Start where field theory begins. Finish at the Lamb shift.

Topics and mapped lessons

• Natural units ℏ = c = 1, the metric (+, -, -, -), and converting to seconds and barns ↗

  With ℏ = c = 1 every quantity is a power of mass; ℏ c = 197.327 MeV fm, ℏ = 6.582 x 10⁻²⁵ GeV s, 1 barn = 100 fm² and (ℏ c)² = 0.3894 GeV² mb restore lengths, times and cross-sections, worked on Advanced QM's SI results: the charged pion's ℏ/(mc) = 1.41 fm, a 1.6 ns lifetime as a 0.41 micro-eV width, and Thomson's 8 pi α²/(3m²) as 0.665 barn. The metric (+, -, -, -) is stated once for the course, with α = e²/(4 π) = 1/137.036 in Heaviside-Lorentz units. A dimensionless action gives scalar and photon fields mass dimension 1 and a Dirac field 3/2, so the coefficient of an operator of dimension n has dimension 4 - n, derived, the counting Unit 2 uses.

• What Advanced QM supplies, quoted: free fields, microcausality and Coulomb-gauge photons ↗

  The course's one review topic, each result quoted in a line by code, restated in ℏ = c = 1 and never re-derived: Noether currents (UAQM-09.01) and the real, complex and Dirac modes with H, P and Q (UAQM-09.02, 09.03, 09.05, 09.07), for Units 1, 2 and 7; D(x - y) and the Pauli-Jordan commutator (UAQM-09.04), for topics 1.05 to 1.07; spin sums and traces (UAQM-04.08, 04.03), for Units 7, 9 and 10; free-field spin-statistics (UAQM-09.09), for Units 7 and 11; Coulomb-gauge photons (UAQM-11.03, 11.04), Unit 8's comparison; and Bethe's one mass subtraction (UAQM-13.04), for Units 6 and 15. What it left undone, time ordering, interactions and a covariant photon, is the course's agenda.

• Invariant normalisation and the Poincare group on Fock space ↗

  Letting UAQM-09.02's box grow without limit turns its modes into continuum operators with [a(p), a-dagger(q)] = (2 π)³ 2Eₚ δ³(p - q), and d³p/((2 π)³ 2Eₚ) is shown equal to d⁴p/(2 π)³ times δ(p² - m²) θ(p⁰), so ⟨p|q⟩ is invariant, derived. UAQM-09.01's theorem applied to Lorentz transformations gives M^{(μ ν)} beside P^μ, and these charges, written in modes, generate a unitary U(Λ, a) with U φ(x) U⁻¹ = φ(Λ x + a), carrying |p⟩ to |Λ p⟩ times a translation phase, the Lorentz algebra of UAQM-05.01 quoted and translations added, derived. Once interactions enter, P and J keep their free form while H and the boosts do not, stated for Unit 2.

• Wigner's little group: mass, spin and helicity label the one-particle states ↗

  P² and W², W^μ the Pauli-Lubanski vector, commute with every Poincare generator and so label irreducible representations, W² = −m² j(j + 1) for a massive particle, derived. Transforming a state boosted from a standard momentum leaves a Wigner rotation in that momentum's little group: SO(3) for m > 0, giving 2j + 1 states, half-integer j entering through the double cover, quoted; ISO(2) for m = 0, whose translation-like generators act trivially unless states carry a continuous label, set aside, leaving one helicity no boost or rotation changes, derived in outline. UAQM-11.04 quoted this; parity pairs helicities h and −h, the photon's two states, and topic 8.05 uses it again.

• The time-ordered product and the Feynman propagator D_F(x - y) ↗

  ⟨0|T φ(x) φ(y)|0⟩ = θ(x⁰ - y⁰) D(x - y) + θ(y⁰ - x⁰) D(y - x), built from UAQM-09.04's vacuum amplitude, is Lorentz invariant although the step functions are not, because a proper orthochronous transformation changes the time order of two points only at spacelike separation, where D(x - y) = D(y - x), derived. There D_F equals D: non-zero and falling as exp(−mr), so the Feynman propagator is an amplitude, not a signal. Time ordering is chosen because Unit 2's Dyson series orders every interaction in time, stated here and shown there.

