University Quantum Field Theory I · Natural Units, Relativistic States and the Feynman Propagator · 1.1

Natural units, the metric, and back to seconds and barns

Every amplitude in this course is written in powers of GeV, yet experiments report seconds, femtometres and barns. Here the conventions are fixed once, ħ = c = 1, the metric (+, −, −, −) and α = e²/(4π), and two habits are built that every later unit leans on: restoring ħ and c to reach a measured number, and counting mass dimensions.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

Relativity measures time and length in one currency with c as the exchange rate; quantum mechanics does the same for energy and frequency, and momentum and wavenumber, with ħ. Setting ħ = c = 1 leaves one of the three mechanical units, so every quantity becomes a power of mass: energies, momenta and masses in GeV, lengths and times in GeV⁻¹, cross-sections in GeV⁻². Nothing is lost.

Because ħ carries J s and c carries m s⁻¹, a result of mass dimension D returns to metres, seconds or barns through exactly one product ħ^a c^b, and ħc = 197.327 MeV fm and ħ = 6.582 × 10⁻²⁵ GeV s follow exactly from the SI definitions. The same bookkeeping applied to the action, which enters quantum theory only as S/ħ and so must be a pure number, gives every field a mass dimension and every coupling the dimension 4 − n left over by the operator it multiplies. The scheme stops in three places.

Dimensions cannot supply pure numbers: the 8π/3 in Thomson's cross-section and the 192π³ in muon decay come only from calculation, and whether a 4π sits in α or in Maxwell's equations is a convention to be stated. Limits need the constants back: the non-relativistic limit is c → ∞ and the classical limit ħ → 0, and neither can be taken on a formula from which c and ħ have been erased. And the counting is the free-field count; Unit 6 shows loops adding logarithms of a scale on top of it.

Simple definition: Natural units set ħ = c = 1, so energy, momentum and mass share the unit GeV while lengths and times are measured in GeV⁻¹; ħc = 197.327 MeV fm and ħ = 6.582 × 10⁻²⁵ GeV s convert back to SI.
Example: The proton's mass, 938.272 MeV, is at once an inverse length, ħc/(mₚc²) = 0.2103 fm, an inverse time, ħ/(mₚc²) = 7.015 × 10⁻²⁵ s, and, squared and inverted, an area of 0.4423 mb.

Length from an inverse energyL = ħc/E, with ħc = 197.327 MeV fm

Derived. The charged pion's 139.570 MeV becomes the length ħ/(m_π c) = 1.414 fm, the reach of the force its exchange carries between nucleons.

E in MeV (1 MeV = 1.602 × 10⁻¹³ J); L in fm (10⁻¹⁵ m); ħc is exact, fixed by the SI-defined h, c and e

Lifetime from a widthτ = ħ/Γ, with ħ = 6.582 × 10⁻²⁵ GeV s

Derived. Hydrogen 2p's 1.596 ns lifetime is a width of 0.412 μeV, which a spectroscopist quotes as Γ/h = 99.7 MHz.

Γ, the decay width, in GeV; τ, the mean lifetime, in s; a line width in Hz is Γ/h = Γ/(2πħ), not Γ/ħ

Cross-section from an inverse mass squaredσ[mb] = 0.3894 × σ[GeV⁻²], from (ħc)² = 0.3894 GeV² mb

Derived. Thomson's 8πα²/(3mₑ²) = 1708 GeV⁻² becomes 0.6652 b, recovering Advanced QM's (8π/3)rₑ² from UAQM-12.09.

σ is an area; 1 b = 10⁻²⁸ m² = 100 fm², so 1 fm² = 10 mb and 1 GeV⁻² = 0.3894 mb

The course metric (+, −, −, −)a⋅b = g_μν a^μ b^ν = a⁰b⁰ − a¹b¹ − a²b² − a³b³

A convention, stated once. On shell p² = m² is positive; x² is positive for timelike and negative for spacelike separation; □ = ∂ₜ² − ∇².

g_μν = diag(1, −1, −1, −1); p^μ = (E, p) in GeV; x^μ = (t, x) in GeV⁻¹, restoring to (ct, x) in m

Fine-structure constant, Heaviside-Lorentzα = e²/(4π) = 1/137.036, so e = √(4πα) = 0.3028

α is quoted, a measurement; e follows from it. Gaussian texts write α = e², so their e is √(4π) smaller. Hydrogen's −α²mₑ/2 restores to −13.606 eV.

SI form α = e²/(4πε₀ħc) with e = 1.602 × 10⁻¹⁹ C; here ε₀ = μ₀ = 1 and e is a pure number

Coupling dimension from a dimensionless action[g] = 4 − n for a term g O in ℒ with [O] = n

Derived. eψ̄γ^μψA_μ has n = 4, so e is a pure number, as α = e²/(4π) demands; Fermi's (ψ̄ψ)(ψ̄ψ) has n = 6, so G_F = 1.166 × 10⁻⁵ GeV⁻².

