Notation, Natural Units & Fundamental Constants

Open this book at any page and you will find an equation like $E^2 = p^2 + m^2$. Read it as an engineer and it is nonsense: you cannot add a momentum squared to a mass squared. Open it a page later and a particle has “a mass of 125 GeV” — an energy — and “a width of 2.5 GeV” — which is somehow a lifetime. Nothing here is sloppy. The whole book runs in a unit system where those statements are exact, and this page is where you learn it, because every other page assumes it.

This page is the toolkit: the unit convention, the two constants that undo it, the numbers from Appendices 1 and 2, and the symbol table for the whole site. Keep it open in a second tab.

The problem: three units for one thing

Write relativistic energy–momentum the way SI demands, and then again the way this book writes it:

The second line is not an approximation and it is not lazy. It is the statement that c and ħ are conversion factors, not physics — and that you may choose units in which they equal 1.

Step 1: set c = 1

Special relativity says space and time are one geometry. If they are, measuring them in different units (metres and seconds) is a historical accident — like measuring a rectangle’s width in inches and its height in centimetres. Setting

declares one second and $2.998 \times 10^8$ metres to be the same length. Immediately:

• velocities become dimensionless — a velocity is $\beta = v/c$, a pure number between $-1$ and $1$, which is why the book writes $\beta$ everywhere;
• $E = mc^2$ becomes $E = m$, so a mass is an energy. The proton is “0.938 GeV”, full stop;
• momentum $p$, energy $E$ and mass $m$ all carry the same unit, which is what makes $E^2 = p^2 + m^2$ legal.

Step 2: set ħ = 1

Quantum mechanics ties energy to frequency by $E = \hbar\omega$ and momentum to inverse wavelength by $p = \hbar k$. Setting

declares an energy and an inverse time to be the same quantity. Now:

• a time is an inverse energy, so a lifetime $\tau$ and a width width Γ, the energy spread of an unstable state, tied to its lifetime by Γτ = ħ; literally the time–bandwidth relation, so a width quoted in MeV is an inverse lifetime. defined in the reference pages — open in glossary $\Gamma = \hbar/\tau$ are the same information written two ways;
• a length is an inverse energy too (via $p = \hbar k$), so “1 GeV” is also a distance scale;
• angular momentum and action become pure numbers, which is why spins are quoted as $1/2$, $1$, $3/2$ rather than as $\hbar/2$.

Everything is a power of energy

Undoing it: the only two constants you need

You will eventually want an answer in metres, seconds or barns. Two numbers do all of that work, and the book prints both in Appendix 2 as exact.

ħc turns an energy into a length

(ħc)² turns an inverse energy squared into a cross-section

import numpy as np

hbar_c   = 197.3269804e-3      # GeV fm   (exact, App. 2)
hbar     = 6.582119569e-25     # GeV s    (exact, App. 2)
hbarc2   = hbar_c**2 / 100 * 1e6   # GeV^2 ub   (1 b = 100 fm^2 = 1e6 ub)
alpha    = 1 / 137.035999166

print(f"hbar*c            = {hbar_c:.10g} GeV fm")
print(f"(hbar*c)^2        = {hbarc2:.10g} GeV^2 ub   (book prints 389.3793271 - typo)")
print(f"Q to resolve 0.84 fm = {hbar_c / 0.84 * 1000:.1f} MeV")

sigma_gev2 = 4 * np.pi * alpha**2 / (3 * 100.0)        # GeV^-2 at sqrt(s)=10 GeV
print(f"sigma(ee->mumu) at 10 GeV = {sigma_gev2 * hbarc2 * 1e3:.4f} nb")  # 1 ub = 1e3 nb

print(f"Z0 lifetime from Gamma = {hbar / 2.4952:.4e} s")
print(f"proton mass       = {0.938272088 * 1.7826619216e-27:.6e} kg")
print(f"300 K             = {300 * 8.617333262e-14 * 1e12:.2f} meV")

hbar*c            = 0.1973269804 GeV fm
(hbar*c)^2        = 389.3793719 GeV^2 ub   (book prints 389.3793271 - typo)
Q to resolve 0.84 fm = 234.9 MeV
sigma(ee->mumu) at 10 GeV = 0.8685 nb
Z0 lifetime from Gamma = 2.6379e-25 s
proton mass       = 1.672622e-27 kg
300 K             = 25.85 meV

Try it: the converter

Everything above, as a knob. Type a number, pick a unit, and watch the same quantity in every other unit — including the natural-unit power of GeV. For an energy it also shows you the length, time and area that energy is.

Appendix 2 — the constants themselves

Transcribed from p. 495. Click any cell with a dotted underline for what the constant is and where the book uses it. Note the exact column: since the 2019 SI redefinition, several of these are not measurements at all.

Orders of magnitude — the map you should carry

Every energy in this book is somewhere on this line, and by ħc every energy is also a distance. Read it in both directions.

The symbols this book uses

Particle physics ran out of letters decades ago. This table is the site-wide convention; every chapter obeys it.

Greek letters, and the overloading problem

Appendix 1 is just the alphabet. The useful version is knowing which glyph is overloaded — because the book will not warn you.

Where this is used

• §1.6 Measurement Units applies all of this to real collisions and decays, and is where barns barn the natural area unit for cross-sections, 1 b = 10⁻²⁸ m² = 100 fm²; sub-multiples mb, μb, nb, pb, fb appear throughout. defined in the reference pages — open in glossary first earn their keep.
• §1.8 Scattering Experiments turns a cross-section into an event rate — the “effective target area” picture.
• The particle data explorer is the same Appendix 3 tables, made searchable; every mass there is in the units defined here.