I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}s\,H(s)\right)\tag{1}. $$ In Breuer and Petruccione, it's said that if the commutator of the Hamiltonian at some time $t$ with itself at some other time $t^\prime$ is a $c$-number function/ a complex function, i.e. $$\left[H(t),\,H(t^\prime)\right]=f(t,t'),$$ then the time evolution is given by $$ U(t)=\exp\left(-\frac{1}{2}\int_0^t\mathrm{d}s\int_0^t\mathrm{d}s^\prime\,\left[H(s),\,H(s^\prime)\right]\Theta(s-s^\prime)\right)$$ $$\times \exp\left(-\mathrm{i}\int_0^t\mathrm{ds}\,H(s)\right),\tag{2} $$ where $\Theta(s-s^\prime)$ is the Heaviside function. No proof or reference is given and I could not find any explanation anywhere, this is why I ask here. Any help would be very appreciated.


You are looking for a mathematical result known as the Magnus expansion. In general, this gives an exact representation of the time-ordered matrix exponential $$V(t) = {\rm T} \exp \left( \int_0^t dt' \, A(t')\right),$$ in terms of an equivalent ordinary exponential $$ V(t) = \exp\left (S(t)\right),$$ where $S(t)$ can be expressed as an infinite series of nested commutators, $S(t) = \sum_{n=1}^\infty S_n(t)$, e.g.\begin{align} S_1 & = \int_0^t dt_1\, A(t_1),\\ S_2 & = \frac{1}{2}\int_0^t dt_1 \int_0^{t_1} dt_2\, [A(t_1),A(t_2)],\\ & \vdots \end{align} The next terms in the expansion involve higher-order commutators like $[A(t_3),[A(t_1),A(t_2)]]$, which obviously vanish when $[A(t_1),A(t_2)]$ is a $c$-number. See Blanes et al., Physics Reports 470 (2009), 151-238 for further details.


OP's formula (2) is a continuum version of

$$\begin{align} &\exp(A_n)\ldots \exp(A_1)\cr &~=~\exp\left(\sum_{i\in\{1, \ldots, n\}} A_i + \frac{1}{2}\sum_{i,j\in\{1, \ldots, n\}}^{i>j} [A_i,A_j]\right),\end{align}\tag{A}$$

or equivalently,

$$\begin{align} & \exp(A_1)\ldots \exp(A_n)\cr &~=~\exp\left(\sum_{i\in\{1, \ldots, n\}} A_i + \frac{1}{2}\sum_{i,j\in\{1, \ldots, n\}}^{i<j} [A_i,A_j]\right),\end{align}\tag{B}$$

which are valid if we assume that

$$ \forall i,j,k~\in~\{1, \ldots, n\}: [[A_i,A_j],A_k]~=~0. \tag{C} $$

Eq. (B) in turn follows by repeated application of the truncated BCH formula:

$$\begin{align} e^Ae^B~=~&e^{A+B+\frac{C}{2}}, \cr \qquad \text{with}\qquad &C~\equiv~[A,B],\cr \qquad \text{if}\qquad & [A,C]~=~0~=~[B,C].\end{align} \tag{D}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.