nLab Dirac interaction picture

Idea

The “interaction picture” in quantum physics is a way to decompose solutions to the Schrödinger equation and more generally the construction of quantum field theories into a free field theory-part and the interaction part that acts as a perturbation of the free theory. Therefore the interaction picture lends itself to the construction of perturbative quantum field theory, and in fact the only mathematically rigorous such construction scheme that is known, namely causal perturbation theory, proceeds this way.

Dynamics in physics affects both observables and, dually, states; this is most well known in quantum physics but applies equally well to classical physics. The different “pictures” of physics differ in how the dynamics is explicitly formalized:

• In the Schrödinger picture, states are propagated through time, while observables are held fixed; the axiomatic formalization of this is given by cobordism category representations in FQFT.

• In the Heisenberg picture, the dependence of observables on time (or more generally spacetime) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the Haag–Kastler axioms of AQFT.

• The Dirac (interaction) picture is a mixture of these two approaches: dynamics is split into a free (or otherwise solvable) part and an interaction (often then treated as a perturbation); one of these is taken to affect the states, the other the observables.

The pictures are named after those physicists who first used or popularised these approaches to quantum physics.

In quantum mechanics

In quantum mechanics, let $\mathcal{H}$ be some Hilbert space and let

$$H = H_{free} + V$$

be Hermitian operator, thought of as a Hamiltonian, decomposed as the sum of a free part (kinetic energy) and an interaction part (potential energy).

For example for a non-relativistic particle of mass $m$ propagating on the line subject to a potential energy $V_{pot} \colon \mathbb{R} \to \mathbb{R}$, then $\mathcal{H} = L^2(X)$ is the Hilbert space of square integrable functions and

$$H = \underset{H_{free}}{\underbrace{\tfrac{- \hbar^2}{2m} \frac{\partial^2}{\partial^2 x}}} + V \,,$$

where $V = V_{pot}(x)$ is the operator of multiplying square integrable functions with the given potential energy function.

Now for

$$\array{ \mathbb{R} &\overset{\vert \psi (-)\rangle }{\longrightarrow}& \mathcal{H} \\ t &\mapsto& \vert \psi(t) \rangle }$$

a one-parameter family of quantum states, the Schrödinger equation for this state reads

$$\frac{d}{d t} \vert \psi(t) \rangle \;=\; \tfrac{1}{i \hbar} H \vert \psi\rangle \,.$$

It is easy to solve this differential equation formally via its Green function: for $\vert \psi \rangle \in \mathcal{H}$ any state, then the unique solution $\vert \psi(-) \rangle$ to the Schrödinger equation subject to $\vert \psi(0) \rangle = \vert \psi \rangle$ is

$$\vert \psi(t)\rangle_S \coloneqq \exp( \tfrac{t}{i \hbar} H ) \vert \psi \rangle \,.$$

(One says that this is the solution “in the Schrödinger picture”, whence the subscript.)

However, if $H$ is sufficiently complicated, it may still be very hard to extract from this expression a more explicit formula for $\vert \psi(t) \rangle$, such as, in the example of the free particle on the line, its expression as a function (“wave function”) of $x$ and $t$.

But assume that the analogous expression for $H_{free}$ alone is well understood, hence that the operator

$$U_{S,free}(t_1, t_2) \coloneqq \exp\left({\tfrac{t_2 - t_1}{i \hbar} H_{free}}\right)$$

is sufficiently well understood. The “interaction picture” is a way to decompose the Schrödinger equation such that its dependence on $V$ gets separated from its dependence on $H_{free}$ in a way that admits to treat $H_{int}$ in perturbation theory.

Namely define analogously

(1)$$\begin{aligned} \vert \psi(t)\rangle_I &\coloneqq \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({ \tfrac{+ t}{i \hbar} H} \right)\vert \psi \rangle \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)\vert \psi \rangle \end{aligned} \,.$$

This is called the solution of the Schrödinger equation “in the interaction picture”, whence the subscript. Its definition may be read as the result of propagating the actual solution $\vert \psi(-)\rangle_S$ at time $t$ back to time $t = 0$, but using just the free Hamiltonian, hence with “the interaction switched off”.

Notice that if the operator $V$ were to commute with $H_{free}$ (which it does not in all relevant examples) then we would simply have $\vert \psi(t)\rangle_I = \exp( \tfrac{t}{i \hbar } V ) \vert \psi\rangle$, hence then the solution (1) in the interaction picture would be the result of “propagating” the initial conditions using only the interaction. Now since $V$ may not be assumed to commute with $H_{free}$, the actual form of $\vert \psi(-) \rangle_{I}$ is more complicated. But infinitesimally it remains true that $\vert \psi(-)\rangle_I$ is propagated this way, not by the plain operator $V$, though, but by $V$ viewed in the Heisenberg picture of the free theory. This is the content of the differential equation (2) below.

