Quantum Mechanics Built On Algebraic Geometry
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Quantum Mechanics Built on Algebraic Geometry
This book presents a novel standpoint concerning contemporary physics, namely, quantum mechanics with a view toward algebraic geometry. As is well-known, algebraic geometry is the study of geometric objects delineated by polynomials, and the polynomial representations are ubiquitous in physics. For this reason, quantum mechanics is also an object of algebraic geometry. An example is the eigenvalue problem. It is a set of polynomial equations and has traditionally been the question of linear algebra. However, the modern method of computational algebraic geometry accurately unravels the information encapsulated in the polynomials. This approach shall not remain as a plaything. It has betokened an innovative style of electronic structure computation. The objects of this new method include the simultaneous determination of the wave-functions and the movements of nuclei, or the prediction of the required structure that shall show the desired property. Accordingly, this book explains the basic ideas of computational algebraic geometry and related topics, such as Groebner bases, primary ideal decomposition, Dmodules, Galois, class field theory, etc. The intention of the author is, nevertheless, not to give an irksome list of abstract concepts. He hopes that the readers shall use algebraic geometry as the active tool of the computations. For this reason, this book abundantly presents the model computations, by which the readers shall learn how to apply algebraic geometry toward quantum mechanics. The readers shall also see the modern computer algebra could facilitate the study when you would like to apply abstract mathematical ideas to definite physical problems.
Geometry Of Time-spaces: Non-commutative Algebraic Geometry, Applied To Quantum Theory
Author: Olav Arnfinn Laudal
language: en
Publisher: World Scientific
Release Date: 2011-03-21
This is a monograph about non-commutative algebraic geometry, and its application to physics. The main mathematical inputs are the non-commutative deformation theory, moduli theory of representations of associative algebras, a new non-commutative theory of phase spaces, and its canonical Dirac derivation. The book starts with a new definition of time, relative to which the set of mathematical velocities form a compact set, implying special and general relativity. With this model in mind, a general Quantum Theory is developed and shown to fit with the classical theory. In particular the “toy”-model used as example, contains, as part of the structure, the classical gauge groups u(1), su(2) and su(3), and therefore also the theory of spin and quarks, etc.
The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory
Author: Damien Calaque
language: en
Publisher: American Mathematical Society
Release Date: 2025-05-29
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