Part IV, Chapter 1 | Page 1 of 3

The Interaction Picture

Bridging the Schrodinger and Heisenberg pictures to isolate interactions in quantum field theory

1.1 Three Pictures of Quantum Mechanics

Quantum mechanics admits multiple equivalent formulations that distribute time dependence differently between states and operators. All three pictures yield identical physical predictions — they are related by unitary transformations — but they differ dramatically in computational convenience for different problems.

In the Schrodinger picture, the full time dependence resides in the state vector. The state $|\psi_S(t)\rangle$ obeys the Schrodinger equation:

$$i\hbar\frac{d}{dt}|\psi_S(t)\rangle = H|\psi_S(t)\rangle$$

while operators $O_S$ are time-independent. The formal solution is:

$$|\psi_S(t)\rangle = e^{-iHt}|\psi_S(0)\rangle$$

In the Heisenberg picture, all time dependence is transferred to the operators. States are frozen: $|\psi_H\rangle = |\psi_S(0)\rangle$. Operators evolve as:

$$O_H(t) = e^{iHt}O_S\,e^{-iHt}$$

which satisfies the Heisenberg equation of motion:

$$\frac{dO_H}{dt} = i[H, O_H(t)]$$

Both pictures give the same expectation values, since:

$$\langle\psi_S(t)|O_S|\psi_S(t)\rangle = \langle\psi_H|O_H(t)|\psi_H\rangle$$

Schrodinger

States evolve, operators fixed

$i\frac{d|\psi\rangle}{dt} = H|\psi\rangle$

Best for: time-dependent perturbation theory in QM

Heisenberg

States fixed, operators evolve

$\frac{dO}{dt} = i[H,O]$

Best for: operator algebra, free fields

Interaction

Both evolve, split by H = H₀ + H_int

$O_I \sim H_0, \; |\psi_I\rangle \sim H_{\text{int}}$

Best for: interacting QFT, scattering

1.2 Constructing the Interaction Picture

The interaction picture is specifically designed for situations where the full Hamiltonian splits into a solvable free part and a perturbative interaction:

$$H = H_0 + H_{\text{int}}$$

We define interaction-picture states by stripping off the free time evolution from the Schrodinger-picture state:

$$\boxed{|\psi_I(t)\rangle \equiv e^{iH_0 t}|\psi_S(t)\rangle}$$

Derivation of state evolution. We differentiate $|\psi_I(t)\rangle$ with respect to time:

$$i\frac{d}{dt}|\psi_I(t)\rangle = i\frac{d}{dt}\bigl(e^{iH_0 t}|\psi_S(t)\rangle\bigr)$$

Applying the product rule:

$$= -H_0 e^{iH_0 t}|\psi_S(t)\rangle + e^{iH_0 t}\bigl(i\frac{d}{dt}|\psi_S(t)\rangle\bigr)$$

Using the Schrodinger equation $i\frac{d}{dt}|\psi_S\rangle = (H_0 + H_{\text{int}})|\psi_S\rangle$:

$$= -H_0 e^{iH_0 t}|\psi_S\rangle + e^{iH_0 t}(H_0 + H_{\text{int}})|\psi_S\rangle = e^{iH_0 t}H_{\text{int}}|\psi_S\rangle$$

Writing $|\psi_S\rangle = e^{-iH_0 t}|\psi_I\rangle$ and defining $H_{\text{int},I}(t) \equiv e^{iH_0 t}H_{\text{int}}e^{-iH_0 t}$, we obtain:

$$\boxed{i\frac{d}{dt}|\psi_I(t)\rangle = H_{\text{int},I}(t)\,|\psi_I(t)\rangle}$$

💡Why the Interaction Picture Matters

The interaction picture is the workhorse of perturbative QFT because it achieves a clean separation of concerns:

Operators evolve with H₀ — they are free-field operators we already know how to handle (creation/annihilation operators with standard commutation relations).
States evolve with H_int — the interaction drives transitions between states, and we can expand perturbatively in the coupling constant.
In the absence of interactions (H_int = 0), interaction-picture states are time-independent, just like the Heisenberg picture.

