Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.
Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.
Quantum entanglement, a cornerstone of quantum mechanics, remains challenging to classify, particularly in multipartite systems. Here, we present a new interpretation of entanglement classification by revealing a profound connection to Galois groups, the algebraic structures governing polynomial symmetries. This approach not only uncovers hidden geometric relationships between entangled quantum states and polynomial roots but also introduces a method for quantifying entanglement in multi-qubit symmetric states. By reframing the classification of GHZ, W, and separable states within the structure of Galois symmetries, we establish a previously unrecognized hierarchy in their entanglement properties. This work bridges the mathematical elegance of Galois theory with the complexities of quantum mechanics, opening pathways for advances in quantum computing and information theory.
Solving problems in quantum mechanics is an essential skill and research activity for scientists, engineers and others. Nowadays, the labour of scientific computation has been greatly eased by the advent of computer algebra packages. These do not merely perform number-crunching, but enable users to manipulate algebraic expressions and equations symbolically. For example, differentiation and integration can now be carried out algebraically by the computer. This book collects standard and advanced methods in quantum mechanics and implements them using REDUCE, a popular computer algebra package. Throughout, sample programs and their output have been displayed alongside explanatory text, making the book easy to follow. Selected problems have also been implemented using two other popular packages, MATHEMATICA and MAPLE, and in the object-oriented programming language C++. Besides standard quantum mechanical techniques, modern developments in quantum theory are also covered. These include Fermi and Bose Operators, coherent states, gauge theory and quantum groups. All the special functions relevant to quantum mechanics (Hermite, Chebyshev, Legendre and more) are implemented. A careful balance is struck between practical computation and the underlying mathematical concepts, making the book well-suited for use with quantum mechanics courses.
Solving problems in quantum mechanics is an essential skill and research activity for physicists, mathematicians, engineers and others. Nowadays, the labor of scientific computation has been greatly eased by the advent of computer algebra packages, which do not merely perform number crunching, but also enable users to manipulate algebraic expressions and equations symbolically. For example, the manipulations of noncommutative operators, differentiation and integration can now be carried out algebraically by the computer algebra package. This book collects standard and advanced methods in quantum mechanics and implements them using SymbolicC++ and Maxima, two popular computer algebra packages. Throughout, the sample programs and their outputs are accompanied with explanatory text of the underlying mathematics and physics explained in detail. Selected problems have also been implemented using two other popular packages Mathematica and Maple while some problems are implemented in C++. Modern developments in quantum theory are covered extensively, beyond the standard quantum mechanical techniques. The new research topics added to this second edition are: entanglement, teleportation, Berry phase, Morse oscillator, Magnus expansion, wavelets, Pauli and Clifford groups, coupled Bose Fermi systems, super-Lie algebras, etc.
In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We have developed a Python program that generates symbolic formulas of one- and two-electron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. We discuss the possibility of the extension of our scheme to treat the models beyond the Born-Oppenheimer approximation.
This paper gives polynomial time quantum algorithms for computing the ideal class group (CGP) under the Generalized Riemann Hypothesis and solving the principal ideal problem (PIP) in number fields of arbitrary degree. These are are fundamental problems in number theory and they are connected to many unproven conjectures in both analytic and algebraic number theory. Previously the best known algorithms by Hallgren [20] only allowed to solve these problems in quantum polynomial time for number fields of constant degree. In a recent breakthrough, Eisenträger et al. [11] showed how to compute the unit group in arbitrary fields, thus opening the way to the resolution of CGP and PIP in the general case. For example, Biasse and Song [3] pointed out how to directly apply this result to solve PIP in classes of cyclotomic fields of arbitrary degree. The methods we introduce in this paper run in quantum polynomial time in arbitrary classes of number fields. They can be applied to solve other problems in computational number theory as well including computing the ray class group and solving relative norm equations. They are also useful for ongoing cryptanalysis of cryptographic schemes based on ideal lattices [5, 10]. Our algorithms generalize the quantum algorithm for computing the (ordinary) unit group [11]. We first show that CGP and PIP reduce naturally to the computation of S-unit groups, which is another fundamental problem in number theory. Then we show an efficient quantum reduction from computing S-units to the continuous hidden subgroup problem introduced in [11]. This step is our main technical contribution, which involves careful analysis of the metrical properties of lattices to prove the correctness of the reduction. In addition, we show how to convert the output into an exact compact representation, which is convenient for further algebraic manipulations.
Abstract The fundamental entity in the algebraic approach to quantum theory is an abstract C*-algebra 𝒜 together with its set of states ℱ, that is, the set of all positive linear functionals on 𝒜. I shall call such a pair an (abstract) quantum system. The theory of C*-algebras and their representations provides the most powerful abstract approach to quantum theory; unlike the propositional lattice approach it also has direct applications in mainstream physics, and in a certain sense is even forced upon us in quantum field theory (OFT) (sec sections (2.11), (2.16)-(2.18)). Mathematically, the subject is a complex of analysis and algebra; it is a rather wild area involving high degrees of infinity. Philosophically it is most strongly associated with opera tionalism; in part this rellects the predilections of its originators, and in part the continuing philosophical appeal of positivism within the mathematics community. (I do not know why this is so.)