Interaction picture

The interaction picture, also known as the Dirac picture or interaction representation, is a formalism in quantum mechanics in which both the state vectors and the dynamical operators evolve in time, providing an intermediate framework between the Schrödinger and Heisenberg pictures.^[1] It was introduced by Paul Dirac in his seminal work The Principles of Quantum Mechanics.^[1] This picture arises by splitting the total Hamiltonian H = H_0 + V(t) into a free (or unperturbed) part H_0, which is typically solvable and time-independent, and a perturbation or interaction part V(t), which may be time-dependent and small in magnitude.^[2] In the interaction picture, the state vectors evolve according to the interaction Hamiltonian V_I(t), defined as V_I(t) = e^{i H_0 t / \hbar} V(t) e^{-i H_0 t / \hbar}, while the operators evolve under the free Hamiltonian as O_I(t) = e^{i H_0 t / \hbar} O_S e^{-i H_0 t / \hbar}, where O_S denotes the time-independent Schrödinger-picture operator.^[1] This contrasts with the Schrödinger picture, where states evolve under the full Hamiltonian and operators are fixed, and the Heisenberg picture, where states are time-independent and operators evolve under the full Hamiltonian.^[3] The time evolution of states in the interaction picture is governed by the Schrödinger-like equation i \hbar \frac{d}{dt} |\psi_I(t)\rangle = V_I(t) |\psi_I(t)\rangle, which can be solved using a time-ordered exponential or perturbative series expansion when V(t) is weak.^[2] The interaction picture is particularly advantageous for time-dependent perturbation theory, enabling the calculation of transition probabilities and amplitudes in systems subject to external fields or weak interactions, such as atomic transitions under electromagnetic radiation.^[3] It forms the basis for the Dyson series, which expresses the time-evolution operator as a perturbative expansion, and has been extended to quantum field theory, where it underpins the S-matrix formalism and Feynman diagram techniques developed by Freeman Dyson in the 1940s and 1950s.^[1] Applications span condensed matter physics, quantum optics, and particle physics, facilitating the analysis of scattering processes and non-equilibrium dynamics.^[4]

Introduction

Motivation and historical development

In the Schrödinger picture of quantum mechanics, the time evolution of state vectors is governed by a time-dependent Schrödinger equation when the Hamiltonian includes interactions that vary with time, such as those arising from external fields or perturbations. This leads to significant computational challenges, as the full Hamiltonian lacks a simple diagonal form in the unperturbed basis, making exact solutions intractable for all but the simplest systems and complicating the separation of intrinsic dynamics from interaction effects.^[5] The interaction picture emerged as a response to these difficulties, with its origins tracing back to Paul Dirac's 1927 work on the quantum theory of radiation, where he proposed an intermediate representation to handle the interaction between atoms and the electromagnetic field by decomposing the evolution into free-field propagation and perturbative scattering processes. This approach was later extended and formalized in the context of relativistic quantum field theory during the 1940s, particularly through Sin-Itiro Tomonaga's 1946 development of a covariant formulation that maintained Lorentz invariance while treating interactions perturbatively,^[6] and Julian Schwinger's 1948 covariant quantum electrodynamics, which explicitly defined the interaction representation as a hybrid framework for calculating scattering amplitudes.^[7] Wolfgang Pauli contributed to the early foundations of quantum electrodynamics in collaboration with Werner Heisenberg around 1929–1930, laying groundwork for handling field interactions that influenced the picture's application in perturbative calculations.^[8] Conceptually, the interaction picture motivates its use by combining features of the Schrödinger and Heisenberg pictures: state vectors evolve under the free Hamiltonian alone, capturing the unperturbed dynamics in a manner akin to the Heisenberg picture, while interaction operators retain explicit time dependence from the full Hamiltonian, facilitating time-dependent perturbation theory for weak interactions. This hybrid structure is particularly advantageous for systems where the interaction is small compared to the free evolution, allowing systematic expansion in powers of the perturbation strength. For instance, in a two-level atom interacting with an electromagnetic field, the free evolution accounts for the atomic energy splitting, while the dipole interaction drives transitions; the interaction picture simplifies analysis by removing the rapid free oscillations, enabling efficient computation of transition probabilities without solving the full time-dependent problem.^[9]^[2]

