68
J. Maruani
this system, there are seven basic quantities/units: length/meter, mass/kilogram,
time/second, electric current/ampere, temperature/kelvin, substance amount/mole,
and light intensity/candela. There are also twenty-two derived quantities (and units),
designed to quantify such entities as planar or solid angle, force and pressure, energy and power, electric charge and circuit properties, magnetic and light properties,
radioactivity, and biocatalysis. Multiples and fractions of these units are defined according to the decimal system.
One may wonder why the plane or solid angle was not retained as a basic quantity, for angle is linked to isotropy, a basic symmetry which entails conservation of
angular momentum. A property like magnetic flux density could also have been selected, although magnetism is related to electricity, contrary to gravitation. It seems
strange that radioactivity is relegated to derived quantities and measured in s −1 , no
more lethal than a music note! On the other hand, one may argue about the theoretical relevance (though not the practical convenience) of such basic units as the
kelvin, mole, or candela, all related to energy through heat, mass, or light.
One may also wonder if different units could not be defined for the three dimensions of space: length, width, and height. For Aristotle, it made sense to distinguish
between going up or down, East or West, South or North. Contrary to the two horizontal directions, the vertical seemed oriented, as time is for us. Medieval painters
represented objects on two dimensions, ignoring perspective display by projective
geometry (and, of course, holographic storing and retrieval by diffraction patterns
of coherent light). Until Descartes’ analytical geometry in the 17th century, it was
not clearly understood that space has three dimensions, with axis representations
and measuring units that are both rotatable (due to space isotropy) and translatable
(due to space homogeneity) [65].
In general relativity theory, the vertical relative to a massive body is different
indeed from the other two space dimensions, due to space curvature resulting from
gravitational attraction; and in special relativity theory, time is already a different
dimension, involving an imaginary axis. Hence, a non-Euclidean metric in the two
theories. Having reduced the vertical to the two horizontal dimensions, one could
further simplify by reducing time to the three spatial dimensions, writing in dimensional terms: c ∼ L/T ∼ 1 (the length associated with 1 second being 1 parsec).
The velocity of light c, assumed to be an upper limit in relativity theory, is one
of the fundamental constants of Nature. The quantum of action , assumed to be a
lower limit in quantum theory, is another fundamental constant. If it is also taken as
dimensionless, one has: ∼ 1 → ML 2 T −1 ∼ ML ∼ 1 → M ∼ L −1 ∼ T −1 . This
is consistent with Eq. (3.4) for the Compton radius when both c ∼ 1 and ∼ 1
(dimensionwise), as well as with mass being identical to energy when c ∼ 1 (and
with time inverse when ∼ 1). In Table 3.2, we show how some universal constants
and physical properties are related in this dimensional system, commonly used in
relativistic quantum mechanics.
This system should not be confused with the natural system of units proposed in
1881 by George Stoney, who derived units for length, time, and mass by normalizing
G, c, and e to unity. This system was extended in 1899 by Max Planck, who derived
also units for temperature by normalizing G, c, , and k B (the Boltzmann constant),
J. Maruani
this system, there are seven basic quantities/units: length/meter, mass/kilogram,
time/second, electric current/ampere, temperature/kelvin, substance amount/mole,
and light intensity/candela. There are also twenty-two derived quantities (and units),
designed to quantify such entities as planar or solid angle, force and pressure, energy and power, electric charge and circuit properties, magnetic and light properties,
radioactivity, and biocatalysis. Multiples and fractions of these units are defined according to the decimal system.
One may wonder why the plane or solid angle was not retained as a basic quantity, for angle is linked to isotropy, a basic symmetry which entails conservation of
angular momentum. A property like magnetic flux density could also have been selected, although magnetism is related to electricity, contrary to gravitation. It seems
strange that radioactivity is relegated to derived quantities and measured in s −1 , no
more lethal than a music note! On the other hand, one may argue about the theoretical relevance (though not the practical convenience) of such basic units as the
kelvin, mole, or candela, all related to energy through heat, mass, or light.
One may also wonder if different units could not be defined for the three dimensions of space: length, width, and height. For Aristotle, it made sense to distinguish
between going up or down, East or West, South or North. Contrary to the two horizontal directions, the vertical seemed oriented, as time is for us. Medieval painters
represented objects on two dimensions, ignoring perspective display by projective
geometry (and, of course, holographic storing and retrieval by diffraction patterns
of coherent light). Until Descartes’ analytical geometry in the 17th century, it was
not clearly understood that space has three dimensions, with axis representations
and measuring units that are both rotatable (due to space isotropy) and translatable
(due to space homogeneity) [65].
In general relativity theory, the vertical relative to a massive body is different
indeed from the other two space dimensions, due to space curvature resulting from
gravitational attraction; and in special relativity theory, time is already a different
dimension, involving an imaginary axis. Hence, a non-Euclidean metric in the two
theories. Having reduced the vertical to the two horizontal dimensions, one could
further simplify by reducing time to the three spatial dimensions, writing in dimensional terms: c ∼ L/T ∼ 1 (the length associated with 1 second being 1 parsec).
The velocity of light c, assumed to be an upper limit in relativity theory, is one
of the fundamental constants of Nature. The quantum of action , assumed to be a
lower limit in quantum theory, is another fundamental constant. If it is also taken as
dimensionless, one has: ∼ 1 → ML 2 T −1 ∼ ML ∼ 1 → M ∼ L −1 ∼ T −1 . This
is consistent with Eq. (3.4) for the Compton radius when both c ∼ 1 and ∼ 1
(dimensionwise), as well as with mass being identical to energy when c ∼ 1 (and
with time inverse when ∼ 1). In Table 3.2, we show how some universal constants
and physical properties are related in this dimensional system, commonly used in
relativistic quantum mechanics.
This system should not be confused with the natural system of units proposed in
1881 by George Stoney, who derived units for length, time, and mass by normalizing
G, c, and e to unity. This system was extended in 1899 by Max Planck, who derived
also units for temperature by normalizing G, c, , and k B (the Boltzmann constant),