In 1918 Hermann Weyl tried to reproduce electromagnetism by adding the
notion of an arbitrary scale or gauge to the metric of general relativity—and
noted the “gauge invariance” of his theory under simultaneous transformation of
the electromagnetic potentials and multiplication of the metric by a
position-dependent factor. Following the
introduction of the Schrödinger equation in quantum mechanics in 1926 it was
almost immediately noticed that the equations for a charged particle in an electromagnetic
field were invariant under gauge transformations in which the wave function was
multiplied by a position-dependent phase factor. The idea then arose that perhaps some kind of
gauge invariance could also be used as the basis for formulating theories of
forces other than the electromagnetism.
After a few earlier attempts, Yang-Mills theories were introduced in
1954 by extending the notion of a phase factor to an element of an arbitrary
non-Abelian group. In the 1970s the
Standard Model then emerged, based entirely on such theories. In mathematical terms, gauge theories can be
viewed as describing fiber bundles in which connections between values of group
elements in fibers at neighboring spacetime points are specified by gauge
potentials—and curvatures correspond to gauge fields. (General relativity is in effect a special
case in which the group elements are themselves related to spacetime
coordinates.)
—Stephen Wolfram, A New Kind of
Science, Notes for Chapter 9, p. 1045.
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. . . The difference between a
neutron and a proton is then a purely arbitrary process. As usually conceived, however, this
arbitrariness is subject to the following limitation: once one chooses what to call a proton, what
a neutron, at one space-time point, one is then not free to make any choices at
other space-time points.
It seems that this is
not consistent with the localized field concept that underlies the usual
physical theories. In the present paper
we wish to explore the possibility of requiring all interactions to be
invariant under independent rotations of the isotopic spin at all space-time
points. …
—Yang and Mills (1954). Used as
epigraph to Part III (Non-Abelian Gauge Theory and QCD) in Gauge Theories in Particle Physics:
A Practical Introduction, second edition, by Ian J. R.
Aitchison and Anthony J. G. Hey.
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Isotopic
spin (isospin; isobaric spin) A quantum
number applied to hadrons (see elementary particles) to distinguish
between members of a set of particles that differ in their electromagnetic
properties but are otherwise apparently identical. For example if electromagnetic interactions
and weak interactions are ignored, the proton cannot be distinguished from the
neutron in their strong interactions:
isotopic spin was introduced to make a distinction between them. The use of the word “spin” implies only an
analogy to angular momentum, to which isotopic spin has a formal resemblance.
—A Dictionary of Physics,
third edition, Oxford University Press, 1996.
------------------
The theory of the addition of
quantum mechanical angular momenta has found an interesting application in the
study of strongly interacting particles.
Basically the proton and neutron are very similar particles. Their masses are nearly equal, they each have
spin ½, and when interacting with themselves or other particles via strong
interactions they behave very
similarly Of course, the one striking
difference between the neutron and proton is that the proton has a charge,
whereas the neutron has none. This means
that they have very different electromagnetic interactions; e.g., an electron will be attracted to a
proton but will feel rather indifferent about a neutron. However, the strong forces are several orders
of magnitude stronger, at small distances, than electromagnetic forces, and in
studying such forces, one can to a first approximation neglect the
electromagnetic forces. Thus from the
point of view of strong interactions, the neutron and proton are practically
identical.
In fact, one can say that the
neutron and proton are the same particle—the nucleon, N—which has an internal
degree of freedom which can take on two possible values—protonliness or
neutronliness. This is analogous to our
thinking of a spin-up electron and a spin-down electron as being the same kind of
particle—the electron—in different states of the internal degree of
freedom—spin. By analogy, one calls the
internal degree of freedom of the nucleon isotopic
spin, or isospin for short. The
“isospin up” state of the nucleon is the proton and the “isospin down” state is
the neutron.
—Gordon Baym, Lectures on Quantum Mechanics, first two
paragraphs of Chapter 16: Isotopic Spin.
