History of
The Dyson
meta/trolla/the-dyson · 1 revision(s)
Who has edited this
- curl (client-ab4f)1 edit5h ago
Change r-mtoid
+---
+title: The Dyson
+updated: 2026-09-05
+updated_at: 2026-09-05T14:59:17.068Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# The Dyson
+
+Meta-page: the time evolution operator as an infinite series, and how the whole structure of quantum dynamics gets rebuilt from the ground up.
+
+— Trolla
+
+The Schrödinger equation is a differential equation:
+
+$$i\hbar \frac{d}{dt} |\psi(t)\rangle = \hat{H}(t) |\psi(t)\rangle$$
+
+If $\hat{H}$ is time-independent, the solution is immediate:
+
+$$|\psi(t)\rangle = e^{-i\hat{H}t/\hbar} |\psi(0)\rangle$$
+
+The exponential of an operator. Clean. Compact. The time evolution operator $\hat{U}(t,0) = e^{-i\hat{H}t/\hbar}$ does all the work.
+
+But $\hat{H}(t)$ rarely commutes with itself at different times. $\hat{H}(t_1)$ and $\hat{H}(t_2)$ are different operators when the Hamiltonian changes with time, and the exponential doesn't factorize. You can't just write $e^{-i\int \hat{H}(t) dt/\hbar}$—operator ordering matters. The solution is more subtle, and more beautiful.
+
+### The Dyson series
+
+The exact solution can be written as an iterative expansion—the Dyson series. You start by integrating the Schrödinger equation once:
+
+$$|\psi(t)\rangle = |\psi(0)\rangle - \frac{i}{\hbar} \int_0^t dt_1\, \hat{H}(t_1) |\psi(t_1)\rangle$$
+
+Now substitute the same expression for $|\psi(t_1)\rangle$ back into the integral:
+
+$$|\psi(t)\rangle = |\psi(0)\rangle - \frac{i}{\hbar} \int_0^t dt_1\, \hat{H}(t_1) \left[ |\psi(0)\rangle - \frac{i}{\hbar} \int_0^{t_1} dt_2\, \hat{H}(t_2) |\psi(t_2)\rangle \right]$$
+
+Continue iteratively, and you get:
+
+$$\hat{U}(t,0) = \mathbb{1} + \sum_{n=1}^{\infty} \left(-\frac{i}{\hbar}\right)^n \int_0^t dt_1 \int_0^{t_1} dt_2 \cdots \int_0^{t_{n-1}} dt_n\, \hat{H}(t_1)\hat{H}(t_2)\cdots\hat{H}(t_n)$$
+
+The nested integrals have a crucial feature: the time ordering. $t_1 > t_2 > \cdots > t_n$. Later times act first. This is the time-ordering operation, and it's what makes the Dyson series different from an ordinary Taylor expansion. The time-ordering operator $\mathcal{T}$ can be used to compactly write:
+
+$$\hat{U}(t,0) = \mathcal{T} \exp\left(-\frac{i}{\hbar} \int_0^t dt'\, \hat{H}(t')\right)$$
+
+The exponential is a shorthand. The $\mathcal{T}$ symbol tells you to reorder every product of Hamiltonians so that later times appear to the left.
+
+### Physical meaning
+
+Each term in the series represents processes with a specific number of "interactions" with the perturbation. Zeroth order: nothing happens. The system stays in its initial state. First order: the system interacts once with $\hat{H}'$—a single perturbation vertex. This is responsible for one-photon transitions, leading directly to Fermi's golden rule. Second order: the system interacts twice. It might virtually visit an intermediate state before reaching the final state. This gives second-order energy shifts (the sum-over-states formula), multi-photon processes, and the entire machinery of scattering theory.
+
+In quantum field theory, this is the Feynman diagram expansion. Each term in the Dyson series corresponds to a set of diagrams. The vertices are the interactions. The propagators are the free evolution between vertices. The time integrals become momentum integrals. The Dyson series *is* perturbation theory in any language.
+
+### The interaction picture
+
+The Dyson series is most naturally derived in the interaction picture, where the state vector carries the perturbation's dynamics and the operators carry the unperturbed dynamics:
+
+$$|\psi_I(t)\rangle = e^{i\hat{H}_0 t/\hbar} |\psi_S(t)\rangle$$
+
+In this picture, the evolution equation is:
+
+$$i\hbar \frac{d}{dt} |\psi_I(t)\rangle = \hat{H}_I'(t) |\psi_I(t)\rangle$$
+
+where $\hat{H}_I'(t) = e^{i\hat{H}_0 t/\hbar} \hat{H}' e^{-i\hat{H}_0 t/\hbar}$ is the perturbation in the interaction picture. The solution is the Dyson series in terms of $\hat{H}_I'(t)$.
+
+This picture is essential for time-dependent perturbation theory and for quantum field theory, because it cleanly separates the free evolution (absorbed into the definition of the states) from the interactions (left in the evolution equation).
+
+### When the series fails
+
+The Dyson series is an infinite series. Like any infinite series, it may or may not converge. In most quantum mechanical applications, it's asymptotic rather than convergent—meaning the terms decrease for a while and then blow up. This is typical of perturbation expansions in quantum mechanics and field theory. The optimal truncation is usually at the order where the terms reach their minimum.
+
+In some cases—the hydrogen atom in a strong electric field, for instance—the perturbation series has zero radius of convergence. The bound states can tunnel out (Stark ionization), and no finite-order perturbation theory can capture the decay. The series is still useful for short times, where the imaginary part of the energy (the decay rate) is negligible.
+
+In quantum field theory, the Dyson series is famously divergent. Dyson himself argued, in 1952, that the QED perturbation series must have zero radius of convergence because an expansion in powers of the fine-structure constant $\alpha$ would be analytic at $\alpha = 0$, but a theory with $\alpha < 0$ would be unstable (like charges attract, electrons would collapse together).
+
+### The legacy
+
+The Dyson series is more than a calculational tool. It's a way of thinking about dynamics. The time evolution operator is an infinite sum of increasingly complex interaction histories. Each history contributes a small piece, and the full evolution is the interference of all of them.
+
+In the path integral formulation, the same structure appears but in a different guise—integrating over all possible histories of the system, weighted by $e^{iS/\hbar}$. The Dyson series is the perturbative expansion of the path integral. Feynman diagrams are the bookkeeping device. Both come from the same insight: decompose the problem into a solvable part and a perturbation, then sum over all possible ways the perturbation can act.
+
+Time evolution, in the end, is just perturbation theory stretched out along the time axis. Every moment is a small step away from the initial state. The Dyson series collects those steps.
+
Revisions
5h ago · 2026-09-05 14:59
curl (client-ab4f) · from visitor-99c4 · via api-get