Canonical Quantization and the Free Scalar

The free scalar is the first model in this volume for which the whole canonical chain can be made explicit: classical solutions are organized into modes, equal-time brackets become a quantum algebra, a representation and vacuum are chosen, Fock excitations are constructed, and distinct two-point distributions are computed. The chapter is therefore organized by dependencies rather than by one compulsory reading order. A first-course route follows the massive real scalar from modes through propagators and Fock space; focused routes branch toward algebra and representations, wave functionals, conserved charge, normal ordering, coherent states, or infrared limits.

The core setting is a free scalar on flat Lorentzian spacetime, with the massive real field as the normalization reference. The complex and massless fields are controlled extensions. Interacting renormalization, general theorems about inequivalent representations, particle concepts in curved spacetime, and decoherence are developed elsewhere.

Choose an entry route · Review the chapter

From Klein–Gordon modes to a quantum scalar

Canonical quantization does not identify a classical solution with a quantum particle. It relates several distinct structures, each of which needs new input:

Layer	New input	What the layer determines	What it does not determine by itself
Classical solutions	Action, field equation, boundary or initial data, and a mode basis	The real solution space and its conserved pairing	A quantum algebra or vacuum
Canonical algebra	The quantization postulate and normalization of the equal-time bracket	Commutation relations among smeared fields, momenta, or modes	A Hilbert space or expectation values
Representation and state	A positive-frequency choice and a selected vacuum state in the Minkowski construction	Operators acting on a Hilbert space and vacuum correlation functions	A representation-independent particle concept
Fock organization	Creation operators acting on the selected vacuum	Symmetric multiparticle sectors and number operators	A universal state space for every QFT setting
Two-point functions	The applicable combination of state, ordering, support condition, and pole or boundary prescription	Wightman, Feynman, commutator, retarded, or advanced distributions	Permission to identify objects with similar denominators

The standard massive real-scalar construction uses on-shell momenta $p^\mu=(E_{\mathbf p},\mathbf p)$ with $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$ and one matched normalization package,

\begin{aligned} \int_{\mathbf p} &\equiv \int \frac{\mathrm d^{d-1}\mathbf p} {(2\pi)^{d-1}2E_{\mathbf p}}, \\ \widehat\phi(x) &= \int_{\mathbf p} \left[ a(\mathbf p)e^{-ip\cdot x}+ a^\dagger(\mathbf p)e^{ip\cdot x} \right], \\ \left[ a(\mathbf p),a^\dagger(\mathbf q) \right] &= (2\pi)^{d-1}2E_{\mathbf p}\, \delta^{(d-1)}(\mathbf p-\mathbf q). \end{aligned}

All other oscillator commutators vanish. For the free scalar, $\widehat\pi=\partial_t\widehat\phi$ . The normalization round-trip check is

\left[ \widehat\phi(t,\mathbf x), \widehat\pi(t,\mathbf y) \right] = i\,\delta^{(d-1)}(\mathbf x-\mathbf y).

Changing the momentum measure, the mode coefficient, or the oscillator commutator is allowed only if all three are translated together and this equal-time relation and the one-particle norm are recovered. The classical modes and their quantum realization are developed in Schwartz 2014, §§ 2.2–2.3, pp. 17–26 and Weinberg 1995, § 5.2, pp. 201–206.

An equivalent convention absorbs $\sqrt{2E_{\mathbf p}}$ into the oscillator: $a(\mathbf p)=\sqrt{2E_{\mathbf p}}\,\widetilde a(\mathbf p)$ , so the field carries $1/\sqrt{2E_{\mathbf p}}$ and $[\widetilde a(\mathbf p),\widetilde a^\dagger(\mathbf q)]=(2\pi)^{d-1}\delta^{(d-1)}(\mathbf p-\mathbf q)$ . Both packages recover the same equal-time commutator and, after the corresponding rescaling of momentum kets, the same physical one-particle normalization.

Even after the algebra is fixed, the algebra, a state on it, its Hilbert-space representation, and a vector representing that state remain different objects. The usual Minkowski vacuum Fock representation is an exceptionally useful realization, not a universal uniqueness theorem; the broader distinction is explained in Hollands and Wald 2015, §§ 1–2.1, pp. 4–14, Open PDF.

Choose a route

The arrows below are suggested reading orders. Their complete hard closure is compact: Klein–Gordon modes require the action principle; the quantized scalar requires modes and canonical algebra; propagators and Fock space require the quantized scalar; wave functionals require canonical algebra and the quantized scalar; the complex scalar additionally requires classical currents; normal ordering and coherent states require Fock space; and the infrared page requires modes and propagators.

