QFT Bridge: S-Matrix

The scattering matrix maps asymptotic incoming states to asymptotic outgoing states. S-Matrix is the canonical nonrelativistic construction from Møller wave operators. This bridge asks what survives when the asymptotic states become multiparticle states of quantum fields.

The shared abstraction is

$$S_{fi} = \langle f,\mathrm{out}|i,\mathrm{in}\rangle.$$

The difference is what the states mean and how amplitudes are normalized.

In and Out States

In nonrelativistic scattering, one often starts from a free incoming state and constructs an outgoing-boundary scattering state:

$$|\psi^{(+)}\rangle = |\phi\rangle + G_0^{(+)}(E)V|\psi^{(+)}\rangle.$$

The potential acts within a fixed-particle Hilbert space.

In QFT, the in and out states are Fock-space states. Particle number can change, so an $S$-matrix element can describe processes such as

$$2\to2, \qquad 1\to3, \qquad 2\to n.$$

The fixed-particle potential picture is therefore only a special limiting intuition.

Normalization Changes

Nonrelativistic scattering often uses amplitudes $f(\theta,\phi)$ defined by

$$\psi(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r}.$$

QFT usually defines an invariant amplitude $\mathcal M$ after factoring out momentum conservation:

$$\langle f|S|i\rangle = \langle f|i\rangle + i(2\pi)^4\delta^{(4)}(P_f-P_i) \mathcal M_{fi}$$

in a common convention. Different texts move factors of $i$, $2\pi$, and normalization into different symbols.

The practical rule is simple: translate complete cross sections or rates, not isolated amplitudes.

LSZ Preview

In potential scattering, the Lippmann–Schwinger equation constructs scattering states from a Hamiltonian and boundary conditions. In QFT, the LSZ reduction formula extracts scattering amplitudes from time-ordered correlation functions by isolating external one-particle poles.

The analogy is:

$$\text{asymptotic free states} \quad \leftrightarrow \quad \text{external particle poles}.$$

The implementation differs because QFT uses fields, relativistic normalization, and multiparticle phase space.

From Correlation Functions to QFT Observables explains why LSZ starts from a connected time-ordered correlator and what remains before it becomes a cross section.

What Carries Over

The following ideas carry over directly:

• amplitudes are complex and interfere;
• unitarity constrains the $S$-matrix;
• resonances are tied to poles;
• total probability is distributed among all allowed channels;
• perturbation theory computes matrix elements order by order.

What Does Not Carry Over Directly

The following features change:

• particle number is no longer fixed;
• state normalization includes relativistic factors;
• Lorentz-invariant phase space replaces simple solid-angle counting;
• interactions come from field operators rather than a fixed potential;
• spin, antiparticles, and crossing symmetry become structural.

Common Mistakes

• Treating the nonrelativistic amplitude $f$ as if it were $\mathcal M$.
• Forgetting that QFT states live in Fock space.
• Ignoring delta-function normalization when comparing $S$-matrix elements.
• Assuming LSZ is just the Lippmann–Schwinger equation in different notation.

Exercises

1. What is the shared meaning of $S_{fi}$ in QM and QFT?

Solution

In both settings, $S_{fi}$ is the amplitude for an incoming asymptotic state $i$ to be detected as an outgoing asymptotic state $f$:

$$S_{fi} = \langle f,\mathrm{out}|i,\mathrm{in}\rangle.$$

The meaning and normalization of the states differ between nonrelativistic quantum mechanics and QFT.

2. Why does particle production force a conceptual change from potential scattering to QFT scattering?

Solution

Potential scattering usually acts in a fixed-particle Hilbert space. QFT scattering acts on Fock space, where states with different particle numbers coexist. Processes such as $2\to3$ or decays cannot be represented as ordinary one-particle scattering from a fixed potential without adding extra structure by hand.

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.