Canonical Quantum Field Theory

Intuition [Beginner]

Why do we need quantum field theory at all? The answer is that relativistic quantum mechanics, as you have studied it so far, does not work. The Klein-Gordon equation has negative probability densities. The Dirac equation predicts antiparticles but cannot describe their creation. Both equations treat particle number as fixed, but in the real world particles are created and destroyed: an electron and positron annihilate into photons, a photon splits into an electron-positron pair, a muon decays into an electron and neutrinos.

Quantum field theory solves these problems by making the field the fundamental object, not the particle. Particles emerge as excitations (quanta) of the field. An electron is a quantum of the electron field. A photon is a quantum of the electromagnetic field. Just as a vibrating string has quantized standing-wave modes (harmonics), a quantum field has quantized modes that we identify as particles.

The logic proceeds in stages:

1. Classical field theory. Start with a field $\varphi (x)$ defined at every point in spacetime. Specify a Lagrangian density $\mathcal{L}$ that determines the dynamics via the Euler-Lagrange equations. This is the continuum mechanics of fields.

2. Canonical quantization. Treat the field as a collection of harmonic oscillators (one for each Fourier mode). Promote the field and its conjugate momentum to operators obeying canonical commutation relations. The ladder operators of each oscillator become creation and annihilation operators for particles in that mode.

3. Fock space. The Hilbert space is built by acting with creation operators on the vacuum $|0⟩$: $a^{†}|0⟩$ is a one-particle state, $a^{†}{a}^{†}|0⟩$ is a two-particle state, and so on. Particle number is not fixed -- it is an eigenvalue of the number operator $\hat{N}={\sum }_k{a}_k^{†}{a}_k$.

4. Interactions and Feynman diagrams. Free fields are exactly solvable. Interactions (non-quadratic terms in $\mathcal{L}$) are treated perturbatively. The perturbation series organizes itself into Feynman diagrams -- pictorial representations of scattering processes where lines are propagating particles and vertices are interactions.

The payoff is enormous: QFT unifies quantum mechanics and special relativity, naturally accommodates particle creation and annihilation, and yields the most precisely tested predictions in all of physics (quantum electrodynamics agrees with experiment to better than one part in $10^{12}$).

Visual [Beginner]

FROM PARTICLES TO FIELDS
=========================

  Classical mechanics        Quantum mechanics         QFT
  ------------------        ------------------        ---------

  point particle             wave function             field operator
  position q(t)             psi(x,t)                  phi-hat(x,t)

  Lagrangian L(q,qdot)      Hilbert space              Fock space
                             |psi>                     |n1,n2,...>

  EOM: Newton/Lagrange      Schrodinger eq.           Heisenberg eq.
                                                      for field operators


CANONICAL QUANTIZATION OF A SCALAR FIELD
==========================================

  Classical field:           phi(x,t) and conjugate pi(x,t) = d(phi)/dt

  Quantize:                  [phi(x), pi(x')] = i*hbar*delta(x-x')
                             (like [q, p] = i*hbar for each point)

  Fourier modes:             phi(x) = sum_k (a_k u_k(x) + a_k^dag u_k^*(x))

  Each mode = harmonic oscillator:
                             a_k^dag |0> = |one particle, momentum k>
                             a_k^dag a_k^dag |0> = |two particles>


FEYNMAN DIAGRAM EXAMPLE: electron-electron scattering
======================================================

       e-              e-
        \              /
         \            /
          \    ~~    /       ~~ = photon propagator
           \  /  \  /
            \/    \/
            /\    /\
           /  \  /  \
          /    \/    \
         /            \
        /              \
       e-              e-

  Two electrons exchange a photon.
  Each vertex: e- emits or absorbs a photon.
  More photons = higher order = smaller contribution.

Concept	Quantum mechanics	Quantum field theory
Fundamental object	Wavefunction $\psi (x,t)$	Field operator $\hat{\varphi }(x,t)$
Particle number	Fixed	Variable (creation/annihilation)
Relativity	Nonrelativistic or ad hoc	Built in from the start
Hilbert space	$L^2({\mathbb{R}}^3)$	Fock space
Scattering	Born series	Feynman diagrams
Vacuum	Empty state $|0⟩$	Fluctuating fields, virtual particles

Worked example [Beginner]

Problem: Consider a free real scalar field with Lagrangian density $\mathcal{L}=\frac{1}{2}({\partial }_{\mu }\varphi )({\partial }^{\mu }\varphi )-\frac{1}{2}{m}^2{\varphi }^2$. Show that the Euler-Lagrange equation gives the Klein-Gordon equation, and count the degrees of freedom.

