B
Natural Units
In particle physics, a widely adopted convention is to work in a system of
units, called natural units, in which
ħ = c = 1.
(B.1)
This avoids having to keep track of untidy factors of ħ and c throughout a
calculation; only at the end is it necessary to convert back to more usual units.
Let us spell out the implications of this choice of c and ħ.
(i) c = 1. In conventional MKS units c has the value
c = 3 × 10
8 m s
−1 .
(B.2)
By choosing units such that
c = 1
(B.3)
since a velocity has the dimensions
[c] = [L][T]
−1
(B.4)
we are implying that our unit of length is numerically equal to our unit of
time. In this sense, length and time are equivalent dimensions:
[L] = [T].
(B.5)
Similarly, from the energy–momentum relation of special relativity
2 2
2 4
E
2 = p c + m c
(B.6)
we see that the choice of c = 1 also implies that energy, mass and momentum
all have equivalent dimensions. In fact, it is customary to refer to momenta
in units of ‘MeV/c’ or ‘GeV/c’; these all become ‘MeV’ or ‘GeV’ when c = 1.
(ii) ħ = 1. The numerical value of Planck’s constant is
ħ = 6.6 × 10
−22 MeV s
(B.7)
and ħ has dimensions of energy multiplied by time so that
[ħ] = [M][L]
2 [T]
−1 .
(B.8)
Setting ħ = 1 therefore relates our units of [M], [L] and [T]. Since [L] and
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