Definite Inner Product Research Articles

A note on Euclidean spaces $\mathbb{R}^n$ and $n$-normed spaces

This article gives us a relation between Euclidean space $\mathbb{R}^n$ and a subspace of $X$ (an $n$-normed space) using properties of determinant of square matrices. Some functionals will be investigated on these spaces. Furthermore, we obtain definition of inner product and $m$-inner product with $m<n$ on the subspace of $X$.

Noncommutative quantum cosmology with perfect fluid

In this paper, we study a noncommutative quantum description of Kantowski–Sachs cosmological model in the presence of perfect fluid matter field which governs the time evolution of the wave function for the universe. The effect of the noncommutative parameter for this cosmological model has also been analyzed through the behavior of the probability density function. It is observed that a multiverse universe model is possible even in the absence of noncommutativity. The effect of noncommutativity is revealed by a decrease in the highest value of the individual peaks. We also discuss on the unitarity of the model and show that it can be restored with a proper definition of inner product as it has been shown to happen in other models of anisotropic quantum cosmologies.

Positive and Negative Solutions of a Class of Fractional Schrödinger Equation

In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and <i>L^p</i> space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions <i>u</i>⁺, <i>u</i>⁻ and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term <i>f</i>, it is proved that functional <I>I</I> satisfies mountain road structure. Secondly, we need to prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition, we first prove that the sequence <i>u_n</i> is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition. Then, we define functional <I>I</I>^± and its inner product form to verify that functional <I>I</I>^± also has mountain path structure and satisfies (<i>C_c</i>) condition. Finally, taking <i>u</i>⁺ and <i>u</i>⁻ as experimental functions respectively, it is verified that they are the solutions of functional <I>I</I>. It is obtained that both <i>u</i>⁺ and <i>u</i>⁻ are the solutions of functional <I>I</I>. Therefore, we get the conclusion.

Translation, modulation and dilation systems in set-valued signal processing

In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p&lt;\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.

Straight-line motion of classical point particles in a three-dimensional lattice

By means of the usual definition of inner product and of the Gauss condition on the sum of squares of three integers in number theory, it can be seen that there exist specific directions in continuous space in which a classical point particle moving in a three-dimensional lattice cannot propagate. When representing all directions for which propagation is possible as points on a unitary sphere, the forbidden directions appear as vacancies on this sphere. By means of a stereographic projection of the allowed direction, it is argued that propagation is not allowed for specific sets of points on the stereographic plane. The present work can be considered as an interdisciplinary lecture for advanced high-school students or to first-year college students.

On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation

Hybrid finite element methods are superior to conventional displacement-based finite element formulations in many aspects and to date many types of hybrid elements have been proposed based on different assumed stress fields. However, some of them are essentially equivalent to each other and there is no general approach available yet to determine their corresponding relationships. In addition, even though the hybrid techniques have many advantages compared to the displacement-based elements, there is an obvious disadvantage, the additional computational cost of the matrix inversions required in constructing element stiffness matrix. In this paper, we show that a linearly independent transformation of the assumed stress field for a given element leads to an equivalent hybrid finite element. This is verified through the proof of equivalence between various existing hybrid elements proposed by different researchers. To improve the computational efficiency, a new inner product including the compliant matrix of material is introduced. With this new definition of inner product, a simplified orthogonalization procedure for the assumed stress modes is then proposed. Based on the orthonorm stress modes obtained using the proposed orthogonalization, an explicit formulation of the hybrid element stiffness matrix is presented. The effectiveness of the present method is validated by numerical examples.

Rigid Particle and its Spin Revisited

The arguments by Pandres that the double valued spherical harmonics provide a basis for the irreducible spinor representation of the three dimensional rotation group are further developed and justified. The usual arguments against the inadmissibility of such functions, concerning hermiticity, orthogonality, behavior under rotations, etc., are all shown to be related to the unsuitable choice of functions representing the states with opposite projections of angular momentum. By a correct choice of functions and definition of inner product those difficulties do not occur. And yet the orbital angular momentum in the ordinary configuration space can have integer eigenvalues only, for the reason which have roots in the nature of quantum mechanics in such space. The situation is different in the velocity space of the rigid particle, whose action contains a term with the extrinsic curvature.

Two-component wave formalism in spherical open systems

We study wave evolution in open dielectric spheres by expanding the wave field and its conjugate momentum?the two components?in terms of relevant quasi-normal modes (QNMs), which are complete under appropriate conditions. We first establish a novel outgoing boundary condition at the surface of a sphere for waves emanating from its interior. A proper definition of inner product for two-component outgoing wavefunctions, involving only the waves inside the sphere and a surface term, can then be defined in general. The orthogonality relation of QNMs and hence a unique expansion in terms of the QNM basis are found, which can be applied to solve for the evolution of waves inside open dielectric cavities. Furthermore, a time-independent perturbation for QNMs can also be developed.

State-Space-Based Approach to Quantum Field Theory for Arbitrary Observers in Electromagnetic Backgrounds

A reformulation of fermionic QFT in electromagnetic backgrounds is presented which uses methods analogous to those of conventional multiparticle quantum mechanics. Emphasis is placed on the (Schrödinger picture) states of the system, described in terms of Slater determinants of Dirac states, and not on the field operator ψ̂(x) (which is superfluous in this approach). The vacuum state “at time τ” is defined as the Slater determinant of a basis for the span of the negative spectrum of the “first quantized” Hamiltonian H(τ), thus providing a concrete realisation of the Dirac Sea. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that the S-matrix follows directly from the definition of inner product between Slater determinants. The process of “Hermitian extension,” inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes and provides an extremely straightforward derivation of the general expectation value of the theory. The concept of “radar time,” advocated by Bondi in his work on k-calculus, is used to generalise the particle interpretation to an arbitrarily moving observer. A definition of particle results, which depends only on the observer's motion and the background present, not on any choice of coordinates or gauge, or of the particle detector. We relate this approach to conventional methods by comparing and contrasting various derivations. Our particle definition can be viewed as a generalisation to arbitrary observers of the approach of G. W. Gibbons (1975, Comm. Math. Phys.44, 245).

Second Quantization of the Dirac Field: Normal Modes in the Robertson–Walker Space-Time

The quantization of the Dirac field in thecontext of the Robertson–Walker spacetime isreconsidered in some of its constitutive elements. Theparticular solutions of the Dirac equation previouslydetermined are used to construct the normal mode solutionsin the case of flat, closed, and open space-time. Theprocedure is based on a general standard definition ofinner product between solutions of the Dirac equation that is applied by making use of anintegral property of the separated time equation. Theopen-space case requires the recurrence relations offunctions associated to solutions of the Diracequation.

Quantum Theory of Gravity. I. The Canonicál Theory

Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface ${x}^{0}=\mathrm{constant}$, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a term. The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface ${x}^{0}=\mathrm{constant}$. The label ${x}^{0}$ itself is irrelevant, and time must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of time. The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.

Quantum Theory of Gravity. I. The Canonical Theory

Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface ${x}^{0}=\mathrm{constant}$, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a term. The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface ${x}^{0}=\mathrm{constant}$. The label ${x}^{0}$ itself is irrelevant, and time must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of time. The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.