Bose Algebras: The Complex and Real Wave Representations

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The mathematics of Bose-Fock spaces is built on the notion of a commutative algebra and this algebraic structure makes the theory appealing both to mathematicians with no background in physics and to theoretical and mathematical physicists. The well-known complex and real wave representations are presented here as natural consequences of the basic mathematical structure. Operators generated by creations and annihilations in a given Bose algebra are shown to give rise to a new Bose algebra of operators yielding the Weyl calculus of pseudo-differential operators. This text has been designed for mathematicians interested in the analysis of infinitely many dimensions or in the mathematics of quantum fields, and also for theoretical physicists who require an effective and rigorous Bose formalism.