• D_F as a four-momentum integral: i/(p² - m² + i ε) and its poles ↗

  Closing the p⁰ contour below for x⁰ > y⁰ and above otherwise turns the integral over d⁴p/(2 π)⁴ of i exp(−ip . (x - y))/(p² - m² + i ε) into topic 1.05's two terms, the poles at Eₚ - i epsilon and −Eₚ + i epsilon each giving one, derived: the i epsilon is the time ordering, equivalently a mass squared m² - i epsilon. (box + m²) D_F = −i δ⁴(x - y) makes D_F a Green's function, and the four placements of the two poles give the retarded, advanced, Feynman and anti-Feynman functions, shown. With the Feynman poles in the second and fourth quadrants, turning the p⁰ contour anticlockwise to the imaginary axis crosses neither, the step Unit 6's Wick rotation takes.

• The retarded propagator, the Pauli-Jordan function and where causality lives ↗

  D_R(x - y) = θ(x⁰ - y⁰) ⟨0|[φ(x), φ(y)]|0⟩, UAQM-09.04's Pauli-Jordan function cut at equal times, satisfies (box + m²) D_R = −i δ⁴(x - y), the δ coming from the step function and the equal-time commutator, derived; it vanishes outside the forward light cone, both its poles lying below the real p⁰ axis. D_F - D_R = D(y - x), a solution of the free equation, shown: the classical field a source radiates is i D_R acting on the source, while amplitudes are built from D_F. Causality lives in commutators, not in amplitudes.

• The complex field's propagator: particles forward and antiparticles backward in time ↗

  ⟨0|T φ(x) φ-dagger(y)|0⟩ describes a particle created at y and absorbed at x when x⁰ > y⁰ and an antiparticle created at x and absorbed at y otherwise, both of positive energy, both carrying charge q from y to x, and one function D_F for both, derived from UAQM-09.03's modes: the Stueckelberg-Feynman reading of negative frequencies, with no sea; ⟨0|T phi φ|0⟩ vanishes by charge conservation. In position space D_F depends only on the invariant interval, (m/(4 π² r)) K₁(mr) at spacelike interval r and 1/(4 π² r²) at equal times for m = 0, derived, its Hankel-function form inside the cone quoted.

• A classical source: Poisson-distributed quanta, vacuum persistence and the Yukawa energy ↗

  For a field driven by a c-number source j, D_R gives the late-time field and the vacuum evolves into a coherent state, extending UAQM-11.07 from one mode to all: quanta Poisson-distributed with mean the integral of |j~(p)|² d³p/((2 π)³ 2Eₚ) on shell, and ⟨0|0⟩ⱼ = exp(-(1/2) integral of j D_F j), whose modulus squared is exp(−mean number), derived. For two static point sources held a long time T the cross term is exp(−iET) with E = -(g²/(4 π r)) exp(−mr): scalar exchange attracts like sources with range 1/m, 1.41 fm for the pion, derived, each source's divergent self-energy dropped. The sources are classical and do not recoil; topic 5.06 finds the same potential in scattering.

Learning outcomes

• Convert any result between natural units and SI with ℏ c = 197.327 MeV fm, ℏ = 6.582 x 10⁻²⁵ GeV s and (ℏ c)² = 0.3894 GeV² mb, and give the mass dimension of a field or a coefficient in L from a dimensionless action.
• Show that d³p/((2 π)³ 2Eₚ) is Lorentz invariant, normalise one-particle states covariantly, and construct the unitary action of translations and Lorentz transformations on Fock space from the Noether charges.
• Use the little group to count the states of a massive particle of spin j and to explain why a massless particle carries one Lorentz-invariant helicity, saying which steps are derived and which are quoted.
• Derive the Feynman propagator as a time-ordered vacuum expectation, evaluate its four-momentum integral by closing the p⁰ contour, and show that the placement of the i epsilon is the time ordering.
• Distinguish the Feynman and retarded propagators by pole placement, support and use, showing that D_F - D_R solves the free equation and that causality is carried by the commutator, not the amplitude.
• Compute the coherent state, the mean number of quanta and the vacuum persistence amplitude for a scalar field driven by a classical source, check |⟨0|0⟩|² against the Poisson probability of no emission, and extract the Yukawa energy of two static sources.

