The Free Scalar Field

This page carries out the canonical quantization of the simplest relativistic field — a single free real scalar $\varphi (x)$ obeying the Klein–Gordon equation. The classical Lagrangian, Euler–Lagrange equation, and Hamiltonian were assembled in Lagrangian and Hamiltonian field theory; here they are promoted to operators. The mode expansion, ladder operators, vacuum energy, and Feynman propagator built below are the concrete realization of the Fock-space inventory and the input to perturbation theory.

Conventions: $\hslash =c=1$, metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$, ${\omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$.

The classical starting point

From classical field theory, the free real scalar has

$$\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2,\pi =\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=\dot{\varphi },\mathcal{H}=\frac{1}{2}{\pi }^2+\frac{1}{2}(\nabla \varphi {)}^2+\frac{1}{2}{m}^2{\varphi }^2,$$

and the Klein–Gordon equation of motion $(\square +m^2)\varphi =0$.

Canonical quantization (Postulate — specializes QFT Postulate 9)

Canonical quantization promotes the field $\varphi (\mathbf{x})$ and its conjugate momentum $\pi (\mathbf{x})$ to operators on a Hilbert space and replaces the classical Poisson brackets by equal-time commutators (Dirac's rule $\{,\}\to -i[,]$):

$$\boxed{[\varphi (\mathbf{x}),\pi (\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y}),[\varphi (\mathbf{x}),\varphi (\mathbf{y})]=[\pi (\mathbf{x}),\pi (\mathbf{y})]=0.}$$

These are imposed at equal times; the fields are in the Heisenberg picture, carrying the full spacetime dependence $\varphi (x)=\varphi (\mathbf{x},t)$, so their unequal-time commutator is a dynamical quantity computed below. The choice of a commutator (rather than an anticommutator) is forced for spin-0 by the spin–statistics theorem; using an anticommutator here would give a theory with no positive-definite energy.

Mode expansion and ladder operators

Because each Fourier mode of a free field is an independent harmonic oscillator, the general operator solution of $(\square +m^2)\varphi =0$ consistent with the canonical commutators is the mode expansion (quoted in preliminaries § Fock Space):

$$\boxed{\varphi (x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}(a_{\mathbf{p}}{e}^{-ip\cdot x}+a_{\mathbf{p}}^{†}{e}^{+ip\cdot x}){|}_{p^0={\omega }_{\mathbf{p}}},}$$

with $\pi (x)=\dot{\varphi }(x)=\int \frac{d^{3}p}{(2\pi {)}^3}(-i)\sqrt{{\omega }_{\mathbf{p}}/2}(a_{\mathbf{p}}{e}^{-ip\cdot x}-a_{\mathbf{p}}^{†}{e}^{+ip\cdot x})$. Because $\varphi$ is real (Hermitian), the coefficient of $e^{+ip\cdot x}$ is the adjoint of the coefficient of $e^{-ip\cdot x}$; there is a single species of excitation (a real scalar is its own antiparticle). Inverting the transform and imposing the field commutators yields the ladder-operator algebra

$$[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q}),[a_{\mathbf{p}},a_{\mathbf{q}}]=[a_{\mathbf{p}}^{†},a_{\mathbf{q}}^{†}]=0.$$

This is the continuum of harmonic-oscillator algebras described in the Fock-space terminology callout: $a_{\mathbf{p}}^{†}$ creates a quantum of momentum $\mathbf{p}$, $a_{\mathbf{p}}$ destroys one.

The Hamiltonian, normal ordering, and vacuum energy

Substituting the mode expansion into $H=\int d^{3}x\mathcal{H}$ and using the commutators gives

$$H=\int \frac{d^{3}p}{(2\pi {)}^3}{\omega }_{\mathbf{p}}\left(a_{\mathbf{p}}^{†}{a}_{\mathbf{p}}+\frac{1}{2}[a_{\mathbf{p}},a_{\mathbf{p}}^{†}]\right)=\int \frac{d^{3}p}{(2\pi {)}^3}{\omega }_{\mathbf{p}}\left(a_{\mathbf{p}}^{†}{a}_{\mathbf{p}}+\frac{1}{2}(2\pi {)}^3{\delta }^3(0)\right).$$

The first term counts particles weighted by their energy; the second is the sum of zero-point energies $\frac{1}{2}{\omega }_{\mathbf{p}}$ over all modes — infinite (the ${\delta }^3(0)$ is the spatial volume, and the $\int d^{3}p{\omega }_{\mathbf{p}}$ diverges in the UV). Since only energy differences are observable, this constant is removed by normal ordering $:\cdot :$, the instruction to place all $a^{†}$ to the left of all $a$:

$$:H:=\int \frac{d^{3}p}{(2\pi {)}^3}{\omega }_{\mathbf{p}}{a}_{\mathbf{p}}^{†}{a}_{\mathbf{p}}.$$

With this convention the vacuum has zero energy. (The zero-point energy is not entirely unphysical — its changes with boundary conditions produce the measurable Casimir force, and it gravitates, which is the cosmological-constant problem.) The momentum operator obtained from the Noether $T^{0i}$ is likewise $:\mathbf{P}:=\int \frac{d^{3}p}{(2\pi {)}^3}\mathbf{p}{a}_{\mathbf{p}}^{†}{a}_{\mathbf{p}}$.