Relation to other pictures

In quantum mechanics, the Schrödinger picture describes the time evolution of physical systems by allowing state vectors to evolve according to the Schrödinger equation while keeping operators time-independent.^[10] In contrast, the Heisenberg picture fixes the state vectors and transfers the time dependence to the operators, which evolve via the Heisenberg equation of motion.^[11] The interaction picture, also known as the Dirac picture and introduced by Paul Dirac, serves as a hybrid representation that combines elements of both: state vectors evolve solely under the unperturbed Hamiltonian, while operators incorporate the effects of interactions.^[10] This hybrid nature arises from decomposing the total Hamiltonian as H = H_0 + V(t), where H_0 is the solvable free Hamiltonian and V(t) represents the interaction term, often time-dependent.^[11] In the interaction picture, the free evolution dictated by H_0 is "factored out" of the states, leaving them to evolve only due to the interaction V(t), whereas operators are transformed to explicitly reflect the interaction dynamics on top of the free evolution.^[10] This conceptual flow bridges the Schrödinger picture's focus on state dynamics and the Heisenberg picture's emphasis on operator evolution, providing a unified framework for analyzing perturbations.^[11] The primary advantage of the interaction picture lies in its simplification of calculations for systems where H_0 is exactly solvable, allowing the interaction V(t) to be treated perturbatively without the complexity of the full Hamiltonian.^[10] It is particularly chosen for scenarios involving time-dependent perturbation theory, such as in quantum optics or scattering processes, where isolating the interaction facilitates iterative solutions and clearer physical interpretation of transition amplitudes.^[11]

Formal definition

Hamiltonian decomposition

In quantum mechanics, the Hamiltonian decomposition forms the basis for the interaction picture by splitting the total Hamiltonian H(t) of a system into two parts: a time-independent unperturbed Hamiltonian H_0 and a perturbation or interaction Hamiltonian V(t), such that

H(t) = H_0 + V(t).

This splitting was first employed by Dirac in his analysis of radiation emission and absorption, where H_0 represented the unperturbed atomic and field energies, and V(t) captured their coupling.^[12] Both H_0 and V(t) act on the same Hilbert space of the system.^[13] The unperturbed Hamiltonian H_0 is chosen to be exactly solvable, meaning its time-evolution operator can be computed analytically, often corresponding to the dominant dynamics of the system in the absence of interactions. For instance, in the case of a free particle, H_0 = \frac{\mathbf{p}^2}{2m}, where \mathbf{p} is the momentum operator and m is the mass. In bound systems, such as a harmonic oscillator, H_0 may include both kinetic energy and a quadratic potential, H_0 = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2, with \omega the angular frequency.^[13]^[10] The interaction term V(t) typically represents a weaker, often time-dependent perturbation that drives transitions or modifications to the unperturbed evolution; its magnitude is assumed small relative to H_0 to ensure the validity of perturbative approximations. A common example arises in atomic physics under an external electromagnetic field, where V(t) = -\mathbf{d} \cdot \mathbf{E}(t), with \mathbf{d} the electric dipole operator and \mathbf{E}(t) the time-varying electric field. Another illustrative case is a two-level quantum system, such as a spin-1/2 particle in a magnetic field, with H_0 = \frac{\hbar \omega_0}{2} \sigma_z and V(t) = \frac{\hbar \omega_1}{2} [\cos(\omega t) \sigma_x + \sin(\omega t) \sigma_y], where \sigma_z, \sigma_x, \sigma_y are Pauli matrices.^[13]^[10] This decomposition is crucial because it isolates the solvable dynamics of H_0 from the complicating effects of V(t), facilitating the application of unitary transformations that define the interaction picture and enable perturbative treatments of time-dependent phenomena.^[13]

