This book offers a non-standard introduction to quantum mechanics and quantum field theory, approaching these topics from algebraic and geometric perspectives. Beginning with fundamental notions of quantum theory and the derivation of quantum probabilities from decoherence, it proceeds to prove the expression for the scattering matrix in terms of Green functions (LSZ formula), along with a similar expression for the inclusive scattering matrix. The exposition relies on recent findings by the author that provide a deeper understanding of the structure of quantum theory and extend beyond its traditional boundaries. The book is suitable for graduate students and young researchers in mathematics and theoretical physics seeking to delve into innovative concepts within quantum theory. The book contains many recent results therefore it should be interesting also to accomplished physicists and mathematicians.

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Albert Schwarz is a Soviet and American mathematician and theoretical physicist, currently Professor Emeritus at UC Davis, USA. He started his long career as a topologist studying the geometry of uniform continuity. This work led him to the notion of the volume invariant of a group, later rediscovered by Milnor as the growth of a group. Schwarz's paper is considered a seminal work in geometric group theory. Investigating topological questions within the calculus of variations, he introduced the concept of the genus of fiber space, which found applications in the topological complexity of algorithms and topological robotics.

Schwarz later switched to mathematical problems of physics, applying methods of various branches of modern mathematics (homotopy topology, differential topology, algebraic geometry, noncommutative geometry, homological algebra, and number theory) to quantum field theory and string theory. Schwarz's papers on topologically non-trivial objects in physics, such as magnetic monopoles, instantons, and Alice strings, were groundbreaking. Later he found a way to apply ideas of physics to topology constructing the first examples of topological quantum field theories. Now such theories play a prominent role both in mathematics and physics. Schwarz's papers where noncommutative geometry was applied to M(atrix) theory sparked a flurry of activity among physicists. His contributions extend across various domains, including the geometry of superconformal manifolds, multiloop contributions to string amplitudes, BV formalism, supergravity, and maximally supersymmetric gauge theories, among others.