The mathematics of the Bose gas and its condensation

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The mathematical study of the Bose gas began in the early twentieth century with the advent of quantum mechanics. The term originates from Indian physicist S. N. Bose, who in 1924 recognized that the statistics governing photons, initially conceptualized by Max Planck in 1900, could be understood by restricting the physical Hilbert space to the symmetric tensor product of single photon states. Einstein later applied this concept to massive particles, leading to the discovery of Bose-Einstein condensation. Initially regarded as a mathematical curiosity, this phenomenon gained experimental relevance with the peculiar properties of liquid helium, first liquefied by Kammerlingh Onnes in 1908. However, a significant mathematical challenge arose: the atoms in liquid helium are not the non-interacting particles assumed in Einstein's theory. The critical question remains whether Bose-Einstein condensation occurs in strongly or even weakly interacting systems, a query that persists today. The first systematic mathematical approach to this issue was introduced by Bogoliubov in 1947. Although his theory was intuitively appealing and correct in many respects, it contained notable gaps and flaws. Renewed interest in the 1950s and 1960s advanced theoretical intuition, yet the mathematical framework did not see significant improvement.