Transformation to interaction picture

The interaction picture is obtained by applying a unitary transformation to the Schrödinger picture, where the total Hamiltonian is decomposed as H = H_0 + V(t), with H_0 being the solvable part and V(t) the interaction.^[2] The unitary operator defining this transformation is U_I(t, t_0) = \exp\left( i H_0 (t - t_0)/\hbar \right), which generates the free evolution under H_0; often, t_0 = 0 is assumed for simplicity.^[10]^[1] In terms of state vectors, the interaction picture state is related to the Schrödinger picture state by |\psi_I(t)\rangle = U_I(t, 0) |\psi_S(t)\rangle, where the subscript S denotes the Schrödinger picture.^[2]^[10] For operators, the transformation is A_I(t) = U_I(t, 0) A_S U_I^\dagger(t, 0), where A_S is the Schrödinger-picture operator, which is time-independent unless it has explicit time dependence.^[1]^[2] The initial condition ensures continuity at t = [0](/page/0): |\psi_I(0)\rangle = |\psi_S(0)\rangle and A_I(0) = A_S(0), since U_I(0, 0) = I.^[10]^[2] To derive this, begin with the Schrödinger equation i[\hbar](/page/H-bar) \frac{d}{dt} |\psi_S(t)\rangle = H |\psi_S(t)\rangle. Substitute |\psi_S(t)\rangle = U_I^\dagger(t, 0) |\psi_I(t)\rangle and differentiate with respect to time, using the fact that U_I satisfies i[\hbar](/page/H-bar) \frac{d}{dt} U_I(t, 0) = H_0 U_I(t, 0). This yields the interaction picture equation i[\hbar](/page/H-bar) \frac{d}{dt} |\psi_I(t)\rangle = V_I(t) |\psi_I(t)\rangle, where V_I(t) = U_I(t, 0) V(t) U_I^\dagger(t, 0).^[2]^[1]^[10]

Time evolution

Evolution of state vectors

In the interaction picture, the time evolution of the state vector |\psi_I(t)\rangle is described by a differential equation analogous to the Schrödinger equation, but driven solely by the interaction Hamiltonian. Specifically, it satisfies

i \hbar \frac{d}{dt} |\psi_I(t)\rangle = V_I(t) |\psi_I(t)\rangle,

where V_I(t) = e^{i H_0 t / \hbar} V(t) e^{-i H_0 t / \hbar} is the interaction Hamiltonian transformed into the interaction picture, or equivalently V_I(t) = U_0^\dagger(t) V(t) U_0(t) with U_0(t) = e^{-i H_0 t / \hbar} the unitary evolution operator generated by the free Hamiltonian H_0.^[10]^[2] This equation isolates the effects of the perturbation V(t), as the free evolution under H_0 has been factored out through the picture transformation. The formal solution to this equation is given by the time evolution operator in the interaction picture,

|\psi_I(t)\rangle = \mathcal{T} \exp\left( -\frac{i}{\hbar} \int_0^t V_I(t') \, dt' \right) |\psi_I(0)\rangle,

where \mathcal{T} is the time-ordering operator that arranges the non-commuting operators V_I(t') in chronological order, with later times to the left.^[10]^[14]^[2] This time-ordered exponential ensures the correct handling of the time-dependent perturbation when operators at different times do not commute, a feature originating from the structure of quantum evolution under time-varying Hamiltonians. This formulation highlights that state vectors in the interaction picture evolve exclusively due to interactions, with the unperturbed dynamics removed to simplify analysis of perturbative effects.^[10] The evolution operator remains unitary provided V_I(t) is Hermitian at all times, preserving the norm of the state vector.^[14] Furthermore, the time-ordered exponential serves as the starting point for perturbative expansions, such as the Dyson series, which approximate the evolution for weak interactions.^[2]

Evolution of operators

In the interaction picture of quantum mechanics, the time evolution of operators combines the free dynamics governed by the unperturbed Hamiltonian H_0 with the overall framework that isolates interaction effects for states, creating a hybrid representation.^[10] For an operator A that is time-independent in the Schrödinger picture (i.e., no explicit time dependence), the corresponding interaction-picture operator A_I(t) is defined as

A_I(t) = e^{i H_0 t / \hbar} A e^{-i H_0 t / \hbar},

where the transformation removes the free evolution from the states and assigns it to the operators.^[15] This form reflects the free propagation under H_0, analogous to the Heisenberg picture but limited to the unperturbed part of the Hamiltonian.^[10] The time derivative of A_I(t) follows a Heisenberg-like equation of motion derived from the unitary transformation:

i \hbar \frac{d A_I(t)}{dt} = [A_I(t), H_0].

