From Evolution Operators to Time-Ordered Products

Interaction-picture quantum mechanics contains the operator skeleton of perturbative quantum field theory. Split the Hamiltonian into a solvable part and an interaction, evolve with a time-ordered exponential, expand that exponential, and insert operators into its ordered products. In QFT, the interaction operators are built from fields at spacetime points, so the same Dyson expansion produces time-ordered field products.

That handoff is structural, not merely notational:

$$\begin{gathered} U_I(t,t_0) = \mathcal T \exp\left[ - \frac{i}{\hbar} \int_{t_0}^{t}dt'\,H_I(t') \right], \\ H_I(t) = \int d^3\mathbf x\, \mathcal H_I(\mathbf x,t). \end{gathered}$$

Interaction Picture owns the picture transformation. Dyson Expansion as Formal Evolution owns the series derivation, Dyson Expansion for Transition Amplitudes owns its use for quantum-mechanical transition paths, and Time Ordering owns the ordering operation. This page owns the bridge from those objects to field products, vacuum correlators, and the perturbative QFT workflow.

Interaction-Picture Evolution in Quantum Mechanics

Let

$$H(t) = H_0+V(t),$$

where $H_0$ is chosen as the exactly solvable part. If

$$U_0(t,t_0) = \exp\left[ - \frac{i}{\hbar} H_0(t-t_0) \right],$$

then the interaction-picture operator corresponding to a Schrödinger-picture operator $O_S$ is

$$O_I(t) = U_0^\dagger(t,t_0) O_S U_0(t,t_0).$$

The interaction becomes

$$V_I(t) = U_0^\dagger(t,t_0) V(t) U_0(t,t_0).$$

The remaining evolution operator satisfies

$$i\hbar \frac{\partial}{\partial t} U_I(t,t_0) = V_I(t)U_I(t,t_0),$$

with

$$U_I(t_0,t_0) = I.$$

This split is exact before any perturbative truncation. The choice of $H_0$ determines which motion is built into the operators and which interaction is expanded.

Dyson Expansion

The formal solution is

$$U_I(t,t_0) = \mathcal T \exp\left[ - \frac{i}{\hbar} \int_{t_0}^{t} dt_1\,V_I(t_1) \right].$$

Expanding gives

$$\begin{aligned} U_I(t,t_0) &= I - \frac{i}{\hbar} \int_{t_0}^{t} dt_1\,V_I(t_1) \\ &\quad+ \frac{1}{2!} \left( - \frac{i}{\hbar} \right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t}dt_2 \\ &\qquad\qquad\times \mathcal T \left[ V_I(t_1)V_I(t_2) \right] + \cdots . \end{aligned}$$

The factor $1/n!$ compensates for integrating over the full $n$-dimensional time cube. Time ordering partitions that cube into regions with definite operator order. Equivalently, one may integrate only over the ordered simplex and omit $1/n!$.

This series is formally exact when it exists. A finite-order perturbative calculation is an approximation in the interaction strength or another declared parameter and need not preserve exact unitarity term by term.

Time Ordering Creates Operator Products

For bosonic operators,

$$\begin{aligned} \mathcal T \left[ A(t_1)B(t_2) \right] &= \Theta(t_1-t_2) A(t_1)B(t_2) \\ &\quad+ \Theta(t_2-t_1) B(t_2)A(t_1). \end{aligned}$$

The purpose is not to sort symbols aesthetically. It preserves the causal composition order of infinitesimal evolution operators when the insertions do not commute.

Once an observable is inserted, matrix elements take the schematic form

$$\begin{aligned} \mathcal A &= \langle f\vert \mathcal T \Bigg\{ O_{1,I}(t_1)\cdots O_{n,I}(t_n) \\ &\qquad\qquad\times \exp\left[ - \frac{i}{\hbar} \int dt\,V_I(t) \right] \Bigg\} \vert i\rangle. \end{aligned}$$

Expanding the exponential produces ordered products containing both the explicitly inserted observables and interaction vertices. This is the direct operator origin of perturbative time-ordered correlation functions.

Equal-time prescriptions require care. Step-function conventions can matter at coincident times, and differentiating a time-ordered product can generate contact terms when derivatives act on the hidden step functions.

