Scattering

Scattering theory is one of the cleanest bridges from quantum mechanics to field theory. Quantum mechanics introduces amplitudes, cross sections, phase shifts, unitarity, and the optical theorem. Field theory reorganizes these ideas around asymptotic particle states, relativistic normalization, Feynman amplitudes, and the S-matrix.

Quantum Mechanics Starting Point

For a short-range potential in three dimensions, the outgoing scattering state has asymptotic form

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

The differential cross section is

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

The Lippmann–Schwinger equation packages the outgoing boundary condition:

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + \frac{1}{E-H_0+i0}V \lvert\psi^{(+)}\rangle.$$

Field-Theory Continuation

In field theory, scattering is expressed through matrix elements of the S-matrix between asymptotic multi-particle states:

$$\langle f\vert S\vert i\rangle.$$

Perturbative QFT computes these amplitudes using fields, propagators, vertices, and external-state rules. The LSZ reduction formula relates time-ordered correlation functions to S-matrix elements under assumptions about asymptotic particle states.

What Carries Over

• Incoming and outgoing states.
• Transition amplitudes.
• Energy and momentum conservation.
• Cross sections and rates.
• Unitarity constraints.
• Optical theorem logic.
• Bound states and resonances as analytic structures.

What Changes

Relativistic scattering changes normalization, phase space, spin sums, identical-particle factors, and the meaning of particle creation. The amplitude computed from Feynman diagrams is not the same object as the nonrelativistic scattering amplitude $f(\theta,\phi)$ without convention-dependent factors.

Density of States in Transition Rates gives the nonrelativistic normalization dictionary and shows how its final-state measure becomes Lorentz-invariant phase space.

Convention Checks

• State the normalization of momentum eigenstates.
• Distinguish $S$, $T$, $\mathcal M$, and $f$.
• Track factors of $2\pi$, volume, and relativistic energy.
• Identify whether states are nonrelativistic one-particle states or asymptotic QFT particle states.
• Use current ratios for nonrelativistic probabilities.

Common Mistakes

• Equating a Feynman amplitude directly with a nonrelativistic scattering amplitude.
• Forgetting identical-particle symmetry factors.
• Treating the Born approximation as always equivalent to tree-level QFT.
• Ignoring long-range Coulomb complications.
• Quoting the optical theorem without matching normalization conventions.

Canonical Links

• Scattering Amplitude
• Differential and Total Cross Sections
• Lippmann–Schwinger Equation
• Optical Theorem
• Bridge to QFT Scattering
• QFT Bridge S-Matrix
• Density of States in Transition Rates
• QFT Bridge Optical Theorem and Unitarity

References

• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• 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, Westview Press, 1995.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.

Exercises

1. Why can two books write different optical theorem prefactors without contradiction?

Solution

The optical theorem relates forward scattering to a total cross section, but the prefactor depends on the normalization of states and the definition of the scattering amplitude. If two conventions define $f$, $T$, or $\mathcal M$ differently, the same unitarity statement appears with different numerical factors.