A Tutorial Review on Fractal Spacetime and Fractional Calculus

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.     

The spaces of trial and generalized functions of infinite number of variables

In this article the spaces of trial and generalized functions of infinite number of variables closely connected with the equipment of Fock space are introduced. The spaces introduced are described in new terminology. The properties of continuity and differentiability of trial functions are studied.     

Quantum Minkowski Space and the q-Analogue of Dirac Equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices. The quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects a q-analogue of Dirac equation follows directly.     

Canonical solutions of variational problems and canonical equations of mechanics

The canonical (non-parametric) solutions of the variational problems for integral functionals are considered and the canonical solutions of variational problems of mechanics in Minkowski spaces are derived. By combining the variational principles of least action, flow, and hyperflow canonically invariant equations for the energy-momentum variable are obtained. From these equations the equations for the action and wave functions as a general solution of the combined variational problems of mechanics are derived. These equations are applicable for describing different types of particles and interactions and are summarized within the approach of general relativity.     

A generalization of the concept of constant mean curvature and canonical time

Some compact spaces of achronal hypersurfaces are constructed in various types of space-time. A variational principle is introduced on these spaces, smooth extremals of which are spacelike hypersurfaces of constant mean curvature. The integrand of the variational principle is shown to be upper semicontinuous and the direct methods of the calculus of variations are applied to obtain aC ⁰ extremal, which is defined to be a spacelike hypersurface of generalized constant mean curvature. The family of such hypersurfaces generated by altering the value of the mean curvature is discussed and the mean curvature itself is shown to have many of the properties of a canonical time coordinate.     

Covariant differential calculus on quantum Minkowski space and theq-analogue of Dirac equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices and quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects aq-analogue of Dirac equation follows directly.     

A Fourier series solution for the transverse vibration response of a beam with a viscous boundary

This paper presents the generalized Fourier series solution for the transverse vibration of a beam subjected to a viscous boundary. The model of the system produces a non-self-adjoint eigenvalue-like problem which does not yield orthogonal eigenfunctions; therefore, such functions cannot be used to calculate the coefficients of expansion in the Fourier series. Furthermore, the eigenfunctions and eigenvalues are complex valued. Nevertheless, the eigenfunctions can be utilized if the space of the operator is extended and a suitable inner product is defined. The methodology presented in this paper utilizes Hilbert space methods and is applicable in general to other problems of this type. As an adjunct to the theoretical discussion, the results from numerical simulations are presented.     

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

The Operator Formulation of Classical Mechanics and Semiclassical Limit

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h ₀. For the later of these two extreme values, introduced operator algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operator framework.     

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat quantum gauge field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley–Wiener–Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation. Received: 29 December 1995 / Accepted: 13 September 1996     

Feynman quantization: An operator approach to path integration over commuting and anticommuting variables

We present an operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval as an extension of the quantization using Feynman path integrals. We define the continuous direct product as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. The extension of the theory to coherent states and anticommuting variables is performed.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

Iterated BRST Cohomology

The iterated BRST cohomology is studied by computing cohomology of the variational complex on the infinite order jet space of a smooth fibre bundle. This computation also provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory.     

Ruelle resonances in kicked quantum spin chain

We study exponential decay of high temperature time correlation functions in a non-integrable quantum spin chain problem, namely Ising spin 1/2 chain kicked with tilted homogeneous magnetic field. For this purpose we define a master propagator over a suitable banach space of quantum observables (quantum many-body analogue of Perron–Frobenius operator) whose leading eigenvalue determines the asymptotic decay of correlations. This is demonstrated with explicit calculation for which a fast algorithm for the construction of the master propagator is developed.     

Spectral Triples and Associated Connes-de Rham Complex for the Quantum <Emphasis Type="Italic">SU</Emphasis>(2) and the Quantum Sphere

In this article, we construct spectral triples for the C^*-algebra of continuous functions on the quantum SU(2) group and the quantum sphere. There have been various approaches towards building a calculus on quantum spaces, but there seem to be very few instances of computations outlined in Chapter 6, [5]. We give detailed computations of the associated Connes-de Rham complex and the space of L₂-forms.The first author would like to acknowledge support from the National Board for Higher Mathematics, India.     

Orientational Order Parameters for Arbitrary Quantum Systems

The concept of quantum-mechanical nematic order, which is important in systems such as superconductors, is based on an analogy to classical liquid crystals, where order parameters are obtained through orientational expansions. This method is generalized to quantum mechanics based on an expansion of Wigner functions. This provides a unified framework applicable to arbitrary quantum systems. The formalism recovers the standard definitions for spin systems. For Fermi liquids, the formalism reveals the nonequivalence of various definitions of the order parameter used in the literature. Moreover, new order parameters for quantum molecular systems with low symmetry are derived, which cannot be properly described with the usual nematic tensors.     

Axioms for quantum theory

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrödinger equations, for phase space amplitudes.     

Path integration in non-relativistic quantum mechanics

A new method for computing path integrals explicitly is developed and applied to problems in non-relativistic quantum mechanics, such as: wave functions, propagators on configuration spaces and on phase space, caustic problems, bound states. Path integrals for paths on curved spaces and for paths on multiply-connected spaces are computed.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

A new spinor calculus

Spinor calculus is developed in a simple, axiomatic manner, without the use of fibre bundles. Tangent vectors of spacetime are identified with self-adjoint endomorphisms of spin spaces. Only one spin space is defined at each point, instead of the two which are required in traditional spinor calculus. The properties of the spin connection and spin curvature are derived. The spinor index formalism and the formalism of complex 3-vectors are discussed.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.