The mathematical study of the Bose gas goes back to the ?rst quarter of
the twentieth century, with the invention of quantum mechanics. The name
refers to the Indian physicist S.N. Bose who realized in 1924 that the
statistics governing
photons(essentiallyinventedbyMaxPlanckin1900)isdetermined(usingmodern
terminology) by restricting the physical Hilbert space to be the
symmetric tensor product of single photon states. Shortly afterwards,
Einstein applied this idea to massive particles, such as a gas of atoms,
and discovered the phenomenon that we now call Bose-Einstein
condensation. At that time this was viewed as a mathematical curiosity
with little experimental interest, however. The peculiar properties of
liquid Helium (?rst lique?ed by Kammerlingh Onnes in 1908) were
eventually viewed as an experimental realization of Bose- Einstein
statistics applied to Helium atoms. The unresolved mathematical pr- lem
was that the atoms in liquid Helium are far from the kind of
non-interacting particles envisaged in Einstein's theory, and the
question that needed to be - solved was whether Bose-Einstein
condensation really takes place in a strongly interacting system -- or
even in a weakly interacting system. That question is still with us,
three quarters of a century later! The ?rst systematic and semi-rigorous
mathematical treatment of the pr- lem was due to Bogoliubov in 1947, but
that theory, while intuitively appealing and undoubtedly correct in many
aspects, has major gaps and some ?aws. The 1950's and 1960's brought a
renewed ?urry of interest in the question, but while
theoreticalintuitionbene?tedhugelyfromthisactivitythemathematicalstructure
did not signi?cantly improve.
