Where Quantum Field Theory Begins
A relativistic wave equation supplies mode solutions and their transformation laws. A quantum-field prediction also needs an operator algebra, a state, an interaction and an observable. The boundary is therefore a change in the specified model and question, not a rule that a certain symbol or energy automatically means “QFT.” This page applies the general quantum-mechanics/field-theory boundary to the calculations in this volume.
A wave equation leaves the quantum construction open
Section titled “A wave equation leaves the quantum construction open”The same Klein–Gordon mode can appear in a classical field, a restricted one-particle description, or the expansion of an operator-valued quantum field. The numerical solution does not become an operator simply because its equation is relativistic. Conversely, numerical mode functions remain part of a quantized field calculation.
Quantum field theory retains the quantum formalism of states, observables, amplitudes and probability rules. It adds field degrees of freedom and their local operator structure rather than discarding those rules. Why Fields Replace Wavefunctions distinguishes mode functions, quantum states, field operators, expectation values and wavefunctionals in detail.
| Starting calculation | Additional input for a field prediction | Where to follow it |
|---|---|---|
| A set of scalar normal modes | Oscillator algebra, normalization and a quantum state | Harmonic Oscillators to Fields and Fock Space to Quantum Fields |
| Positive- and negative-frequency Dirac columns | Fermionic mode operators, particle/antiparticle assignment and vacuum | Spinors to Fermion Fields |
| A classical source inverse | Operator ordering and a state for the two-point function | Propagators to Correlators |
| An in/out mode mixing coefficient | Initial occupation and the relevant bosonic or fermionic algebra | Pair Creation |
| A many-point correlation function | External-pole, state-normalization and scattering assumptions | Scattering to LSZ |
| Matter in a prescribed electromagnetic potential | Gauge-field dynamics, constraints and photon states | Gauge Covariance to Gauge Theory |
The table identifies missing information. It does not claim that every field theory has a preferred particle description or a useful scattering matrix in every background.
Particle number and locality answer different questions
Section titled “Particle number and locality answer different questions”An interacting process that changes the total number of asymptotic particles cannot be represented within one fixed-number sector alone. The variable-number argument and Fock-space bridge make that limitation precise. Conserved charge can still label a sector containing different numbers of particle–antiparticle pairs.
Free and suitably controlled background problems can retain invariant or approximately decoupled sectors. Their one-particle solutions do not become invalid merely because a fuller theory permits additional processes. Use Antiparticle Decoupling for the dynamical assumptions behind such a restriction.
Fock space is also used in ordinary many-body quantum mechanics. By itself it does not specify relativistic locality, a field equation, an interacting vacuum or a gauge theory. Those are further structures. For fermionic fields, spacelike anticommutation and the commutativity of even local observables have distinct roles; the volume’s free-field checks do not replace a general spin–statistics proof.
Likewise, a nonzero spacelike vacuum correlation is not the same as a controllable signal. Locality and Causality Warnings separates the homogeneous correlation, commutator and retarded response.
Three calculations make the added input visible
Section titled “Three calculations make the added input visible”A detector outcome. A Wightman function becomes an excitation probability only after the detector gap, initial state, spatial coupling and switching are specified. A finite switched vacuum response need not indicate particles already present before the interaction.
A coherent pair pulse. Matching a scalar mode across a changing background gives complex coefficients. The initial vacuum and bosonic algebra turn their moduli into occupations and pair-number probabilities. The phase can matter during a second pulse even when it is absent from the first occupation number.
An exchange amplitude. Adding a dynamical photon field produces an amplitude with currents and external states for both charged species. A prescribed Coulomb potential emerges only after a stated reduction; target recoil, real photon emission and loop effects are not generated by renaming that potential.
The three Bridge Capstones carry out these bounded calculations. They are useful checkpoints because each names both the extra input and the observable it permits.
Where to continue beyond the checkpoint
Section titled “Where to continue beyond the checkpoint”Continue to a systematic field-theory treatment when the goal requires renormalized loop amplitudes, infrared-safe charged scattering, dynamical backreaction, non-Abelian gauge fields or interacting nonperturbative states. The LSZ owner already explains why the isolated stable-pole argument needs qualification in charged infrared sectors.
For a learning sequence use the Bridge Roadmap. For source selection use the Reading List. For a practical calculation that remains within wave mechanics, use When Relativistic QM Is Useful to state the intended accuracy and the omitted effects.
References
Section titled “References”- Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002. Spinor wavefunctions and fermionic field conventions.
- Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006–2007. Author-hosted notes. Free fields, correlators and dynamical QED.
- Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995. doi:10.1017/CBO9781139644167. Particle representations, fields, locality and scattering.