= Dyson series
{c}
{title2=$U_I(t,t_0)$}
{wiki}
The <Dyson series> solves <interaction picture> evolution by keeping noncommuting <Hamiltonian operators> in chronological order. With $\hbar=1$, integrating $i\partial_tU_I=H_I(t)U_I$ and iterating gives
$$
U_I(t,t_0)=I-i\int_{t_0}^t dt_1\,H_I(t_1)
+(-i)^2\int_{t_0}^t dt_1\int_{t_0}^{t_1}dt_2\,H_I(t_1)H_I(t_2)+\cdots.
$$
<Time ordering> puts later times on the left. Integrating over all time labels instead of the ordered region introduces $1/n!$. The formal <scattering operator> limit is
$$
S=T\exp\!\left(-i\int d^4x\,\mathcal H_{\mathrm{int}}(x)\right)
=\sum_{n\geq0}\frac{(-i)^n}{n!}\int d^4x_1\cdots d^4x_n\,T\!\left\{\prod_{j=1}^n\mathcal H_{\mathrm{int}}(x_j)\right\}.
$$
An ordinary <matrix exponential> suffices when all the $H_I(t)$ commute. For <quantum fields>, the displayed expansion requires a prescription for singular products and need not converge. <Quantum Field Theory by David Tong>[Section 3.1.1, Dyson's formula and chronological expansion]{ref}
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