But first notice that this will indeed be useful: If an explicit expression for the “state in the interaction picture” (1) is known, then the assumption that also the operator $\exp\left({\tfrac{t}{i \hbar} H_{free}}\right)$ is sufficiently well understood implies that the actual solution

$$\vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I$$

is under control. Hence the question now is how to find $\vert \psi(-)\rangle_I$ given its value at some time $t$. (It is conventional to consider this for $t \to \pm \infty$, see (3) below.)

Now it is clear from the construction and using the product law for differentiation, that $\vert \psi(-)\rangle_S$ satisfies the following differential equation:

(2)$$\frac{d}{d t} \vert \psi(t) \rangle \;=\; V_I(t) \vert \psi(t)\rangle_I \,,$$

where

$$V_I(t) \coloneqq \exp\left( -\tfrac{t}{i \hbar} H_{free} \right) V \exp\left( +\tfrac{t}{i \hbar} H_{free} \right)$$

is known as the interaction term $V$ “viewed in the interaction picture”. But in fact this is just $V$ “viewed in the Heisenberg picture”, but for the free theory. By our running assumption that the free theory is well understood, also $V_I(t)$ is well understood, and hence all that remains now is to find a sufficiently concrete solution to equation (2). This is the heart of working in the interaction picture.

Solutions to equations of the “parallel transport”-type such as (2) are given by time-ordering of Heisenberg picture operators, denoted $T$, applied to the naive exponential solution as above. This is known as the Dyson formula:

$$\vert \psi(t)\rangle_I \;=\; T\left( \exp\left( \int_{t_0}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \vert \psi(t_0)\rangle \,.$$

Here time-ordering means

$$T( V_I(t_1) V_I(t_2) ) \;\coloneqq\; \left\{ \array{ V_I(t_1) V_I(t_2) &\vert& t_1 \geq t_2 \\ V_I(t_2) V_I(t_1) &\vert& t_2 \geq t_2 } \right. \,.$$

(This is abuse of notation: Strictly speaking time ordering acts on the tensor algebra spanned by the $\{V_I(t)\}_{t \in \mathbb{R}}$ and has to be followed by taking tensor products to actual products. )

In applications to scattering processes one is interested in prescribing the quantum state/wave function far in the past, hence for $t \to - \infty$, and computing its form far in the future, hence for $t \to \infty$.

The operator that sends such “asymptotic ingoing-states” $\vert \psi(-\infty) \rangle_I$ to “asymptotic outgoing states” $\vert \psi(+ \infty) \rangle_I$ is hence the limit

(3)$$S \;\coloneqq\; \underset{t \to \infty}{\lim} T\left( \exp\left( \int_{-t}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \,.$$

This limit (if it exists) is called the scattering matrix or S-matrix, for short.

In quantum field theory

In perturbative quantum field theory the broad structure of the interaction picture in quantum mechanics (above) remains a very good guide, but various technical details have to be generalized with due care:

1. The algebra of operators in the Heisenberg picture of the free theory becomes the Wick algebra of the free field theory (taking into account “normal ordering” of field operators) defined on microcausal functionals built from operator-valued distributions with constraints on their wave front set.

2. The time-ordered products in the Dyson formula have to be refined to causally ordered products and the resulting product at coincident points has to be defined by point-extension of distributions – the freedom in making this choice is the renormalization freedom (“conter-terms”).

3. The sharp interaction cutoff in the Dyson formula that is hidden in the integration over $[t_0,t]$ has to be smoothed out by adiabatic switching of the interaction (making the whole S-matrix an operator-valued distribution).

Together these three point are taken care of by the axiomatization of the “adiabatically switched S-matrix” according to causal perturbation theory.

The analogue of the limit $t \to \infty$ in the construction of the S-matrix (now: adiabatic limit) in general does not exist in field theory (“infrared divergencies”). But in fact it need not be taken: The field algebra in a bounded region of spacetime may be computed with any adiabatic switching that is constant on this region. Moreover, the algebras assigned to regions of spacetime this way satisfy causal locality by the causal ordering in the construction of the S-matrix. Therefore, even without taking the adiabatic limit in causal perturbation theory one obtains a field theory in the form of a local net of observables. This is the topic of locally covariant perturbative quantum field theory.

Related concepts

• interaction

• interacting field algebra

• interacting field theory

• S-matrix

• Møller operator

References

For instance

• Eberhard Zeidler, section 7.19.3 of Quantum field theory. A bridge between mathematicians and physicists – volume I Springer (2009) (web)