Beyond the Free Field Assumption: Deriving LSZ via the EOM

[Anaresil]

TL;DR

I derived the LSZ reduction formula directly from the interacting field equations instead of standard free-field assumptions. I discuss the subtle math of the infinite-time limit and request community insight on the justification.

Motivation
Standard derivation of the LSZ reduction formula often rely on the assumption of an "in" or "out" free-field limit from the outset. While elegant, this approach can obscure the connection between the actual dynamics of the interacting field $\varphi(x)$ and the resulting creation/annihilation operators. Here, I attempt to derive the relation by working directly with the interacting equation of motion.

The setup
Consider an interacting scalar theory with the equation of motion:
\begin{equation}
\label{EOM_field_2}
\left( \partial^2 + m^2 \right) \varphi(x) = \mathcal{L}'_{int}\left[ \varphi(x) \right]
\end{equation}
We express the field in terms of time-dependent operators $a(\mathbf{k}, t)$ and $a^\dagger(\mathbf{k}, t)$ via Fourier-like decomposition:
\begin{equation}
\label{field_expansion_2}
\varphi(x) = \int \frac{\text{d}^3k}{(2\pi)^3} \frac{1}{2\omega}
\left[ a(\mathbf{k}, t)e^{ikx} + a^\dagger(\mathbf{k}, t)e^{-ikx} \right]
\end{equation}
where $\omega = \sqrt{\left|\mathbf{k}\right|^2 + m^2}$.

By substituting the mode expansion Eq.(\ref{field_expansion_2}) into the equation of motion Eq.(\ref{EOM_field_2}), we project out the specific mode $\mathbf{k}$. After some algebra manipulation, the time evolution of the creation and annihilation operators are governed by:
\begin{equation}
\label{EOM_operators_2}
\begin{aligned}
\left( \partial_t^2 + 2 i \omega \partial_t \right) a(\mathbf{k}, t) &= b(\mathbf{k}, t),\\
\left( \partial_t^2 - 2 i \omega \partial_t \right) a^\dagger(\mathbf{k}, t) &= b^\dagger(\mathbf{k}, t)
\end{aligned}
\end{equation}
where $b(\mathbf{k}, t)$ is defined by the Fourier transform of the interaction term:
\begin{equation}
\label{interaction_term_expansion_2}
\mathcal{L}'\left[\varphi(x)\right] =
\int \frac{\text{d}^3k}{(2\pi)^3} \frac{1}{2\omega}
\left[
b(\mathbf{k}, t)e^{ikx} + b^\dagger(\mathbf{k}, t)e^{-ikx}
\right]
\end{equation}
These equations explicitly show that in an interacting theory, $a(\mathbf{k}, t)$ is not a constant. Its dynamics are driven by the interaction source $b(\mathbf{k}, t)$. This provides a dynamical picture of how the field's interaction term redistributes energy between different momentum modes, moving beyond the static representation found in the free-field theory.

Derivation via Inversion
By peforming the inverse Fourier transform on $\varphi(x)$ and its time derivative $\partial_t \varphi(x)$, one can isolate $a(\mathbf{k}, t)$. Specifically, using the completeness of plane waves, we have
\begin{equation}
\begin{aligned}
\int \text{d}^3 x e^{-ikx} \varphi(x)
&= \frac{1}{2 \omega}
\left[
a(\mathbf{k}, t) + a^\dagger(-\mathbf{k}, t)
\right],\\
\int \text{d}^3 x e^{-ikx} \partial_t \varphi(x)
&= \frac{1}{2\omega}
\left(
\left[ \left( \partial_t + i \omega \right) a(\mathbf{k}, t) \right] +
\left[ \left( \partial_t - i\omega \right) a^\dagger(-\mathbf{k}, t)\right] e^{-2i\omega t}
\right)
\end{aligned}
\end{equation}
To isolate $a(\mathbf{k}, t)$ and $a^\dagger(\mathbf{k}, t)$, we solve the system of equations formed by the field and its time derivative. By combining the equations above, we. obtain:
\begin{equation}
\label{ladder_operators_2}
a(\mathbf{k}, t)
= -i \int \text{d}^3 x e^{-ikx}
\left(\partial_t + i \omega\right) \varphi(x)
+ \frac{i}{2\omega}
\left[
\partial_t a(\mathbf{k}, t) + \left(\partial_t a^\dagger(-\mathbf{k}, t)\right)e^{-2i\omega t}
\right]
\end{equation}
Taking the time derivative of $a(\mathbf{k}, t)$, we arrive at the dynamical evolution:
\begin{equation}
\label{time_derivative_operators_2}
\partial_t a(\mathbf{k}, t)
= -i \int \text{d}^3 x e^{-ikx}
\left( \partial^2 + m^2 \right)\varphi(x) + \frac{i}{2 \omega}
\left[
\partial_t^2 a(\mathbf{k}, t)
+ \partial_t
\left(
\partial_t a^\dagger(-\mathbf{k}, t)
e^{- 2 i \omega t}
\right)
\right]
\end{equation}