Equivalently,

\frac{d A_I(t)}{dt} = \frac{i}{\hbar} [H_0, A_I(t)].

This equation indicates that operators evolve solely due to the commutator with H_0, independent of the interaction Hamiltonian V.^[15] The interaction effects are instead captured in the evolution of the states, making the interaction picture particularly useful for perturbation theory where free dynamics are well-understood.^[10] If the operator A commutes with H_0, i.e., [A, H_0] = 0, the evolution simplifies further, and A_I(t) becomes time-independent: A_I(t) = A. In this case, there is no free evolution, and the operator remains fixed, allowing expectation values to reflect only the interaction-driven changes in the state.^[15] For operators that do not commute with H_0, such as position or momentum in a free-particle system, the full free evolution applies, rotating the operator in the interaction frame according to the explicit exponential form. This distinction highlights the hybrid nature of the picture: operators experience the deterministic free dynamics of H_0, while states incorporate the perturbative influence of V_I(t).^[10] In cases where operators depend on the interactions—such as when considering composite operators involving the perturbation—the evolution retains the Heisenberg-like structure in the interaction frame but must account for the transformed interaction V_I(t) = e^{i H_0 t / [\hbar](/page/H-bar)} V e^{-i H_0 t / [\hbar](/page/H-bar)}. However, the core equation remains governed by H_0, ensuring consistency with the picture's design to separate free and interacting components.^[15] This framework facilitates computations in systems like quantum field theory or time-dependent perturbations, where operator evolution under H_0 simplifies matrix elements and transition amplitudes.^[10]

Evolution of density matrices

In the interaction picture, the density matrix \rho_I(t) is defined through a similarity transformation of the Schrödinger-picture density matrix \rho_S(t) using the free-evolution operator U_0(t) = \exp(-i H_0 t / \hbar), yielding \rho_I(t) = U_0^\dagger(t) \rho_S(t) U_0(t). This transformation removes the dynamics due to the unperturbed Hamiltonian H_0, focusing the evolution on the interaction term. Equivalently, since the full time-evolution operator in the Schrödinger picture is U_S(t, 0) = U_0(t) U_I(t, 0), where U_I(t, 0) is the interaction-picture evolution operator, the density matrix evolves as \rho_I(t) = U_I(t, 0) \rho_I(0) U_I^\dagger(t, 0), with \rho_I(0) = \rho_S(0).^[16]^[17] The time evolution of \rho_I(t) satisfies the von Neumann equation restricted to the interaction Hamiltonian:

i \hbar \frac{d}{dt} \rho_I(t) = [V_I(t), \rho_I(t)],

where V_I(t) = U_0^\dagger(t) V(t) U_0(t) is the interaction Hamiltonian in the interaction picture, and V(t) is the perturbation in the Schrödinger picture. This equation arises from substituting the interaction-picture definitions into the full von Neumann equation i \hbar \dot{\rho}_S = [H_0 + V, \rho_S] and using the unitarity of U_0(t). The formal solution can be expressed using the time-ordered exponential in the superoperator sense, where the Liouvillian \mathcal{L}_I(t) \rho = [V_I(t), \rho]/\hbar:

\rho_I(t) = \mathcal{T} \exp\left( -i \int_0^t \mathcal{L}_I(t') \, dt' \right) \rho_I(0),

with \mathcal{T} denoting time ordering to account for the non-commutativity of V_I(t) at different times. This form highlights the perturbative expansion in powers of the interaction, analogous to the Dyson series for state vectors but acting on the operator space.^[18]^[16] The interaction-picture formalism for density matrices is essential for mixed states, where pure-state descriptions fail, such as in open quantum systems subject to environmental decoherence or initial statistical ensembles. It facilitates the incorporation of relaxation and dephasing effects through master equations, simplifying calculations in contexts like quantum optics and condensed-matter spectroscopy. For pure states, where \rho_I(0) = |\psi_I(0)\rangle \langle \psi_I(0)|, the evolution reduces to \rho_I(t) = U_I(t, 0) |\psi_I(0)\rangle \langle \psi_I(0)| U_I^\dagger(t, 0) = |\psi_I(t)\rangle \langle \psi_I(t)|, recovering the pure-state dynamics.^[17]^[18]

