CORNELIUS LANCZOS
COLLECTED PUBLISHED PAPERS
WITH COMMENTARIES
Published by
North Carolina State
University
College of Physical and Mathematical Sciences and Department
of Physics
Raleigh, North Carolina 276958202 USA [1998]
Library of Congress Catalog Card Number: 9867928
ISBN (for the Collection of Volumes IVI): 092949301X
Copyright 1997 and 1999 North Carolina State University
Printed by the OOK Press, Veszprém, Hungary
[See Web: www.physics.ncsu.edu/lanczos]
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Published in cooperation with and support from the Eötvös Physical Society, Hungary, with additional support for printing from the Swiss Academy of Sciences SAS and the Swiss National Science Foundation.
Cornelius Lanczos (18931974) was a physicist and mathematician who had a profound impact on the foundations of twentieth century science. His papers cover a vast array of disciplines, including general relativity, quantum mechanics, scientific computation, applied mathematics and numerical analysis. This Collection provides documentation (a) that Lanczos was indeed one of the twentieth century's most versatile and innovative scientific minds, and (b) that many of Lanczos's ideas are still of interest to presentday research in physics and applied mathematics. This Collection will be of special interest to theoretical physicists, numerical analysts and science historians. The Cornelius Lanczos Collected Published Papers with Commentaries (Lanczos Collection or CLCPPC hereinafter referred to as the Collection) represents the second phase of a twopart celebration of the life and work of Cornelius Lanczos. The first phase of this celebration occurred in December 1993 when North Carolina State University's College of Physical and Mathematical Sciences hosted the Cornelius Lanczos International Centenary Conference. [See the Proceedings of the Cornelius Lanczos International Centenary Conference, J. D. Brown, M. T. Chu, D. C. Ellison and R. J. Plemmons, eds. (Philadelphia, PA: Society for Industrial and Applied Mathematics [SIAM], 1994.]
This Collection is the culmination of an eightyear effort involving 15 editors, 4 translators and 40 other contributors from departments of physics, mathematics, mathematical sciences, computer science, electrical engineering, statistics, multidisciplinary studies and English at universities and research institutes and centers in the United States, Australia, Canada, Germany, Hungary, Ireland, Mauritius, New Zealand, Sweden, Switzerland, and the United Kingdom. A list of these editors and contributors is given on the following pages. The first three sets of this Collection were distributed to the Library of Congress, the Swiss National Library in Bern, and the Albert Einstein Archives in the The Jewish National and University Library at The Hebrew University of Jerusalem, Israel.
This sixvolume, over 3,200page library edition printed on acidfree paper in B5 format (168 mm x 238 mm) has handsome hardback covers with gold stamped letters. This Collection features:
General editor: William R. Davis; editors: Moody T. Chu, Patrick Dolan, James R. McConnell, Larry K. Norris, Eduardo Ortiz, Robert J. Plemmons, Don Ridgeway, B. K. P. Scaife, William J. Stewart, James W. York, Jr.; associate editors: Wesley O. Doggett, Barbara M. Gellai, André A. Gsponer; consulting editor: Carmine A. Prioli; translators: József Illy, Laurent Choquet, Don Ridgeway, Judith Kontság Meskó; with a foreword by George Marx, President of the Eötvös Physical Society of Hungary.