Prerequisite thread

• Advanced QM's quantised free fields: the real and complex Klein-Gordon fields and the Dirac field expanded in box modes with normal-ordered H, P and Q, the vacuum amplitude D(x - y), and the Pauli-Jordan commutator that vanishes at spacelike separation
• Lagrangian field theory as Advanced QM Unit 9 set it up: the field Euler-Lagrange equation, conjugate momenta, Noether currents, the energy-momentum tensor and its conserved charges
• Lorentz transformations as Advanced QM Unit 5 treated them: the proper orthochronous group, the rotation and boost generators J and K with their commutators, and spinors that change sign under a rotation through 2 pi
• The Coulomb-gauge radiation field of Advanced QM Unit 11: photons with two helicities, Wigner's classification quoted for them, and the coherent state a classical current drives in one mode
• Special relativity in four-vector form, with invariant intervals, the light cone and four-momentum, and the SI values of ℏ, c and e with the electron and pion masses
• Complex analysis through residues and closing contours by Jordan's lemma, with Fourier transforms in four dimensions and the delta and step functions handled as distributions

Laboratory directions

Problem practice

Carrying results between natural units and SI to a measurable number, evaluating propagator integrals by contours with the i epsilon kept until the end, and checking every measure, normalisation and time-ordered product for Lorentz invariance, saying of each result whether it was derived here or quoted from Advanced QM by code

Capstone: For a free real scalar of mass m, derive D_F twice, from the time-ordered vacuum expectation and from the four-momentum integral with the contour closed each way, and show that the i epsilon is the time order. Build D_R from the Pauli-Jordan function, prove (box + m²) D_R = −i δ⁴(x - y) and show that D_F - D_R = D(y - x). Then switch on two static point sources a distance r apart for a long time T, derive their energy from the vacuum persistence amplitude, and evaluate it in MeV at r = 1 fm for the pion's mass and g²/(4 π) = 1; for a source pulse of duration tau find the mean number of quanta and check |⟨0|0⟩|² = exp(−N-bar). Finally say which result used D_F and which D_R, which inputs were quoted from Advanced QM and by which code, and why a D_F that is non-zero outside the light cone sends no signal.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• Model interactions: φ⁴, scalar Yukawa and Yukawa theory, and their coupling dimensions ↗

  Adding -(λ/4!) φ⁴, −g ψ-dagger psi phi with psi complex of mass M and phi real of mass m, or −g phi ψ-bar psi with psi a Dirac field gives the course's three models. By topic 1.01's counting lambda and the Dirac Yukawa g are dimensionless while the scalar Yukawa g is a mass, so its relative strength g/E fades at high energy, derived; φ → −φ forbids φ⁴ any change of particle number by an odd amount, and conserved psi number keeps psi and ψ-bar stable and lets phi decay, to psi ψ-bar first, only if m > 2M, derived. The scalar Yukawa potential is unbounded below, unseen order by order, so the model serves trees, decays and widths, never its vacuum, stated.