Key equations and properties

Expectation values

In the interaction picture, the expectation value of an operator A at time t is computed as

\langle A \rangle_I(t) = \langle \psi_I(t) | A_I(t) | \psi_I(t) \rangle,

where |\psi_I(t)\rangle is the state vector and A_I(t) is the operator in this picture. Equivalently, for a mixed state described by the density operator \rho_I(t), it is given by

\langle A \rangle_I(t) = \mathrm{Tr} \left[ \rho_I(t) A_I(t) \right].

This quantity is invariant across pictures, matching the Schrödinger picture expectation value exactly, because the transformation between pictures is implemented by a unitary operator that preserves traces and inner products.^[19]^[20] The time evolution of this expectation value follows from the general quantum mechanical formula for the derivative of an expectation value. In the interaction picture, it takes the form

\frac{d \langle A \rangle}{dt} = \frac{i}{\hbar} \langle [H_0 + V_I(t), A_I(t)] \rangle + \left\langle \frac{\partial A_I(t)}{\partial t} \right\rangle,

where H_0 is the free Hamiltonian, V_I(t) is the interaction Hamiltonian, and the partial derivative accounts for any explicit time dependence in A. Since A_I(t) evolves solely under H_0, the commutator with H_0 reflects the free dynamics. This can be separated to emphasize contributions from the interaction:

\frac{d \langle A \rangle}{dt} = \frac{i}{\hbar} \langle [H_0, A_I(t)] \rangle + \left\langle \frac{i}{\hbar} [V_I(t), A_I(t)] + \frac{\partial A_I(t)}{\partial t} \right\rangle.

The second term isolates the perturbative effects of the interaction on the observable's average behavior.^[19] For position x and momentum p operators in a system where H_0 = p^2 / 2m, the above equations yield an analog of Ehrenfest's theorem: the expectation values satisfy

\frac{d \langle x \rangle}{dt} = \frac{\langle p \rangle}{m}, \quad \frac{d \langle p \rangle}{dt} = \left\langle -\frac{\partial V_I(t)}{\partial x} \right\rangle,

combining classical free-particle trajectories from H_0 with force terms from the interaction, thus illustrating the quantum-to-classical correspondence within this frame.^[21] In practice, computing expectation values in the interaction picture is particularly convenient for perturbative treatments, as the free evolution is exactly solvable and interactions can be expanded in series, facilitating approximations for weakly coupled systems.^[20]

Schwinger–Tomonaga equation

The Schwinger–Tomonaga equation provides a covariant generalization of the time-evolution equation in the interaction picture, applicable to relativistic quantum field theories. In this framework, the state is described by a functional \Psi[\phi; \sigma] depending on field configurations \phi on a spacelike hypersurface \sigma, with the evolution parameterized by deformations of \sigma. The equation takes the form

i \hbar \frac{\delta \Psi[\phi; \sigma]}{\delta \sigma(x)} = H_{\rm int}(x) \Psi[\phi; \sigma],