[ℒ] = 4 since [S] = 0 and [d⁴x] = −4; [φ] = [A_μ] = [∂_μ] = 1 and [ψ] = 3/2 build n

Two exchange rates set to one, and one unit left

Each constant converts one kind of quantity into another that physics already treats as the same thing: c turns a time into a length, x⁰ = ct, and ħ turns a frequency into an energy, E = ħω, and a wavenumber into a momentum, p = ħk. Put both equal to 1 and seconds, metres and joules collapse into a single unit, here the GeV. A quantity's mass dimension is the power of GeV it then carries: [E] = [p] = [m] = 1; [x] = [t] = −1; a cross-section has [σ] = −2; a decay width [Γ] = 1; a velocity and the action S have dimension 0. The two exchange rates are exact consequences of the 2019 SI, which fixes h, c and e: ħc = (1.054571817 × 10⁻³⁴ J s)(299 792 458 m s⁻¹) ÷ (1.602176634 × 10⁻¹³ J MeV⁻¹) = 197.327 MeV fm, and ħ = 6.582 × 10⁻²⁵ GeV s. They are derived, carry no uncertainty, and with the barn, 10⁻²⁸ m², they are all the conversion data this course needs. Throughout, ħ = c = 1 and the metric is (+, −, −, −).

Back to metres and seconds: one product of ħ and c

A natural-unit result of mass dimension D is a number times GeV^D. To reach SI, multiply by the one product ħ^a c^b that turns GeV^D into the unit wanted; ħ carries J s and c carries m s⁻¹, so matching metres and seconds fixes a and b. A length is ħc/E, a time ħ/E. On Advanced QM's numbers: the charged pion's quoted mass, 139.570 MeV, is the length ħ/(m_π c) = 197.327/139.570 fm = 1.414 fm, the range of pion exchange between nucleons. Hydrogen's 2p state lives 1.596 ns, as UAQM-12.02 computed and beam-foil timing confirms, so its width is Γ = ħ/τ = 6.582 × 10⁻¹⁶ eV s ÷ 1.596 × 10⁻⁹ s = 0.412 μeV. Mind the 2π: as a spectral line the same width is Γ/h = 1/(2πτ) = 99.7 MHz, never Γ/ħ. Two checks: the width is 1.2 × 10⁻⁷ of the 3.40 eV binding, so the state is sharp, and a tenth of the measured 1057.8 MHz Lamb shift, which is why Unit 15's shift can be resolved at all.

Cross-sections in barns, and Thomson's 0.665 b

An area is (ħc)²/E². The barn is 10⁻²⁸ m² = 100 fm², so 1 fm² = 10 mb and (ħc)² = (0.197327 GeV fm)² = 0.038938 GeV² fm² = 0.3894 GeV² mb: a cross-section of 1 GeV⁻² is 0.3894 mb. Thomson's cross-section, quoted from UAQM-12.09 (it is the ω → 0 limit of the Compton cross-section that Unit 10 derives), reads σ_T = 8πα²/(3mₑ²) once ħ = c = 1. With mₑ = 0.51099895 MeV, mₑ² = 2.6112 × 10⁻⁷ GeV², and σ_T = 4.4612 × 10⁻⁴ ÷ 2.6112 × 10⁻⁷ GeV⁻² = 1708 GeV⁻² = 665.2 mb = 0.6652 b. The SI route agrees: the classical radius α/mₑ restores to αħc/(mₑc²) = 2.818 fm, and (8π/3)rₑ² = 0.6652 b. Check the dimensions, [α²/m²] = −2, an area, and the mass dependence: for a proton the same formula is smaller by (mₑ/mₚ)² = 2.97 × 10⁻⁷, giving 197 nb, which is why X-rays scatter from a crystal's electrons rather than from its nuclei.

The metric (+, −, −, −), stated once for the course

This course uses g_μν = diag(1, −1, −1, −1) everywhere. A contraction is a⋅b = a⁰b⁰ − a¹b¹ − a²b² − a³b³, so p² = E² − |p|², and a particle on its mass shell has p² = m², positive. An interval x² = t² − |x|² is positive for timelike separation and negative for spacelike, the region where UAQM-09.04's microcausality makes field commutators vanish. Derivatives carry a lower index, ∂_μ = (∂ₜ, ∇), so ∂^μ = (∂ₜ, −∇), □ = ∂_μ∂^μ = ∂ₜ² − ∇², the Klein-Gordon equation of UAQM-03.01 reads (□ + m²)φ = 0, and a positive-energy plane wave is e^(−ip⋅x) with p⋅x = p_μx^μ = Et − p⃗⋅x⃗. Limiting case: restoring c, E = √(|p|²c² + m²c⁴) ≈ mc² + |p|²/(2m) when |p| ≪ mc, so e^(−ip⋅x) becomes a Schrödinger plane wave times the rest-energy phase e^{(−imc²t/ħ)}. Texts on the mostly-plus metric write p² = −m² and propagators with p² + m²; translating flips the sign of every contraction, and no measurable number changes.