1.3 Operator Evolution in the Interaction Picture

Interaction-picture operators are defined by:

$$\boxed{O_I(t) = e^{iH_0 t}\,O_S\,e^{-iH_0 t}}$$

Notice this is the same transformation rule as the Heisenberg picture, but with H₀ instead of H. The equation of motion follows immediately:

$$\frac{dO_I}{dt} = i[H_0, O_I(t)]$$

This means interaction-picture field operators satisfy the free field equations. For a scalar field $\phi_I(x)$:

$$(\partial^2 + m^2)\phi_I(x) = 0$$

so the mode expansion takes the familiar free-field form:

$$\phi_I(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\Big(a_{\mathbf{p}}\,e^{-ip\cdot x} + a_{\mathbf{p}}^\dagger\,e^{ip\cdot x}\Big)$$

All the technology of free-field quantization — commutation relations, propagators, Fock space — carries over directly to the interaction picture. The interacting theory's complexity is entirely encoded in the state evolution.

1.4 The Time Evolution Operator

The state evolution equation $i\frac{d}{dt}|\psi_I\rangle = H_{\text{int},I}(t)|\psi_I\rangle$ is solved by introducing the time evolution operator U(t, t₀):

$$|\psi_I(t)\rangle = U(t,t_0)\,|\psi_I(t_0)\rangle$$

Substituting into the evolution equation, U must satisfy:

$$i\frac{\partial}{\partial t}U(t,t_0) = H_{\text{int},I}(t)\,U(t,t_0), \qquad U(t_0,t_0) = \mathbb{1}$$

U(t, t₀) has important algebraic properties that follow from its definition:

Unitarity: $U^\dagger(t,t_0)U(t,t_0) = \mathbb{1}$, preserving probability
Composition: $U(t_2,t_0) = U(t_2,t_1)U(t_1,t_0)$ for any intermediate time t₁
Inverse: $U^{-1}(t,t_0) = U(t_0,t) = U^\dagger(t,t_0)$

These properties ensure the physical consistency of the formalism. The challenge is solving for U explicitly — this leads to the Dyson series, developed on the next page.

1.5 Connection to the S-Matrix

The S-matrix is defined as the time evolution operator taken over all of time:

$$\boxed{S = U(+\infty, -\infty) = \lim_{\substack{t\to+\infty \\ t_0\to-\infty}} U(t, t_0)}$$

The physical setup is: at $t \to -\infty$, particles are far apart and non-interacting (free states). They approach, interact, and at $t \to +\infty$ they are again free. The S-matrix maps the initial asymptotic state to the final asymptotic state:

$$|\text{out}\rangle = S\,|\text{in}\rangle$$

The scattering amplitude for a specific process is:

$$S_{fi} = \langle f|S|i\rangle = \delta_{fi} + i(2\pi)^4\delta^{(4)}(p_f - p_i)\,\mathcal{M}_{fi}$$

where $\delta_{fi}$ is the no-scattering (forward) contribution and $\mathcal{M}_{fi}$ is the invariant amplitude that encodes all the non-trivial physics. Observable cross sections and decay rates are computed from $|\mathcal{M}_{fi}|^2$.

💡The Adiabatic Hypothesis

Taking $t \to \pm\infty$ requires care. We assume the interaction is "turned on" adiabatically: replace $H_{\text{int}} \to e^{-\epsilon|t|}H_{\text{int}}$ and take $\epsilon \to 0^+$ at the end. This ensures the asymptotic states are genuine free-particle states and avoids infrared divergences in the formal definition of S. The physical justification is that real particles are widely separated (and hence non-interacting) in the asymptotic past and future.

Key Concepts

The Schrodinger, Heisenberg, and interaction pictures are unitarily equivalent formulations of quantum mechanics
In the interaction picture, operators evolve with the free Hamiltonian H₀ while states evolve with H_int
Interaction-picture fields satisfy free field equations and retain their free-field mode expansions
The time evolution operator U(t, t₀) is unitary and satisfies a first-order differential equation driven by H_int,I(t)
The S-matrix S = U(+$\infty$, -$\infty$) maps asymptotic "in" states to "out" states
Physical scattering amplitudes are extracted from S_fi = $\delta_{fi} + i(2\pi)^4\delta^{(4)}(p_f - p_i)\mathcal{M}_{fi}$

Next: The Dyson Series→

Quantum Field Theory Course