# What is the best way of evaluating time-ordered integrals numerically?

In time-dependent perturbation theories, one encounters multi-dimensional time-ordered integrals $$(-i)^n\int_0^tdt_1\int_0^{t_1} dt_2 \cdots \int_0^{t_{n-1}}dt_n f(t_1,t_2,\cdots,t_n)$$

What is the best way of numerically evaluating such multi-dimensional time-ordered integrals? Monte Carlo integration is the first way that comes into my mind. How does Monte Carlo deal with the time-ordering?

The time-ordered integral can be transformed to a normal multi-dimensional integral over a rectangular volume by changing variables. Setting $$t=\beta$$, and using the following change of variables: $$\begin{cases} t_1 = y_1\\ t_2 = y_2\frac{y_1}{\beta}\\ t_3 = y_3\frac{y_2y_1}{\beta^2}\\ \vdots\\ t_n = y_n\frac{y_{n-1}\cdots y_1}{\beta^{n-1}} \end{cases}$$ and the corresponding Jacobian $$\left(\frac{y_1}{\beta}\right)^{n-1}\left(\frac{y_2}{\beta}\right)^{n-2}\cdots \left(\frac{y_{n-1}}{\beta}\right)^{1},$$ the integral $$(-i)^n\int_0^tdt_1\int_0^{t_1} dt_2 \cdots \int_0^{t_{n-1}}dt_n f(t_1,t_2,\cdots,t_n)$$ becomes $$(-i)^n\int_0^\beta dy_1 \int_0^{\beta} dy_2 \cdots \int_0^{\beta} dy_n \, f(y_1, \frac{y_2y_1}{\beta},\cdots,\frac{y_ny_{n-1}\cdots y_1}{\beta^{n-1}}) \left(\frac{y_1}{\beta}\right)^{n-1}\left(\frac{y_2}{\beta}\right)^{n-2}\cdots \left(\frac{y_{n-1}}{\beta}\right)^{1}.$$
• Its a bit confusing that $\beta$ and $t$ is used together in the final expression although beta has to take on the value of t. Perhaps add that $\beta=t$, unless I am misunderstanding something. Jul 2, 2022 at 17:35
• is there a typo in your $t_3$? Shouldn't this be $y_3 y_2 y_1/\beta^2$ rather than $y_3 y_2 y_2/\beta^2$ as you have it now? And would you have a source to this? Nov 19, 2022 at 19:53