Bridge to QFT Scattering

Nonrelativistic scattering theory teaches many ideas that survive in quantum field theory: amplitudes, cross sections, asymptotic states, unitarity, resonances, and the optical theorem. What changes is the kinematics, normalization, role of particle number, and the way interactions are represented.

This page is a bridge. It does not replace a full relativistic scattering treatment, but it prepares the translation from potential scattering to field-theoretic amplitudes.

What Carries Over

The following ideas transfer directly in spirit:

• Scattering is described by transitions between asymptotic in and out states.
• Amplitudes are complex and interfere.
• Cross sections are probabilities per incident flux.
• Unitarity constrains amplitudes.
• The forward amplitude is tied to total probability through the optical theorem.
• Resonances appear as poles and rapid phase variation.
• Perturbation theory computes amplitudes order by order.

The nonrelativistic formula

$$\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2$$

is not the QFT formula, but it teaches the central logic: an amplitude plus flux and final-state counting produces an observable cross section.

What Changes

In relativistic scattering, the basic constraints include Lorentz invariance and relativistic energy-momentum conservation:

$$p_1+p_2 = p_3+\cdots+p_n.$$

The number of particles can change. A process such as

$$2\to3$$

has no analogue in fixed-particle potential scattering unless additional internal channels are introduced by hand.

Interactions are not encoded primarily by a potential $V(\mathbf r)$. They are encoded by a Hamiltonian or Lagrangian density built from quantum fields. The perturbative expansion produces diagrams and invariant matrix elements rather than wavefunctions scattered by a fixed external potential.

Amplitude Conventions

In nonrelativistic potential scattering, one common asymptotic convention defines

$$\psi(\mathbf r) \sim e^{ikz} + f(\theta) \frac{e^{ikr}}{r}.$$

The amplitude $f$ has dimensions of length, and

$$\frac{d\sigma}{d\Omega}=|f(\theta)|^2.$$

In relativistic QFT, one usually defines an invariant amplitude $\mathcal M$ through an $S$-matrix convention with momentum-conserving delta functions. For two-to-two scattering in the center-of-mass frame, a common convention gives

$$\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \overline{|\mathcal M|^2},$$

where

$$s=(p_1+p_2)^2.$$

The overline indicates appropriate spin averages and sums when needed. Different normalization conventions move factors between state normalization, $\mathcal M$, and phase space.

Phase Space

In potential scattering, the final state for elastic two-body scattering is often described by a solid angle $d\Omega$ at fixed energy. In QFT, final states are counted by Lorentz-invariant phase space.

For an $n$-particle final state, the schematic phase-space measure is

$$d\Phi_n(P;p_1,\ldots,p_n) = (2\pi)^4 \delta^{(4)} \left( P-\sum_{j=1}^n p_j \right) \prod_{j=1}^n \frac{d^3p_j}{(2\pi)^3 2E_j}.$$

This measure replaces the simpler angular counting used in elementary potential scattering. It is where thresholds, masses, and multiparticle kinematics enter.

Born Approximation and Tree-Level Exchange

The first Born approximation gives

$$f_{\mathrm B}(\mathbf q) = - \frac{m}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r).$$

It says that, at leading order, the scattering amplitude is controlled by the Fourier transform of the interaction.

In perturbative QFT, tree-level exchange produces momentum-space denominators with similar interpretive content. For example, a Yukawa potential has Fourier transform proportional to

$$\frac{1}{\mathbf q^2+\mu^2}.$$

Relativistically, this becomes part of a propagator structure. The analogy is useful, but it should not be pushed too far: QFT amplitudes include relativistic normalization, spin, antiparticles, crossing, and field-theoretic conservation laws.

LSZ Preview

In potential scattering, the Lippmann-Schwinger equation builds scattering states from free states plus outgoing boundary conditions:

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

In QFT, scattering amplitudes are extracted from time-ordered correlation functions by isolating external particle poles. This is the role of LSZ reduction.

The shared idea is that asymptotic free particles are related to interacting dynamics by a limiting procedure. The implementation is different because QFT uses fields and relativistic multiparticle states rather than fixed-number wavefunctions in a potential.

Unitarity and the Optical Theorem

The nonrelativistic optical theorem reads

$$\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im}f(0)$$

in a standard convention. Its conceptual content is unitarity:

$$S^\dagger S=1.$$

In QFT, the same condition relates the imaginary part of a forward invariant amplitude to a sum over all allowed intermediate on-shell states. Diagrammatically, this leads to cutting rules.

The formula changes because state normalization and phase space change. The conservation-of-probability principle does not.

Resonances and Poles

Nonrelativistic resonances are associated with poles of the analytically continued $S$-matrix. This idea carries over strongly to QFT. Particle resonances are unstable excitations associated with complex pole locations.

Near an isolated resonance, a denominator of the form

$$E-E_R+i\Gamma/2$$

in nonrelativistic scattering is analogous in spirit to a relativistic denominator involving mass, invariant energy, and width. But relativistic resonance analysis must account for spin, thresholds, channels, and energy-dependent widths.

Breit–Wigner Form owns this local parameterization and distinguishes Breit–Wigner parameters from complex pole parameters before translating the denominator to invariant energy $s$.

Normalization Warnings

Many errors in moving from quantum mechanics to QFT are normalization errors. Watch for:

• plane-wave normalization conventions,
• factors of $2E$ in relativistic states,
• delta functions in energy versus four-momentum,
• flux factors,
• identical-particle symmetry factors,
• spin averages and sums,
• powers of $2\pi$.

When comparing formulas, compare complete cross sections or decay rates, not isolated pieces of notation.

Common Mistakes

• Identifying the nonrelativistic amplitude $f$ directly with the invariant amplitude $\mathcal M$.
• Forgetting that relativistic final states require phase-space integration.
• Treating a potential as the fundamental interaction in QFT.
• Ignoring particle production and channel thresholds.
• Using the optical theorem without matching normalization conventions.
• Reading every Feynman diagram as a literal classical trajectory rather than as a term in an amplitude.

Cross-Links

• Scattering Amplitude
• Differential and Total Cross Sections
• Lippmann-Schwinger Equation
• First Born Approximation
• Optical Theorem
• Resonances
• Breit–Wigner Form
• Inelastic Scattering Preview
• Path Integrals
• Why Dynamics Matters for QFT

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• 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.

Exercises

1. Why does $d\sigma/d\Omega=|f|^2$ not directly become the QFT two-to-two cross-section formula?

Solution

The nonrelativistic formula assumes a specific potential-scattering normalization and elastic two-body kinematics. In QFT, states are relativistically normalized, amplitudes carry momentum-conserving delta functions by convention, and cross sections include flux and Lorentz-invariant phase space. These ingredients produce factors such as

$$\frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|}$$

in a common two-to-two convention.

2. Identify the shared principle behind the nonrelativistic and QFT optical theorems.

Solution

Both follow from unitarity:

$$S^\dagger S=1.$$

In nonrelativistic scattering, this relates the imaginary part of the forward amplitude to the total cross section. In QFT, the same condition relates the imaginary part of a forward invariant amplitude to a sum over all allowed on-shell intermediate states, with relativistic phase-space factors.

3. In what sense is the Born approximation a useful bridge to tree-level QFT amplitudes, and where does the analogy stop?

Solution

The Born approximation shows that leading scattering can be controlled by a momentum-space interaction, such as the Fourier transform of a potential. Tree-level QFT amplitudes similarly contain momentum-space propagators and vertices. The analogy stops when relativistic normalization, spin, particle creation, antiparticles, crossing, gauge symmetry, and field-theoretic phase space become essential.