Free Fields
Free fields show how particle content emerges from a field equation and its quantization. Their actions are quadratic, so independent normal modes provide a calculable starting point. The profiles compare neutral and charged scalars, fermions, and electromagnetic waves in flat spacetime, with the state and boundary conditions stated where needed.
Enter this chapter
Section titled “Enter this chapter”Begin with Real Scalar Field to connect a wave equation, a mass shell, and a two-point function. If your question concerns physical photon polarizations, enter through Maxwell Theory, where constraints are central even without interactions.
Topics
Section titled “Topics”- Real Scalar Field — Relate the Klein–Gordon equation to neutral spin-zero particles and check Gaussian factorization of a four-point function.
- Complex Scalar Field — Distinguish particle and antiparticle modes and derive the charge that counts their number difference.
- Dirac Field — Identify spinor degrees of freedom, the fermion mass shell, and the role of anticommutation relations.
- Maxwell Theory — Separate gauge descriptions from physical transverse polarizations and examine electromagnetic propagation.
How the ideas fit together
Section titled “How the ideas fit together”The equation of motion determines the allowed modes; the field’s reality, transformation law, and quantization determine how those modes become particle species. A complex scalar has more independent degrees of freedom than a real scalar. Its conserved charge distinguishes particles from antiparticles, even when their masses agree. Spinor components require a different relation between field components and physical states.
For a free scalar in its Gaussian vacuum, higher correlation functions reduce to pairings of two-point functions. Subtracting those disconnected pairings leaves a vanishing connected four-point function. This is the baseline against which an interaction is detected; it does not mean that vacuum correlations vanish. The mode and pairing constructions are developed in Srednicki 2007, §§3 and 8.
Maxwell theory adds another distinction: components of a potential need not be independent physical oscillators. Gauge constraints remove redundant descriptions. Consequently, counting components in a Lagrangian is not enough to count particles.
Where to go next
Section titled “Where to go next”Continue to Scalar and Fermion Interactions to see how scattering and vacuum structure change when the action is no longer quadratic. For quantization in greater detail, follow the Library links in each profile.
References
Section titled “References”- Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§3 and 8. DOI.