Quantum field theory treats fields—not fixed particle coordinates—as the basic dynamical objects. Particles then appear as quantised excitations of field modes, and Fock space allows the particle number to change while the theory remains linear at the level of states.
Guide 37 quantised one harmonic oscillator. Quantum field theory repeats that construction for infinitely many coupled spatial modes. A free scalar field decomposes into independent momentum oscillators. Each mode has creation and annihilation operators. Their occupation numbers build Fock space. Interactions then couple those modes and make particle creation, annihilation and scattering possible.
This guide develops canonical quantisation of a real scalar field, occupation-number states, field commutators, propagators, time ordering, Dyson expansion and Wick’s theorem. It uses natural units ℏ=c=1 and metric signature (+,−,−,−). Different field-normalisation conventions move factors of 2E and (2π)³ between operators and measures; consistency matters more than memorising one convention.
Classical field → canonical variables → oscillator modes → creation/annihilation operators → Fock space → propagators → Wick contractions → perturbative scattering.
1. From particle coordinate to field
Ordinary quantum mechanics may describe one particle by wavefunction ψ(x,t). A relativistic field theory instead starts with a classical field φ(x,t) defined at every spatial point.
For a real scalar field, a standard Lagrangian density is
ℒ=(1/2)∂_μφ∂^μφ−(1/2)m²φ².
The action is
S=∫d⁴x ℒ.
Applying the Euler–Lagrange equation gives the Klein–Gordon equation
(□+m²)φ=0.
2. Canonical momentum field
The field’s canonical momentum is
π(x,t)=∂ℒ/∂(∂_tφ)=∂_tφ.
The Hamiltonian density is
ℋ=(1/2)π²+(1/2)(∇φ)²+(1/2)m²φ².
Thus
H=∫d³x ℋ.
The field theory is an infinite-dimensional Hamiltonian system whose canonical coordinates are φ(x) and π(x).
3. Canonical quantisation
Promote the classical fields to operators and impose equal-time commutation relations
[φ(t,x),π(t,y)]=iδ³(x−y)
with
[φ(t,x),φ(t,y)]=0;[π(t,x),π(t,y)]=0.
The Dirac delta replaces the discrete Kronecker delta because there is one canonical degree of freedom per spatial point before mode decomposition.
4. Fourier modes are harmonic oscillators
Fourier transform the free field into momentum modes. Each momentum p has relativistic frequency
E_p=√(p²+m²).
The Hamiltonian becomes a continuum sum of independent harmonic oscillators labelled by p.
This is the bridge from Guide 37: quantising a free field means quantising every momentum oscillator at once.
5. Mode expansion
A convenient real-scalar expansion is
φ(x)=∫ d³p/(2π)³ · 1/√(2E_p) [a_p e^{-ip·x}+a_p† e^{ip·x}]
where p⁰=Ep.
The oscillator commutator is
[a_p,a_q†]=(2π)³δ³(p−q).
All [a,a] and [a†,a†] commutators vanish.
6. Vacuum and one-particle states
The vacuum |0⟩ satisfies
a_p|0⟩=0
for every momentum p.
A one-particle momentum state is created by
|p⟩∝a_p†|0⟩.
Under the present operator normalisation, the precise delta-function normalisation of |p⟩ depends on whether one inserts a √(2Ep) factor into the state definition. Relativistic texts choose conventions for convenience; physical amplitudes must be treated consistently.
7. Fock space
Fock space is the direct sum
ℱ=ℂ|0⟩ ⊕ H_1 ⊕ H_2 ⊕ H_3 ⊕ …
where Hn is the n-particle sector with bosonic symmetrisation for a scalar boson.
An interaction can move amplitude among different sectors. Particle number is therefore not a fixed dimension of the state space.
“Second quantisation” is historical language for this occupation-number/field-operator formulation. It does not mean quantum mechanics itself is quantised a second time in some universal hierarchy.
