Interest rates in quantum finance: Caps, swaptions and bond options

The prices of the main interest rate options in the financial markets, derived from the Libor (London Interbank Overnight Rate), are studied in the quantum finance model of interest rates. The option prices show new features for the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated.Black’s caplet formula for quantum finance is given an exact path integral derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived in the framework of quantum finance. The approximate Libor option prices are derived using the volatility expansion.The BGM-Jamshidian (Gatarek et al. (1996) [1], Jamshidian (1997) [2]) result for the Libor swaption prices is obtained as the limiting case when all the Libors are exactly correlated. A path integral derivation is given of the approximate BGM-Jamshidian approximate price.     

Correlations for a Family of Two-Ququart Quantum States

In this paper, a quantum correlations analytical treatment for a family of two-ququart states are presented. For this purpose, a family of two-ququart quantum states, as well as a set of von Neumann measurements are proposed. A quantum discord analytical treatment is derived for this family of two-ququart states. Finally, the classical mutual information, the entanglement of formation and quantum discord are illustrated in this family of two-ququart quantum states.

     

Cosmological perturbations and quantum fields in curved space

We identify the quantum theory of cosmological perturbations with the quantum field theory in curved spacetime with emphasis on its field concept. We materialize this idea by using a coherent state as a quantum analogue of a nontrivial classical field configuration. We present analytic results in a de Sitter universe for the massless and massive minimal free scalar fields. Some new features on the spectrum of perturbations are obtained for the massive case. We also show how such quantum field theories can be derived from quantum gravity using the semiclassical approximation. A physical degree of freedom is picked up from three scalar perturbations in the quantum gravity scalar system and its Schrödinger equation is derived. Peculiar features of quantum fields at imaginary time and its possible implications on boundary conditions for the wave function of the universe are also discussed.     

Quantum Maps with Memory from Generalized Lindblad Equation

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.     

Quantum transport theory for abelian plasmas

The gauge invariant relativistic quantum equations of motion for the fermion and photon Wigner operators are derived from QED. In the mean field (Hartree) approximation, we extract the generalized quantum Vlasov and mass-shell constraint equations for fermions. In addition, a complete spinor decomposition is performed. A systematic method for computing quantum corrections to all orders in h is developed. First order quantum (spin) corrections are computed explicitly. Finally, the relations between gauge dependent and independent definitions of the photon Wigner function and their corresponding transport equations are discussed.     

An Alternative Derivation of Quantum Mechanical Transition State Reaction Rate Theory

The quantum mechanical transition state reaction rates for both microcanonical and macrocanonical expressions are derived from a recently developed classical reaction rate theory. The newly derived quantum transition-state-like rates are good for both unimolecular reaction and bimolecular reaction depending on the choices of the transition state dividing surfaces. In a special choice of the transition dividing surface for bimolecular reaction, the macrocanonic rate derived in this paper is reduced to the flux-flux autocorrelation formalism derived from quantum bimolecular reactive scattering by Miller and to those obtained by Yamamo to using Ku to linear response an alysis.     

Quantum Einstein’s equations and constraints algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.     

Simple WZW currents

A complete classification of simple currents of WZW theories is obtained. The proof is based on an analysis of the quantum dimensions of the primary fields. Simple currents are precisely the primaries with unit quantum dimension; for WZW theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible.     

Dressed ion acoustic solitary waves in quantum plasmas with two polarity ions and relativistic electron beams

A theory for dressed quantum ion acoustic waves (QIAWs), which includes higher-order corrections when QIAWs are investigated by the reductive perturbation method, is presented for unmagnetized plasmas containing positive and negative ions and weakly relativistic electron beams. The properties of the QIAWs are investigated using a quantum hydrodynamic model, from which a Korteweg–de Vries equation is derived using the reductive perturbation method. An equation including higher-order dispersion and nonlinearity corrections is also derived, and the physical parameter space is discussed for the importance of these corrections.     

