Given the rapid pace of development in economics and finance, a concise and up-to-date introduction to mathematical methods has become a prerequisite for all graduate students, even those not specializing in quantitative finance. This book offers an introductory text on mathematical methods for graduate students of economics and finance–and leading to the more advanced subject of quantum mathematics. The content is divided into five major sections: mathematical methods are covered in the first four sections, and can be taught in one semester. The book begins by focusing on the core subjects of linear algebra and calculus, before moving on to the more advanced topics of probability theory and stochastic calculus. Detailed derivations of the Black-Scholes and Merton equations are provided – in order to clarify the mathematical underpinnings of stochastic calculus. Each chapter of the first four sections includes a problem set, chiefly drawn from economics and finance. In turn, section five addresses quantum mathematics. The mathematical topics covered in the first four sections are sufficient for the study of quantum mathematics; Black-Scholes option theory and Merton’s theory of corporate debt are among topics analyzed using quantum mathematics.

This book provides an introduction to how the mathematical tools from quantum field theory can be applied to economics and finance. Providing a range of quantum mathematical techniques for designing financial instruments, it demonstrates how a range of topics have quantum mechanical formulations, from asset pricing to interest rates.

The word "quantum" is from the Latin for "how much" and in this book mathematician David Orrell shows how it applies to the world of economic transactions. Written in clear and accessible language, the book covers the essential mathematics behind topics such as quantum cognition, option pricing, and quantum game theory, and delves into the nuts and bolts of quantum mechanics, the principles of quantum economic modelling, and the basics of quantum computation. On the way the reader will learn how quantum interference can be used to model cognitive dissonance, how a quantum walk goes further than a random walk, and how financial entanglement explains the rate of mortgage default. It is aimed at anyone who wants to understand the quantum ideas working their way into economics and finance, without getting drowned in wave equations. As interest in quantum computing grows, many companies from established banks to startups are looking at ways to perform financial simulations using quantum algorithms. But what if we should be using quantum models anyway - because the monetary system has quantum properties of its own, and because they work? The field is developing rapidly, and this second edition contains many updates including new material on quantum logic and quantum agent-based models, and a guest chapter from Ramy Aboushelbaya and Marko Mayr of Quantum Dice on quantum hardware. David Orrell is an applied mathematician with extensive experience in mathematical modelling, and the author of a dozen books on science and economics.

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).

This book presents various theories and algorithms to create a quantum computer. The concept of the classical and quantum computers, and the concept of circuits and gates are reviewed. The example of the Deutsch and the Deutsch-Josca algorithm is discussed to illustrate some key features of quantum computing. The Grover algorithm, considered to be of major milestone of the subject, is discussed in detail to exemplify the techniques used in computer algorithms. The role of quantum superposition (also called quantum parallelism) and of quantum entanglement is discussed in order to understand the key advantages of a quantum over a classical computer.

This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics is dominated by stochastic calculus. The present book offers a formulation that is completely independent of that approach. As such many results emerge from the ideas developed by the author. This work will be of interest to physicists and mathematicians working in the field of finance, to quantitative analysts in banks and finance firms and to practitioners in the field of fixed income securities and foreign exchange. The book can also be used as a graduate text for courses in financial physics and financial mathematics.

A decade after the financial crisis, there is a growing consensus that economics has failed and needs to go back to the drawing board. David Orrell argues that it has been trying to solve the wrong problem all along. Economics sees itself as the science of scarcity. Instead, it should be the science of money (which plays a surprisingly small role in mainstream theory). And money is a substance that turns out to have a quantum nature of its own. Just as physicists learn about matter by studying the exchange of particles at the subatomic level, so economics should begin by analysing the nature of money-based transactions. Quantum Economics therefore starts with the meaning of the phrase 'how much' – or, to use the Latin word, quantum. From quantum physics to the dualistic properties of money, via the emerging areas of quantum finance and quantum cognition, this profoundly important book reveals that quantum economics is to neoclassical economics what quantum physics is to classical physics – a genuine turning point in our understanding.

Professor Kok Khoo Phua is the Founding Director and Emeritus Professor of the Institute of Advanced Studies (IAS) at Nanyang Technological University (NTU) and Adjunct Professor of Department of Physics both at Nanyang Technological University (NTU) and National University of Singapore (NUS). He is the Chairman and Editor-in-Chief of World Scientific Publishing Co Pte Ltd.When he was elected a Fellow of the American Physical Society (APS) in 2009, the citation read: 'For tireless efforts to strengthen scientific research throughout Asia and promote international physics education and scholarly exchanges, and for enriching science and education through the World Scientific Publishing Company he founded.'This unique volume on the occasion of his 80th birthday is a compilation of tributes from his friends who have known him for decades along with scientific articles that celebrate his visionary approach to promote science worldwide.

The purpose of the Special Issue “Quantitative Methods in Economics and Finance” of the journal Risks was to provide a collection of papers that reflect the latest research and problems of pricing complex derivates, simulation pricing, analysis of financial markets, and volatility of exchange rates in the international context. This book can be used as a reference for academicians and researchers who would like to discuss and introduce new developments in the field of quantitative methods in economics and finance and explore applications of quantitative methods in other business areas.

Control of the molecular alignment or orientation by laser pulses / Arne Keller -- Quantum computing and devices : A short introduction / Zhigang Zhang, Viswanath Ramakrishna and Goong Chen -- Dynamics of mixed classical-quantum systems, geometric quantization and coherent states / Hans-Rudolf Jauslin and Dominique Sugny -- Quantum memories as open systems / Robert Alicki -- Two mathematical problems in quantum information theory / Alexander S. Holevo -- Dissipatively induced bipartite entanglement / Fabio Benatti -- Scattering in nonrelativistic quantum field theory / Jan Derezinski -- Mathematical theory of atoms and molecules / Volker Bach