The Interaction Picture and the Dyson Series

The canonical quantization pages built free fields, whose dynamics are exactly solvable. Real physics comes from interactions — extra terms in the Lagrangian that couple fields. This page sets up the perturbative treatment: the split $H=H_0+H_{\text{int}}$, the interaction picture, and the Dyson series for the time-evolution operator and the S-matrix. It is the relativistic counterpart of time-dependent perturbation theory in QM and the entry point to Wick's theorem and Feynman rules.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Splitting the Hamiltonian

Write the full Hamiltonian as a solvable free part plus an interaction:

$$H=H_0+H_{\text{int}},H_{\text{int}}=-\int d^{3}x{\mathcal{L}}_{\text{int}}(x)$$

(for interactions with no derivative couplings, ${\mathcal{H}}_{\text{int}}=-{\mathcal{L}}_{\text{int}}$). The canonical example is ${\varphi }^4$ theory,

$$\mathcal{L}=\underset{{\mathcal{L}}_0}{\underbrace{\frac{1}{2}(\partial \varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2}}\underset{{\mathcal{L}}_{\text{int}}}{\underbrace{-\frac{\lambda }{4!}{\varphi }^4}},$$

and the physically central example is QED, ${\mathcal{L}}_{\text{int}}=-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$, the coupling of the Dirac current to the photon. Perturbation theory is an expansion in the small dimensionless coupling ($\lambda$, or $\alpha =e^2/4\pi \approx 1/137$ in QED).

The interaction picture

Neither the Schrödinger picture (states evolve with full $H$) nor the Heisenberg picture (operators evolve with full $H$) is convenient here. The interaction picture is the hybrid in which operators evolve with the free Hamiltonian $H_0$ and states carry the residual evolution due to $H_{\text{int}}$:

$${\varphi }_I(t,\mathbf{x})=e^{i{H}_0(t-t_0)}\varphi (\mathbf{x})e^{-i{H}_0(t-t_0)},|\psi (t){⟩}_I=e^{i{H}_0(t-t_0)}|\psi (t){⟩}_S.$$

The payoff: interaction-picture fields are free fields, so they carry exactly the mode expansions and propagators already derived. All the interacting dynamics is pushed into the evolution of the state, governed by

$$i\frac{d}{dt}|\psi (t){⟩}_I=H_I(t)|\psi (t){⟩}_I,H_I(t)=e^{i{H}_0(t-t_0)}{H}_{\text{int}}{e}^{-i{H}_0(t-t_0)},$$

where $H_I(t)$ is the interaction Hamiltonian built from free (interaction-picture) fields.

The time-evolution operator

Define the interaction-picture propagator $U(t,t_0)$ by $|\psi (t){⟩}_I=U(t,t_0)|\psi (t_0){⟩}_I$. It satisfies $i{\partial }_{t}U=H_I(t)U$ with $U(t_0,t_0)=1$. Because $H_I(t)$ at different times does not commute with itself, the naive $e^{-i\int H_I}$ is wrong; the correct solution is built iteratively. Integrating and re-substituting gives the series

$$U(t,t_0)=1+(-i){\int }_{t_0}^{t}d{t}_1{H}_I(t_1)+(-i{)}^2{\int }_{t_0}^{t}d{t}_1{\int }_{t_0}^{t_1}d{t}_2{H}_I(t_1)H_I(t_2)+\cdots ,$$

where in the $n$-th term the times are ordered, $t>t_1>t_2>\cdots >t_n>t_0$.

The Dyson series (Standard machinery)

The nested time-ordered integrals are unwieldy. Dyson's observation: using the time-ordering operator $T$, each nested integral can be rewritten over the full hypercube (all $t_i$ from $t_0$ to $t$), because the $n!$ orderings are identical after $T$-ordering, giving a $1/n!$:

$$\boxed{U(t,t_0)=\sum _{n=0}^{\infty }\frac{(-i{)}^n}{n!}{\int }_{t_0}^{t}d{t}_1\cdots d{t}_{n}T\{H_I(t_1)\cdots H_I(t_n)\}=T\exp \left(-i{\int }_{t_0}^t{H}_I(t^{\prime })d{t}^{\prime }\right).}$$

This is the Dyson series — the same $T\exp$ quoted in the preliminaries and in QED/historical.md §0.6.

The S-matrix as a Dyson series

The S-matrix connects the far past to the far future, so it is $U$ with $t_0\to -\infty$, $t\to +\infty$:

$$\boxed{S=T\exp \left(-i{\int }_{-\infty }^{\infty }{H}_I(t)dt\right)=T\exp \left(i\int d^{4}x{\mathcal{L}}_{\text{int}}(x)\right).}$$

The final form is manifestly Lorentz-covariant: the interaction enters through the spacetime integral of the scalar ${\mathcal{L}}_{\text{int}}$. Expanding order by order,

$$S=1+i\int d^{4}x{\mathcal{L}}_{\text{int}}(x)+\frac{i^2}{2!}\int d^{4}x{d}^{4}yT\{{\mathcal{L}}_{\text{int}}(x){\mathcal{L}}_{\text{int}}(y)\}+\cdots ,$$

and each term is a time-ordered product of free fields sandwiched between initial and final Fock states. Evaluating those matrix elements is exactly what Wick's theorem automates, and the resulting terms are what Feynman diagrams picture.

The adiabatic switching subtlety

Taking $t_0\to -\infty$ assumes the interaction "switches off" so that asymptotic states are free — the content of Postulate 10a. This is implemented formally by the adiabatic switching ${\mathcal{L}}_{\text{int}}\to e^{-ϵ|t|}{\mathcal{L}}_{\text{int}}$, $ϵ\to 0^{+}$. It fails for long-range forces — Coulomb in QED (soft-photon IR divergences) and confinement in QCD — where the honest asymptotic states are dressed or composite. In those cases the Dyson series still computes correct inclusive observables once IR-safe quantities are formed (see collider observables); the LSZ formula provides the $H$-independent alternative route to the same S-matrix.

Summary

• Interaction picture: fields are free, states evolve under $H_I(t)$.
• Dyson series: $U=T\exp (-i\int H_I)$ resums the time-ordered perturbation expansion.
• S-matrix: $S=T\exp (i\int d^{4}x{\mathcal{L}}_{\text{int}})$, expanded in the coupling.

Where this leads

• Evaluating the matrix elements of time-ordered free-field products: Wick's theorem and contractions.
• Picturing and bookkeeping the terms: Feynman diagrams and rules.
• Turning correlators into observables: the LSZ reduction formula and the S-matrix, cross sections and decay rates.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 4.1–4.2.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 3.5, 8.
• Srednicki, Quantum Field Theory, Ch. 8–9.