Vectors In Hilbert Space Research Articles

Space-Time Code. III

The quantum (q) concept of space-time dynamics described previously is extended from free to interacting systems. The idea is developed that the world is not a plenum of q objects but a plexus of q processes. The q mode of description by Hilbert-space vectors with a special rule of interpretation (stators) ordinarily used for physical objects is used instead for physical processes acting upon objects, including free propagation. The geometrical object proposed as stator for a q network of processes is a natural generalization of a tensor, a Feynman-diagram amplitude, and a chain in a complex, called a plexor. The notation, algebra, and geometry of plexors is illustrated. A strong Mach principle is a self-evident feature of such a q-process theory of space-time, in which it is meaningless to speak of empty space-time or of space-time relations between noninteracting systems.

Photon Spin Formalism

A new approach to quantum electrodynamics is considered, in which photon spin terms are incorporated phenomenologically into the equations of classical electrodynamics in such a way as to yield quantized equations in the Heisenberg picture, for which relativistic requirements are satisfied. Photon momentum and energy analyses in the new formalism are then compared to the corresponding analyses in conventional field-theoretic formalisms, and questions regarding mathematical and physical consistency are discussed. Different notations are used to distinguish between physical and Hilbert space vectors, noting that the Hilbert space of the vector wavefunction has a 3-dimensional subspace resembling physical space, mathematically, in a way that allows interesting results, which can be described by two formalisms when auxiliary notation is necessary to analyze relationships involving both kinds of vectors.

Time dependence of elementary processes

There is a number of inequivalent descriptions of time-dependent processes, which give approximately the same time dependence in standard cases. However, substantial differences occur for other situations. Also, looking for finer details makes only sense if one has an adequate description. Moreover, unique prescriptions are of importance for the study of weak interactions. We therefore analyze typical experimental situations and their connection with quantum theoretic probabilities. The most important cases are considered in detail. If stable, possibly composite particles are observed initially and finally, everything can be based on the exact formulation of multichannel scattering theory. If unstable particles are actually detected, suitable Hilbert space vectors have to be defined. Various properties of the resulting probabilities are pointed out. Usual approaches are critically discussed.

Space-Time Position Operators

A formalism is presented giving the development of $\ensuremath{\psi}(\overline{x})$ in time with the interpretation that $\ensuremath{\psi}(\overline{x})$ is the probability amplitude for observing an event at a space-time point $\overline{x}$. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which $\ensuremath{\psi}(\overline{x})$ is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schr\"odinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of ${P}_{\ensuremath{\mu}}{P}^{\ensuremath{\mu}}$ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.

Unified Model for Stark Broadening

Abstract Stark broadening of spectral lines is considered as a semi-classical many-body problem. Starting from a Liouville equation for a distribution function depending on the atomic Hilbert space vector and the coordinates and velocities of the classical plasma particles, BBGKY hierarchy techniques are used to derive a complete line profile for the electron contribution. The line shape formula is expressed in terms of the atomic time evolution operator for a collision with a single plasma electron. This operator is approximated by a strictly unitary exponential expression, yielding more accuracy than second order perturbation theory and being valid also in the quasi-static limit. The resulting line shape expression covers the whole frequency domain from the impact regime to the quasi-static regime. The results of the impact and quasi-static approximations are recovered as special cases for small and large distances from the line center. A numerical application to Lyman - α shows very good agreement with an experiment of Boldt and Cooper.

Ergodic states in a non commutative ergodic theory

Using the Godement mean ℳ of positive-type functions over a groupG, we study “ℳ-abelian systems” {\(\mathfrak{A}\), α} of aC*-algebra\(\mathfrak{A}\) and a homomorphic mapping α of a groupG into the homomorphism group of\(\mathfrak{A}\). Consideration of the Godement mean off(g)U g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) π(α g (A)) withA e\(\mathfrak{A}\) and π a covariant representation of the system {\(\mathfrak{A}\), α} for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over\(\mathfrak{A}\). Finally we investigate the “discrete spectrum” of covariant representations of {\(\mathfrak{A}\), α} (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).

On the Gram Matrix

The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).

Direct and iterative methods for the solution of linear operator equations in Hilbert space

Introduction. During the last two decades the problem of justifying existing methods and finding new ones for the solution of the equation Lu = f, where f is a given vector in Hilbert space and L is a given operator, has been investigated by many authors(2). This investigation led to the development of a group of direct methods of Ritz, Galerkin, least squares, and moments; and the iterative methods of steepest descent or gradient method, the method with minimal residuals, and their variants. These methods have a long history and have been studied and extensively applied by earlier authors to integral and differential equations. Only recently, however, the Hilbert space operator theory has been used for their study. Thus, using the variational principle, Mikhlin [16] proved the convergence of Ritz method for a self-adjoint positive definite operator L and the convergence of the method of least squares for an invertible operator L. In case L is of the form L = A + B, where A is self-adjoint and positive definite and B satisfies some additional conditions, the method of Galerkin was investigated by Mikhlin [16], Polsky [19], and others [9; 13]. For similar differential operators the method of moments was studied by Kravchuk [11], Polsky [19], and Zdanov [22]. The convergence and the estimate of error of the gradient method for selfadjoint and positive definite operators L have been studied by Kantorovich [7] and Hayes [6]. The latter considered also a slightly more general problem. In case L is a finite, symmetric, and positive definite matrix the method with minimal residuals was developed by Krasnoselsky and Krein [10]. A powerful method related to direct methods was also developed by Murray [18]. The purpose of this paper is to extend the study and the applicability of these methods to a larger class of linear operator equations(3) than those considered by the above authors and to present these seemingly distinct methods in a more unified manner. In our investigation we do not use the variational principle

Properties of solutions of an infinite system of ordinary linear differential equations of the first order with auxiliary boundary conditions

where y(x) (yi(x)) is a vector in Hilbert space, i.e., E= converges and A (x) (A jj(x)) is a matrix each of whose elements is a Lebesgue summable function on X:O < x <1. Furthermore, the matrix A (x) is limited, in the sense defined by Hilbert, by a Lebesgue summable function 4(x) on X. In ?2 preliminary definitions are given and the system of notation used throughout the paper is explained. In ?3 the properties of matrices of functions which satisfy system (1) and the corresponding adjoint system are considered. The adjoint system is given by dzi 0 (2) dx= Aai(X)Za (i = 1,2, ... dx a=1