where H_{\rm int}(x) is the interaction Hamiltonian density at spacetime point x, and the functional derivative \delta / \delta \sigma(x) describes infinitesimal changes in the hypersurface location.^[7] This equation arises from extending the non-relativistic interaction picture to field theory by considering evolution along arbitrary spacelike surfaces, ensuring Lorentz invariance. The derivation involves defining local time parameters for field points on the hypersurface and imposing commutation relations that hold simultaneity only on \sigma, leading to the functional derivative form through variational principles on the action. In the special case of flat hypersurfaces parameterized by a global time \alpha (where \sigma: x^0 = \alpha), it simplifies to i \hbar \partial_\alpha \Psi_\alpha = \int H_{\rm int}(x) \delta(x^0 - \alpha) \, d^3\mathbf{x} \, \Psi_\alpha, mirroring the standard quantum mechanical evolution.^[22] Historically, Sin-Itirō Tomonaga introduced the core idea in 1946 to resolve inconsistencies in early quantum electrodynamics arising from non-covariant equal-time quantization. Julian Schwinger built upon this in 1948, fully developing the formalism for quantum electrodynamics and demonstrating its role in handling radiative corrections through renormalization, which paved the way for the theory's predictive success.^[7] In the absence of dynamical fields—reducing to fixed particle positions—the equation reverts to the ordinary interaction picture equation of quantum mechanics, i \hbar \partial_t |\Psi(t)\rangle = H_{\rm int}(t) |\Psi(t)\rangle. Today, it underpins covariant perturbation theory, enabling manifestly invariant expansions essential for modern quantum field calculations.^[7]

Applications

Perturbation theory

In the interaction picture, time-dependent perturbation theory is employed to compute transition probabilities between unperturbed eigenstates of the free Hamiltonian H_0, treating the interaction V(t) as a small perturbation. The total Hamiltonian is decomposed as H = H_0 + V(t), where the eigenstates |n\rangle and eigenvalues E_n of H_0 provide a solvable basis. The state vector in the interaction picture evolves according to i\hbar \frac{d}{dt} |\psi_I(t)\rangle = V_I(t) |\psi_I(t)\rangle, with initial condition |\psi_I(0)\rangle = |\psi(0)\rangle. Expanding |\psi_I(t)\rangle = \sum_n c_n(t) e^{-i E_n t / \hbar} |n\rangle, assuming the system starts in state |i\rangle so c_i(0) = 1 and c_n(0) = 0 for n \neq i, yields the perturbative solution for the coefficients to first order:

c_k^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle k | V_I(t') | i \rangle e^{i \omega_{ki} t'} \, dt',

where \omega_{ki} = (E_k - E_i)/\hbar and V_I(t) = e^{i H_0 t / \hbar} V(t) e^{-i H_0 t / \hbar}.^[23] For systems coupled to a continuum of final states, the first-order transition probability |c_f(t)|^2 leads to Fermi's golden rule in the long-time limit. Considering a harmonic perturbation V(t) = V e^{-i \omega t} + V^\dagger e^{i \omega t}, the transition rate from initial state |i\rangle to a final state |f\rangle with density of states \rho(E_f) is

\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \delta(E_f - E_i - \hbar \omega),

describing absorption (\omega > 0) or stimulated emission (\omega < 0) processes, such as in atomic transitions driven by electromagnetic fields. This rate arises from evaluating the integral in c_f^{(1)}(t) for large t, where the delta function enforces energy conservation.^[23] The validity of this perturbative approach requires the interaction strength to satisfy \|V\| \ll the relevant energy scales of H_0, such as level spacings |E_k - E_i|, ensuring higher-order corrections remain small and secular divergences are avoided over the timescale of interest.^[23] This condition aligns with the adiabatic theorem, which guarantees negligible transitions if the perturbation varies slowly compared to the inverse energy gaps, connecting the perturbative regime to adiabatic evolution when \dot{V}/|E_k - E_i|^2 \ll 1. A representative example is the interaction of a two-level atom with a monochromatic field, where H_0 describes the atomic levels separated by \hbar \omega_0, and V(t) = -\vec{d} \cdot \vec{E}_0 \cos(\omega t) is the dipole coupling. The first-order rate for excitation from ground state |g\rangle to excited state |e\rangle yields \Gamma_{g \to e} = (\pi / 2\hbar^2) |\langle e | \vec{d} \cdot \vec{E}_0 | g \rangle|^2 \delta(\omega_0 - \omega), quantifying absorption at resonance \omega = \omega_0, while the reverse process governs emission.^[23]

