Bridge Concepts
Relativistic wave equations supply modes, currents, representation data, and inverse operators. Quantum field theory adds a field algebra, a state, local observables, and dynamics for the participating fields. This chapter makes that transition through specific calculations: mode counting, one-particle matrix elements, fermionic locality, occupied-state correlations, source response, scattering reduction, and photon exchange.
The bridge retains quantum states and the Born rule. It changes which degrees of freedom and observables are needed for the problem. No one-particle equation alone determines a vacuum, an interacting correlator, or a photon-counting probability.
Enter this chapter
Section titled “Enter this chapter”Start with Why Fixed Particle Number Fails and Why Fock Space Is Necessary to understand the possible state sectors. Then read Why Fields Replace Wavefunctions for the distinction between a state, a numerical mode function, an operator field, and a field configuration in a functional integral.
For the scalar branch, bring the Klein–Gordon inner product and invariant mass-shell measure. For the spinor branch, bring free Dirac spinors and their spin sums. For the electromagnetic branch, bring minimal coupling. Each leaf lists the additional capability it actually uses.
Three routes through the transition
Section titled “Three routes through the transition”Construct a field and test locality. Start with Harmonic Oscillators to Fields to count standing and traveling modes. Continue to Fock Space to Quantum Fields to recover a one-particle wave amplitude from a field matrix element. Then Spinors to Fermion Fields shows how spin sums and anticommutators produce fermionic locality. You should be able to explain why a spacelike correlation can be nonzero even when the causal bracket vanishes.
Calculate correlations or scattering. Begin with Propagators to Correlators: an occupied oscillator has different fluctuations from the vacuum even though both correlators satisfy the same source equation. Next choose Generating Functionals for source insertions and initial-state response, or Scattering to LSZ for the connection between field correlations and external particles. The source-functional route is useful, but is not required to define the scattering operator.
Make the electromagnetic field dynamical. Read Gauge Covariance to Gauge Theory to see how Maxwell dynamics and Gauss’s law constrain the photon degrees of freedom. After the spinor-field and scattering pages, finish with Relativistic QM to QED. Its one-photon-exchange calculation recovers the signed Coulomb potential and explains how a quantum target becomes an approximately fixed source.
Where these calculations lead
Section titled “Where these calculations lead”The general constructions are developed in Fock Space and Occupation Number, the oscillator-to-fields introduction, Green Functions from QM to QFT, and Field Path Integrals. Use those pages when you need the underlying algebra rather than its relativistic application.
Free-field locality is a useful check, but a general spin–statistics theorem requires further assumptions. Simple LSZ reduction requires suitable stable asymptotic particles. A tree-level QED calculation does not include radiative corrections or resolve the charged infrared problem. Continue to NRQED for systematic low-energy corrections, or to the Schwinger–Keldysh bridge for expectation values in a prepared initial state.
References
Section titled “References”- Beisert, Niklas. Quantum Field Theory I. ETH Zurich lecture notes, autumn semester 2025. Lecture notes. Particle poles and scattering reduction.
- Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). doi:10.1017/9781139540940. Field correlations, gauge dynamics, and effective descriptions.
- Tong, David. Quantum Field Theory. University of Cambridge lecture notes (2006), chapters 2, 5, and 6. Free scalar fields, Dirac fields, and QED.