8. Discrete-box version
To avoid delta-function normalisation temporarily, place the field in a finite periodic box of volume V.
Momentum becomes discrete and
[a_k,a_l†]=δ_kl.
A multimode occupation state is
|n_1,n_2,…⟩=∏_k (a_k†)^{n_k}/√(n_k!) |0⟩.
The continuum theory is recovered as V→∞ while sums become integrals.
9. Hamiltonian and vacuum energy
For the free scalar field, the Hamiltonian has oscillator form
H=∫d³p/(2π)³ E_p[a_p†a_p + formal vacuum term].
Every momentum oscillator contributes a zero-point energy Ep/2, producing a formally divergent total vacuum energy in infinite volume.
In nongravitational scattering calculations one often normal orders H and subtracts the common vacuum constant. In gravity, absolute energy density itself gravitates, so vacuum energy becomes a deeper physical issue rather than a harmless offset.
10. Normal ordering
Normal ordering, written :O:, places all creation operators to the left of annihilation operators for bosons.
For example,
:aa†:=a†a.
Using [a,a†]=1,
aa†=:aa†:+1.
That extra commutator is the simplest prototype of a contraction in Wick’s theorem.
11. Complex scalar fields and antiparticles
A complex scalar field needs independent particle and antiparticle operators:
φ(x)∼a_p e^{-ipx}+b_p†e^{ipx}.
a† creates particles; b† creates antiparticles carrying opposite conserved U(1) charge.
Relativistic field theory naturally accommodates antiparticles because positive- and negative-frequency solutions are reorganised as annihilation and creation operators acting on a stable vacuum.
12. Fermionic fields
Fermionic field modes use anticommutation relations rather than commutators:
{b_p,b_q†}∝δ³(p−q).
Because (b†)²=0 for one identical mode, Fermi occupation is 0 or 1.
Spin-statistics connects integer-spin relativistic fields to bosonic commutators and half-integer-spin fields to fermionic anticommutators under the theorem’s locality and relativistic assumptions.
13. Heisenberg field creates and annihilates excitations
Because φ(x) contains both a and a† terms, acting with the field operator on a state can lower or raise occupation.
This is fundamentally different from a fixed-particle coordinate operator. The field is an operator-valued distribution whose action changes the particle-content decomposition of the state.
Strictly speaking, φ(x) at one exact spacetime point is too singular to be an ordinary bounded operator. Mathematically controlled statements smear the field against smooth test functions.
14. Microcausality
For a relativistic bosonic scalar field, locality requires
[φ(x),φ(y)]=0
when x−y is spacelike.
This does not mean every correlation function vanishes outside the light cone. Vacuum correlations can be nonzero. The vanishing commutator means operations at spacelike separation cannot be used to create causal influence through that local observable algebra.
15. Time ordering
The time-ordering operator T places later-time bosonic operators to the left:
T{φ(x)φ(y)}=θ(x⁰−y⁰)φ(x)φ(y)+θ(y⁰−x⁰)φ(y)φ(x).
For fermionic operators, exchanging order also introduces minus signs.
Time ordering is central because perturbative evolution in the interaction picture is built from products of interaction Hamiltonians at different times.
16. Feynman propagator
The free scalar Feynman propagator is the vacuum time-ordered two-point function
Δ_F(x−y)=⟨0|T{φ(x)φ(y)}|0⟩.
In the convention used here,
Δ_F(x)=∫d⁴p/(2π)⁴ · i e^{-ip·x}/(p²−m²+iε).
The iε prescription specifies how poles are displaced and therefore how the time-ordering boundary condition is implemented.
17. Propagator as Green function
Apply the Klein–Gordon operator to the momentum representation:
(□+m²)e^{-ipx}=−(p²−m²)e^{-ipx}.
Therefore, distributionally,
(□+m²)Δ_F(x)=−iδ⁴(x)
under the stated convention.
The propagator is thus an inverse of the differential operator with a specific causal/time-ordering prescription.