• The interacting theory: the Heisenberg field, the true vacuum and correlation functions ↗

  In φ⁴ the Heisenberg field obeys (box + m²) φ = -(λ/6) φ³ and keeps its equal-time commutator with π = dₜ φ, derived, but no expansion with time-independent a(p) solves it, and the ground state |Ω⟩ of the full H is not the free |0⟩: at first order Hᵢₙₜ mixes |0⟩ with states of two and four quanta of zero total momentum, shown. The n-point functions ⟨Ω|T φ(x₁) ... φ(xₙ)|Ω⟩ are defined as what perturbation theory computes. That m and the normalisation of phi need no longer be the measured mass and one-particle amplitude, and that particles and scattering must be recovered from these functions rather than assumed, is stated and left to Unit 4.

• Interaction-picture fields and U(t, t₀) as a time-ordered exponential ↗

  With H = H₀ + Hᵢₙₜ, the interaction-picture field exp(iH₀(t - t₀)) φ(t₀, x) exp(−iH₀(t - t₀)) is topic 1.02's free field, and the Heisenberg field is U-dagger(t, t₀) φ_I(x) U(t, t₀) with U = exp(iH₀(t - t₀)) exp(−iH(t - t₀)), derived. Quantum Mechanics II's Dyson series (UQMII-12.02), quoted, solves i dU/dt = H_I(t) U as U = T exp(−i times the integral of H_I from t₀ to t), H_I the spatial integral of the interaction density written in φ_I, with U(t₁, t₂) U(t₂, t₃) = U(t₁, t₃), shown. Without derivative couplings the canonical momentum is unchanged and Hᵢₙₜ = −Lᵢₙₜ, shown; derivative couplings are taken up in topic 2.04.

• The S-operator T exp(i integral of Lᵢₙₜ d⁴x) is Lorentz invariant ↗

  For couplings without derivatives S = U(+infinity, −infinity) is T exp(i integral of Lᵢₙₜ d⁴x), each order an integral of time-ordered products of a scalar density over all spacetime. Frames disagree on the order only of spacelike-separated vertices, where the densities, local in the free field, commute by UAQM-09.04's and 09.07's microcausality, quoted, so U(Λ) S U-dagger(Λ) = S, derived. With derivative couplings Hᵢₙₜ differs from −Lᵢₙₜ by non-covariant contact terms that the covariant T* product absorbs, stated; QED's interaction in Unit 9 has none. That S between free states gives physical amplitudes is flagged as an assumption, settled by Unit 4's LSZ formula.

• Projecting onto the interacting vacuum: T taken to infinity times (1 - i ε) ↗

  Expanding |0⟩ in eigenstates |n⟩ of the full H, exp(−iHT)|0⟩ is a sum of exp(−iEₙ T)|n⟩⟨n|0⟩, and taking T to infinity along T(1 - i ε) damps each term by exp(−ε Eₙ T), so the ground state outlasts the rest and |Ω⟩ = lim U(0, −T)|0⟩/(exp(−iE₀ T)⟨Ω|0⟩), derived. This assumes ⟨Ω|0⟩ non-zero, true in a box of volume V though falling exponentially with V, and a gap above E₀, both stated; where massless quanta close the gap, Gell-Mann and Low's adiabatic switching of the coupling is the named substitute. The same tilt in the time integrals of the expansion supplies topic 1.06's i epsilon in every propagator, shown.

• The Gell-Mann-Low formula for correlation functions in the interacting vacuum ↗

  Writing each Heisenberg field as U-dagger φ_I U, merging adjacent evolution operators and using topic 2.05 at both ends, the unknown phases and overlaps cancel against the same expression with no fields, giving ⟨Ω|T φ(x₁) ... φ(xₙ)|Ω⟩ = ⟨0|T φ_I(x₁) ... φ_I(xₙ) exp(−i integral of H_I dt)|0⟩/⟨0|T exp(−i integral of H_I dt)|0⟩, T taken to infinity times (1 - i ε), derived: the formula every diagram in the course evaluates. Haag's theorem, that in infinite volume no unitary operator joins an interacting field at one time to a free one, is stated, topic 2.05's finite box being the working answer; the path-integral route, UQMII-11.02 applied to fields, is named only.

