Confusion Regarding the Propagator

Question

To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$ will be at position $x_f$ at time $t_f$, and the probability itself is given by $|G^+|^2$. Now for a free particle, $$G^+=\theta(t_x-t_y)\sqrt{\dfrac{m}{2\pi i(t_f-t_i)}}exp\left[\dfrac{im(x_f-x_i)^2}{2(t_f-t_i)}\right],$$ which means $$|G^+|^2=\theta(t_f-t_i)\dfrac{m}{2\pi(t_f-t_i)}.$$ I'm confused because this doesn't seem to give a normalizable probability density function? Have I misunderstood what $G^+$ is?