New longitudinal waves in electron-positron-ion quantum plasmas

A general quantum dispersion equation for electron-positron(hole)-ion quantum plasmas is derived and studied for some interesting cases. In an electron-positron-ion degenerate Fermi gas, with or without the Madelung term, a new type of zero sound waves are found. Whereas in an electron-hole-ion plasmas a new longitudinal quantum waves are revealed, which have no analogies in quantum electron-ion plasmas. The excitation of these quantum waves by a low-density monoenergetic straight electron beam is examined. Furthermore, the Korteweg-de Vries (KdV) equation for novel quantum waves is derived and the contribution of the Madelung term in the formation of the KdV solitons is discussed.     

Supersymmetry algebras that include topological charges

We show that in supersymmetric theories with solitons, the usual supersymmetry algebra is not valid; the algebra is modified to include the topological quantum numbers as central charges. Using the corrected algebra, we are able to show that in certain four dimensional gauge theories, there are no quantum corrections to the classical mass spectrum. These are theories for which Bogomolny has derived a classical bound; the argument involves showing that Bogomolny's bound is valid quantum mechanically and that it is saturated.     

Quantum Teardrops

Algebras of functions on quantum weighted projective spaces are introduced, and the structure of quantum weighted projective lines or quantum teardrops is described in detail. In particular the presentation of the coordinate algebra of the quantum teardrop in terms of generators and relations and classification of irreducible *-representations are derived. The algebras are then analysed from the point of view of Hopf-Galois theory or the theory of quantum principal bundles. Fredholm modules and associated traces are constructed. C*-algebras of continuous functions on quantum weighted projective lines are described and their K-groups computed.     

Classical-like description of quantum states and propagator for particles in time-independent and dispersing δ potentials

Probability distributions that determine completely the bound state in the problem of a quantum particle in one or two delta-function wells are derived within the recently developed tomographic representation of quantum states, where a state is characterized by a positive probability distribution. The quantum propagator for the Schrödinger equation is obtained for the problem of two dispersing delta-function wells.     

Loss of nonclassical properties of quantum states in linear phase-insensitive processes with arbitrary time-dependent parameters

Conditions of disappearance of different “nonclassical” properties (usual and high-order squeezing, sub-Poissonian statistics, negativity of s-parametrized quasidistributions) are derived for a quantum oscillator, whose evolution is governed by the standard master equation of quantum optics with arbitrary time-dependent coefficients.     

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.     

Propagation of electrostatic energy through a quantum plasma

The theory of propagation of electrostatic energy through an infinite, homogeneous electron–ion quantum plasma is presented. Simple expressions for the energy flow, energy density, and energy velocity of longitudinal oscillation waves in the system are derived using the linearized quantum hydrodynamic theory for the electron fluid, which incorporates the important quantum statistical pressure and electron diffraction force, while the optical response of the ion particles is characterized by the classical frequency‐dependent dielectric function, ?_ion. Both cases of plasmon (high‐frequency) and quantum ion‐acoustic (low‐frequency) waves are considered.     

Transformation Properties of Dynamical Systems at the Quantum Level

Based on the generating functional of Green function for a dynamical system, the general equations of transformation properties at the quantum level are derived. In some cases they can be reduced to the quantum Noether theorem. In some other cases they can be reduced to momentum theorem or angular momentum theorem etc. at the quantum level. An example is presented and it shows that the classical conservation laws don’t always preserve in quantum theories. PACS: 11.10.E     

Quantum-classical correspondence for motion on a plane with deficit angle

A particle constrained to move on a cone and bound to its tip by harmonic oscillator and Coulomb-Kepler potentials is considered both in the classical as well as in the quantum formulations. The SU(2) coherent states are formally derived for the former model and used for showing some relations between closed classical orbits and quantum probability densities. Similar relations are shown for the Coulomb-Kepler problem. In both cases a perfect localization of the densities on the classical solutions is obtained even for low values of quantum numbers.     

Quantum Mechanics from Focusing and Symmetry

A foundation of quantum mechanics based on the concepts of focusing and symmetry is proposed. Focusing is connected to c-variables—inaccessible conceptually derived variables; several examples of such variables are given. The focus is then on a maximal accessible parameter, a function of the common c-variable. Symmetry is introduced via a group acting on the c-variable. From this, the Hilbert space is constructed and state vectors and operators are given a definite interpretation. The Born formula is proved from weak assumptions, and from this the usual rules of quantum mechanics are derived. Several paradoxes and other issues of quantum theory are discussed.