• Normal ordering and the contraction: T φ(x) φ(y) = :φ(x) φ(y): + D_F(x - y) ↗

  Splitting the free field into φ^+, holding annihilators, and φ^-, holding creators, gives φ(x) φ(y) = :φ(x) φ(y): + [φ^+(x), φ^-(y)], the commutator being UAQM-09.04's c-number D(x - y), quoted; treating the other time order alike, T φ(x) φ(y) = :φ(x) φ(y): + D_F(x - y), derived. Normal-ordered products have zero vacuum expectation value, and the contraction of two fields is defined as D_F. For a complex field ⟨0|T phi φ|0⟩ vanishes while ⟨0|T phi φ-dagger|0⟩ is topic 1.08's propagator, so only phi with φ-dagger contracts, shown; the anticommuting case waits for topic 7.02.

• Wick's theorem for time-ordered products, proved by induction ↗

  T(φ₁ ... φₙ) equals :φ₁ ... φₙ: plus every way of replacing pairs of fields by contractions, the rest normal-ordered, proved by induction on n: with x₁ the latest time, φ₁^- already stands on the left and moving φ₁^+ right through the (n - 1)-field result yields exactly its contractions with the other fields. In the vacuum only full contractions survive: none for an odd number of fields, (2n - 1)!! for 2n of them, derived. UAQM-10.05's four-operator rule, quoted, is the same sum over pairings with the sign anticommuting fields add, recovered in topic 7.02; the generating form, whose vacuum value is topic 1.09's exp(-(1/2) integral of j D_F j), is stated.

• Wick's theorem at work: the free four-point function and a first decay amplitude ↗

  ⟨0|T φ₁ φ₂ φ₃ φ₄|0⟩ = D_F(x₁ - x₂) D_F(x₃ - x₄) + D_F(x₁ - x₃) D_F(x₂ - x₄) + D_F(x₁ - x₄) D_F(x₂ - x₃), derived. Contracting fields with external states, φ_I^+(x)|k⟩ = exp(−ik . x)|0⟩, the first-order scalar Yukawa element is ⟨p, q|iT|k⟩ = −ig (2 π)⁴ δ⁴(k - p - q), so M = −g for φ → psi ψ-bar, of mass dimension 1 as topic 2.01 requires, derived, its use as a physical amplitude assumed until Unit 4. Normal ordering Hᵢₙₜ forbids contractions within a vertex; in φ⁴ the dropped terms shift m² by (λ/2) D_F(0) and add a constant, shown, so the course leaves Hᵢₙₜ unordered and lets Unit 6's counterterms do that job.

Learning outcomes

• Determine the mass dimensions of the couplings of φ⁴, the scalar Yukawa model and Yukawa theory with Dirac fermions, and state each model's symmetry and the selection rules it imposes on processes.
• Construct interaction-picture fields and the evolution operator U(t, t₀) as a time-ordered exponential, and show that the S-operator T exp(i integral of Lᵢₙₜ d⁴x) is Lorentz invariant, identifying the step that needs microcausality.
• Derive the Gell-Mann-Low formula for ⟨Ω|T φ(x₁) ... φ(xₙ)|Ω⟩, listing where the overlap ⟨Ω|0⟩, the gap above the vacuum and the tilt T(1 - i ε) are used, and state what Haag's theorem says about the derivation.
• Prove Wick's theorem for time-ordered products of free scalar fields by induction on the number of fields, and count the (2n - 1)!! full contractions of 2n fields.
• Evaluate vacuum expectation values and first-order matrix elements by Wick contraction, obtaining the free four-point function and M = −g for φ → psi ψ-bar, and show what normal ordering Hᵢₙₜ removes in φ⁴.

