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Dimensionless physical constants

Fundamental physical constants

In physics, dimensionless or fundamental physical constants are, in the strictest sense, universal physical constants that are independent of
systems of units and hence are dimensionless quantities. However, the term may also be used (for example, by NIST)
to refer to any dimensioned universal physical constant,
such as the speed of light (free
)or the gravitational constant. While
both mathematical constants and fundamental physical constants are dimensionless, the latter are determined only by physical measurement and not defined by any combination of pure
mathematical constants. The list of fundamental physical constants decreases when physical theory advances and shows how some previously fundamental constant can be computed in terms of others. The list increases when experiments measure new effects.

Physicists try to make their theories simpler and more elegant by reducing the number
of physical constants appearing in the mathematical expression of their theories. This is
accomplished by defining the units of measurement in such a way that several of the most
common physical constants, such as the speed of light, among others, are normalized to
unity. The resulting system of units, known as natural units,
has a fair following in the literature on advanced physics because it considerably
simplifies many equations.

Some physical constants, however, are dimensionless numbers which cannot be eliminated
in this way. Their values have to be ascertained experimentally. A classic example is the fine structure constant,


where e is the elementary chargeh-bar
is the reduced Planck’s constantc
is the speed of light in a vacuum, and epsilon
is the permittivity of free space.

In simple terms, the fine structure constant determines how strong the electromagnetic
force is. Nobody knows why it has the value it does.

A long-sought goal of theoretical physics is to reduce the number of fundamental
constants that need to be put in by hand, by calculating some from first principles. The
reduction of chemistry to physics was an enormous step in this direction, since properties
of atoms and molecules can now be calculated from the Standard Model, at least in
principle. A successful Grand Unified Theory or Theory of Everything might reduce the number of
fundamental constants further, ideally to zero. However, this goal remains elusive.

According to Michio Kaku (1994: 124-27), the Standard Model of particle physics contains 19 arbitrary dimensionless
constants that describe the masses of the particles and the strengths of the various
interactions. This was before it was discovered that neutrinos
can have nonzero mass, and his list includes a quantity called the theta angle which seems to be zero. After the discovery of
neutrino mass, and leaving out the theta angle, John Baez (2002) noted that the
new Standard Model requires 25 arbitrary fundamental
constants, namely:

If we take gravity into account we need at least one more fundamental constant, namely

This gives a total of 26 fundamental physical constants. There are presumably more
constants waiting to be discovered which describe the properties of dark matter. If dark energy
turns out to be more complicated than a mere cosmological
, even more constants will be needed.

In his book Just Six Numbers, Martin Rees considers the following numbers:

These constants constrain any plausible fundamental physical theory, which must either
be able to produce these values from basic mathematics, or accept these constants as
arbitrary. The question then arises: how many of these constants emerge from pure
mathematics, and how many represent degrees of freedom for
multiple possible valid physical theories, only some of which can be valid in our
Universe? This leads to a number of interesting possibilities, including the possibility
of multiple universes with different values of
these constants, and the relation of these theories to the anthropic principle.

Note that Delta = 3; being simply an integer, most physicists would not consider this a
dimensionless physical constant of the usual sort.

Some study of the fundamental constants has bordered on numerology.
For instance, the physicist Arthur Eddington argued
that for several mathematical reasons, the fine structure constant had to be exactly
1/136. When its value was discovered to be closer to 1/137, he changed his argument to
match that value. Experiments since his day have
shown that his arguments are still wrong; the constant is about 1/137.036.

The mathematician Simon Plouffe has made an extensive
search of computer databases of mathematical formulae, seeking formulae giving the mass
ratios of the fundamental particles.

See also:

CKM matrix

fine structure
Physical cosmologyMaki-Nakagawa-Sakata
Standard ModelWeinberg angleCabibbo angle


John D. Barrow, 2002. The
Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of
the Universe
. Pantheon Books. ISBN 0-375-42221-8.

John D. Barrow and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford
Univ. Press.
Michio Kaku, 1994. Hyperspace: A Scientific Odyssey Through Parallel
Universes, Time Warps, and the Tenth Dimension
. Oxford University Press.
Martin Rees, 1999. Just
Six Numbers
: The Deep Forces that Shape the Universe. London: Phoenix. ISBN 0-7538-1022-0

External articles


Physical Constants from NIST

Values of fundamental
CODATA, 2002.

Variable fundamental constants

ints on varying a and the promise of
Phys.Lett. B585: 29-34.

Scientific American Magazine (June 2005 Issue) Inconstant Constants – Do the inner workings of nature change with time?

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