Why isn't the free particle particle a function of the absolute value of the difference of the time?

Question

The one-dimensional free particle Lagrangian is given by $$ \mathcal{L} = \frac{m}{2}\dot x^2. $$ Since the Lagrangian is translation-invariant, one usually argues that the propagator can only be a function of $|x - y|$. By the same logic, shouldn't one be able to argue that the propagator can only be a function of $|t_f - t_i|$? However, the propagator is given by $$ \sqrt{\frac{m}{2\pi i \hbar (t_f - t_i)} }\exp\left(\frac{im(x_f - x_i)^2}{2\hbar (t_f - t_i)}\right), $$ so my question is why doesn't this logic doesn't hold?

Answer 1

Time evolution in QM is unitary $U(t)^{\dagger}=U(-t)$, so the free propagator is only expected to be invariant under combined time-reversal $t\to -t$ and complex conjugation.

See also this related Phys.SE post.