Dyson series expansion

The Dyson series expansion represents the perturbative solution to the time evolution operator in the interaction picture, providing a systematic way to compute the dynamics under a time-dependent interaction Hamiltonian V_I(t). Formulated by Freeman Dyson as part of unifying early quantum electrodynamics approaches, it expresses the evolution operator U_I(t, t_0) as a power series in the interaction strength. The operator takes the form of a time-ordered exponential:

U_I(t, t_0) = \mathcal{T} \exp\left( -\frac{i}{\hbar} \int_{t_0}^t V_I(t') \, dt' \right),

where \mathcal{T} denotes the time-ordering operator, which arranges operators in chronological order (earlier times to the left) to account for non-commutativity. This formal solution satisfies the differential equation i \hbar \frac{d}{dt} U_I(t, t_0) = V_I(t) U_I(t, t_0) with initial condition U_I(t_0, t_0) = \mathbb{I}. Expanding the exponential yields the Dyson series:

U_I(t, t_0) = \sum_{n=0}^\infty \frac{(-i)^n}{\hbar^n} \int_{t_0}^t dt_1 \int_{t_0}^t dt_2 \cdots \int_{t_0}^t dt_n \, \mathcal{T} \left\{ V_I(t_1) V_I(t_2) \cdots V_I(t_n) \right\}.

The n=0 term is the identity, while higher orders contribute corrections proportional to powers of the interaction. The time-ordered product \mathcal{T} \{ V_I(t_1) \cdots V_I(t_n) \} can be explicitly implemented using Heaviside step functions \theta(t_i - t_j) to enforce t_1 > t_2 > \cdots > t_n, symmetrized over permutations, or equivalently through nested integrals that impose strict ordering. The n-th order term is thus:

\frac{(-i)^n}{n! \hbar^n} \sum_{P} \int_{t_0}^t dt_1 \cdots \int_{t_0}^t dt_n \, \theta(t_{P(1)} - t_{P(2)}) \cdots \theta(t_{P(n-1)} - t_{P(n)}) V_I(t_{P(1)}) \cdots V_I(t_{P(n)}),

where the sum is over n! permutations P, and the $1/n! factor avoids overcounting identical orderings. Alternatively, it is often written without permutations as:

\frac{(-i)^n}{\hbar^n} \int_{t_0}^t dt_n \int_{t_0}^{t_n} dt_{n-1} \cdots \int_{t_0}^{t_2} dt_1 \, V_I(t_1) V_I(t_2) \cdots V_I(t_n),

with the understanding that the product is time-ordered. This structure ensures causality in the expansion.^[23] For time-dependent interactions, the series serves as a perturbative tool when V_I(t) is small relative to the free Hamiltonian, allowing truncation at low orders for approximate solutions; convergence holds rigorously in such regimes, as the norms of higher-order terms diminish. In scattering theory, taking the adiabatic limit t \to \infty and t_0 \to -\infty yields the S-matrix, whose perturbative expansion is directly the Dyson series, enabling computation of transition amplitudes between asymptotic states.^[24] A representative example is the second-order term, which captures virtual intermediate processes:

U_I^{(2)}(t, t_0) = \frac{(-i)^2}{\hbar^2} \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 \, V_I(t_1) V_I(t_2) + \frac{(-i)^2}{\hbar^2} \int_{t_0}^t dt_2 \int_{t_2}^t dt_1 \, V_I(t_2) V_I(t_1),

but in the time-ordered form, it simplifies to accounting for all pairwise orderings, corresponding to propagations via off-shell intermediate states that contribute to energy non-conservation temporarily before final resolution. Dyson's series also connects to the path integral formulation, where its terms correspond to discretized worldlines summed in Feynman's approach, laying the foundation for diagrammatic expansions in quantum field theory.