18. Propagator is not a literal classical particle path
A Feynman propagator is a two-point Green function/correlation amplitude.
Internal lines in Feynman diagrams represent factors of propagators inside perturbative integrals. They should not be interpreted naively as directly observable particles following hidden spacetime trajectories between vertices.
“Virtual particle” language can be useful bookkeeping intuition, but the invariant object is the perturbative amplitude built from field correlation functions.
19. Interaction picture and Dyson series
Split H=H₀+HI, with H₀ exactly solvable.
The interaction-picture evolution operator is
U_I(t,t₀)=T exp[-i∫_{t₀}^{t}dt' H_I(t')].
Expanding the exponential gives the Dyson series:
U_I=1−i∫H_I +(−i)²/2! ∫∫T{H_IH_I}+…
Perturbation theory is an ordered expansion in powers of the interaction coupling.
20. Wick’s theorem
Wick’s theorem rewrites a time-ordered product of free fields as a normal-ordered product plus all possible contractions.
For a free scalar field, a contraction between φ(x) and φ(y) is the Feynman propagator:
contraction(φ(x)φ(y))=Δ_F(x−y).
The theorem converts operator algebra into combinatorics of pairings.
21. Worked four-field Wick expansion
For the free vacuum expectation of four fields, all normal-ordered terms vanish because annihilation operators eventually hit |0⟩.
Therefore
⟨0|Tφ₁φ₂φ₃φ₄|0⟩=Δ₁₂Δ₃₄+Δ₁₃Δ₂₄+Δ₁₄Δ₂₃.
There are three perfect pairings of four labelled objects.
This pairing combinatorics is the simplest origin of diagrammatic lines in free Gaussian field theory.
22. Number of pairings
The number of complete pairings of 2n labelled fields is
(2n−1)!!=(2n)!/(2^n n!).
For 2n=6, the number is
5!!=15.
Gaussian free theories are manageable because every higher correlation function reduces to sums of two-point functions through this pairing rule.
23. λφ⁴ interaction
A simple interacting scalar model has
ℒ=ℒ_free−λφ⁴/4!.
Each interaction insertion supplies four fields at one spacetime point.
Wick’s theorem contracts those fields with external operators or with one another. The combinatorial 4! in the Lagrangian is chosen so the basic four-leg vertex receives the simple perturbative rule −iλ under standard conventions.
24. Tree versus loop diagrams
A tree diagram has no closed internal momentum loop. Its momenta are largely fixed by external momentum conservation.
A loop diagram contains unconstrained internal momentum that must be integrated over.
Loop integrals frequently diverge at large momentum and require regularisation and renormalisation. The physical content lies in finite predictions after parameters are defined through a renormalisation prescription.
25. Momentum conservation at vertices
Translation invariance makes the spacetime integral over an interaction vertex produce a momentum-conserving delta function:
(2π)⁴δ⁴(Σp_in−Σp_out).
This is Noether’s theorem visible inside perturbative diagrammatics.
If the background is not translation invariant, exact momentum conservation can be modified even though other symmetries remain.
26. Correlation functions are central objects
The n-point Green functions
G_n(x₁,…,x_n)=⟨0|T{φ(x₁)…φ(x_n)}|0⟩
contain the information from which perturbative scattering amplitudes can be extracted using reduction formulas such as LSZ under suitable asymptotic-particle assumptions.
QFT is therefore naturally a theory of operator correlation functions rather than merely a theory of pictorial Feynman diagrams.
27. Path integral viewpoint
The same correlation functions can be generated from a functional integral
Z[J]=∫𝒟φ exp{i[S[φ]+∫Jφ]}.
Functional derivatives with respect to source J generate time-ordered n-point functions.
Canonical quantisation and path integrals are complementary formulations. Path integrals make symmetries and perturbative diagrams especially transparent, while canonical language makes Hilbert space and operators explicit.
28. Effective field theory
A quantum field theory used at energy scale E need not remain fundamental to arbitrarily high energies.