Prerequisite thread

• Unit 1's toolkit: natural units with the mass dimensions of scalar and Dirac fields, invariant normalisation of one-particle states, and the Feynman propagator D_F(x - y) as a time-ordered vacuum expectation with its four-momentum form i/(p² - m² + i ε) and its complex-field version
• The free real, complex and Dirac fields as Advanced QM quantised them (UAQM-09.02, 09.03, 09.07): mode expansions, normal ordering, the vacuum amplitude D(x - y) and microcausality, the vanishing of the commutator at spacelike separation (UAQM-09.04)
• Lagrangian field theory from UAQM-09.01: the field Euler-Lagrange equation, canonical momenta, the Legendre transform to a Hamiltonian density, and Noether charges from continuous symmetries
• Quantum Mechanics II's interaction picture and Dyson series (UQMII-12.02), with first-order transition amplitudes (UQMII-12.03)
• Second quantisation from Advanced QM Unit 10: ladder operators on Fock space, normal-ordered one- and two-body operators, and the four-operator contraction rule of UAQM-10.05
• Complex analysis to contour integrals and the damping of exp(−iEt) along a ray in the complex t plane, and the combinatorics of pairings with proof by induction

Laboratory directions

Problem practice

Carrying a correlation function from its Heisenberg-picture definition through the interaction picture to sums of contractions of free fields, saying at each step which assumption is used, the overlap with the free vacuum, the gap, the tilted time contour or microcausality, and counting contractions exactly before Unit 3 turns them into diagrams

Capstone: For φ⁴ theory, start from the Heisenberg-picture definition of ⟨Ω|T φ(x₁) φ(x₂) φ(x₃) φ(x₄)|Ω⟩ and derive the Gell-Mann-Low formula, keeping a ledger of where the overlap ⟨Ω|0⟩, the gap, the tilt T(1 - i ε) and microcausality enter, and say which step fails for a massless field and what replaces it. Expand numerator and denominator to first order in λ, reduce each to free contractions by Wick's theorem, and count the full contractions of the numerator's eight fields as 7!! = 105 = 24 + 72 + 9 by how many external points join the vertex, leaving their reading as diagrams, symmetry factors and the fate of the vacuum terms to Unit 3. Then, in the scalar Yukawa model, obtain M = −g for φ → psi ψ-bar by contracting with external states, check its mass dimension, show that ψ → psi φ has the same vertex but a δ function with no support, and say what normal ordering Hᵢₙₜ removes in each model.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• Position-space Feynman rules for φ⁴: D_F per line and −i lambda per vertex ↗

  Expanding topic 2.06's numerator to first order in -(λ/4!) φ⁴ and contracting by topic 2.08 gives the numerator of the two-point function as D_F(x - y), plus the tadpole -(i λ/2) times the integral over z of D_F(x - z) D_F(0) D_F(z - y), plus D_F(x - y) times a figure-eight bubble that topic 3.03 cancels against the denominator, derived. Drawing the terms reads back the rules: D_F per line, −i lambda and an integral over all spacetime per vertex, the 1/n! of the exponential cancelled by relabelling vertices. The interaction is not normal ordered, so tadpoles stay, topic 2.09's mass shift being the alternative; D_F(0) diverges and waits for topic 6.05.

• Symmetry factors: where the 1/4! fails to cancel ↗

  Relabelling the n vertices and the four fields at each permutes contractions, n!(4!)ⁿ ways, and the contractions giving one diagram form a single orbit whose stabiliser is the diagram's automorphism group, so the diagram comes with 1/S, S the number of interchanges of lines and vertices that leave it unchanged, derived. S is computed for the tadpole (2), the figure-eight bubble (8), the one-loop four-point bubble (2) and the two-loop sunset (6), each checked by counting contractions directly; a tree has S = 1 if its external points are labelled, shown, but trees with source legs, which topic 3.09 iterates, keep the interchanges of identical legs and branches, S = 6 for three on one vertex.

