Inquantum mechanics, theinteraction picture (also known as theinteraction representation orDirac picture afterPaul Dirac, who introduced it)^[1][2] is an intermediaterepresentation between theSchrödinger picture and theHeisenberg picture. Whereas in the other two pictures either thestate vector or theoperators carry time dependence, in the interaction picture both carry part of the time dependence ofobservables.^[3] The interaction picture is useful in dealing with changes to thewave functions and observables due to interactions. Most field-theoretical calculations^[4] use the interaction representation because they construct the solution to the many-bodySchrödinger equation as the solution tofree particles in presence of some unknown interacting parts.

Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.

The interaction picture is a special case ofunitary transformation applied to theHamiltonian and state vectors.

Haag's theorem says that the interaction picture doesn't exist in the case of interactingquantum fields.

Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation) to those same operators and state vectors in the Schrödinger picture.

To switch into the interaction picture, we divide the Schrödinger pictureHamiltonian into two parts:

Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so thatH_0,S is well understood and exactly solvable, whileH_1,S contains some harder-to-analyze perturbation to this system.

If the Hamiltonian hasexplicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms withH_1,S, leavingH_0,S time-independent:

We proceed assuming that this is the case. If thereis a context in which it makes sense to haveH_0,S be time-dependent, then one can proceed by replacing${\displaystyle {\mathrm{e}}^{\pm \mathrm{i}{H}_{0,\text{S}}t/\hslash }}$ by the correspondingtime-evolution operator in the definitions below.

Let${\displaystyle |{\psi }_{\text{S}}(t)⟩={\mathrm{e}}^{-\mathrm{i}{H}_{\text{S}}t/\hslash }|\psi (0)⟩}$ be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture,${\displaystyle |{\psi }_{\text{I}}(t)⟩}$, is defined with an additional time-dependent unitary transformation.^[5]

An operator in the interaction picture is defined as

Note thatA_S(t) will typically not depend ont and can be rewritten as justA_S. It only depends ont if the operator has "explicit time dependence", for example, due to its dependence on an applied external time-varying electric field. Another instance of explicit time dependence may occur whenA_S(t) is a density matrix (see below).

For the operator${\displaystyle H_0}$ itself, the interaction picture and Schrödinger picture coincide:

${\displaystyle H_{0,\text{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{H}_{0,\text{S}}{\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash }=H_{0,\text{S}}.}$

This is easily seen through the fact that operatorscommute with differentiable functions of themselves. This particular operator then can be called${\displaystyle H_0}$ without ambiguity.

For the perturbation Hamiltonian${\displaystyle H_{1,\text{I}}}$, however,

${\displaystyle H_{1,\text{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_{0,\text{S}}t/\hslash }{H}_{1,\text{S}}{\mathrm{e}}^{-\mathrm{i}{H}_{0,\text{S}}t/\hslash },}$

where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [H_1,S,H_0,S] = 0.

It is possible to obtain the interaction picture for a time-dependent HamiltonianH_0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated byH_0,S(t), or more explicitly with a time-ordered exponential integral.

Thedensity matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, letρ_I andρ_S be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probabilityp_n to be in the physical state |ψ_n⟩, then

Transforming theSchrödinger equation into the interaction picture gives

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}|{\psi }_{\text{I}}(t)⟩=H_{1,\text{I}}(t)|{\psi }_{\text{I}}(t)⟩,}$

which states that in the interaction picture, a quantum state is evolved by the interaction part of the Hamiltonian as expressed in the interaction picture.^[6] A proof is given in Fetter and Walecka.^[7]

If the operatorA_S is time-independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution forA_I(t) is given by

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}{A}_{\text{I}}(t)=[A_{\text{I}}(t),H_{0,\text{S}}].}$

In the interaction picture the operators evolve in time like the operators in theHeisenberg picture with the HamiltonianH' =H₀.

The evolution of thedensity matrix in the interaction picture is

${\displaystyle \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}{\rho }_{\text{I}}(t)=[H_{1,\text{I}}(t),{\rho }_{\text{I}}(t)],}$

in consistency with the Schrödinger equation in the interaction picture.