Operators Become Fields

In a scalar field theory, choose a free Hamiltonian $H_0$ and an interaction

$$H_{\rm int} = \int d^3\mathbf x\, \mathcal H_{\rm int} \left( \phi,\pi,\nabla\phi \right).$$

The interaction-picture field

$$\phi_I(x) = U_0^\dagger(t,t_0) \phi_S(\mathbf x) U_0(t,t_0)$$

obeys the free field equation because its time dependence is generated by $H_0$. The interaction-picture state carries the remaining dynamics through

$$H_I(t) = \int d^3\mathbf x\, \mathcal H_I(\mathbf x,t).$$

Hence

$$\begin{aligned} U_I(t,t_0) &= \mathcal T \exp\Bigg[ - \frac{i}{\hbar} \int_{t_0}^{t}dt' \\ &\qquad\qquad\times \int d^3\mathbf x\, \mathcal H_I(\mathbf x,t') \Bigg]. \end{aligned}$$

Writing $d^4x=dt\,d^3\mathbf x$, this is often abbreviated as a spacetime integral. For interactions without time derivatives, one commonly has $\mathcal H_{\rm int}=-\mathcal L_{\rm int}$, giving the equivalent Lagrangian-density sign convention. Derivative interactions, constraints, and gauge systems require a more careful Hamiltonian analysis.

Example: Scalar Quartic Interaction

For

$$\mathcal L_{\rm int} = - \frac{\lambda}{4!} \phi^4,$$

the interaction Hamiltonian density is

$$\mathcal H_{\rm int} = \frac{\lambda}{4!} \phi^4.$$

The first-order interaction-picture evolution is

$$U_I^{(1)} = - \frac{i\lambda}{4!\hbar} \int d^4x\, \phi_I^4(x).$$

The second-order term contains

$$\frac{1}{2!} \left( - \frac{i\lambda}{4!\hbar} \right)^2 \int d^4x\,d^4y\, \mathcal T \left[ \phi_I^4(x)\phi_I^4(y) \right].$$

The fields at $x$ and $y$ are operator-valued distributions. Their products at coincident points are not automatically finite operators; regularization, composite-operator definitions, and renormalization enter in a full QFT treatment.

From Time-Ordered Products to Correlators

An interacting vacuum $n$-point function is

$$G_n(x_1,\ldots,x_n) = \langle\Omega\vert \mathcal T \left[ \phi_H(x_1)\cdots\phi_H(x_n) \right] \vert\Omega\rangle.$$

The Heisenberg fields contain the full dynamics. Perturbation theory rewrites this object in terms of interaction-picture fields and the free reference vacuum.

Under the usual adiabatic and vacuum-overlap assumptions, the Gell-Mann–Low structure is schematically

$$\begin{aligned} G_n &= \frac{ \langle0\vert \mathcal T \left\{ \phi_I(x_1)\cdots\phi_I(x_n) e^{-(i/\hbar)\int dt\,H_I(t)} \right\} \vert0\rangle }{ \langle0\vert \mathcal T \left\{ e^{-(i/\hbar)\int dt\,H_I(t)} \right\} \vert0\rangle }. \end{aligned}$$

The denominator normalizes the interacting-vacuum expectation and cancels disconnected vacuum-to-vacuum factors in perturbation theory. It is not optional decoration.

This formula hides important assumptions: the interaction is switched or projected so that the free reference vacuum connects to the desired interacting vacuum; vacuum degeneracy and level crossings are controlled; regulators are present; and the perturbative construction is meaningful. Haag-type issues and nonperturbative vacuum structure prevent one from reading the formula as a naive equality between globally defined free and interacting Fock representations.

Wick’s Theorem Preview

Time ordering and normal ordering are different operations:

• $\mathcal T$ orders operators by time labels and includes fermionic permutation signs;
• normal ordering places creation operators to the left of annihilation operators relative to a chosen free vacuum;
• Wick’s theorem rewrites a time-ordered product of free or Gaussian fields as normal-ordered products plus all contractions.

For a free real scalar field, define the contraction convention by

$$\Delta_F(x-y) = \langle0\vert \mathcal T \left[ \phi_I(x)\phi_I(y) \right] \vert0\rangle.$$

Then the free four-point vacuum correlator is

$$\begin{aligned} \langle0\vert \mathcal T \left[ \phi_1\phi_2\phi_3\phi_4 \right] \vert0\rangle &= \Delta_{12}\Delta_{34} \\ &\quad+ \Delta_{13}\Delta_{24} + \Delta_{14}\Delta_{23}, \end{aligned}$$

where $\phi_j=\phi_I(x_j)$ and $\Delta_{ij}=\Delta_F(x_i-x_j)$.

Each contraction supplies a free two-point function. Interaction-density insertions supply vertices. Expanding the Dyson exponential and enumerating contractions is the operator origin of Feynman-diagram combinatorics. A diagram is bookkeeping for a term in this expansion, not a literal spacetime photograph of particles following classical paths.

