Interactions, Time Ordering, and Perturbation Theory

Time ordering makes perturbation theory calculable by expressing interacting evolution as ordered integrals of free-field correlation functions. The Dyson expansion fixes the ordering and combinatorics; Wick’s theorem reduces those ordered products to Feynman functions. The result is an expansion around a chosen free Hamiltonian and state, subject to vacuum-projection, asymptotic-state, ultraviolet, infrared, and late-time limitations.

Required background. Quantum Mechanics supplies Hilbert-space evolution and noncommuting operators; Classical Field Theory supplies interaction Lagrangians and Hamiltonians; Special Relativity supplies relativistic causality and four-momentum conservation. Helpful background. Calculus and Linear Algebra supplies operator exponentials and iterated integrals; Waves, Green Functions, and Geometric Optics supplies Green-function expansions. Review Unitary Dynamics and Pictures of Motion if the picture change below is unfamiliar.

The interaction picture and Dyson ordering

Start with finite volume and a regulator in a setting where the self-adjoint Hamiltonians and their unitary propagators are defined. This makes the change of picture an operator statement. The continuum QFT expressions below are then used as regulated, renormalized perturbative expansions; they do not prove that free and interacting continuum fields share an exact Hilbert-space representation.

For a time-independent split $H=H_0+H_{\mathrm{int}}$, choose a reference time $t_0$ and define

$$\begin{aligned} O_I(t) &=e^{iH_0(t-t_0)}O_S e^{-iH_0(t-t_0)},\\ |\Psi(t)\rangle_I &=e^{iH_0(t-t_0)}|\Psi(t)\rangle_S,\\ H_I(t) &=e^{iH_0(t-t_0)}H_{\mathrm{int}} e^{-iH_0(t-t_0)}. \end{aligned}$$

Thus interaction-picture operators evolve with $H_0$, while states obey

$$i\frac{d}{dt}|\Psi(t)\rangle_I =H_I(t)|\Psi(t)\rangle_I.$$

For $t\geq t_0$, the solution with $U_I(t_0,t_0)=1$ is

$$U_I(t,t_0) =T\exp\left[-i\int_{t_0}^{t}dt'\,H_I(t')\right].$$

This compact expression is defined by the Dyson series,

$$\begin{aligned} U_I(t,t_0) ={}&1-i\int_{t_0}^{t}dt_1\,H_I(t_1)\\ &+\frac{(-i)^2}{2!} \int_{t_0}^{t}dt_1\int_{t_0}^{t}dt_2\, T\!\left\{H_I(t_1)H_I(t_2)\right\} +O(H_I^3). \end{aligned}$$

Time ordering is forced by noncommutativity: the later interaction Hamiltonian stands to the left, so the earlier-time operator acts first on a ket. Differentiating the ordered exponential recovers the evolution equation and is a useful sign check.

Coordinate time order can reverse under a Lorentz transformation only for spacelike-separated vertices. For local bosonic interactions, microcausality makes their densities commute there, so the reordered product is unchanged. This is why the Dyson expression can be Lorentz covariant even though it is written with an inertial time coordinate. Tong 2006–2007, §§3.1–3.1.1, printed pp. 50–53, PDF derives the interaction-picture evolution and ordered series.

Example: a quartic scalar interaction

For

$$\mathcal L_{\mathrm{int}} =-\frac{\lambda}{4!}\phi^4, \qquad H_I(t)=\int d^3x\,\frac{\lambda}{4!}\phi_I^4(x),$$

where $H_I=-\int d^3x\,\mathcal L_{\mathrm{int}}$ because this interaction contains no time derivatives. Derivative interactions require a fresh Legendre transform and need not obey that shortcut.

Each Dyson term contains time-ordered products of free fields. Normal ordering, denoted $:\!\cdots\!:$, moves annihilation operators to the right of creation operators; its vacuum expectation value vanishes. Wick’s theorem rewrites a time-ordered product as its normal-ordered product plus every possible contraction. A contraction is precisely the Feynman function from Correlation Functions and Propagators:

$$\operatorname{Contr}\!\left[\phi_I(x)\phi_I(y)\right] =\langle0|T\phi_I(x)\phi_I(y)|0\rangle =G_{\mathrm F}(x-y).$$

For four fields, the combinatorics are already visible:

$$\begin{aligned} T\{\phi_1\phi_2\phi_3\phi_4\} ={}&:\!\phi_1\phi_2\phi_3\phi_4\!:\\ &+\sum_{6\ \mathrm{single\ pairs}} G_{\mathrm F}(i,j):\!\phi_k\phi_l\!:\\ &+\sum_{3\ \mathrm{pairings}} G_{\mathrm F}(i,j)G_{\mathrm F}(k,l). \end{aligned}$$

The six single contractions choose one of the $\binom42$ pairs; the three double contractions partition four labels into two unordered pairs. Tong 2006–2007, §3.3, printed pp. 56–60, PDF proves the general theorem and develops these combinatorics.

Vertices, contractions, external wavefunctions, symmetry factors, and momentum-conserving delta functions can then be represented by diagrams. A Feynman diagram is a term in an expansion, not a literal spacetime history of particles.