• Vacuum bubbles exponentiate and cancel against the denominator ↗

  Every diagram factors into a piece joined to the external points times disconnected vacuum bubbles, and with nᵢ copies of connected bubble Vᵢ weighted 1/nᵢ! for their interchange the sum over all sets is exp of the sum of the Vᵢ, derived combinatorially; the Gell-Mann-Low denominator is the same exponential, so correlation functions are sums of bubble-free diagrams. Each connected bubble carries (2 π)⁴ δ⁴(0), the volume of spacetime, so the exponent is −i times the elapsed time times topic 2.05's E₀, itself proportional to the volume of space, shown: a constant energy density, divergent, unobservable except through gravity, and dropped.

• Connected diagrams and the four-point function at order lambda ↗

  Through order lambda the four-point function is topic 2.09's three products of propagators, the same products with a tadpole on one line, and one connected term, −i lambda times the integral over z of the four D_F from z to the external points, symmetry factor 1, derived. Subtracting every product of lower functions defines the connected four-point function, at this order that vertex alone, shown: a local interaction integrated over all spacetime, each D_F covering both time orders of its ends. That only connected diagrams with every external point attached reach scattering is stated here and shown in Unit 4.

• Momentum-space Feynman rules: conservation at every vertex and one integral per loop ↗

  Inserting topic 1.06's Fourier integral for each D_F and integrating each vertex position gives (2 π)⁴ δ⁴ of the momenta meeting there, so the rules become i/(p² - m² + i ε) per line, −i lambda per vertex, an integral over d⁴l/(2 π)⁴ for each momentum conservation leaves free and one overall delta function, derived. The free momenta number L = I - V + 1, and with 4V = 2I + E a connected φ⁴ diagram has V = L + E/2 - 1, so each loop costs one more λ, derived. The tadpole's loop is independent of the external momentum, shown; loop integrals are evaluated in Units 5 and 6.

• Feynman rules for the scalar Yukawa model: arrows for psi number ↗

  Contracting −g ψ-dagger psi φ, where only psi with ψ-dagger and phi with phi contract, gives the rules, derived: a vertex −ig with no 1/3!, since its three fields differ, joining an arrow in, an arrow out and a phi line, i/(p² - M² + i ε) along each arrow from topic 1.08's ⟨0|T psi ψ-dagger|0⟩, and an undirected i/(p² - m² + i ε). Arrows never end inside a diagram, so psi number is conserved and an arrow against the momentum flow is an antiparticle; a closed psi loop carries no 1/2, its arrows telling apart lines a real field would interchange, shown. External states wait for topic 4.05, the Dirac case for 7.04.

• Reading Feynman rules off a Lagrangian, with derivative couplings and two-point vertices ↗

  A vertex factor is i times its term's coefficient in L times the number of ways of attaching identical lines, recovering −i lambda and −ig, derived in general from topics 3.01 and 3.06. A derivative on a line of incoming momentum p brings −i p_μ, a rule that needs topic 2.04's covariant T* product, its agreement with the Hamiltonian route, Matthews' theorem, quoted. A quadratic term is a two-point vertex: (1/2) δ_Z (d φ)² - (1/2) δₘ φ² gives i(p² δ_Z - δₘ), derived, the form Unit 6's counterterms take, and Unit 9's QED vertex is read off the same way, with no new combinatorics.

• One-particle-irreducible diagrams, the self-energy and the full propagator ↗

  Diagrams that stay connected when any one internal line is cut are one-particle irreducible; their two-point sum, external propagators removed, defines −i Σ(p²), and the geometric series of insertions gives i/(p² - m₀² - Σ(p²) + i ε), derived, with Sigma at order lambda the momentum-independent tadpole. A constant insertion resums to the free propagator with a shifted mass, a check, so the physical mass is the zero of p² - m₀² - Σ(p²), not the Lagrangian's m₀. The residue there is topic 4.03's Z, and the imaginary part Sigma acquires above threshold is left to topic 5.09.

