An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity, and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of this unbounded transformation is explicitly evaluated. The mapping fails for a critical value of the coupling constants.

The Euclidean action with acceleration has been analyzed in Ref. 1, and referred to henceforth as Paper I, for its Hamiltonian and path integral. In this paper, the state space of the Hamiltonian is analyzed for the case when it is pseudo-Hermitian (equivalent to a Hermitian Hamiltonian), as well as the case when it is inequivalent. The propagator is computed using both creation and destruction operators as well as the path integral. A state space calculation of the propagator shows the crucial role played by the dual state vectors that yields a result impossible to obtain from a Hermitian Hamiltonian. When it is not pseudo-Hermitian, the Hamiltonian is shown to be a direct sum of Jordan blocks.

An index-linked coupon bond is defined that pays coupons whose values are stochastic, depending on a market defined index. This is an asset class distinct from the existing coupon bonds. The index-linked coupon bond is an example of a sukuk, which is an instrument that implements one of the cornerstones of Islamic finance (Askari et al., 2012): that an investor must share in the risk of the issuer in order to earn profits from the investment. The index-linked coupon bond is defined using the mathematical framework of quantum finance (Baaquie, 2004, 2010). The coupons are stochastic, with the quantum of coupon payments depending on a publicly traded index that is chosen to reflect the primary drivers of the revenues of the issuer of the bond. The index ensures there is information symmetry; regarding the quantum of coupon being paid; between issuer and investor. The dependence of the coupon on the index is designed so that the variation of the index mirrors the changing fortunes of the issuer, with the coupon's quantum increasing for increasing values of the index and conversely, decreasing with a fall of the index.

Financial instruments have a random evolution and can be described by a stochastic process. It is shown that another approach for modeling financial instruments considered as a (classical) random system is by employing the mathematics that results from the formalism of quantum mechanics. Financial instruments are described by the elements of a linear vector state space and its evolution is determined by a Hamiltonian operator. It is further shown that interest rates can be described by a random function which is mathematically equivalent to a two dimensional Euclidean quantum field.

The economic crisis of 2008 has shown that the capital markets need new theoretical and mathematical concepts to describe and price financial instruments. Focusing almost exclusively on interest rates and coupon bonds, this book does not employ stochastic calculus - the bedrock of the present day mathematical finance - for any of the derivations. Instead, it analyzes interest rates and coupon bonds using quantum finance. The Heath-Jarrow-Morton and the Libor Market Model are generalized by realizing the forward and Libor interest rates as an imperfectly correlated quantum field. Theoretical models have been calibrated and tested using bond and interest rates market data. Building on the principles formulated in the author's previous book (Quantum Finance, Cambridge University Press, 2004) this ground-breaking book brings together a diverse collection of theoretical and mathematical interest rate models. It will interest physicists and mathematicians researching in finance, and professionals working in the finance industry.

Quantum Chromodynamics is the theory of strong interactions: a quantum field theory of colored gluons (Yang-Mills gauge fields) coupled to quarks (Dirac fermion fields). Lattice gauge theory is defined by discretizing spacetime into a four-dimensional lattice - and entails defining gauge fields and Dirac fermions on a lattice. The applications of lattice gauge theory are vast, from the study of high-energy theory and phenomenology to the numerical studies of quantum fields. This book examines the mathematical foundations of lattice gauge theory from first principles. It is indispensable for the study of Dirac and lattice gauge fields and lays the foundation for more advanced and specialized studies.

We study the low energy behavior of the four-dimensional nonlinear sigma model with anomaly using the 2+ expansion and renormalization group methods. It is shown that the theory has a non-trivial ir stable fixed point, in addition to the usual trivial fixed point. If pions happen to exist in the non-trivial phase, their propagator would scale at low energies with anomalous exponents.

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 ...

Merton has proposed a model of the contingent claims on a firm as an option on the firms value, and the model is based on a generalization of the Black-Scholes stochastic equation. Merton's model can be used to price any contingent claim on the firm. A risk-sharing oscillator model for the pricing of corporate coupon bonds is proposed that leads to stochastic coupons, with the dynamics of the contingent claims being determined by the quantum oscillator. The oscillator model allows for the exact derivation of many results using quantum mathematics. The price of the risk-sharing coupon bonds and the stochastic coupons is derived exactly using the Feynman path integral.

An option pricing formula, for which the price of an option depends on both the value of the underlying security as well as the velocity of the security, has been proposed in Baaquie and Yang (2014). The FX (foreign exchange) options price was empirically studied in Baaquie et al., (2014), and it was found that the model in general provides an excellent fit for all strike prices with a fixed model parameters; unlike the Black-Scholes option price Hull and White (1987) that requires the empirically determined implied volatility surface to fit the option data. The option price proposed in Baaquie and Cao Yang (2014) did not fit the data during the crisis of 2007 & 2008. We make a hypothesis that the failure of the option price to fit data is an indication of the market's large deviation from its near equilibrium behavior due to the market's instability. Furthermore, our indicator of market's instability is shown to be more accurate than the option's observed volatility. The market prices of the FX option for various currencies are studied in the light of our hypothesis.

The U(1) Kac-Moody character functions are derived from a path integral expression. It is shown that point-split regularization of the Virasoro generator also exactly regularizes the path integral. An exact derivation of the Weyl-Kac denominator is then given for an arbitrary Lie group using the semiclassical approximation and the results of the U(1) calculation.

Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrodinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.