Fock space and one-particle states

The vacuum $|0⟩$ is defined by $a_{\mathbf{p}}|0⟩=0$ for all $\mathbf{p}$; it is the unique Poincaré-invariant state of Postulate 3. Acting with creation operators builds the Fock space of the inventory page:

$$|\mathbf{p}⟩=\sqrt{2{\omega }_{\mathbf{p}}}{a}_{\mathbf{p}}^{†}|0⟩,|{\mathbf{p}}_1,\dots ,{\mathbf{p}}_n⟩=\sqrt{2{\omega }_{{\mathbf{p}}_1}}\cdots \sqrt{2{\omega }_{{\mathbf{p}}_n}}{a}_{{\mathbf{p}}_1}^{†}\cdots a_{{\mathbf{p}}_n}^{†}|0⟩.$$

The factor $\sqrt{2{\omega }_{\mathbf{p}}}$ gives the relativistically normalized states $⟨\mathbf{p}|\mathbf{q}⟩=2{\omega }_{\mathbf{p}}(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, whose normalization is Lorentz-invariant (the measure $\frac{d^{3}p}{(2\pi {)}^{3}2{\omega }_{\mathbf{p}}}$ is the invariant one from Wigner's classification). One checks $H|\mathbf{p}⟩={\omega }_{\mathbf{p}}|\mathbf{p}⟩$ with ${\omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$ — the state is a relativistic particle of mass $m$. The bosonic symmetry $a_{\mathbf{p}}^{†}{a}_{\mathbf{q}}^{†}=a_{\mathbf{q}}^{†}{a}_{\mathbf{p}}^{†}$ makes the multi-particle states automatically symmetric under exchange, realizing Bose statistics.

Microcausality

With the fields now operators at all times, the physically essential quantity is the unequal-time commutator. A direct computation from the mode expansion gives a c-number (proportional to the identity):

$$[\varphi (x),\varphi (y)]=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{2{\omega }_{\mathbf{p}}}(e^{-ip\cdot (x-y)}-e^{+ip\cdot (x-y)})\equiv i\Delta (x-y).$$

The function $\Delta (x-y)$ is Lorentz-invariant and, for spacelike separation $(x-y{)}^2<0$, vanishes — the two exponentials can be mapped into each other by a (proper orthochronous) Lorentz boost when there is no invariant time-ordering, and they cancel:

$$\boxed{[\varphi (x),\varphi (y)]=0\text{whenever}(x-y{)}^2<0.}$$

This is microcausality / local commutativity — Postulate 6 — derived here rather than assumed: measurements of the field at spacelike-separated points do not interfere. It is precisely the requirement that the commutator (not the anticommutator) vanish outside the light cone that ties integer spin to Bose statistics; the Dirac field shows the fermionic mirror image.

The Feynman propagator

The central object of perturbation theory is the Feynman propagator, the time-ordered two-point function (using the time-ordering operator):

$$D_F(x-y)=⟨0|T\varphi (x)\varphi (y)|0⟩=\theta (x^0-y^0)⟨0|\varphi (x)\varphi (y)|0⟩+\theta (y^0-x^0)⟨0|\varphi (y)\varphi (x)|0⟩.$$

Evaluating with the mode expansion (only the $a{a}^{†}$ cross-term survives between vacua) and combining the two time-orderings into a single Lorentz-invariant contour integral yields the momentum-space propagator:

$$\boxed{D_F(x-y)=\int \frac{d^{4}p}{(2\pi {)}^4}\frac{i{e}^{-ip\cdot (x-y)}}{p^2-m^2+iϵ}.}$$

Two features are essential:

• It is a Green's function of the Klein–Gordon operator, $({\square }_x+m^2)D_F(x-y)=-i{\delta }^4(x-y)$. This is why the external-leg operators $(\square +m^2)$ in the LSZ formula cancel propagators and amputate diagrams.
• The $iϵ$ prescription shifts the poles at $p^0=\pm {\omega }_{\mathbf{p}}$ off the real axis so that positive-energy modes propagate forward in time and negative-energy modes backward — the Feynman (causal) boundary condition. The same $ϵ$ is the Wick-rotation seed connecting to the Euclidean path integral.

The propagator is the line in a Feynman diagram; the vertices come from interactions added to $\mathcal{L}$.

The complex scalar field and antiparticles

A complex scalar $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ has $\mathcal{L}={\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi$ and two independent sets of ladder operators:

$$\varphi (x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}(a_{\mathbf{p}}{e}^{-ip\cdot x}+b_{\mathbf{p}}^{†}{e}^{+ip\cdot x}),$$

with $[a,a^{†}]=[b,b^{†}]=(2\pi {)}^3{\delta }^3$ and all other commutators zero. Now $a^{†}$ creates a particle and $b^{†}$ a distinct antiparticle of the same mass. The $U(1)$ Noether charge becomes

$$Q=\int \frac{d^{3}p}{(2\pi {)}^3}(a_{\mathbf{p}}^{†}{a}_{\mathbf{p}}-b_{\mathbf{p}}^{†}{b}_{\mathbf{p}})=N_a-N_b,$$

counting particles minus antiparticles: the field $\varphi$ lowers $Q$ by one (destroys a particle or creates an antiparticle), ${\varphi }^{\ast }$ raises it. Gauging this $U(1)$ turns $Q$ into electric charge and $\varphi$ into the charged-scalar field of scalar QED. Antiparticles here appear as a derived consequence of relativistic quantization — the concrete version of the Wigner–Weinberg claim that antiparticles are forced by locality plus positive energy.

Summary

Where this leads

• Spin $\frac{1}{2}$, with anticommutators, antiparticles, and the Dirac equation: The Dirac Field.
• Spin 1 and gauge redundancy: The Free Vector Field.
• Interactions: adding ${\mathcal{L}}_{\text{int}}$ and expanding the Dyson series; the propagator becomes the internal line of Feynman diagrams.
• Discrete symmetries acting on $\varphi$: C, P, T on fields.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 2.3–2.4.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 5.2.
• Srednicki, Quantum Field Theory, Ch. 3, 8.