Natural
units
Particle
physicists have a habit that on the face of it seems to violate
physics. They set the values of natural constants such as the speed
of light and Planck's constant to the number one. How can they do
this? It's because the relationship between units by which we measure
length, time and energy is a matter of choice. For example, we know
what a second is, and we know what a meter is. But how is many meters
make a second? Is that like asking how many apples are there in
an orange? Maybe. But not quite, because the constants of Nature
like Planck's constant and the speed of light represent natural
relationships between different units of measurement, as explained
below.
Speed
of light
The
measured value of the speed of light
is
so
why would physicists want to pretend that instead c=1?
What they are really doing is choosing a relationship between a
unit of time, the second, and a unit of space, the meter, so that
these two units are not independent but related. The natural constant
of relation is the speed of light, so that
If
we relate meters and seconds so that one second is equal to 300
million meters, then c=1. It's very simple. Now notice that in this
system of units, mass and energy have the same units, because the
relationship E = m c^{2} in units with c=1
just reduces to E = m.
Planck's
constant
Planck's
constant has units of energy x time. The preferred unit
of energy in physics is the the electron
volt (or eV for
short) is the amount of work necessary to move one
electron across a potential of one
volt. This is a very handy unit for particle physicists
because particle experiments use electric potentials to accelerate
electrons and other particles with that same charge, and bang them
into each other. In units with c=1, the mass of the electron is
0.5 MeV. Written in terms of eV, Planck's constant takes the value
The
version of Planck's constant
that physicists normally use is called "hbar" and is
equal to
In
c=1 units, time is described
in meters, the value of Planck's constant becomes
But
physicists aren't happy to stop there. We like to make all calculations
as simple as possible. So the next step is set (this version of)
Planck's constant equal to one to set the relationship between energy
units and length units so that
The
size of an atom is roughly 1010 meters. Atomic physicists use a
unit called the Ångstrom, where 1 Å = 1010 meters.
Written in these terms we get the relationship
Notice
that in these new units, increasing energy means decreasing length.
Distances scales that are much smaller than the size of an atom
have mass scales associated with them that are much larger than
2000 eV.
That is typical behavior for quantum mechanics.
The de Broglie relation for waveparticle duality
also
shows that in quantum physics, it is necessary to use a large energy
or momentum scale to probe a small distance scale. That's why particle
accelerators are like microscopes. When the particle accelerator
energy gets bigger, the distance scale being probed gets smaller.
Gravity
Can
we play the same game with Newton's gravitational constant? Not
really, because there aren't any new units to relate that aren't
already related. If we consider spacetimes with gravity in higher
dimensions, Newton's constant has units that depend on the dimension
of spacetime. The value currently measured is, of course, for d=4.
In
natural units, Newton's constant has units of L^{d2}, or
M^{2d}.
Thus one can express Newton's constant in units of time, length
or mass, as desired.
The
Planck length is thought to be the natural distance scale at which
quantum gravitational effects become strong enough to notice.