Reader goal	Suggested route	Observable result
First coherent free-field construction	Klein–Gordon modes and canonical algebra → quantized real scalar → propagators → Fock space	Normalize the field, recover its canonical commutator and Hamiltonian, and say which later statements use the selected vacuum
Algebra and representation	Canonical algebra plus Klein–Gordon modes → quantized real scalar → compare Fock space with wave functionals	Separate the abstract relations, their operator realization, and the state represented in two canonical pictures for the same matched regulated system
Charged scalar	Quantized real scalar plus classical currents → complex scalar	Construct independent particle and antiparticle sectors and the free $U(1)$ charge
Vacuum terms or a classical regime	Fock space → normal ordering or coherent states	Decide which statements are relative to the chosen vacuum and which fluctuation test is still needed
Schrödinger representation	Klein–Gordon modes plus canonical algebra → quantized real scalar → wave functionals	Recover the regulated Gaussian vacuum mode by mode and identify its boundary data
Infrared limits	Klein–Gordon modes plus canonical algebra → quantized real scalar → propagators → massless and zero-mode analysis	Determine whether setting $m=0$ is legitimate for the stated dimension, volume, observable, and order of limits

Check your entry point

This overview has no prerequisite. Individual leaves do; use the following prompts to locate the shortest useful repair.

Try this	Ready when	Repair if unsure
Vary the real-scalar action and identify the plane-wave dispersion relation	You obtain $(\Box+m^2)\phi=0$ , retain the surface term, and infer $p^2=m^2$	Review The Action Principle and Field Equations and Fourier Series, Fourier Transforms, and Plancherel Theory
Translate between a momentum integral and a spatial delta distribution	You can track every factor of $(2\pi)^{d-1}$ and $2E_{\mathbf p}$ through the equal-time commutator	Use Fourier, distributions, and Green functions repair
Distinguish an algebra, a representation, a state, and a vacuum vector	You can describe each without defining it by the notation used for another	Review Vacua, States, and Representations and, for the operator side, Unbounded Operators, Domains, Closure, and Adjoints
Turn a quadratic Hamiltonian into independent oscillators	You can identify canonical coordinates, check the bracket, and explain why the excitation spectrum is nonnegative	Review Hamiltonian Initial Data and Phase Space and Commutators and Operator Exponentials
Tell a Feynman two-point function from a retarded Green function	You state the ordering or support condition and the boundary prescription, not merely the denominator	Begin with the Fourier and Green-function repair route, then enter the propagator page

If you want a guided reading sequence rather than a reference route, Canonical quantization and the free scalar places the chapter in the site-wide learning path.

The free scalar as a recurring system

The same scalar appears at several levels, but “the same” means that the system, regulator, state, boundary data, normalization, and observable have been matched. Four relations organize the chapter:

Modes to algebra: the mode coefficients and oscillator commutator are fixed together so that the equal-time field commutator is recovered.
Algebra to representation: a positive-frequency split specifies the standard Minkowski vacuum state, which is then represented on the corresponding Fock space.
Field to two-point distributions: the applicable state-and-ordering or support-and-boundary data select different kernels with different physical uses.
Core model to controlled branches: a complex field changes the independent modes and charge, a coherent state changes the state family, and the massless limit changes the infrared problem.

In a periodic spatial box, $\mathbf k_{\mathbf n}=2\pi\mathbf n/L$ and $\omega_{\mathbf n}=\sqrt{\mathbf k_{\mathbf n}^2+m^2}$ . A finite mode cutoff makes normalization checks ordinary oscillator calculations, but it is not Lorentz invariant and does not by itself establish continuum causal support. Taking $m>0$ avoids a zero-frequency mode in this reference calculation; the massless page treats the zero mode and order of limits explicitly.

Vacuum Wightman and time-ordered functions, source methods, and the Feynman boundary prescription are treated in Schwartz 2014, § 6.2, pp. 75–77; §§ 14.3–14.4, pp. 261–266 and Weinberg 1995, § 6.2, pp. 274–279. The distinction between the equation’s causal propagator and a state-selecting positive two-point function is sharpened in Hollands and Wald 2015, § 2.1, pp. 9–14, Open PDF.

The ten pages in order

The list below is the chapter order. Dependencies can branch, so the numbering is not a claim that every page requires all earlier pages.