Effective field theory includes every interaction compatible with the relevant symmetries, organised by powers of E/Λ where Λ is a higher-energy cutoff or new-physics scale.
Higher-dimension operators are suppressed at low energy but become important as E approaches Λ.
This turns “nonrenormalisable” interactions into controlled approximations rather than automatic failures when a finite validity range is stated.
29. Common misconception: fields are only convenient wavefunctions for a fixed number of particles
Quantum fields act on Fock space and can change occupation number. They are operator-valued distributions whose excitations are interpreted as particles in regimes where a particle description is appropriate.
30. Common misconception: a Feynman diagram is a literal microscopic movie
A diagram is a term in a perturbative expansion. Internal lines encode propagators and momentum integrals; vertices encode interaction factors. Observable predictions arise after summing the relevant amplitudes and squaring/averaging according to the process.
31. Common misconception: normal ordering solves every vacuum divergence
Normal ordering removes selected free-field vacuum constants in specific operator expressions. Interacting QFT still contains ultraviolet divergences requiring regularisation and renormalisation, and gravity makes absolute vacuum energy physically consequential.
32. Worked synthesis problem
A free scalar theory contains six field operators inside a vacuum time-ordered expectation value.
Step 1: Wick structure. A nonzero Gaussian vacuum expectation requires complete pairing.
Step 2: Pairing count. The number is (6−1)!!=5·3·1=15.
Step 3: Each term. Every pairing contributes a product of three Feynman propagators ΔF(xi−xj).
Step 4: Interacting theory. If these fields arise from Dyson-expanded interaction vertices, spacetime integrals and coupling constants multiply the contractions.
Step 5: Diagram meaning. The 15 pairings can be represented graphically, but the mathematics is the Wick expansion of operator correlation functions.
33. Practice set
- Write the free real-scalar Lagrangian density.
- What is the canonical momentum field?
- State the equal-time canonical commutator.
- What is E_p for a relativistic scalar particle?
- Why does a free field reduce to harmonic oscillators?
- What space contains variable particle-number sectors?
- Define normal ordering conceptually.
- What is microcausality?
- Define the Feynman propagator.
- What does the iε prescription control?
- State Wick’s theorem conceptually.
- How many complete pairings exist for 2n fields?
Answers
(1/2)∂_μφ∂^μφ−(1/2)m²φ².π=∂_tφ.[φ(t,x),π(t,y)]=iδ³(x−y).√(p²+m²).- Fourier transformation diagonalises the quadratic free Hamiltonian into independent momentum modes.
- Fock space.
- Move creation operators left of annihilation operators, with the appropriate fermionic signs in fermion theories.
- Local bosonic fields commute at spacelike separation.
- The vacuum time-ordered two-point function
⟨0|Tφ(x)φ(y)|0⟩. - How poles are bypassed and therefore the time-ordering/causal boundary condition of the Green function.
- Rewrite time-ordered products as normal-ordered products plus every possible contraction.
(2n−1)!!.
Sources and further study
[1] David Tong, Lectures on Quantum Field Theory. Graduate-level notes covering canonical scalar-field quantisation, propagators, Dyson expansion, Wick’s theorem and Feynman diagrams.
[2] Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley (1995). A standard perturbative field-theory reference.
[3] Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press (1995). A systematic relativistic and symmetry-based treatment of quantum fields.
[4] Rudolf Haag, Local Quantum Physics, Springer. A mathematically deeper account of locality and operator-algebraic quantum field theory.
Continue through Quantum Mathematics
Guide 49: Berry Phases, Berry Curvature, Chern Numbers and Quantum Geometry develops geometric state bundles. Guide 50: Topological Bands, Quantum Hall Mathematics, Edge States and Bulk–Boundary Correspondence applies those bundles to lattice bands. Guide 52: Lindblad Master Equations, Liouvillian Spectra, Steady States and Dissipative Quantum Dynamics replaces closed unitary evolution with Markovian open-system semigroups.