• Tree diagrams solve the classical field equation; loops count powers of ℏ ↗

  Iterating the classical equation (box + m²) φ = J - (λ/6) φ³ in powers of the source generates exactly the tree diagrams, symmetry factors included, derived: with D_F it is the tree part of ⟨0|φ(x)|0⟩_J/⟨0|0⟩_J, topic 1.09's set-up with φ⁴ switched on, with D_R the causal classical field, the two differing only in boundary conditions. Restoring ℏ, D_F carries ℏ from the canonical commutator and each vertex 1/ℏ from the Dyson exponent, so a diagram's internal lines and vertices give ℏ^{(I - V)} = ℏ^{(L - 1)}, derived: loops are quantum corrections. UQMII-11.03's stationary-phase reading is named only.

Learning outcomes

• Expand a correlation function of φ⁴ or the scalar Yukawa model to a stated order with Wick's theorem, draw every term as a position-space diagram, and translate any diagram back into its integral with the correct weight.
• Compute the symmetry factor of a diagram both by counting the contractions that produce it and by counting its automorphisms, and show that the two agree for the tadpole, the figure-eight, the one-loop bubble and the sunset.
• Show that vacuum bubbles exponentiate and cancel against the Gell-Mann-Low denominator, and explain why each connected bubble is proportional to the volume of spacetime.
• Write momentum-space Feynman rules for a theory read off its Lagrangian, derivative couplings and two-point counterterm vertices included, and count the loops and powers of the coupling in any diagram.
• Define one-particle-irreducible diagrams and the self-energy, resum them into the full propagator, and locate the physical mass at its pole rather than at the Lagrangian's mass parameter.
• Show that tree diagrams reproduce the iterated solution of the classical field equation with a source, and that restoring ℏ gives the internal lines and vertices of a connected L-loop diagram a factor ℏ^{(L - 1)}.

Prerequisite thread

• Unit 1's Feynman propagator: D_F(x - y) as the time-ordered vacuum expectation of two free fields, its four-momentum integral i/(p² - m² + i ε), and the complex field's ⟨0|T φ(x) φ-dagger(y)|0⟩ with particles forward and antiparticles backward in time
• Unit 2's Gell-Mann-Low formula for correlation functions in the interacting vacuum, as a ratio of free-vacuum expectations of time-ordered products containing exp(−i times the integral of H_I), with the assumptions behind it
• Wick's theorem for time-ordered products of free fields as topic 2.08 proved it: normal-ordered products plus all contractions, only full contractions surviving in the vacuum, and (2n - 1)!! of them for 2n fields
• The course's model interactions, -(λ/4!) φ⁴ and the scalar Yukawa coupling −g ψ-dagger psi φ, with the mass dimensions of their couplings and topic 2.09's first-order decay amplitude
• Lagrangian field theory as UAQM-09.01 gave it: the Euler-Lagrange field equation from a density, here with a nonlinear term and an external source, and the Green's functions of box + m² from topics 1.06 and 1.07
• Fourier analysis in four dimensions, with integrals of plane waves giving delta functions and (2 π)⁴ δ⁴(0) read as a spacetime volume, and the combinatorics of permutations, orbits and exponential series

Laboratory directions

Problem practice

Expanding correlation functions of φ⁴ and the scalar Yukawa model with Wick's theorem to a stated order, drawing and weighting every diagram, checking each symmetry factor by direct count, and moving between contractions, position-space diagrams, momentum-space integrals and the Lagrangian, saying for each rule where it was derived

Capstone: Expand the φ⁴ two-point function to order λ² with Wick's theorem. Sort every term of the Gell-Mann-Low numerator into connected diagrams and diagrams carrying vacuum bubbles, and show that the bubbles cancel against the denominator at this order. For each connected diagram, two tadpoles in series, a tadpole on a tadpole and the sunset, find the symmetry factor twice, by counting contractions and by counting automorphisms, write its momentum-space integral with its loops counted, and say whether it is one-particle irreducible. Then show that resumming the order-λ self-energy already contains the first diagram, say which of the three can depend on the external momentum and so could change the residue at the pole as well as its position, give each diagram's power of ℏ, and say which rule used in the calculation was derived in this unit and which was taken from Units 1 and 2.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge