Canonical Quantization and the Free Scalar
The free scalar field turns infinitely many wave modes into quantum oscillators. This chapter carries that construction from classical solutions through the canonical algebra to vacuum, particle, and propagator descriptions, while separating conclusions that depend on the chosen representation or infrared limit.
Enter this chapter
Section titled “Enter this chapter”Begin with The Klein–Gordon Field and Its Modes to fix the solutions and normalization before quantizing. The wave-functional page provides a second representation of the same regulated system.
Topics
Section titled “Topics”- The Klein–Gordon Field and Its Modes — Resolves classical solutions into mass-shell modes.
- Canonical Quantization: Algebra, Representation, and State — Separates the commutation algebra from a Hilbert-space representation and state.
- Quantizing the Real Scalar Field — Matches oscillator normalization to the field’s equal-time commutators.
- Scalar Propagators, Ordered Correlators, and Sources — Builds ordered and causal two-point functions from one mode expansion.
- Fock Space, Vacuum, and Particle Number — Constructs the vacuum and symmetric multiparticle sectors.
- Complex Scalars and Conserved Charge — Introduces independent particle and antiparticle oscillators and their charge.
- Normal Ordering and Vacuum Terms — States what subtraction relative to a free vacuum changes.
- Schrödinger Wave Functionals — Describes finitely regulated states as wave functions of field coordinates.
- Coherent States and the Classical Limit — Identifies when coherent-state expectations follow classical solutions.
- Massless Scalars, Zero Modes, and Infrared Limits — Tests zero modes and the order of mass, volume, and infrared limits.
How the ideas fit together
Section titled “How the ideas fit together”A positive-frequency choice selects annihilation operators and their vacuum; it thereby fixes the free two-point function. Reordering that same pair of fields produces different correlators, while taking their commutator isolates the causal algebraic information. The field equation alone cannot choose among these objects. See Tong 2006–2007, §§ 2.2 and 2.7.
The free-scalar comparison table follows this same system through canonical, functional, spectral, and Euclidean descriptions with their state and boundary assumptions. Agreement requires those data to match; it is not a universal equivalence between arbitrary formulations.
Review the chapter
Section titled “Review the chapter”Why does setting the mass to zero in a finite-volume vacuum wave function need a separate zero-mode check?
Solution
The spatially constant mode has frequency equal to the mass. Its normalized oscillator ground state spreads without bound as the mass tends to zero. At zero mass it becomes a free-particle coordinate with no normalizable oscillator vacuum, so nonzero-mode formulas do not settle the full state.
Where to go next
Section titled “Where to go next”Continue to Functional Integrals and Free Gaussian Fields to reproduce the regulated scalar covariance by integration.
References
Section titled “References”- David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, 2006–2007, Chapter 2. Author’s online text.