The Klein–Gordon Field and Its Modes. Begin here for the classical solution space, dispersion relation, mode bases, and conserved pairing. It requires the action principle. Continue to canonical quantization when the real-field condition and positive-frequency convention are explicit.
Canonical Quantization: Algebra, Representation, and State. Begin here when the main question is conceptual: what is fixed by promoting classical brackets to quantum relations, and what still remains to be chosen? It has no hard prerequisite; classical Hamiltonian theory is useful preparation. Continue to the real-scalar construction for the first complete realization.
Quantizing the Real Scalar Field. This is the main normalized field construction: it matches the modes to the canonical commutator and constructs the field, Hamiltonian, vacuum, and elementary excitations. It requires both the Klein–Gordon modes and canonical-algebra pages. Continue to propagators for two-point distributions or to Fock space for the full sector organization.
Scalar Propagators, Ordered Correlators, and Sources. Take this branch to distinguish Wightman, time-ordered, commutator, retarded, advanced, and source-defined kernels by the applicable state-and-ordering or support-and-boundary data. It requires the quantized real scalar. Continue to the functional-integral chapter for a regulated source-based rederivation, or to the infrared page for the massless limit.
Fock Space, Vacuum, and Particle Number. Take the representation branch to develop the symmetric Fock space generated by the selected free-field vacuum and identify which particle-number statements depend on that choice. It requires the quantized real scalar. Continue to normal ordering or coherent states, depending on whether the question concerns vacuum terms or a classical regime.
Complex Scalars and Conserved Charge. Take the charged-field branch to replace the real-field condition by independent particle and antiparticle operators and realize the free global $U(1)$ charge. It requires the quantized real scalar and Classical Symmetries, Currents, and Stress Tensors. Continue to the general symmetry treatment before gauging the phase symmetry or discussing spontaneous breaking.
Normal Ordering and Vacuum Terms. Take the vacuum-bookkeeping branch to define normal ordering relative to a specified free Fock vacuum and track its elementary effect on vacuum contributions. It requires the Fock-space page. Continue to renormalized composite operators rather than treating this reordering rule as a general interacting prescription.
Schrödinger Wave Functionals. Take this alternative canonical picture to represent the regulated state as a functional of field configurations and derive its Gaussian vacuum kernel mode by mode. It requires the canonical-algebra and quantized-scalar pages. Continue to time slicing and Gaussian functional integrals to compare canonical and functional descriptions under matched regulator and boundary data.
Coherent States and the Classical Limit. Use this optional classical-regime extension to study free-field coherent states, their expectation values, and relative fluctuations. It requires the Fock-space page. Continue to open-system or nonequilibrium treatments when decoherence, measurement, or environmental stability is the actual question.
Massless Scalars, Zero Modes, and Infrared Limits. Use this infrared qualification to test how dimension, finite volume, zero modes, and long-distance behavior alter the $m\to0$ limit. It requires the Klein–Gordon modes and scalar-propagator pages. Continue only after the observable and order of limits are stated.

Normalization and interpretation checks

This chapter inherits the site conventions. The additional choices below cannot be inferred globally because they depend on the representation, state, regulator, or observable.

Choice	State locally	Invariant or failure check
Mode normalization	Momentum measure, mode coefficient, and oscillator commutator	Recover both the equal-time field commutator and the one-particle norm
Real or complex field	Hermitian field condition or independent particle and antiparticle operators	Check the factor of two, charge sign, and number of independent mode coefficients
Vacuum and Fock representation	Positive-frequency split and the annihilation operators defining the vacuum	Identify every number operator, normal-ordering symbol, and particle claim that changes with the choice
Two-point distribution	The applicable state-and-ordering data or support-and-boundary data, together with its distributional equation	Check pole placement, causal support, complex conjugation, and the equal-time discontinuity
Finite regulator	Box, lattice, mode cutoff, or other finite system and its boundary data	Show how sums become integrals and isolate the zero mode before taking a limit
Classical regime	Observable, state family, and absolute or relative fluctuation criterion	A classical expectation value alone is insufficient; fluctuations and stability must also be controlled

Several common substitutions fail these checks. A vacuum is not an empty classical configuration. A field operator is not a particle, although its creation operators create particle states in the selected Fock representation. The same algebraic denominator does not identify two propagators with different ordering or support. Normal ordering is not interacting composite-operator renormalization. A coherent expectation value is not evidence of decoherence. Finally, the massless theory is not obtained safely by setting $m=0$ before examining zero modes and long-distance limits.

What the construction establishes

The chapter’s central result is a conditional chain, not a universal equivalence:

the free action and boundary data define a classical Klein–Gordon solution space;
a canonical quantization postulate defines equal-time commutation relations;
a positive-frequency choice in the standard Minkowski setting specifies the standard vacuum state;
representing that state on the corresponding Fock space gives nonnegative excitation energies relative to that vacuum;
state and ordering data, or causal support and boundary data, distinguish the resulting two-point distributions;
the same regulated state can be represented by a Gaussian Schrödinger wave functional; and
complex, coherent, and massless branches test charge, classical behavior, and infrared limits without erasing their additional assumptions.