Asymptotic Intgration
Integraing both sides with respect to time over $(-\infty, +\infty)$, we obtain:
\begin{equation}
\label{integration_2}
\begin{aligned}
\int_{-\infty}^{+\infty} \text{d}t \partial_t a(\mathbf{k}, t)
= &
- i \int \text{d}^4 x e^{-ikx}\left(\partial^2 + m^2\right)\varphi(x) \\
& + \frac{i}{2\omega}
\left[
\int_{-\infty}^{+\infty}
\text{d}t \partial_t
\left(
\partial_t a(\mathbf{k}, t)
\right)
+
\int_{-\infty}^{+\infty}
\text{d}t \partial_t
\left(
\partial_t a^\dagger(-\mathbf{k}, t) e^{-2i\omega t}
\right)
\right]
\end{aligned}
\end{equation}
The second integral evaluates to
\begin{equation}
\label{second_integral_2}
\int_{-\infty}^{+\infty}
\text{d}t \partial_t
\left(
\partial_t a(\mathbf{k}, t)
\right)
= \partial_t a(\mathbf{k}, t, +\infty)
- \partial_t a(\mathbf{k}, t, -\infty)
\end{equation}
In our derivation, the validity of this step relies on the Asymptotic Condition. To interpret this physically, consider the standard setup of a particle collider experiment:
1. Preparation (The "In" State): Long before the collision ($t \to -\infty$), our incoming particles are well-separated in the vacuum. In this region, they are essentially free particles; the interaction potential $\mathcal{L}'_{int}$ is negligible because the wave packets are spatially distant. Then $\partial_t a(\mathbf{k}, -\infty) = 0$ and $\partial_t a^\dagger(\mathbf{k}, -\infty) = 0$ since $b(\mathbf{k}, \infty) = 0$ and $b^\dagger(\mathbf{k}, -\infty) = 0$;

2. Interaction (The "Collision" Region): As the particles approach the interaction point (the "collision zone"), the overlap of their wave functions triggers intense dynamical interactions, captured by the non-zero source term $(\partial^2 + m^2)\varphi(x) \neq 0$;

3. Detection (The "Out" State): Long after the collision ($t \to +\infty$), the resulting products disperse into the vacuum, effectively becoming free particles again. Then $\partial_t a(\mathbf{k}, +\infty) = 0$ and $\partial_t a^\dagger(\mathbf{k}, +\infty) = 0$ since $b(\mathbf{k}, +\infty) = 0$ and $b^\dagger(\mathbf{k}, +\infty) = 0$.

This physical reality justifies our mathematical assumption that $\partial_t a(\mathbf{k}) \to 0$ as $t \to \pm\infty$. In these regions, the dynamics of the creation and annihilation operators freeze out because the "source" $b(\mathbf{k}, t)$ of the interaction vanishes. Therefore, the total change in the operator is entirely attributed to the integrated effect of the interaction during the collision window:
$$\partial_t a(\mathbf{k}, +\infty) - \partial_t a(\mathbf{k}, -\infty) = 0$$
This makes the LSZ formula not just a formal identity, but a dynamical map that links the "free" incoming state to the "free" outgoing state through the time-integrated history of the interaction field.
Similarly, the third integral evaluates to
\begin{equation}
\label{third_integral_2}
\int_{-\infty}^{+\infty}
\text{d}t \partial_t
\left(
\partial_t a^\dagger(-\mathbf{k}, t) e^{-2i\omega t}
\right)
=
\partial_t a^\dagger(-\mathbf{k}, +\infty) e^{-2i\omega (+\infty)}
-
\partial_t a^\dagger(-\mathbf{k}, -\infty) e^{-2i\omega (-\infty)}
\end{equation}
This integral is also identical to zero since the Asymptotic Condition $\partial_t a^\dagger(-\mathbf{k}, +\infty) = \partial_t a^\dagger(-\mathbf{k}, \infty) = 0$.
Finally, our calculation leaves
\begin{equation}
\label{in_out_annihilation_2}
a(\mathbf{k}, +\infty) - a(\mathbf{k}, -\infty) =
-i \int \text{d}^4 x e^{ikx} \left(\partial^2 + m^2 \right) \varphi(x)
\end{equation}
and by the same procedure, we write:
\begin{equation}
\label{in_out_creation_2}
a^\dagger(\mathbf{k}, +\infty) - a^\dagger(\mathbf{k}, -\infty) =
i \int \text{d}^4 x e^{-ikx} \left(\partial^2 + m^2 \right) \varphi(x)
\end{equation}
Let's return to the scattering amplitude,
\begin{equation}
\label{amplitude_2}
\langle f | i \rangle
=
\langle 0 \left| T
\left\{
a(\mathbf{k}_{1'}, +\infty)
a(\mathbf{k}_{2'}, +\infty)
\cdots
a^\dagger(\mathbf{k}_1, -\infty)
a^\dagger(\mathbf{k}_2, -\infty)
\cdots
\right\}
\right| 0 \rangle
\end{equation}
Let us use Eq. \ref{in_out_annihilation_2} and Eq. \ref{in_out_creation_2} in Eq. \ref{amplitude_2}. The time-ordering symbol automatically moves all $a(\mathbf{k}_{i'}, -\infty)$’s to the right, where they annihilate $|0\rangle$. Similarly, all $a^\dagger(\mathbf{k}_{i}, +\infty)$’s move to the left, where they annihilate $\langle 0 |$. We are left with
\begin{equation}
\label{LSZ_2}
\begin{aligned}
\langle f | i \rangle
=
i^{n + n'}
\int
&\text{d}^4 x_1 e^{-i k_1 x_1}\left(\partial_1^2 + m^2\right) \cdots \\
&\text{d}^4 x_{1'} e^{i k_{1'} x_{1'}}\left(\partial_{1'}^2 + m^2\right) \cdots \\
&\times \langle 0 \left| T
\varphi(x_1) \cdots \varphi(x_{1'}) \cdots
\right| 0 \rangle
\end{aligned}
\end{equation}
This is the Lehmann-Symanzik-Zimmermann reduction formula, or LSZ formula for short.

 

[berkeman]

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