Heaviside-Lorentz charge: α = e²/(4π)

Electromagnetism needs one more choice. The course uses Heaviside-Lorentz units with ε₀ = μ₀ = 1: Maxwell's equations carry no 4π, the photon Lagrangian is −¼F_μνF^μν, and a charge e has the Coulomb potential e/(4πr). The fine-structure constant, e²/(4πε₀ħc) in SI, becomes α = e²/(4π) = 1/137.036, a measured value quoted here, so the electron's charge is the pure number e = √(4πα) = 0.3028, derived. Gaussian texts write α = e², with an e smaller by √(4π) = 3.545; mixing the two conventions plants a stray 4π in every rate. Check against measurement: the electron-proton Coulomb energy is −α/r, the Bohr radius 1/(αmₑ) restores to ħc/(αmₑc²) = 52.92 pm, and the ground-state energy −α²mₑ/2 = −13.606 eV becomes −13.598 eV with the reduced mass in place of mₑ, the measured ionisation energy of hydrogen.

A dimensionless action fixes the dimensions of fields

The action enters a quantum amplitude only as the phase S/ħ, so with ħ = 1 it is a pure number, [S] = 0. As [d⁴x] = −4, the Lagrangian density has [ℒ] = 4, an energy density: 1 GeV⁴ restores to 2.085 × 10³⁷ J m⁻³. The scalar kinetic term ½∂_μφ∂^μφ, with [∂_μ] = 1, gives 2[φ] + 2 = 4, so [φ] = 1; the photon's −¼F_μνF^μν gives [A_μ] = 1 the same way; ψ̄iγ^μ∂_μψ gives 2[ψ] + 1 = 4, so [ψ] = 3/2, matching SI, where ψ†ψ is a density in m⁻³. The mass terms ½m²φ² and mψ̄ψ then return [m] = 1, as they must. Any other term is a coefficient g times an operator O of dimension n, so [g] = 4 − n, derived. QED's eψ̄γ^μψA_μ has n = 4 and e comes out a pure number, as α = e²/(4π) requires; Fermi's four-fermion operator has n = 6, so G_F = 1.166 × 10⁻⁵ GeV⁻² has dimension −2. Unit 2 applies the counting to φ⁴ and the Yukawa theories.

Change one variable at a time

Make the relationship visible.

Interactive model

Mantissa a in E = a × 10ᵏ GeV1.40

Power of ten k−1

Start at the pion, a = 1.40 and k = −1: the length row reads 1.41 fm. Then a = 4.12 and k = −16, hydrogen 2p's 0.41 μeV width: the time row reads its 1.6 ns lifetime. Step k by one and the length dot crosses one tick while the area dot crosses two.

log₁₀(E/GeV)-0.85

log₁₀(length/fm)0.15

log₁₀(time/s)-23.33

log₁₀(area/b)-1.70

Live interpretationlog₁₀(E/GeV): −0.85. log₁₀(length/fm): 0.15. log₁₀(time/s): −23.33. log₁₀(area/b): −1.70

Catch the common trap

Explain before calculating.

The anomalous magnetic moment acts like Pauli's term κψ̄σ^μνψF_μν added to the QED Lagrangian (UAQM-06.04), where σ^μν = (i/2)[γ^μ, γ^ν] is a matrix of pure numbers. With ħ = c = 1 and a dimensionless action, what mass dimension must κ carry?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyThe ρ⁰ meson shows up as a peak in π⁺π⁻ pairs at 775.3 MeV with full width Γ = 149.1 MeV. Find its mean lifetime in seconds and the distance cτ that light covers in that time, in fm. Compare cτ with the proton's measured charge radius, 0.841 fm, and say whether the ρ⁰ is a sharp particle.

A width has mass dimension 1 and a lifetime −1, so τ = 1/Γ in natural units. Restore ħ to reach seconds: τ = ħ/Γ = 6.582 × 10⁻²² MeV s ÷ 149.1 MeV = 4.415 × 10⁻²⁴ s.
A length needs ħc instead: cτ = ħc/Γ = 197.327 MeV fm ÷ 149.1 MeV = 1.323 fm.
Check the two conversions against each other: cτ/τ = 1.323 × 10⁻¹⁵ m ÷ 4.415 × 10⁻²⁴ s = 3.00 × 10⁸ m s⁻¹, the speed of light, as it must be.
Compare: 1.323/0.841 = 1.57, so in the ρ⁰'s mean lifetime light crosses only about 1.6 proton radii. Its width is Γ/m = 149.1/775.3 = 0.19 of its mass, not small, so it is a broad resonance rather than a sharp particle.