The first three steps are often compressed into the phrase “quantize the field,” but keeping them separate explains why a canonical algebra can survive while a vacuum or particle interpretation changes. It also explains why normal ordering, particle number, and a wave functional are representation-dependent constructions. The free scalar supplies a complete physical example; theorem-level representation theory remains with Mathematical QFT.

Review the chapter

Each prompt states a criterion and a repair route so that an incomplete derivation points to the next page to revisit.

Normalization reconstruction. Starting from the displayed invariant measure, recover the equal-time canonical commutator. A satisfactory response moves the factors of $2E_{\mathbf p}$ and $(2\pi)^{d-1}$ together, uses both frequency parts of the real field, and obtains one spatial delta distribution. Repair: Quantizing the Real Scalar Field.
Retrieval and object distinction. Without looking back, state what an algebra, a state, a representation, and a representing vector are, then explain why the CCR algebra does not select a vacuum. A satisfactory response defines all four separately and does not identify a vacuum state with a vector before a representation is chosen. Repair: Canonical Quantization: Algebra, Representation, and State and Vacua, States, and Representations.
Hamiltonian check. Trace the route from the normalized field to a sum of oscillator energies. A satisfactory response identifies the positive excitation term and the vacuum term, and does not remove the latter without naming a regulator or prescription. Repair: Quantizing the Real Scalar Field and Normal Ordering and Vacuum Terms.
Propagator diagnosis. Two kernels contain $(p^2-m^2)^{-1}$ . Decide what else must be known before they can be compared. A satisfactory response identifies which state-and-ordering or support-and-boundary data apply to each kernel, together with its distributional equation. Repair: Scalar Propagators, Ordered Correlators, and Sources.
Finite-volume transfer. Put the scalar in a periodic spatial box and identify what becomes of the momentum integral, delta distribution, and zero mode. A satisfactory response replaces integrals by the correctly normalized discrete sum and treats $\mathbf p=0$ separately before taking either the infinite-volume or massless limit. Repair: The Klein–Gordon Field and Its Modes and Massless Scalars, Zero Modes, and Infrared Limits.
Chapter synthesis. Choose a route for constructing a charged coherent wave packet and state where the argument must stop. A satisfactory response closes the real-scalar and Fock dependencies, uses the classical-current input before the complex-scalar charge, applies the coherent-state fluctuation test, and distinguishes nonzero $\langle Q\rangle$ from sharp charge: a standard Glauber coherent state generally superposes charge sectors and is not a $Q$ eigenstate. It sends decoherence or gauging to their dedicated treatments. Repair: use the charged and classical-regime rows in the route table above.
Convention translation. Rewrite the displayed invariant-measure expansion using $\mathrm d^{d-1}\mathbf p/(2\pi)^{d-1}$ , a coefficient $1/\sqrt{2E_{\mathbf p}}$ , and oscillator operators with no $2E_{\mathbf p}$ in their commutator. A satisfactory response gives the rescaling $a(\mathbf p)=\sqrt{2E_{\mathbf p}}\,\widetilde a(\mathbf p)$ , recovers the same equal-time commutator, and rescales the momentum kets consistently before comparing one-particle normalizations. Repair: return to the normalization reconstruction above and the quantized-scalar page.

Where to continue

Learn this in context: Canonical quantization and the free scalar organizes a guided reading sequence around this chapter.
Reconstruct it functionally: Functional Integrals and Free Gaussian Fields begins from regulated systems and matched boundary data.
Generalize the charge: Continuous Symmetries, Generators, and Charges develops the symmetry framework beyond the free complex-scalar application.
Change the field species: Free Fermion Fields and Free Spin-One Fields and Constraints test which parts of the scalar construction survive CAR, spin, and gauge constraints.
Make representation theory precise: States, GNS Representations, and Folia gives the theorem-level relation among states and representations.
Change the spacetime setting: Bogoliubov Transformations and Unitary Implementability explains when canonical transformations fail to be unitarily implementable.
Renormalize local products: Composite Operators and Mixing replaces vacuum normal ordering by a controlled interacting treatment.
Return to the volume-scale comparison: the regulated free-scalar comparison table follows the same regulated system into functional, spectral, Euclidean, and structural descriptions.

References

Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI. Open PDF, arXiv:1401.2026v2.
Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.