate no laws of physics for the air to rush to one side of the room.

*A rock breaks a window and fragments fly everywhere. It would violate no laws of physics for the fragments to fly off the floor in just the right directions to join together into a whole pane of glass that kicks the rock away.

Air on one side of the room is easiest to see. Molecules bounce around randomly. Some are very likely to bounce toward the empty side of the room. Since there is nothing to bounce them back except the far wall, they spread out.
If you took every molecule at its final position (including the molecules in the walls) and reversed their direction very precisely, it would aim them just right to reverse their trajectories over the course of many many bounces. They would all bounce back to one side.
Almost all states that a collection of molecules find themselves in through random motion lead to air spread out even in the future. Very very few lead to air on one side of the room. It wouldn't happen in the age of many universes unless you arranged for it to happen.
That is the classical picture of positions and trajectories. Quantum mechanics makes it so you can't even theoretically arrange for it to happen. Molecules have uncertainty of position and momentum. They do not have a precise position and momentum in their final state. They have a final wave function.
There is a time-reversed version of that final wave function. If you started a system in that state, all the molecules would be in their final positions with reversed momenta up to limits of the uncertainty principle. It would be consistent with the laws of physics if they wound up back on one side of the room. But it isn't guaranteed. You cannot predict a final state. You can only predict probabilities. And air on one side is a very improbable outcome.
A runner running backwards is a similar very unlikely outcome, though it is a little harder to see. People can run backwards, but they use different muscles than running forward. When running forward, they push the ground backward to push themselves forward. To run backward requires pushing the ground backward, something like kicking the ground.
In normal running, you would start from rest and push to accelerate to full speed. Time-reversed running would be starting from full speed backward and pushing forward to slow yourself down to a stop. Such motion would violate no laws of physics, but it would be very unnatural.
The thing about time reversal symmetry is that on a microscopic scale, things are not improbable when run backwards or forward. Two molecules can bounce off each other in either direction. It is very natural either way.
The only thing that makes many things run one way is the laws of probability, and they really only matter when you have multiple molecules. Things naturally run in the direction of increased disorder, increased entropy, air flowing from one side to spread out. People have looked for deeper reasons why time seems to flow only one way in the larger world even though the microscopic world is fully reversible. Nothing has been found.
To be fair, when things run forward normally the final state you get, with the final position and velocity of every molecule specified, is also wildly improbable. There are a vast number of similar states that could have been the final state. That particular final state would not be repeated if you ran from similar, or even the same, starting state over and over for the age of many universes.
Many many final states look the same in the larger world. Air spreads out uniformly to the whole room in almost all of these states. The difference between them is in the way the air has been stirred. There are so few states where the air winds up on one side of the room that it has never been seen by chance.