Let $|\Omega\rangle$ be the normalized interacting vacuum and $\phi_H$ the Heisenberg field. If the free vacuum has nonzero overlap with a unique interacting vacuum that can be reached by adiabatic switching, the Gell-Mann–Low relation gives

$$\langle\Omega|T\{\phi_H(x_1)\cdots\phi_H(x_n)\}|\Omega\rangle = \frac{ \langle0|T\{\phi_I(x_1)\cdots\phi_I(x_n) \exp[-i\int d^4x\,\mathcal H_I(x)]\}|0\rangle} {\langle0|T\exp[-i\int d^4x\,\mathcal H_I(x)]|0\rangle}.$$

An $i\epsilon$ deformation, equivalently idealized switching such as $\mathcal H_I(t)\mapsto e^{-\epsilon|t|}\mathcal H_I(t)$ followed by $\epsilon\to0^+$, performs the vacuum projection. The denominator removes the same vacuum-to-vacuum factor from the numerator, so disconnected vacuum bubbles cancel. Degenerate vacua, gapless spectra, vanishing vacuum overlap, and uncontrolled adiabatic limits require a different treatment; the displayed ratio is not an identity without these hypotheses. Tong 2006–2007, §§3.7–3.7.1, printed pp. 75–77, PDF states the vacuum projection and normalized ratio explicitly.

From correlators to scattering

If the theory has stable asymptotic particle states and interactions become negligible sufficiently far in the past and future, one defines

$$S=U_I(+\infty,-\infty).$$

Reduction formulas extract scattering amplitudes from the one-particle poles of time-ordered correlators, including their residues. This requires more than a formal Dyson series: the asymptotic states and limits must exist. The assumptions can fail for confinement, unstable states, long-range interactions, or genuinely time-dependent backgrounds.

At first order in $\lambda$, the connected $2\to2$ matrix element contains one quartic vertex:

$$\langle p_3p_4|S^{(1)}|p_1p_2\rangle_{\mathrm{conn}} =-i\lambda\, (2\pi)^4\delta^{(4)}(p_1+p_2-p_3-p_4),$$

up to the chosen external-state normalization. The factor $4!$ in the Lagrangian cancels the number of ways to attach four distinguishable external slots to the vertex.

Tong 2006–2007, §3.5.3, printed p. 69, PDF performs this first-order quartic contraction and obtains the same vertex factor.

Tong 2006–2007, §§3.7–3.7.2, printed pp. 75–80, PDF develops vacuum-bubble cancellation and the reduction mechanics used here.

In–out amplitudes versus causal expectation values

The Feynman functions above organize in–out vacuum amplitudes: they compare a prepared state in the remote past with a postselected state in the remote future. A different question is the expectation value of an observable at finite time in the same specified initial state. That problem uses an in–in, or closed-time-path, contour with forward and backward evolution; it is not obtained by replacing every Feynman function with a retarded one in an in–out diagram.

This distinction matters in gravitational settings, where cosmological perturbations and semiclassical backreaction are often initial-value expectation-value problems rather than scattering experiments. Continue to Schwinger–Keldysh and In-In Dynamics for causal finite-time observables, or to Perturbative Algebraic Interactions when no global scattering construction is available.

What the expansion does—and does not—control

Perturbation theory requires more than a numerically small symbol $\lambda$:

• the dimensionless expansion parameter can include kinematic logarithms or ratios of scales;
• loop integrals require a regulator and renormalization;
• massless particles can create infrared divergences that cancel only in suitably inclusive observables or require resummation;
• secular terms can invalidate a fixed-order expansion at late times; and
• the series may have zero radius of convergence while still yielding accurate optimally truncated predictions.

Unitarity is exact for Hermitian $H_I$ before truncation. A finite subset of diagrams need not display exact unitarity, gauge invariance, or positivity unless the approximation is organized consistently.

Exercises

Recover the second Dyson term

Derive the first two terms of the Dyson series by iterating the integral equation

$$U_I(t,t_0) =1-i\int_{t_0}^{t}dt_1\,H_I(t_1)U_I(t_1,t_0).$$

Solution

One substitution gives

$$U_I =1-i\int_{t_0}^{t}dt_1H_I(t_1) -\int_{t_0}^{t}dt_1\int_{t_0}^{t_1}dt_2\, H_I(t_1)H_I(t_2)+O(H_I^3).$$

The triangular integration region has $t_1\gt t_2$. Combining it with the complementary region converts the second term to

$$\frac{(-i)^2}{2!} \int_{t_0}^{t}dt_1dt_2\, T\{H_I(t_1)H_I(t_2)\}.$$

For Hermitian $H_I$, this same decomposition checks unitarity through second order. Writing $U_I=1+A_1+A_2+O(H_I^3)$, one has $A_1^\dagger=-A_1$, while the two time-ordering triangles give

$$A_2+A_2^\dagger+A_1^\dagger A_1=0.$$

Therefore $U_I^\dagger U_I=1+O(H_I^3)$. Dropping one ordering triangle would fail this check even though the remaining integrals look perturbatively small.

Count the quartic tree contraction

At first order in $\lambda$, consider the connected contribution in which four external fields are each contracted once with the four fields at one $\phi^4$ vertex. Count the contractions and explain the coefficient of the resulting vertex.

Solution

Choose which of the four vertex fields contracts with the first external field, then the second, and so on. The number of bijections is

$$4\times3\times2\times1=4!.$$

The first-order Dyson factor and interaction density supply

$$(-i)\frac{\lambda}{4!}\times4!=-i\lambda.$$

The spacetime integral over the vertex produces the four-momentum delta function after Fourier transformation. This count applies to the connected tree attachment described in the question; self-contractions and disconnected pieces belong to different terms.

The loop terms generated by this method lead directly to Renormalization and Effective Field Theory.

References

• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, Michaelmas Terms 2006 and 2007, §§3.1–3.7.2, printed pp. 50–80. Official author PDF.