There’s a nice post at The Eternal Universe illustrating discrete symmetries like time reversal. The father of this idea, Eugene Wigner, actually gave a very elegant characterization in his 1931 book:
The following four operations, carried out in succession on an arbitrary state, will result in the system returning to its original state. The ﬁrst operation is time inversion, the second time displacement by t, the third again time inversion, and the last on again time displacement by t. (Wigner  1959, p.326.)
I’ve illustrated this assumption below, using the analogy:
- Reversal of time = Flipping of toy car;
- Evolution through time = Forward motion of car.
Wigner’s claim is then that whatever time reversal means, the initial state of the car below is the same as its ﬁnal state.
The beauty is, if you write down this assumption in terms of quantum mechanical operators (and assume energy is positive), then you can quickly see an important property of time reversal — its antiunitarity (roughly, it involves conjugation). In terms of operators, Wigner assumed that:
This immediately implies that
But this is equivalent to the statement,
Now, can’t be unitary, since then it would follow that , and we could cancel the i to get that . That’s false if energy is always positive. But Wigner’s theorem says that if a symmetry operator is not unitary, then it must be antiunitary — so T must be antiunitary.
Wigner’s argument seems to have been dropped from most modern textbooks. Perhaps the reason is that not all physical interactions are T-reversible, and Wigner’s assumption (illustrated here with cars) takes for granted that they are. On the other hand, all known physical interactions are -reversible. So, a version of this argument still works if we reinterpret Wigner’s “time reversal” as “ reversal.”
- Conferences for Philosophers of Physics, Summer 2010
- Constructing the Ultimate Machine