Scattering Conventions and Connections

Two scattering formulas can look different and still predict the same experiment. State normalization, wave vector versus momentum, and the definition of a transition operator move factors between intermediate expressions. This chapter helps you compare complete conventions and then follow the S-matrix into relativistic field theory.

Helpful background. The scattering amplitude fixes the position-space convention, while the S-matrix and T-matrix explain the operator language.

Compare definitions before coefficients

Use the scattering convention dictionary when a sign or normalization differs between sources. Work through the definitions in this order:

Definition to compare	Why it matters
Wave vector $\mathbf k$ or momentum $\mathbf p=\hbar\mathbf k$	Changes the Fourier phase, integration measure, and delta-function normalization.
Unit incident wave or delta-normalized state	Changes matrix-element prefactors before an amplitude is extracted.
Time dependence and outgoing boundary condition	Fixes the signs of the radiation condition and resolvent prescription.
Single-channel or channel-changing flux	Determines whether a speed ratio is needed in the cross section.
Particle statistics and spin preparation	Determines exchange interference, spin averages, and final-state counting.

For example, $e^{i\mathbf k\cdot\mathbf r}$ and $e^{i\mathbf p\cdot\mathbf r/\hbar}$ describe the same phase. But $d^3p=\hbar^3d^3k$, so continuum-normalized kets cannot be relabeled without adjusting their normalization. The dictionary carries these factors through to amplitudes and observables.

An overall rescaling of the incident wave also rescales both incident and scattered current. Their ratio is unchanged. An amplitude convention should therefore state which incident normalization has already been divided out.

Follow the S-matrix into field theory

The QFT S-matrix bridge starts from the shared idea of an overlap between incoming and outgoing asymptotic states. Relativistic quantum field theory adds multiparticle states, particle creation, relativistic normalization, and phase-space factors. The invariant amplitude $\mathcal M$ is not simply the nonrelativistic length-valued amplitude $f$ under a new name.

Read the broader QFT scattering introduction first for the kinematic picture. The S-matrix bridge then connects asymptotic states with correlation functions and the LSZ construction. Stable asymptotic particles are assumed in the usual formulation; unstable resonances and long-range infrared effects need further care.

For two specific comparisons, Born approximation and tree level explains matching a relativistic interaction to a nonrelativistic potential, and the optical-theorem bridge follows probability conservation through different normalizations.

Exercises

Restore momentum units. A Gaussian Born amplitude contains $e^{-a^2q^2/4}$, where $a$ is a length and $q$ is the magnitude of a wave-vector transfer. Rewrite the exponential using physical momentum transfer $Q=\hbar q$.

Solution

It becomes $e^{-a^2Q^2/(4\hbar^2)}$. The exponent must be dimensionless. Replacing $q$ by a momentum without restoring $\hbar$ changes the formula rather than just its notation.

A changed wave normalization. Multiply an entire scattering wavefunction by a nonzero constant $c$. How do the incident and scattered currents change, and what happens to their ratio?

Solution

Each current is quadratic in the wavefunction and acquires $|c|^2$. The ratio is unchanged, so the physical cross section is unchanged. Intermediate matrix elements may change under a continuum-state normalization convention, which is why the full conversion to flux must be compared.

References

• Peskin, M. E., and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley (1995). Relativistic state normalization, amplitudes, and LSZ reduction.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Nonrelativistic amplitudes, operators, and flux conventions.
• Weinberg, S. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press (1995). Asymptotic states and the relativistic S-matrix.