Solution:

Step 1: The Euler-Lagrange equation for fields is:

$$\frac{\partial \mathcal{L}}{\partial \varphi }-\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}\right)=0$$

Step 2: Compute each piece:

$$\frac{\partial \mathcal{L}}{\partial \varphi }=-m^2\varphi$$

$$\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}={\partial }^{\mu }\varphi$$

$$\frac{\partial }{\partial x^{\mu }}\left({\partial }^{\mu }\varphi \right)={\partial }_{\mu }{\partial }^{\mu }\varphi =\square \varphi$$

where $\square ={\partial }_{\mu }{\partial }^{\mu }=\frac{1}{c^2}{\partial }_t^2-{\nabla }^2$ is the d'Alembertian.

Step 3: Substitute into the Euler-Lagrange equation:

$$-m^2\varphi -\square \varphi =0⟹(\square +m^2)\varphi =0$$

This is the Klein-Gordon equation in covariant form. In natural units ($\hslash =c=1$), the plane-wave solutions are $\varphi \sim e^{-i(Et-\mathbf{k}\cdot \mathbf{x})}$ with $E^2={\mathbf{k}}^2+m^2$ -- the relativistic energy-momentum relation.

Step 4: The field $\varphi$ is real-valued (one real component at each spacetime point), so it has one degree of freedom per spatial point. After quantization, this describes a spin-0 particle (scalar boson) and its antiparticle. For a complex scalar field, there are two real degrees of freedom, allowing distinct particle and antiparticle.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Classical field theory. A classical field theory is specified by a Lagrangian density $\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi )$ depending on fields and their derivatives. The action is:

$$S[\varphi ]=\int d^{4}x\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi )$$

Stationarity of the action ($\delta S=0$) yields the Euler-Lagrange equations:

$$\frac{\partial \mathcal{L}}{\partial {\varphi }^a}-{\partial }_{\mu }\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }^a)}=0$$

Noether's theorem: for every continuous symmetry of $\mathcal{L}$, there is a conserved current $j^{\mu }$ with ${\partial }_{\mu }{j}^{\mu }=0$. Space-time symmetries yield the energy-momentum tensor $T^{\mu \nu }$; internal symmetries yield charge currents.

Canonical quantization of the real scalar field. The free real scalar field with $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2$ has the mode expansion:

$$\hat{\varphi }(x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\left(a_{\mathbf{p}}{e}^{-ip\cdot x}+a_{\mathbf{p}}^{†}{e}^{ip\cdot x}\right)$$

where $E_{\mathbf{p}}=\sqrt{|\mathbf{p}{|}^2+m^2}$ and $p\cdot x=E_{\mathbf{p}}t-\mathbf{p}\cdot \mathbf{x}$. The creation and annihilation operators satisfy:

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

The Fock space is built from the vacuum $|0⟩$ (defined by $a_{\mathbf{p}}|0⟩=0$ for all $\mathbf{p}$):

$$|{\mathbf{p}}_1,\dots ,{\mathbf{p}}_n⟩=a_{{\mathbf{p}}_1}^{†}\cdots a_{{\mathbf{p}}_n}^{†}|0⟩$$

These are $n$-particle momentum eigenstates. The Hamiltonian is:

$$\hat{H}=\int \frac{d^{3}p}{(2\pi {)}^3}{E}_{\mathbf{p}}{a}_{\mathbf{p}}^{†}{a}_{\mathbf{p}}+\left(\text{infinite zero-point energy}\right)$$

Dirac field quantization. The free Dirac Lagrangian is:

$${\mathcal{L}}_{\text{Dirac}}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi$$

The Dirac field is expanded as:

$$\hat{\psi }(x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\sum _{s=1,2}\left(b_{\mathbf{p}}^s{u}^s(p)e^{-ip\cdot x}+d_{\mathbf{p}}^{s†}{v}^s(p)e^{ip\cdot x}\right)$$

where $b_{\mathbf{p}}^s$ annihilates a fermion and $d_{\mathbf{p}}^{s†}$ creates an antifermion. The equal-time anticommutation relations are:

$$\{{\hat{\psi }}_{\alpha }(\mathbf{x}),{\hat{\psi }}_{\beta }^{†}({\mathbf{x}}^{\prime })\}={\delta }_{\alpha \beta }{\delta }^{(3)}(\mathbf{x}-{\mathbf{x}}^{\prime })$$

The use of anticommutators (rather than commutators) for fermion fields is required by causality (microcausality): measurements at spacelike-separated points must not interfere. This is the spin-statistics connection emerging from consistency conditions.

