The LSZ Reduction Formula

The Feynman rules compute time-ordered correlation functions of fields; experiments measure S-matrix elements between physical particle states. The Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is the bridge between the two. The preliminaries state the formula and its significance; this page derives it and shows exactly how it amputates external legs and supplies wavefunction factors.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The problem LSZ solves

The Dyson series and Feynman rules naturally produce the time-ordered $n$-point function

$$G^{(n)}(x_1,\dots ,x_n)=⟨\Omega |T\varphi (x_1)\cdots \varphi (x_n)|\Omega ⟩,$$

where $|\Omega ⟩$ is the interacting vacuum. But the observable is the transition amplitude $⟨p_1\cdots p_n;\text{out}|q_1\cdots q_m;\text{in}⟩$ between asymptotic multi-particle states. LSZ expresses the latter in terms of the former. Its virtue is that it needs only correlators as input — no Hamiltonian, no explicit asymptotic-state construction — so it applies whether the correlators come from perturbation theory, the lattice, or the bootstrap (contrast the Møller-operator route of Postulate 10a).

Interpolating fields and the spectral input

Pick any local field $\varphi (x)$ with a non-zero one-particle matrix element, $⟨\Omega |\varphi (x)|\mathbf{p}⟩\ne 0$ — an interpolating field for the species. Two exact (non-perturbative) facts about the two-point function drive the derivation, both from the Källén–Lehmann spectral representation:

1. The full propagator has a single-particle pole at $p^2=m^2$ (the physical mass) with residue $Z$, the field-strength renormalization: $$\int d^{4}x{e}^{ip\cdot x}⟨\Omega |T\varphi (x)\varphi (0)|\Omega ⟩\stackrel{p^2\to m^2}{\to }\frac{iZ}{p^2-m^2+iϵ}+(\text{regular}).$$
2. Multi-particle states contribute a smooth continuum starting at the threshold $p^2\ge (2m{)}^2$, with no pole.

The field may be normalized so $Z=1$ at tree level, but $Z\ne 1$ once interactions dress the propagator; LSZ tracks the $\sqrt{Z}$ factors explicitly.

The reduction formula (Theorem)

Reducing one incoming particle: the key lemma expresses the creation of an asymptotic in-state by an integral of the interpolating field against a wavepacket. Applying the Klein–Gordon operator $(\square +m^2)$ isolates the on-shell pole — off-shell it annihilates the free wave; on-shell it extracts the residue. Iterating over all external particles gives the full LSZ reduction formula:

$$\boxed{\begin{array}{rl}& ⟨p_1\cdots p_n;\text{out}|q_1\cdots q_m;\text{in}⟩={\left(\frac{i}{\sqrt{Z}}\right)}^{n+m}\prod _{i=1}^n\int d^4{x}_i{e}^{i{p}_i\cdot x_i}({\square }_{x_i}+m^2)\\ & \times \prod _{j=1}^m\int d^4{y}_j{e}^{-i{q}_j\cdot y_j}({\square }_{y_j}+m^2)⟨\Omega |T\varphi (x_1)\cdots \varphi (x_n)\varphi (y_1)\cdots \varphi (y_m)|\Omega ⟩.\end{array}}$$

Each external particle contributes: a Fourier factor $e^{\pm ip\cdot x}$, a Klein–Gordon operator $(\square +m^2)$, and a factor $i/\sqrt{Z}$.

How it amputates diagrams

The mechanism is transparent in momentum space. The correlator $G^{(n+m)}$, as a sum of Feynman diagrams, has a full propagator on each external leg — a factor $\sim iZ/(p^2-m^2)$ near the mass shell. The external-leg operator $(\square +m^2)$ becomes $-(p^2-m^2)$ in momentum space, which cancels the external propagator pole:

$$(\text{external }\square +m^2)\times \frac{iZ}{p^2-m^2}⟶\text{finite, }\propto \sqrt{Z}.$$

So LSZ:

1. removes the external-leg propagators (amputation),
2. puts the external momenta on-shell ($p^2=m^2$),
3. supplies a factor $\sqrt{Z}$ per leg (unity at tree level).

What survives is exactly the amputated, connected, on-shell amplitude — the $i\mathcal{M}$ read off by the Feynman rules. For particles with spin, the external Klein–Gordon operators are replaced by the appropriate wave operators (Dirac operator $i/\partial -m$ for fermions, and external spinors $u,\bar{u}$ / polarizations $ϵ$ replace the scalar leg factors).

Consequences

• No Hamiltonian required. Correlators are the only input; this is the $H$-independent route to the S-matrix promised in the preliminaries.
• Field-redefinition invariance. Replacing $\varphi \to \varphi +c{\varphi }^2+\cdots$ (any local interpolating field with the same quantum numbers) leaves on-shell S-matrix elements unchanged — only the residue $Z$ and off-shell Green's functions change. This underlies the freedom of operator bases in effective field theory and the statement that "the fundamental field is conventional."
• Bridges rules to observables. LSZ is the implicit Step between the Feynman-rule computation of an amputated diagram and the cross-section formula; it is what licenses reading $i\mathcal{M}$ off an amputated on-shell diagram.

Summary

• LSZ converts time-ordered correlators into S-matrix elements.
• Each external leg: Fourier factor $\times$ wave operator $\times i/\sqrt{Z}$.
• Effect: amputate legs, go on-shell, attach $\sqrt{Z}$ and spin wavefunctions, leaving $i\mathcal{M}$.

Where this leads

• Assembling $\mathcal{M}$ into rates: the S-matrix, cross sections and decay rates.
• The residue $Z$ as a renormalization constant: renormalization and counterterms.
• Field-redefinition freedom in EFT: effective field theory.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 7.2.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 10.3.
• Srednicki, Quantum Field Theory, Ch. 5, 13.