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Quantum mechanics is one of the most fascinating, and at the same time most controversial, branches of contemporary science. Disputes have accompanied this science since its birth and have not ceased to this day. Uncommon Paths in Quantum Physics allows the reader to contemplate deeply some ideas and methods that are seldom met in the contemporary literature. Instead of widespread recipes of mathematical physics, based on the solutions of integro-differential equations, the book follows logical and partly intuitional derivations of non-commutative algebra. Readers can directly

Mathematical physics. --- Noncommutative algebras. --- Quantum theory -- Mathematics. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Quantum theory. --- Noncommutative algebras --- Algebras, Noncommutative --- Non-commutative algebras --- Algebra --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics

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Geometry, Differential. --- Noncommutative algebras. --- Mathematical physics. --- Physical mathematics --- Physics --- Algebras, Noncommutative --- Non-commutative algebras --- Algebra --- Differential geometry --- Mathematics --- Mathematical analysis

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"The algebraic theory of automata was created by Schutzenberger and Chomsky over 50 years ago and there has since been a great deal of development. Classical work on the theory to noncommutative power series has been augmented more recently to areas such as representation theory, combinatorial mathematics and theoretical computer science. This book presents to an audience of graduate students and researchers a modern account of the subject and its applications. The algebraic approach allows the theory to be developed in a general form of wide applicability. For example, number-theoretic results can now be more fully explored, in addition to applications in automata theory, codes and non-commutative algebra. Much material, for example, Schutzenberger's theorem on polynomially bounded rational series, appears here for the first time in book form. This is an excellent resource and reference for all those working in algebra, theoretical computer science and their areas of overlap"--

Machine theory --- Noncommutative algebras --- Automates mathématiques, Théorie des --- Algèbres non commutatives --- Automates mathématiques, Théorie des --- Algèbres non commutatives --- Machine theory. --- Noncommutative algebras. --- Automates. --- Algèbres non commutatives. --- Algebras, Noncommutative --- Non-commutative algebras --- Algebra --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Logic, Symbolic and mathematical --- Recursive functions --- Robotics --- Algèbres non commutatives.

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Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics.

Quantum field theory --- Factorization (Mathematics) --- Factors (Algebra) --- Geometric quantization. --- Noncommutative algebras. --- Mathematics. --- Algebras, Noncommutative --- Non-commutative algebras --- Algebra --- Geometry, Quantum --- Quantization, Geometric --- Quantum geometry --- Geometry, Differential --- Quantum theory --- Mathematics --- Relativistic quantum field theory --- Field theory (Physics) --- Relativity (Physics)

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Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: 1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result; 2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation; 3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin; 4) give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type); 5) quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature concerning a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications. The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool for beginners in Algebra.

Noncommutative algebras --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Calculus --- Noncommutative algebras. --- Algebras, Noncommutative --- Non-commutative algebras --- Mathematics. --- Nonassociative rings. --- Rings (Algebra). --- Topological groups. --- Lie groups. --- Differential geometry. --- History. --- Topological Groups, Lie Groups. --- History of Mathematical Sciences. --- Non-associative Rings and Algebras. --- Differential Geometry. --- Annals --- Auxiliary sciences of history --- Differential geometry --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Math --- Science --- Topological Groups. --- Algebra. --- Global differential geometry. --- Geometry, Differential --- Mathematical analysis