Electromagnetic field quantization. The free EM Lagrangian is:

$${\mathcal{L}}_{\text{EM}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$$

where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. Gauge invariance ($A_{\mu }\to A_{\mu }+{\partial }_{\mu }\lambda$) requires gauge fixing. In Coulomb gauge ($\nabla \cdot \mathbf{A}=0$), the physical degrees of freedom are the two transverse polarizations:

$$\hat{\mathbf{A}}(x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2|\mathbf{p}|}}\sum _{\lambda =1,2}\left(c_{\mathbf{p}\lambda }{\mathbf{ϵ}}_{\lambda }(\mathbf{p})e^{-ip\cdot x}+c_{\mathbf{p}\lambda }^{†}{\mathbf{ϵ}}_{\lambda }^{\ast }(\mathbf{p})e^{ip\cdot x}\right)$$

where $c_{\mathbf{p}\lambda }^{†}$ creates a photon with momentum $\mathbf{p}$ and polarization $\lambda$.

Interacting theory and perturbation theory. The full QED Lagrangian is:

$${\mathcal{L}}_{\text{QED}}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m_e)\psi -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$$

The interaction term $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ couples the electron field to the photon field. The S-matrix is expanded as:

$$S=T\exp \left(-i\int d^{4}x{\mathcal{H}}_{\text{int}}\right)=1+(-i)\int d^{4}x{\mathcal{H}}_{\text{int}}+\frac{(-i{)}^2}{2!}\int d^{4}x{d}^{4}yT\{{\mathcal{H}}_{\text{int}}(x){\mathcal{H}}_{\text{int}}(y)\}+\cdots$$

Each term in this expansion corresponds to Feynman diagrams with increasing numbers of vertices.

Feynman rules for QED. The translation from diagrams to amplitudes uses:

• Electron propagator: $\frac{i(/p+m)}{p^2-m^2+iϵ}$
• Photon propagator: $\frac{-i{g}_{\mu \nu }}{p^2+iϵ}$ (Feynman gauge)
• Vertex factor: $-ie{\gamma }^{\mu }$
• External lines: $u^s(p)$ for incoming electron, ${\bar{u}}^s(p)$ for outgoing electron, $v^s(p)$ for incoming positron, ${\bar{v}}^s(p)$ for outgoing positron, ${ϵ}_{\mu }^{\lambda }(p)$ for photons
• Integrate over undetermined loop momenta with $\int \frac{d^{4}k}{(2\pi {)}^4}$

Key results [Intermediate+]

1. Spin-statistics theorem. Consistency of QFT (positive-definite Hilbert space, causality) requires that integer-spin fields are quantized with commutators (bosons) and half-integer-spin fields with anticommutators (fermions). Violating this leads to violations of causality or negative probabilities.

2. CPT theorem. Any Lorentz-invariant, local QFT with a Hermitian Hamiltonian is invariant under the combined operations of charge conjugation (C), parity (P), and time reversal (T). This theorem implies that particles and antiparticles have equal masses and lifetimes.

Advanced treatment [Master]

Path integral formulation. An equivalent formulation of QFT uses the path integral:

$$Z=\int \mathcal{D}\varphi e^{iS[\varphi ]}$$

For correlation functions:

$$⟨0|T\{\varphi (x_1)\cdots \varphi (x_n)\}|0⟩=\frac{1}{Z}\int \mathcal{D}\varphi \varphi (x_1)\cdots \varphi (x_n)e^{iS[\varphi ]}$$

The path integral is defined by analytic continuation to Euclidean time ($t\to -i\tau$), which converts the oscillatory integrand into a damped one:

$$Z_E=\int \mathcal{D}\varphi e^{-S_E[\varphi ]}$$

This makes the integral mathematically better defined and connects QFT to statistical mechanics (the Euclidean action plays the role of $-\beta H$).

