COMMON ANALYTIC BASIS OF THE ζ-FUNCTION AND THE DIMENSIONAL REGULARIZATION SCHEMES

1989 ◽  
Vol 04 (09) ◽  
pp. 863-867
Author(s):  
DAKSH LOHIYA
Keyword(s):  
Analytic Continuation ◽  
Dimensional Regularization ◽  
Zeta Functions ◽  
Infinite Dimensional ◽  
Analytic Methods

The analytic continuation invoked in the theory of generalized zeta functions associated with infinite-dimensional operators is shown to be equivalent in structure to the basic analytic methods deployed in dimensional regularization.

scholarly journals Analytic continuation of multiple zeta-functions and their values at non-positive integers

2001 ◽  
Vol 98 (2) ◽  
pp. 107-116 ◽  
Author(s):  
Shigeki Akiyama ◽  
Shigeki Egami ◽  
Yoshio Tanigawa
Keyword(s):  
Analytic Continuation ◽  
Zeta Functions ◽  
Multiple Zeta ◽  
Positive Integers

scholarly journals On the Local Regularization of Inverse Problems of Volterra Type

1995 ◽  
Author(s):  
Patricia K. Lamm
Keyword(s):  
Tikhonov Regularization ◽  
Volterra Operator ◽  
Dimensional Regularization ◽  
Convergence Theory ◽  
True Solution ◽  
Infinite Dimensional ◽  
Local Regularization ◽  
Regularization Parameters ◽  
Regularization Problem ◽  
Local Procedure

Abstract We consider a local regularization method for the solution of first-kind Volterra integral equations with convolution kernel. The local regularization is based on a splitting of the original Volterra operator into “local” and “global” parts, and a use of Tikhonov regularization to stabilize the inversion of the local operator only. The regularization parameters for the local procedure include the standard Tikhonov parameter, as well as a parameter that represents the length of the local regularization interval. We present a convergence theory for the infinite-dimensional regularization problem and show that the regularized solutions converge to the true solution as the regularization parameters go to zero (in a prescribed way). In addition, we show how numerical implementation of the ideas of local regularization can lead to the notion of “sequential Tikhonov regularization” for Volterra problems; this approach has been shown in (Lamm and Eldén, 1995) to be just as effective as Tikhonov regularization, but to be much more efficient computationally.

scholarly journals WIGNER ROTATIONS, BELL STATES, AND LORENTZ INVARIANCE OF ENTANGLEMENT AND VON NEUMANN ENTROPY

2004 ◽  
Vol 02 (02) ◽  
pp. 183-200 ◽  
Author(s):  
CHOPIN SOO ◽  
CYRUS C. Y. LIN
Keyword(s):  
Lorentz Invariance ◽  
Zeta Functions ◽  
Bell States ◽  
Von Neumann Entropy ◽  
Lorentz Transformations ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Einstein Podolsky Rosen ◽  
Definition Of ◽  
Reduced Density

We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.

A Generalization of Epstein Zeta Functions

1949 ◽  
Vol 1 (4) ◽  
pp. 320-327 ◽  
Author(s):  
S. Minakshisundaram
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
General Class ◽  
Zeta Functions ◽  
Epstein Zeta Function ◽  
Image Position

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.

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