Fermionic Time Ordering

Fermionic fields introduce signs. For a Dirac field,

$$\begin{aligned} \mathcal T \left[ \psi(x)\bar\psi(y) \right] &= \Theta(x^0-y^0) \psi(x)\bar\psi(y) \\ &\quad- \Theta(y^0-x^0) \bar\psi(y)\psi(x). \end{aligned}$$

The minus sign records the odd exchange of fermionic operators. For many insertions, every permutation contributes its fermionic parity. Omitting these signs changes amplitudes and loop factors.

Grassmann path integrals encode the same antisymmetry algebraically, but the operator origin remains the exchange sign in time-ordered products.

Time Ordering and Relativistic Locality

Coordinate-time ordering may appear noncovariant because two spacelike-separated events can reverse temporal order under a Lorentz transformation. Microcausality supplies the resolution for local fields: bosonic local observables commute at spacelike separation, while fermionic fields obey the corresponding graded relation. Reversing a spacelike ordering therefore does not change a properly defined local time-ordered correlator except for the required fermionic grading.

This statement does not remove ultraviolet singularities at coincident points. Products of local fields are distributions, and composite operators require renormalized definitions. Gauge-dependent fields and constrained systems add further qualifications.

S-Matrix Limit

When asymptotically free in- and out-states exist, one writes

$$S = U_I(+\infty,-\infty) = \mathcal T \exp\left[ - \frac{i}{\hbar} \int_{-\infty}^{+\infty} dt\,H_I(t) \right].$$

Matrix elements of $S$ between asymptotic states generate scattering amplitudes after external-state normalization and reduction procedures are handled.

This limit is not universal. Confining theories, finite-volume systems, bound-state questions, curved spacetimes, thermal states, and theories without conventional asymptotic particles require other observables or boundary prescriptions. Time-ordered correlation functions remain meaningful more broadly than a particle S-matrix.

Operator and Path-Integral Views

The operator and path-integral formulations agree only after their boundary and ordering prescriptions are matched.

Operator language	Path-integral language
interaction-picture Dyson exponential	expansion of $e^{iS_{\rm int}/\hbar}$
time-ordered operator insertions	real-time source derivatives or insertions
free-vacuum contractions	inverse of the regulated quadratic kernel
interaction Hamiltonian density	interaction action and vertices
vacuum normalization denominator	division by $Z[0]$
fermionic exchange signs	Grassmann integration signs

Correlation Functions in Path Integrals owns the operator-to-path-integral correlator comparison. From Sources in QM to Generating Functionals in QFT owns the functional-derivative bridge.

What Carries Over and What Changes

Quantum-mechanical structure	QFT continuation	New qualification
$H=H_0+V$	free plus interacting field Hamiltonian	infinitely many modes and regulators
$U_I=\mathcal T e^{-(i/\hbar)\int V_I}$	spacetime interaction-density exponential	local operator products
ordered operator products	ordered field products	bosonic or fermionic grading
matrix elements	vacuum, thermal, or in-out correlators	state and contour dependence
Gaussian contractions	free propagator lines	distributions and pole prescriptions
perturbative terms	Feynman diagrams	symmetry factors and renormalization
long-time transition operator	S-matrix	asymptotic-state assumptions

The algebraic continuity is real. So are the new field-theoretic difficulties.

Common Mistakes

• Treating the interaction picture as a perturbative approximation rather than an exact representation before truncation.
• Dropping $\mathcal T$ when interaction Hamiltonians at different times fail to commute.
• Confusing time ordering with normal ordering.
• Applying Wick’s theorem without specifying a free or Gaussian reference state.
• Forgetting the normalization denominator in the Gell-Mann–Low expression.
• Assuming $\mathcal H_{\rm int}=-\mathcal L_{\rm int}$ for derivative interactions without checking canonical momenta.
• Omitting fermionic permutation signs.
• Calling every time-ordered two-point function a retarded response function.
• Treating fields as ordinary functions rather than operator-valued distributions.
• Interpreting diagrams as literal particle trajectories.
• Taking the S-matrix limit when the theory has no appropriate asymptotic particle states.
• Ignoring regulators, renormalization, contact terms, and vacuum assumptions hidden by compact notation.

Canonical Boundaries

• Interaction Picture owns the Hamiltonian split and picture transformation.
• Dyson Expansion as Formal Evolution owns the integral-equation and ordered-series derivation.
• Dyson Expansion for Transition Amplitudes owns ordered quantum-mechanical transition paths and intermediate-state sums.
• Time Ordering owns the basic bosonic ordering definition.
• Correlation Functions in Path Integrals distinguishes time-ordered, connected, Euclidean, and response functions.
• From Propagators in QM to Propagators in QFT owns two-point functions and pole prescriptions.
• From Sources in QM to Generating Functionals in QFT owns source derivatives and connected generators.
• From Path Integrals in QM to Field Path Integrals owns the change from coordinate histories to field histories.
• From Correlation Functions to QFT Observables owns the map from ordered products to spectra, response, scattering, and Euclidean measurements.