Renormalization. Loop diagrams in perturbation theory produce ultraviolet divergences -- integrals over loop momenta that diverge. Renormalization is the procedure of absorbing these divergences into redefinitions of the parameters (masses, couplings) of the theory. A theory is renormalizable if a finite number of counterterms suffices to render all observables finite. QED, the electroweak theory, and QCD are renormalizable. The renormalization group describes how the effective couplings change with energy scale:

$$\mu \frac{d\lambda }{d\mu }=\beta (\lambda )$$

In QED, the fine-structure constant increases (slowly) with energy: the vacuum polarizes, screening the charge at low energies. In QCD, the coupling decreases at high energy (asymptotic freedom), making perturbation theory reliable for hard processes.

LSZ reduction formula. The Lehmann-Symanzik-Zimmermann formula relates S-matrix elements (scattering amplitudes) to time-ordered correlation functions:

$$⟨f|S|i⟩=\left[\prod _{\text{in}}\int d^4{x}_i{e}^{i{p}_i\cdot x_i}({\square }_{x_i}+m^2)\right]\left[\prod _{\text{out}}\int d^4{y}_j{e}^{-i{k}_j\cdot y_j}({\square }_{y_j}+m^2)\right]⟨0|T\{\varphi (x_1)\cdots \varphi (y_1)\cdots \}|0⟩$$

This is the bridge between the abstract field theory (correlation functions) and observable quantities (cross-sections).

Gauge theories and BRST quantization. Gauge invariance introduces redundancy in the description. The Faddeev-Popov procedure introduces ghost fields (anticommuting scalar fields) to properly account for gauge fixing in the path integral. The BRST symmetry (Becchi-Rouet-Stora-Tyutin) is a global fermionic symmetry of the gauge-fixed action that encodes the original gauge invariance and provides a powerful tool for proving renormalizability and unitarity.

Anomalies. A classical symmetry may fail to survive quantization. The most famous example is the axial anomaly: the classical $U(1{)}_A$ symmetry of massless QED (chiral symmetry) is broken by the quantum measure in the path integral, leading to ${\partial }_{\mu }{j}_5^{\mu }=\frac{e^2}{16{\pi }^2}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }$. Anomalies in gauge symmetries would render the theory inconsistent; their cancellation constrains the particle content of the Standard Model.

Effective field theories. Not all QFTs need to be renormalizable. An effective field theory (EFT) is a low-energy approximation to a more fundamental theory, containing all operators consistent with the symmetries, organized by dimension. Higher-dimensional operators are suppressed by powers of $E/\Lambda$, where $\Lambda$ is the cutoff scale. The Fermi theory of weak interactions is an EFT valid below the $W$ boson mass. Chiral perturbation theory describes low-energy QCD.

Connections [Master]

• 12.11.01 -- Relativistic quantum mechanics (Klein-Gordon, Dirac) provides the classical field equations that are quantized in this unit.
• 12.09.01 -- The creation/annihilation operators and Fock space of many-body quantum mechanics are formally identical to those of QFT; the new ingredient is Lorentz invariance.
• 12.08.01 -- Scattering theory in nonrelativistic QM is the precursor to the S-matrix formalism of QFT; the LSZ formula generalizes the Born series.
• 12.13.01 -- Quantum electrodynamics is the specific QFT obtained by coupling the Dirac field to the electromagnetic field.
• 09.07.01 -- Continuum mechanics introduces the Lagrangian density formalism and the Euler-Lagrange equations for fields; QFT applies the same framework to quantum fields.
• 13.01.01 -- General relativity can be formulated as a classical field theory of the metric; attempts to quantize it (quantum gravity) follow the canonical or path-integral approach described here.

Bibliography [Master]

1. Peskin, M. E. and Schroeder, D. V. An Introduction to Quantum Field Theory, Westview Press, 1995. The standard graduate textbook, covering canonical quantization, path integrals, renormalization, and QED.

2. Weinberg, S. The Quantum Theory of Fields, Vol. 1: Foundations, Cambridge University Press, 1995. A deep, axiomatically motivated treatment emphasizing the S-matrix approach and the logical necessity of QFT.

3. Srednicki, M. Quantum Field Theory, Cambridge University Press, 2007. A modern pedagogical treatment with a logical ordering that derives QFT from the requirements of Lorentz invariance and cluster decomposition.

4. Zee, A. Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010. Emphasizes physical intuition and the path integral approach.

5. Schwartz, M. D. Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. A modern comprehensive treatment with emphasis on phenomenological applications.

6. Itzykson, C. and Zuber, J.-B. Quantum Field Theory, Dover, 2006. A comprehensive reference covering both perturbative and nonperturbative methods.