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.

Use $\hbar=c=1$ in the relativistic formulas below. Nonrelativistic potential formulas retain $\hbar$ explicitly; $m$ there is the particle mass for a fixed target or the reduced mass for relative motion. S-matrix conventions gives the precise normalization translation.

Amplitudes, Flux, and Final States

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

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

This expression uses relativistically normalized one-particle states and distinguishable final particles. Identical final states require the corresponding symmetry factor when integrating over the complete labeled phase space. 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

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

Here $\mathbf q=\mathbf k\prime-\mathbf k$ is a wave-vector transfer, so the projectile momentum change is $\hbar\mathbf q$. At leading order, the 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

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

Here $\mu$ denotes inverse screening length, not reduced mass. In natural units it can be matched to a mediator mass. 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 develops this local parameterization and distinguishes Breit–Wigner parameters from complex pole parameters before translating the denominator to invariant energy $s$.

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.

References

• 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.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.