Scattering and Propagators
A Green function solves a wave equation with a specified source and boundary condition. A scattering amplitude connects specified incoming and outgoing states. A cross section additionally requires normalization, flux, and final-state counting. This chapter follows those steps while separating wave mechanics in a fixed potential from scattering among dynamical particles. It ends with the pole-residue idea behind LSZ reduction.
Enter this chapter
Section titled “Enter this chapter”Use Relativistic Phase Space for the invariant one-particle measure and Free Dirac Spinors for spinor normalization. The Klein–Gordon Equation and Covariant Dirac Equation supply the wave operators. For scattering applications, bring the amplitude and flux ideas from First Born Approximation and Cross Sections.
Begin with Relativistic Normalization. It converts covariant momentum states, delta-normalized states, and unit box spinors. Keeping the measure and the matrix element in the same convention is necessary for every later rate calculation.
Path through source response and propagators
Section titled “Path through source response and propagators”| Page | Main calculation or decision |
|---|---|
| Relativistic Green Functions | Define a two-point unit-source inverse; connect it to evolution, stationary resolvents, and ordered background expansions. |
| Klein–Gordon Propagators | Construct scalar Fourier inverses, evaluate their poles, and check the static Yukawa limit. |
| Dirac Propagators | Build the matrix kernels by factorization and verify projectors, spin sums, and equal-time source jumps. |
| Retarded, Advanced, and Feynman Propagators | Choose boundary data and identify the homogeneous mass-shell terms that distinguish the inverses. |
The source equation is the quickest sign check. The unit-source scalar and spinor inverses differ from their conventional field correlators by explicit factors of . The retarded kernel contains the modes needed for a general initial-value problem; projecting onto positive energy first changes that problem.
For a concrete forced scalar example, continue to Solving Scalar Initial-Value Problems. For the causality interpretation, use Locality and Causality Warnings. Spacelike correlation and a retarded response have different support requirements.
Path through a fixed Coulomb source
Section titled “Path through a fixed Coulomb source”Mott Scattering derives the leading Dirac cross section from a unit box state, the golden-rule rate, incident flux, and a spin trace. Relativistic Coulomb Scattering then derives the scalar comparison and explains the common long-range logarithmic phase.
These pages distinguish three levels of treatment:
- A Born amplitude uses the leading potential matrix element.
- An exact external-field solution treats the specified potential beyond that truncation, with appropriate origin and asymptotic data.
- A dynamical-target process includes recoil and the target’s own quantum states, with a different conservation and flux problem.
The Rutherford limit is a useful check, but equality of one cross-section magnitude does not establish equality of amplitudes or Coulomb phases. The accepted Coulomb Scattering owner supplies the detailed nonrelativistic long-range construction.
From an invariant amplitude to a rate
Section titled “From an invariant amplitude to a rate”Invariant Phase Space Preview combines covariant external states with invariant incident flux. Its two-body delta-function reduction gives a calculation that can be checked independently of the dynamics. It keeps final-state symmetry factors and initial spin averages explicit.
Optical Theorem Preview then translates unitarity into that same normalization. It connects the imaginary forward amplitude to the sum over all open channels and explains why a real first Born amplitude is compatible with a nonzero leading cross section. The general theorem and partial-wave proof remain on Optical Theorem.
The finite forward-amplitude formula assumes a scattering problem for which the stated quantities exist. It cannot be tested by substituting zero angle into the unscreened Coulomb Born expression.
The external-pole transition to fields
Section titled “The external-pole transition to fields”LSZ Preview uses an isolated stable-particle pole to explain external-leg amputation. It distinguishes unit-residue pole removal from full-propagator amputation and checks the result under an interpolating-field rescaling. Its spinor example explains why an on-shell numerator of rank two cannot be treated as an invertible four-by-four matrix.
This is a preparation for field-theory reduction. An interacting correlator, vacuum, particle spectrum, and scattering limits are additional input; a free wave-equation inverse does not provide them. For the broader map, use From Correlation Functions to QFT Observables and QFT Bridge: S-Matrix.
Checks to carry between calculations
Section titled “Checks to carry between calculations”Before comparing two formulas, identify their source equation or state overlap, their Fourier convention, and their boundary condition. Before interpreting an amplitude as a probability, supply the flux, phase-space measure, spin preparation, and identical-state counting. Before interpreting a negative-frequency term as an antiparticle, specify the quantum-field mode assignment.
A useful completed exercise path is to verify the equal-time Dirac jump, recover Rutherford from Mott, integrate massless two-body phase space, check the next-order optical-theorem imaginary part, and finally verify LSZ invariance under a field rescaling. Each calculation tests a different part of the transition from wave equations to scattering observables.
References
Section titled “References”- Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964. Wave-equation propagators and external-field scattering.
- Greiner, Walter. Relativistic Quantum Mechanics: Wave Equations. 3rd edition, Springer, 2000. doi:10.1007/978-3-662-04275-5. Scalar and spinor scattering.
- Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940. Relativistic amplitudes, phase space, and unitarity.