For a general operator${\displaystyle A}$, the expectation value in the interaction picture is given by

${\displaystyle ⟨A_{\text{I}}(t)⟩=⟨{\psi }_{\text{I}}(t)|A_{\text{I}}(t)|{\psi }_{\text{I}}(t)⟩=⟨{\psi }_{\text{S}}(t)|e^{-i{H}_{0,\text{S}}t}{e}^{i{H}_{0,\text{S}}t}{A}_{\text{S}}{e}^{-i{H}_{0,\text{S}}t}{e}^{i{H}_{0,\text{S}}t}|{\psi }_{\text{S}}(t)⟩=⟨A_{\text{S}}(t)⟩.}$

Using the density-matrix expression for expectation value, we will get

${\displaystyle ⟨A_{\text{I}}(t)⟩=\mathrm{Tr}({\rho }_{\text{I}}(t)A_{\text{I}}(t)).}$

The term interaction representation was invented by Schwinger.^[8][9]In this new mixed representation the state vector is no longer constant in general, but it is constant if there is no coupling between fields. The change of representation leads directly to the Tomonaga–Schwinger equation:^[10][9]

${\displaystyle ihc\frac{\mathrm{\partial }\mathrm{\Psi }[\sigma ]}{\mathrm{\partial }\sigma (x)}=\hat{H}(x)\mathrm{\Psi }(\sigma )}$

${\displaystyle \hat{H}(x)=-\frac{1}{c}{j}_{\mu }(x)A^{\mu }(x)}$

Where the Hamiltonian in this case is the QED interaction Hamiltonian, but it can also be a generic interaction, and${\displaystyle \sigma }$ is a spacelike surface that is passing through the point${\displaystyle x}$. The derivative formally represents a variation over that surface given${\displaystyle x}$ fixed. It is difficult to give a precise mathematical formal interpretation of this equation.^[11]

This approach is called the 'differential' and 'field' approach by Schwinger, as opposed to the 'integral' and 'particle' approach of the Feynman diagrams.^[12][13]

The core idea is that if the interaction has a small coupling constant (i.e. in the case of electromagnetism of the order of the fine structure constant) successive perturbative terms will be powers of the coupling constant and therefore smaller.^[14]

The purpose of the interaction picture is to shunt all the time dependence due toH₀ onto the operators, thus allowing them to evolve freely, and leaving onlyH_1,I to control the time-evolution of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term,H_1,S, being added to the Hamiltonian of a solved system,H_0,S. By utilizing the interaction picture, one can usetime-dependent perturbation theory to find the effect ofH_1,I,^{[15]: 355ff} e.g., in the derivation ofFermi's golden rule,^{[15]: 359–363} or theDyson series^{[15]: 355–357} inquantum field theory: in 1947,Shin'ichirō Tomonaga andJulian Schwinger appreciated that covariant perturbation theory could be formulated elegantly in the interaction picture, sincefield operators can evolve in time as free fields, even in the presence of interactions, now treated perturbatively in such a Dyson series.

For a time-independent HamiltonianH_S, whereH_0,S is the free Hamiltonian,

Evolution of:	Picture ( v t e )
	Schrödinger (S)	Heisenberg (H)	Interaction (I)
Ket state	${\displaystyle |{\psi }_{\mathrm{S}}(t)⟩=e^{-i{H}_{\mathrm{S}}t/\hslash }|{\psi }_{\mathrm{S}}(0)⟩}$	constant	${\displaystyle |{\psi }_{\mathrm{I}}(t)⟩=e^{i{H}_{0,\mathrm{S}}t/\hslash }|{\psi }_{\mathrm{S}}(t)⟩}$
Observable	constant	${\displaystyle A_{\mathrm{H}}(t)=e^{i{H}_{\mathrm{S}}t/\hslash }{A}_{\mathrm{S}}{e}^{-i{H}_{\mathrm{S}}t/\hslash }}$	${\displaystyle A_{\mathrm{I}}(t)=e^{i{H}_{0,\mathrm{S}}t/\hslash }{A}_{\mathrm{S}}{e}^{-i{H}_{0,\mathrm{S}}t/\hslash }}$
Density matrix	${\displaystyle {\rho }_{\mathrm{S}}(t)=e^{-i{H}_{\mathrm{S}}t/\hslash }{\rho }_{\mathrm{S}}(0)e^{i{H}_{\mathrm{S}}t/\hslash }}$	constant	${\displaystyle {\rho }_{\mathrm{I}}(t)=e^{i{H}_{0,\mathrm{S}}t/\hslash }{\rho }_{\mathrm{S}}(t)e^{-i{H}_{0,\mathrm{S}}t/\hslash }}$

• L.D. Landau;E.M. Lifshitz (1977).Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.).Pergamon Press.ISBN 978-0-08-020940-1.
• Townsend, John S. (2000).A Modern Approach to Quantum Mechanics (2nd ed.). Sausalito, California: University Science Books.ISBN 1-891389-13-0.

• Bra–ket notation
• Schrödinger equation
• Haag's theorem

• Quantum mechanics

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