References

• F. J. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Physical Review 75, 486–502, 1949, doi:10.1103/PhysRev.75.486.
• F. J. Dyson, “The S matrix in quantum electrodynamics,” Physical Review 75, 1736–1755, 1949, doi:10.1103/PhysRev.75.1736.
• G. C. Wick, “The evaluation of the collision matrix,” Physical Review 80, 268–272, 1950, doi:10.1103/PhysRev.80.268.
• M. Gell-Mann and F. Low, “Bound states in quantum field theory,” Physical Review 84, 350–354, 1951, doi:10.1103/PhysRev.84.350.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

1. Show that the second-order Dyson term written over an ordered triangle equals the time-ordered full-square expression.

Solution

The ordered-triangle form is

$$\left( - \frac{i}{\hbar} \right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, V_I(t_1)V_I(t_2).$$

The full square splits into the regions $t_1\gt t_2$ and $t_2\gt t_1$. Time ordering maps both regions to the later-time operator on the left. Exchanging dummy labels in the second region makes its integral equal to the first. Therefore

$$\begin{aligned} &\frac{1}{2!} \left( - \frac{i}{\hbar} \right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t}dt_2 \\ &\qquad\times \mathcal T \left[ V_I(t_1)V_I(t_2) \right] \\ &= \left( - \frac{i}{\hbar} \right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, V_I(t_1)V_I(t_2). \end{aligned}$$

2. For the scalar quartic interaction, identify the number of field insertions at perturbative order $n$ before external operators are added.

Solution

Each interaction insertion contributes

$$\frac{\lambda}{4!} \int d^4x\, \phi_I^4(x).$$

At order $n$, the Dyson expansion contains $n$ interaction vertices and therefore $4n$ field operators from the interaction alone. External fields in an $m$-point correlator add another $m$ insertions, so Wick’s theorem acts on $4n+m$ free-field operators.

Odd total field number gives zero in the free scalar vacuum when the reference Gaussian has vanishing one-point function.

3. Derive the free scalar four-point function from Wick pairings.

Solution

Every field must be paired with one other field for a nonzero free-vacuum expectation. Four labeled fields have three distinct complete pairings:

$$(12)(34), \qquad (13)(24), \qquad (14)(23).$$

Each pair contributes one free time-ordered two-point function. Hence

$$\begin{aligned} \langle0\vert \mathcal T \left[ \phi_1\phi_2\phi_3\phi_4 \right] \vert0\rangle &= \Delta_{12}\Delta_{34} \\ &\quad+ \Delta_{13}\Delta_{24} + \Delta_{14}\Delta_{23}. \end{aligned}$$

4. Why does the denominator in the Gell-Mann–Low expression cancel vacuum bubbles?

Solution

Wick expansion of the numerator includes contractions entirely among interaction vertices, disconnected from every external insertion. These vacuum subdiagrams factor from the part connected to the external fields. The denominator is the same Dyson vacuum expectation with no external insertions, so its perturbative expansion contains precisely the vacuum-to-vacuum factors.

Dividing removes those common factors and normalizes the vacuum expectation. Connectedness among the external insertions is a further distinction: connected correlators are generated by a logarithm or selected diagrammatically after vacuum normalization.

5. Compare a fermionic time-ordered two-point function with a retarded anticommutator function.

Solution

The time-ordered product is

$$\begin{aligned} \mathcal T \left[ \psi(x)\bar\psi(y) \right] &= \Theta(x^0-y^0) \psi(x)\bar\psi(y) \\ &\quad- \Theta(y^0-x^0) \bar\psi(y)\psi(x). \end{aligned}$$

It contains both temporal orderings and is suited to in-out perturbation theory.

A retarded fermionic Green function instead has the schematic structure

$$G_R(x,y) \propto \Theta(x^0-y^0) \left\langle \{ \psi(x),\bar\psi(y) \} \right\rangle,$$

with convention-dependent factors. It vanishes when $x^0\lt y^0$ and describes causal response or propagation of a disturbance. Time ordering and retarded support answer different questions.

6. State two reasons the formal limit $S=U_I(+\infty,-\infty)$ may not define the useful observable of a QFT.

Solution

First, the theory may not possess asymptotically free particle states. Confinement, a finite box, a thermal medium, curved spacetime, or persistent long-range interactions can invalidate the ordinary in-out particle picture.

Second, the physical question may concern vacuum structure, bound states, finite-time response, thermal correlation functions, or local observables rather than scattering. In those cases, time-ordered, retarded, Euclidean, or contour-ordered correlators with an appropriate state prescription are more fundamental than an S-matrix element.