I'm studying Ryder's quantum field theory book, in chapter 5 section 2 Ryder tries to derive Dyson's series using Feynman's path integral.
On page 160 we have the following definition of Feynman's path integral:
$$\tag{5.14} \langle q_ft_f|q_it_i\rangle=\lim_{n\rightarrow\infty}\bigg(\frac{m}{i\hbar\tau}\bigg)^{(n+1)/2}\int\prod_{1}^n\,dq_j\exp\bigg\{\frac{i\tau}{\hbar}\sum_0^n\bigg[\frac{m}{2}\bigg(\frac{q_{j+1}-q_j}{\tau}\bigg)^2-V\bigg(\frac{q_j+q_{j+1}}{2}\bigg)\bigg]\bigg\}$$
the book then does the Taylor expansion
$$\tag{5.16}\exp\bigg[\frac{-i}{\hbar}\int_{t_i}^{t_f}V(x,t)\,dt\bigg]=1-\frac{i}{\hbar}\int_{t_i}^{t_f}V(x,t)\,dt-\frac{1}{2!\hbar^2}\bigg[\int_{t_i}^{t_f}V(x,t)\,dt\bigg]^2+…$$
The first term gives the free propogator $K_0(x_ft_f,x_it_i)$, the second term gives first order perturbation $$\tag{5.20} K_1(x_ft_f,x_it_i)=-\frac{i}{\hbar}\int_{\infty}^{-\infty}\,dt\int_{-\infty}^\infty \,dx \,K_0(x_ft_f,xt)V(x,t)K_0(xt,x_it_i)$$
I get these two terms. But how did we get the 2nd order perturbation?
$$\tag{5.22} K_2(x_ft_f,x_it_i)=\bigg(\frac{-i}{\hbar}\bigg)^2\int^\infty_{-\infty}\,dt_1\int_{-\infty}^\infty\,dt_1\int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_1\, K_0(x_ft_f,x_2t_2)V(x_2,t_2)\times K_0(x_2t_2,x_1t_1)V(x_1,t_1)K_0(x_1t_1;x_it_i).$$
I mean, if we use equation 5.14, the two $V$'s should occur at equal time, why shouldn't it something like
$$K_2(x_ft_f,x_it_i)=?-\frac{1}{2!\hbar}\int_{\infty}^{-\infty}\,dt\int_{-\infty}^\infty \,dx \,K_0(x_ft_f,xt)V(x,t)^2K_0(xt,x_it_i)$$
Best Answer
The two $V$'s do not have to occur at equal time. Note that in Eq. 5.14, the second order perturbation in $V$ is \begin{align} \lim_{n\rightarrow\infty}{\bigg(\frac{m}{2\pi i\hbar\tau}\bigg)^{(n+1)/2}\sum_{j_2>j_1}} & {\int{\prod_{j_2}^n\,dq_j\exp\bigg\{\frac{i\tau}{\hbar}\sum_{j=j_2}^{n}\bigg[\frac{m}{2}\bigg(\frac{q_{j+1}-q_j}{\tau}\bigg)^2\bigg]\bigg\}}\bigg({-i\tau \over \hbar}V\bigg(\frac{q_{j_2}+q_{j_2+1}}{2}\bigg)\bigg)} \\ & {\int{\prod_{j_1}^{j_2-1}\,dq_j\exp\bigg\{\frac{i\tau}{\hbar}\sum_{j=j_1}^{j_2-1}\bigg[\frac{m}{2}\bigg(\frac{q_{j+1}-q_j}{\tau}\bigg)^2\bigg]\bigg\}}\bigg({-i\tau \over \hbar}V\bigg(\frac{q_{j_1}+q_{j_1+1}}{2}\bigg)\bigg)} \\ & {\int{\prod_{1}^{j_1-1}\,dq_j\exp\bigg\{\frac{i\tau}{\hbar}\sum_{j=1}^{j_1-1}\bigg[\frac{m}{2}\bigg(\frac{q_{j+1}-q_j}{\tau}\bigg)^2\bigg]\bigg\}}} \end{align} This is just $$\bigg(\frac{-i}{\hbar}\bigg)^2\int^{t_f}_{t_i}\,dt_2\int_{t_i}^{t_2}\,dt_1\int_{-\infty}^\infty dx_2\int_{-\infty}^\infty dx_1\, K_0(x_ft_f,x_2t_2)V(x_2,t_2)\times K_0(x_2t_2,x_1t_1)V(x_1,t_1)K_0(x_1t_1;x_it_i)$$