Comparisons and contrasts

With Schrödinger picture

In the Schrödinger picture of quantum mechanics, the time evolution of state vectors is governed by the full Hamiltonian H = H_0 + V, where H_0 is the unperturbed part and V is the interaction term, leading to the equation i \hbar \frac{d}{dt} |\psi_S(t)\rangle = (H_0 + V) |\psi_S(t)\rangle. Operators, excluding those with explicit time dependence, remain fixed in time, while the states carry all the dynamical information. This framework is particularly straightforward for systems where the full Hamiltonian can be solved exactly, but it becomes cumbersome when V introduces rapid or complex variations. In contrast, the interaction picture transforms both states and operators to isolate the effects of H_0 and V. The states evolve more slowly, solely under the interaction Hamiltonian in the interaction picture, V_I(t) = e^{i H_0 t / \hbar} V e^{-i H_0 t / \hbar}, satisfying i \hbar \frac{d}{dt} |\psi_I(t)\rangle = V_I(t) |\psi_I(t)\rangle. Operators acquire time dependence from the unperturbed evolution, such as the position operator x_I(t) = e^{i H_0 t / \hbar} x e^{-i H_0 t / \hbar} for a free particle where H_0 = p^2 / 2m. This decomposition simplifies analysis when H_0 is exactly solvable, allowing the interaction to be treated perturbatively. The interaction picture offers advantages for weak or time-dependent interactions V(t), as it reduces the state evolution to the perturbation alone, facilitating approximations like time-dependent perturbation theory. However, measurements require transforming back to the Schrödinger picture, adding computational overhead. It is preferable to switch to the interaction picture when exact solutions for H_0 are available, as this simplifies numerical simulations by evolving states only under the interaction while handling the free evolution analytically.

With Heisenberg picture

In the Heisenberg picture, quantum states are time-independent, while operators evolve according to the full Hamiltonian H = H_0 + V, where H_0 is the unperturbed (free) Hamiltonian and V is the interaction term. The time evolution of an operator \hat{A}_H(t) is governed by the unitary transformation \hat{A}_H(t) = e^{i H t / \hbar} \hat{A}(0) e^{-i H t / \hbar}, reflecting the complete dynamics encoded in H.^[25] By contrast, the interaction picture adopts a hybrid approach, splitting the evolution between the free and interaction parts of the Hamiltonian. Here, operators \hat{A}_I(t) evolve under H_0 as \hat{A}_I(t) = e^{i H_0 t / \hbar} \hat{A}_S e^{-i H_0 t / \hbar}, where \hat{A}_S is the time-independent Schrödinger-picture operator. States, meanwhile, evolve solely under the interaction dynamics via the unitary operator U_I(t, t_0) satisfying the Schrödinger-like equation i \hbar \frac{d}{dt} U_I(t, t_0) = \hat{V}_I(t) U_I(t, t_0), with \hat{V}_I(t) = e^{i H_0 t / \hbar} V e^{-i H_0 t / \hbar}. This separation facilitates perturbative treatments where H_0 is exactly solvable and V is treated as a small perturbation.^[25] A key difference arises in the equation of motion for operators. In the Heisenberg picture, the time derivative follows the full commutator \frac{d \hat{A}_H}{dt} = \frac{i}{\hbar} [H, \hat{A}_H] = \frac{i}{\hbar} [H_0 + V, \hat{A}_H]. In the interaction picture, it is given solely by the free evolution: \frac{d \hat{A}_I}{dt} = \frac{i}{\hbar} [H_0, \hat{A}_I], where interaction effects on observables are captured through the state evolution under V_I(t).^[25] For illustration, consider the momentum operator \hat{p} in a free-particle scenario ( V = 0): its evolution in both pictures is identical, \hat{p}_H(t) = \hat{p}_I(t) = \hat{p}, as no interactions alter the commutator structure. However, introducing a weak potential V does not change the operator evolution in the interaction picture, where \hat{p}_I(t) remains constant for H_0 = p^2 / 2m since [H_0, \hat{p}] = 0; instead, the expectation value \langle \hat{p}_I(t) \rangle evolves perturbatively due to the state dynamics under V_I(t), approximating force-like effects without requiring the full exponential evolution of the Heisenberg picture.^[25] The interaction picture's utility shines in regimes where the Heisenberg picture's full evolution is intractable, such as quantum field theory, where free-field operators evolve simply under H_0, allowing interactions to be handled perturbatively via Dyson series expansions. This approach, formalized in the interaction representation, enables efficient computation of scattering processes while preserving the locality and causality inherent to the Heisenberg picture.^[26]