In the real world, it seems that traveling backwards in time is impossible, but do we have a theorem in physics that would imply this fact?
Some people (including Feynman) describe antiparticles as moving in the opposite direction of time coordinate axis. For example, the Dirac field involves an integral of the term
$$a(\pmb p,\sigma)e^{ip^\mu x_\mu}+a^{c\dagger}(\pmb p,\sigma)e^{-ip^\mu x_\mu}$$
multiplied with some extra factors($\eta^{00}=-1$ for the metric). The first term is interpreted as annihilation of an electron propagating forward in space-time, and consequently the second term should create a positron traveling back in time. It does make a bit sense, but seems to be against our intuition.
If we apply the time reversal to an arbitrary field $\psi_l(x)$, the effect is just taking $x$ to $\mathscr Px$, and multiplying it by a matrix $Q_{ll'}$, where
$$\mathscr P=diag(-1,-1,-1,1)$$
is the space-inversion transformation. Now does the field obtained evolve against the positive direction of time? Is it possible that there is no positive direction for time at all?
One could argue that due to the second law of thermodynamics, entropy never decrease with time, so that there must be a positive direction. However, what if the systems going backwards in our world adopt a different percentage and claim that they are going forward and we are going backwards? Besides, although we have quantum statistics, I'm not fully persuaded by such a statistical theory that entropy is well defined on a microscopic scale.
It seems that nothing is able to forbid the existence of a system with time reversed, but also we're unable to detect it (or we did but we didn't know that). I'm looking for someone who has an explanation for this.
Best Answer
There are two aspects to this. First we can clarify the quantum field theory a bit. Secondly we need to distinguish the use of the concept "time" in discussions of quantum unitary evolution (Schrodinger's equation) from the concept of "time" in complicated evolution leading to thermodynamic irreversibility. These concepts are related (which is why they have the same name) but the relationship is quite subtle.
In the Feynman path integral method it is not that a positron goes backward in time; it is rather than a positron going forwards in time contributes to the equations just as would an electron going backwards in time. So we can think of a positron going forwards in time as if it was an electron going backwards in time, at least for the purposes of writing down the Feynman propagator and thus finding solutions to Schrodinger's equation.
But in these calculations "time" is a parameter and the evolution is unitary. That means really the behaviour establishes connections between what goes on at timelike-separated regions of spacetime, without really caring whether the evolution is going one way or the other. It is only by invoking the wider meaning of "time" (broadly speaking, the thermodynamic meaning) that we get a sense of direction. For that you have to look at larger numbers of worldlines all weaving together in complicated patterns, and you find that in the limit of large numbers of processes, the entropy gets bigger in one direction, and that is the direction we call the future.
If I understood correctly, I think the question asks whether there could be stuff evolving backwards in time with entropy decreasing as it goes. It seems to me that if one expresses this idea in more detail, one might end up with a scenario identical to the one we observe and all we have succeeded in doing is attach different words to it. That would be like someone who says "a circle is triangular" and then we say "no it is not" and then they say "yes it is, because I have defined 'triangular' in a new way which makes it observationally indistinguishable from 'circular'". Obviously that kind of word-play does not aid understanding so there is no point in doing it. If, on the other hand, one suggests that matter could travel backwards in time while retaining some sort of memory of the future then this would be contrary to the patterns of the physical world as they have been discovered up till now, because it would be contrary to the second law of thermodynamics and it would reverse the direction of cause and effect.
But the question now touches on how the second law of thermodynamics relates to statistical mechanics and ultimately quantum field theory. This remains something of a puzzle I think. There are theorems such as Boltzmann's H-theorem which go a long way to establishing the connection, but it is not fully resolved as far as I am aware.