Answerτ = 4.41 × 10⁻²⁴ s and cτ = 1.32 fm, about 1.6 proton charge radii; with Γ/m = 0.19 the ρ⁰ is a broad resonance.

MediumTOTEM measured the total proton-proton cross-section at √s = 13 TeV as 110.6 mb. Express it in GeV⁻² and in fm². Then model the collision as a black disk of radius R, for which partial waves give σₜₒₜ = 2πR², half absorbed and half diffracted (quoted from scattering theory), find R in GeV⁻¹ and in fm, and compare it with the pion's ħ/(m_π c).

A cross-section has mass dimension −2, so divide by (ħc)² = 0.3894 GeV² mb: σ = 110.6 mb ÷ 0.3894 mb GeV² = 284.0 GeV⁻².
For fm², use 1 b = 100 fm², so 1 mb = 0.1 fm²: σ = 11.06 fm².
Black disk: R = √(σ/2π) = √(284.0/6.283) GeV⁻¹ = 6.724 GeV⁻¹. A length restores with one power of ħc: R = 6.724 × 0.197327 fm = 1.327 fm. The fm² route gives √(11.06/6.283) fm = 1.327 fm, the same, as it must.
Compare: ħ/(m_π c) = 197.327/139.570 fm = 1.414 fm, so R m_π = 0.94. At 13 TeV the protons interact strongly out to about one pion Compton wavelength.
Check the model: a black disk scatters elastically exactly half of σₜₒₜ, but TOTEM's elastic 31.0 mb is 0.28 of it, so the disk is grey at its edge and R is a size scale rather than a sharp radius.

Answerσ = 284.0 GeV⁻² = 11.06 fm²; the black-disk radius is R = 6.72 GeV⁻¹ = 1.33 fm, 0.94 of the pion's 1.41 fm.

HardFermi's interaction joins four fermion fields at one point with strength G_F. Use mass dimensions alone to fix how a lepton's width for decay to eνν̄ depends on G_F and on its mass m, neglecting the final masses. Then, from the measured lifetimes τ_μ = 2.1970 μs and τ_τ = 290.3 fs and the masses m_μ = 105.658 MeV and m_τ = 1776.9 MeV, predict the branching fraction for τ⁻ → e⁻ν̄ₑν_τ and compare it with the measured 17.82%.

[ψ] = 3/2, so the four-fermion operator has n = 4 × 3/2 = 6 and its coefficient G_F has dimension 4 − 6 = −2: an inverse mass squared, measured as 1.1664 × 10⁻⁵ GeV⁻².
A width has dimension 1. The amplitude is linear in G_F, so the rate goes as G_F², dimension −4, and with massless final states the only other scale is m. So Γ = C G_F² m⁵, with C a pure number that dimensions cannot supply.
The same operator drives μ → eνν̄ and τ → eνν̄, so C and G_F cancel in the ratio: Γ(τ → eνν̄)/Γ(μ → eνν̄) = (1776.9/105.658)⁵ = 16.817⁵ = 1.345 × 10⁶.
Restore ħ to turn lifetimes into widths. The muon decays this way essentially every time, so Γ_μ = ħ/τ_μ = 6.582 × 10⁻²⁵ GeV s ÷ 2.1970 × 10⁻⁶ s = 2.996 × 10⁻¹⁹ GeV, and Γ(τ → eνν̄) = 1.345 × 10⁶ × 2.996 × 10⁻¹⁹ GeV = 4.030 × 10⁻¹³ GeV.
The τ's total width is ħ/τ_τ = 6.582 × 10⁻²⁵ GeV s ÷ 2.903 × 10⁻¹³ s = 2.267 × 10⁻¹² GeV, so the branching fraction is 4.030 × 10⁻¹³ ÷ 2.267 × 10⁻¹² = 0.178.
Checks: Γ/m is 2.8 × 10⁻¹⁸ for the muon and 1.3 × 10⁻¹² for the τ, so both are sharp particles. With the quoted C = 1/(192π³), the muon width gives τ_μ = 2.187 μs, 0.4% below the measurement: the one-loop QED correction, which cancels in the ratio.

AnswerΓ = C G_F² m⁵, so B(τ → eνν̄) = (τ_τ/τ_μ)(m_τ/m_μ)⁵ = 0.178, against the measured 0.1782 ± 0.0004.