The Dyson Series
The time-dependent version of perturbation theory.
The Static Case
Time-independent perturbation theory is about energy levels. You have a Hamiltonian that does not change with time, you know its eigenstates and eigenvalues, and you add a small perturbation. The corrections are numbers. They are stationary. They do not evolve. They simply are.
This is useful, but it is limited. Most real physical processes involve time dependence. A field is turned on. A laser pulse hits an atom. A system is prepared in one state and measured in another. Between preparation and measurement, the Hamiltonian changes. The energy levels move. The eigenstates evolve. The static theory cannot describe this.
The Time-Dependent Problem
In time-dependent perturbation theory, the Hamiltonian is $H(t) = H_0 + V(t)$, where $V(t)$ is explicitly time-dependent. The unperturbed Hamiltonian $H_0$ is still time-independent, so its eigenstates $|n\rangle$ and eigenvalues $E_n$ are well-defined. But the full state $|\psi(t)\rangle$ evolves according to the time-dependent Schrödinger equation, and the perturbation $V(t)$ drives transitions between the unperturbed states.
The question is simple: if the system starts in state $|i\rangle$ at time $t = 0$, what is the probability of finding it in state $|f\rangle$ at time $t$? The answer is given by perturbation theory, but the machinery is different. You cannot expand the energy levels. You must expand the time evolution itself.
The Dyson Series
The time evolution operator $U(t, 0)$ satisfies
$$i\hbar \frac{\partial}{\partial t} U(t, 0) = H(t) U(t, 0), \quad U(0, 0) = \mathbb{1}$$
In the interaction picture — where you factor out the free evolution generated by $H_0$ — the evolution operator $U_I(t, 0)$ satisfies
$$i\hbar \frac{\partial}{\partial t} U_I(t, 0) = V_I(t) U_I(t, 0)$$
where $V_I(t) = e^{iH_0 t/\hbar} V(t) e^{-iH_0 t/\hbar}$ is the perturbation in the interaction picture. The solution is an integral equation:
$$U_I(t, 0) = \mathbb{1} - \frac{i}{\hbar} \int_0^t dt_1 , V_I(t_1) U_I(t_1, 0)$$
Iterating this equation gives the Dyson series:
$$U_I(t, 0) = \mathbb{1} + \left(-\frac{i}{\hbar}\right) \int_0^t dt_1 , V_I(t_1) + \left(-\frac{i}{\hbar}\right)^2 \int_0^t dt_1 \int_0^{t_1} dt_2 , V_I(t_1) V_I(t_2) + \cdots$$
Each term adds one more factor of $V_I$ and one more integral. The nested integration domain $0 < t_n < \cdots < t_1 < t$ ensures that the operators are time-ordered: later times appear to the left. This is why the Dyson series is often written with the time-ordering operator $T$:
$$U_I(t, 0) = T \exp\left(-\frac{i}{\hbar} \int_0^t dt' , V_I(t')\right)$$
The exponential is symbolic. It is shorthand for the Dyson series, and it should not be interpreted as an ordinary exponential, because the $V_I(t)$ at different times do not commute.
The First Term
The first-order term gives the transition amplitude from $|i\rangle$ to $|f\rangle$:
$$A_{i \to f}^{(1)}(t) = -\frac{i}{\hbar} \int_0^t dt' , \langle f | V_I(t') | i \rangle$$
If $V(t) = V e^{-i\omega t} + V^\dagger e^{i\omega t}$ — a periodic perturbation, like an electromagnetic wave — the integral can be evaluated exactly. The result is proportional to
$$\frac{e^{i(\omega_{fi} - \omega)t} - 1}{\omega_{fi} - \omega}$$
where $\omega_{fi} = (E_f - E_i)/\hbar$. The transition probability is the square of this amplitude, which gives a sharply peaked function centered at $\omega_{fi} = \omega$. In the long-time limit, it becomes a delta function:
$$P_{i \to f}(t) \approx \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 , t , \delta(E_f - E_i - \hbar\omega)$$
This is Fermi's golden rule. It says: the transition rate is proportional to the square of the matrix element and to the density of final states at the resonance energy. The delta function enforces energy conservation. The perturbation provides the energy $\hbar\omega$, and the final state must absorb it exactly.
The Second Term
The second-order term involves two interactions with the perturbation. The system can go from $|i\rangle$ to an intermediate state $|m\rangle$ and then to $|f\rangle$. The amplitude is
$$A_{i \to f}^{(2)}(t) = \left(-\frac{i}{\hbar}\right)^2 \sum_m \int_0^t dt_1 \int_0^{t_1} dt_2 , \langle f | V_I(t_1) | m \rangle \langle m | V_I(t_2) | i \rangle$$
The intermediate state $|m\rangle$ is virtual. It does not need to satisfy energy conservation. The energy can be violated by an amount $\Delta E$ for a time $\Delta t \sim \hbar/\Delta E$. This is the uncertainty principle in action, and it is one of the most important physical consequences of the second-order term. Virtual transitions enable processes that would be forbidden at first order: two-photon absorption, Raman scattering, and the Casimir effect all arise from second-order (or higher) terms in the Dyson series.
The Connection
The Dyson series is the time-dependent analogue of the time-independent perturbation series. In the static limit, where $V(t) = V$ is time-independent and you wait long enough that the system has reached its perturbed eigenstates, the Dyson series reduces to the Rayleigh-Schrödinger series. The two are the same theory, viewed from different angles. One is about energy levels. The other is about transitions. Both are about starting from a known solution and adding corrections.
The Dyson series is also the starting point for quantum field theory. In QFT, the interaction picture evolution operator is the S-matrix, and the Dyson series becomes the Feynman diagram expansion. Each term in the series corresponds to a class of processes, and each process can be drawn as a diagram. The Feynman rules tell you how to compute the amplitude of each diagram. The Dyson series is the parent of the Feynman diagram.
The Meta-Comment
This page is about the time-dependent version of perturbation theory. It is a meta-page because it describes not just the physics but the structure of the theory itself — how the series is built, how the terms are organized, how the static theory emerges as a limit. The Dyson series is a construction. It is built from the ground up, term by term, starting from the identity and adding interactions. It is a record of how the perturbation acts on the system, one interaction at a time.
The series is infinite. It does not always converge. But like its time-independent cousin, it captures the physics that can be extracted from a perturbative expansion. The terms that it misses — the non-perturbative effects, the bound states that emerge from the continuum, the phases that cannot be seen order by order — are the territory of other methods.
This is perturbation theory's domain: the territory between the exactly solvable and the unsolvable, the corrections that carry the known to the unknown, term by term, integral by integral, order by order.