Hermitian of the Dyson series

Question

If I have some time-dependent interaction Hamiltonian $H_I(\tau)$, with the interaction beginning in the asymptotic past ($\tau$$\rightarrow$$-\infty$) and ending in the asymptotic future ($\tau$$\rightarrow$$\infty$), then the first two terms in the Dyson series will read:

\begin{equation} U=U^{(1)}+U^{(2)}+\cdots \end{equation}

where

\begin{equation} U^{(1)}=-i\int_{-\infty}^{\infty}d\tau H_I(\tau) \end{equation} \begin{equation} U^{(2)}=-\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau'\Theta(\tau-\tau')H_I(\tau)H_I(\tau'). \end{equation}

The interaction Hamiltonian is Hermitian $H_I^{\dagger}(\tau)=H_I(\tau)$. I want to calculate the first two terms of the series expansion of $U^{\dagger}$. I know that

\begin{equation} U^{(1)\dagger}=i\int_{-\infty}^{\infty}d\tau H_I(\tau). \end{equation}

My question is, when I write down $U^{(2)\dagger}$, do I reverse the time ordering (i.e, replace $\Theta(\tau-\tau')$ with $\Theta(\tau'-\tau)$), and write down

\begin{equation} U^{(2)\dagger}=-\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau'\Theta(\tau'-\tau)H_I(\tau)H_I(\tau'), \end{equation}

or do I simply keep the term the same (since there is no factor of $i$. and the Hamiltonians are already Hermitian)?

Answer 1

Technically, when computing the hermitian conjugate of your $U^{(2)}$, you should leave the integration variables unchanged and only swap the Hamiltonians since $$(A B)^\dagger = B^\dagger A^\dagger,$$ and $H$ is assumed to be hermitian. You could then relabel $\tau_1 \leftrightarrow \tau_2$ to explicitly see that this is not equivalent to simply exchanging $\tau_1$ with $\tau_2$: $$\left(\int_0^\infty d\tau_1 \int_0^{\tau_1} d \tau_2 H(\tau_1) H(\tau_2) \right)^\dagger = \int_0^\infty d\tau_1 \int_0^{\tau_1} d \tau_2 H(\tau_2) H(\tau_1) = \int_0^\infty d\tau_2 \int_0^{\tau_2} d \tau_1 H(\tau_1) H(\tau_2).$$

A highly recommended exercise is to use your series expansion to second order to explicitly show that $U^\dagger U = 1$, where it will be helpful to think about the limits of integration as the triangular areas obtained when dividing the first quadrant of the coordinate space $(\tau_1,\tau_2)$ by the line $\tau_1 = \tau_2$.