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 $$ \sqrt{\frac{m}{2\pi i \hbar t} }\exp\left(\frac{im(x_f - x_i)^2}{2\hbar t}\right), $$ so my question is why doesn't this logic doesn't hold?

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?