Symmetry in Physics. The terms “symmetry” and
“symmetry law” occur frequently in present-day
physics, and we now comment on this occurrence.
We have stated above that, in a spatial setting, an
impression of symmetry arises if a design is not changed
by a group of automorphisms of the space, and we
add to this that, as far as design is concerned, the
underlying space is a setting for it, and any auto-
morphism of the space is a certain change of this
setting. Quite generally, in mathematics and mathe-
matically controlled science, the imperviousness to
changes within a setting is technically called
“invariance” (relative to these changes). In this sense
a law of symmetry is a particular case of a law of
invariance, and, to begin with, the two are not
coextensive because symmetry involves a connotation
of space, whereas invariance is more comprehensive.
However, in physics many invariances involve space
variables, or at least space data, and in this way the
terms symmetry and invariance have drawn ever closer
together and have become almost synonymous.
Thus, a present-day physicist may view even the
nineteenth-century law of conservation of energy as
a symmetry law. The nineteenth century envisaged
various forms of energy, mechanical, thermal, electri-
cal, etc., and admitted that they may be transformed
into each other, that is, change over into each other.
But the law maintained (and continues to maintain)
that throughout such changes of form, the total amount
of energy in a closed physical system remains constant,
that is invariant. To this and the following see the first
half of Eugene P. Wigner, Symmetry and Reflection.
It is not always easy to decide how two physical
laws of symmetry relate to each other. It can be stated,
for instance, that Newton's force of gravitational
attraction (inverse square of the distance) obeys two
laws of symmetry, a spatial and a temporal. According
to the spatial law, the force of attraction is invariant
for all orthogonal transformations of space, not distin-
guishing between points of origin or directions in
space. According to the temporal law, it is invariant
in time.
As just stated, these two laws of symmetry are sepa-
rate, and have equal standing. However, the theory
of relativity erases the separateness, and geology casts
doubt upon their equal standing. In fact, the general
theory of relativity, by the very formulation of the
phenomenon of gravitation, fuses the two laws of
symmetry into one, so that they cannot be separated.
On the other hand, in geology, the temporal invariance
is part of the general law of uniformitarianism which
seems to assert that, say, in the rise of the solar system,
or at least in the evolution of the earth, the familiar
laws of “classical” physics are “eternal,” that is
temporally invariant, and thus have always been before
what they are presently. This seems to make the
temporal symmetry of the gravitational force hier-
archically prior, and thus superior to the spatial one.
In basic physics of the twentieth century, as the
century progressed, the number of invariance proper-
ties has shown a tendency to increase (Wigner, p. 60).
There has been many a period of anguish when two
leading symmetry laws were seemingly in an irrecon-
cilable clash with each other. Occasionally such a
period of anguish was followed by a period of relief
when, to the soothing accompaniment of a Nobel prize,
the clash was somehow composed.
On the other hand, the twentieth century also undid
at least one law of the preceding century. It was the
law of similitude. Although not very important, it was
treated with respect. It asserted
... that physical experiments can be scaled; that the abso-
lute magnitude of objects be irrelevant from the point of
view of their behavior on the proper scale. The existence
of atoms, of an elementary charge, and of a limiting velocity
spelled the doom of this principle
(Wigner, p. 5).
We might add that the elementary charge, that is the
magnitude of the electron, seems to be the most im-
placable of the instruments of doom.
This doom reached beyond the law of similitude,
which is a not-too-important law of physics. It also
enveloped the serene vision of Leibniz that space is
nothing but order and relations, and perhaps also
with predestined harmony ensuing; it somehow also
enveloped the creeds of the eighteenth and nineteenth
centuries—“naive” creeds in the eighteenth century
and less naive ones in the nineteenth—that, in spite
of all vehemence and violence in man, everything will
in the end turn out to be continuous, controllable, and
adjustable to scale.
But these creeds overlooked the electron. James
Clerk Maxwell, at the height of Victorianism, tried to
forget about the electron, by creating a magnificent
field theory in which everything can be adjusted to
scale, but it was not enough for physics; the electron
simply had to be considered and taken very seriously
indeed. Magnetism is a close kin to electricity. When
Thales of Miletus, almost twenty-six centuries ago, saw
a magnet in action, he exclaimed that the world is full
of Gods. And so it is even today.