An introduction to how the mathematical tools from quantum field theory can be applied to economics and finance, this book provides a wide range of quantum mathematical techniques for designing financial instruments. The ideas of Lagrangians, Hamiltonians, state spaces, operators and Feynman path integrals are demonstrated to be the mathematical underpinning of quantum field theory and are employed to formulate a comprehensive mathematical theory of asset pricing as well as of interest rates, which are validated by empirical evidence. Numerical algorithms and simulations are applied to the study of asset pricing models as well as of nonlinear interest rates. A range of economic and financial topics is shown to have quantum mechanical formulations, including options, coupon bonds, nonlinear interest rates, risky bonds and the microeconomic action functional. This is an invaluable resource for experts in quantitative finance and in mathematics who have no specialist knowledge of quantum field theory.

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.

At the end of the 19th century Sigmund Freud discovered that our acts and choices are not only decisions of our consciousness, but that they are also deeply determined by our unconscious (the so-called "Freudian unconscious"). During a long correspondence between them (1932-1958) Wolfgang Pauli and Carl Gustav Jung speculated that the unconscious could be a quantum system. This book is addressed both to all those interested in the new developments of the age-old enquiry in the relations between mind and matter, and also to the experts in quantum physics that are interested in a formalisation of this new approach. The description of the "Bilbao experiment" adds a very interesting experimental inquiry into the synchronicity effect in a group situation, linking theory to a quantifiable verification of these subtle effect

Risk free forward interest rates (Diebold and Li, 2006 [1]; Jamshidian, 1991 [2]) and their realization by US Treasury bonds as the leading exemplar have been studied extensively. In Baaquie (2010), models of risk free bonds and their forward interest rates based on the quantum field theoretic formulation of the risk free forward interest rates have been discussed, including the empirical evidence supporting these models. The quantum finance formulation of risk free forward interest rates is extended to the case of risky forward interest rates. The examples of the Singapore and Malaysian forward interest rates are used as specific cases. The main feature of the quantum finance model is that the risky forward interest rates are modeled both a) as a stand-alone case as well as b) being driven by the US forward interest rates plus a spread having its own term structure above the US forward interest rates.

The statistical theory of commodity prices has been formulated by Baaquie (2013). Further empirical studies of single (Baaquie et al., 2015) and multiple commodity prices (Baaquie et al., 2016) have provided strong evidence in support the primary assumptions of the statistical formulation. In this paper, the model for spot prices (Baaquie, 2013) is extended to model futures commodity prices using a statistical field theory of futures commodity prices. The futures prices are modeled as a two dimensional statistical field and a nonlinear Lagrangian is postulated. Empirical studies provide clear evidence in support of the model, with many nontrivial features of the model finding unexpected support from market data.

The theory of commodity pricing is one of the foundations of economic theory and applications. A mathematical model is proposed, from first principles and based on the formalism of statistical physics, for describing the prices of commodities. Both spot and futures prices are analyzed. The calibration and predictions of the model, based on market data, provide strong evidence in support of the model. Consider the behavior of market prices. As can be seen from Figure 8.1, the price of silver and gold appear to have a random time evolution. Furthermore, the two prices seem to be positively correlated; in contrast, the price of gold and oil seem to be negatively correlated. Market data seems to indicate that commodity prices are stochastic variables, and it is this feature of market prices that leads to its statistical modelling. Statistical microeconomics takes the commodity prices as random stochastic processes, and in particular, aims to explain the auto- and cross-correlation of commodity prices ...

A statistical generalization is made of microeconomics in the spirit of going from classical to statistical mechanics. The price and quantity of every commodity1 traded in the market, at each instant of time, is considered to be an independent random variable: all prices and quantities are considered to be stochastic processes, with the observed market prices being a random sample of the stochastic prices. The dynamics of market prices is determined by an action functional and, for concreteness, a specific model is proposed. The model can be calibrated from the unequal time correlation of the market commodity prices. A perturbation expansion for the correlation functions is defined in powers of the inverse of the total budget of the aggregate consumer and the propagator for the market prices is evaluated.

A review is made of the statistical generalization of microeconomics by Baaquie (Baaquie 2013 Phys. A 392, 4400-4416. (doi:10.1016/j.physa.2013.05.008)), where the market price of every traded commodity, at each instant of time, is considered to be an independent random variable. The dynamics of commodity market prices is given by the unequal time correlation function and is modelled by the Feynman path integral based on an action functional. The correlation functions of the model are defined using the path integral. The existence of the action functional for commodity prices that was postulated to exist in Baaquie (Baaquie 2013 Phys. A 392, 4400-4416. (doi:10.1016/j.physa.2013.05.008)) has been empirically ascertained in Baaquie et al. (Baaquie et al. 2015 Phys. A 428, 19-37. (doi:10.1016/j.physa.2015.02.030)). The model's action functionals for different commodities has been empirically determined and calibrated using the unequal time correlation functions of the market commodity prices using a perturbation expansion (Baaquie et al. 2015 Phys. A 428, 19-37. (doi:10.1016/j.physa.2015.02.030)). Nine commodities drawn from the energy, metal and grain sectors are empirically studied and their auto-correlation for up to 300 days is described by the model to an accuracy of R2>0.90 using only six parameters.

A model of the firm is proposed that considers the firm to be a stochastic and random entity described by an action functional and the Feynman path integral. The action functional is postulated based on the profit maximization principle. The Cobb-Douglas production function and the Solow-Swan model for capital input are employed to define a specific model for the firm's action functional. An option is defined on the profit of a firm in the framework of the statistical model. The option's price can be studied empirically. A profit and loss sharing system of wages is defined as an extension of fixed wages.