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 fields and test locality. Read the motivation, then From Harmonic Oscillators to Fields → From Fock Space to Quantum Fields → From Spinors to Fermion Fields. The checkpoints are independent traveling modes, a vacuum-to-particle wave amplitude, and the scalar or spinor causal bracket. Generic oscillator and Fock constructions remain at their accepted owners.
Calculate correlations, response, and scattering. Continue from the field construction to From Propagators to Correlators. Then choose Generating Functionals for source insertions and the distinction between vacuum boundary data and initial-state response, or From Scattering to LSZ for external particles and observables. The latter applies the existing LSZ pole mechanism; source-function differentiation is a useful way to obtain correlations, not a prerequisite for defining the scattering operator.
Make the electromagnetic field dynamical. Read From Gauge Covariance to Gauge Theory after minimal coupling and the field/state distinction. Add the spinor-field and scattering branches before From Relativistic QM to QED. The final calculation recovers the signed Coulomb potential from one-photon exchange between distinguishable quantum particles.
The question answered by each page
Section titled “The question answered by each page”| Page | Main task | Check that prevents a common confusion |
|---|---|---|
| Why Fields Replace Wavefunctions | Identify the new operators and state data | A field expectation value does not specify the whole quantum state |
| Harmonic Oscillators to Fields | Count standing and traveling modes of a real field | Reality does not identify opposite-momentum annihilation operators |
| Fock Space to Quantum Fields | Assemble a local scalar field and extract a one-particle amplitude | The field operator and its vacuum-to-particle matrix element are different objects |
| Spinors to Fermion Fields | Use the CAR and spin sums to obtain graded locality | Numerical spinors do not themselves obey operator anticommutation relations |
| Propagators to Correlators | Specify a state and an ordering | An occupied mode changes a correlator without changing its delta-source equation |
| Path Integrals to Generating Functionals | Choose source-functional boundary data | A vacuum in/out derivative need not be a causal expectation value |
| Scattering to LSZ | Select the connected interaction, external poles, and measurement data | A nonzero free four-point function need not represent scattering |
| Gauge Covariance to Gauge Theory | Add dynamics and impose electromagnetic constraints | Four potential components do not give four physical photon polarizations |
| Relativistic QM to QED | Match a dynamical amplitude to a static potential | Covariant target normalization must be removed before taking an external-source limit |
Keep the technical owners in view
Section titled “Keep the technical owners in view”The oscillator owner and its Reference field bridge provide the normal-mode quantization used here. The many-body Fock-space chapter owns the general number-sector construction. The new scalar and spinor pages specialize that algebra to relativistic local fields.
For kernels and source functionals, use the canonical Dynamics Green-function translation, source-functional bridge, and field-path-integral construction. The relativistic applications retain the scalar and Dirac source normalizations of the propagator chapter.
The general phase-to-gauge bridge owns the symmetry distinctions. The application here starts where the electromagnetic field becomes dynamical. It does not repeat the local-phase proof or global holonomy analysis.
Readiness checks for further field theory
Section titled “Readiness checks for further field theory”You should now be able to:
- Extract a numerical one-particle amplitude from a field matrix element while preserving the chosen state normalization.
- Distinguish spacelike correlations from a causal commutator or graded anticommutator.
- Give the state, ordering, and delta-source convention of a propagator before using it.
- Choose retarded initial-state response or vacuum in/out boundary conditions according to the physical question.
- Separate connected interactions, external pole normalization, and cross-section counting.
- Explain why Gauss’s law constrains photon degrees of freedom and why charged sectors require infrared care.
- Recover the sign and normalization of a static Coulomb interaction from a dynamical two-body amplitude.
The constructions remain bounded. Free-field locality checks are not a general proof of the spin–statistics theorem; a simple LSZ pole assumes suitable stable asymptotic particles; a tree-level QED match does not include loops or resolve the charged infrared problem. For controlled low-energy extensions, continue to NRQED. For prepared-state nonequilibrium response, continue to the Schwinger–Keldysh bridge.
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.