William K. Atkins*  André A. Gsponer*  Don Ridgeway* 
William M. Baker*  Kevin S. Hammon*  Theodore J. Rivlin* 
R. N. Bracewell*  Nick Higham*  Vladimir Rosenhaus* 
J. David Brown*  JeanPierre Hurni*  Axel Ruhe* 
John Butcher*  Kang C. Jea*  Youcef Saad* 
P. Choudhury*  David W. Kammler*  B. K. P. Scaife 
Moody T. Chu  Gerald H. Katzin*  Lawrence F. Shampine* 
C. W. Clenshaw*  H. G. Khajah*  G. W. Stewart* 
James W. Cooley*  Arkady Kheyfets*  William J. Stewart 
William R. Davis*  George Marx*  Peter Szekeres* 
P. Deuflhard*  James R. McConnell*  John Todd* 
Ranier Dick*  Larry K. Norris*  B. L. van der Waerden* 
Wesley O. Doggett  Robert L. Nowack*  D. S. Watkins* 
Patrick Dolan*  Dianne P. O'Leary*  Ralph A. Willoughby* 
Ronald O. Fulp*  Eduardo L. Ortiz*  James W. York, Jr.* 
Barbara M. Gellai*  Chris C. Paige*  David M. Young* 
Ian Gladwell*  Beresford N. Parlett*  Joseph D. Zund* 
Gene Golub*  Robert J. Plemmons  
David Greene*  Carmine A. Prioli* 
The following topics of the Lanczos Collection are described below:
Origin and Purpose of the Collection
Brief Characterization of
Lanczos's Contributions and Research
The EinsteinLanczos Correspondence
(19191955)
Commentaries on the Published
Papers of Lanczos
The Introductions
The Translations
The Contents and Structure of the
Collection
The Cornelius Lanczos Collected Published Papers with Commentaries represents the second phase of a twopart celebration of the life and work of Cornelius Lanczos (18931974), a physicist and mathematician who had a profound impact on the foundations of twentiethcentury science. This Collection also provides documentation (a) that Lanczos was one of the twentieth century's most versatile and innovative scientific minds and (b) that many of Lanczos's ideas are still of interest to presentday research in physics and applied mathematics. The first phase of this celebration occurred in December 1993 when North Carolina State University's College of Physical and Mathematical Sciences hosted the Cornelius Lanczos International Centenary Conference. The proceedings of the Conference were published by SIAM in 1994. Cornelius Lanczos visited NCSU several times during the period 19621968, holding two Visiting Professorships, and remained in correspondence until his death in 1974 with NCSU faculty members, particularly Professor William R. Davis, the General Editor of the Collection.
Brief Characterization of Lanczos's Contributions and Research
Cornelius Lanczos's research covered a vast array of disciplines, including general relativity, quantum mechanics, applied mathematics and scientific computation. He produced seminal and discovery papers in these disciplines. He pioneered investigations in a large number of research areas and problems, many of which are still of widespread interest today. Some of the more wellknown contributions and related papers and books are cited below. (To identify these references see the list of Lanczos's publications in the Proceedings of the Cornelius Lanczos International Centenary Conference SIAM]> previously referenced.)
In general relativity, these investigations include:
The written correspondence between Albert Einstein and Cornelius Lanczos spanned the years 1919 to 1955. Mostly handwritten in German, the letters number about 200 pages, all of which had to be carefully studied before Part 1 and most of the introductory sections of Part 2 of the Collection could be completed. The initial responsibility for this study was undertaken by Barbara Gellai. During the last four years of preparation of the Collection, this study of the EinsteinLanczos Correspondence has also been part of the responsibility of William R. Davis with essential help provided by Don Ridgeway. The editors are very grateful to the Albert Einstein Archives, The Jewish National and University Library, The Hebrew University of Jerusalem, Israel, for making this correspondence available to them. However, it must be pointed out that under the restrictions of the Albert Einstein Archives, only a very limited amount of the material could be quoted. Consequently, the editors needed to be highly selective in the materials chosen for direct quotation and translation. Nevertheless, a substantial amount of the material written in Part 1 and the introductory section of Part 2 reflects insights and information gained by the editors from the correspondence that is not directly quoted.
During the eightyear period of preparation of this Collection, the editors assembled fiftyfour reviewed commentaries which discuss the results of Lanczos's papers, his ideas and methods, and topics that were of interest to him. In addition to these discussions, a number of these commentaries also examine Lanczos's work and thought as they relate to or have influenced research areas of current interest in theoretical physics and applied mathematics. Except for revisions to make the commentaries compatible with the overall format of this Collection, papers have been published essentially as they were submitted by the authors. William R. Davis and Moody T. Chu took the primary responsibility for collecting the commentaries for Part 2  Physics and Part 3  Mathematics, respectively.
The editors and other contributors have written sixteen introductions for each major section of Parts 2, 3 and 4 of the Collection. These introductions generally remark upon the major results of the papers which are covered in the section and provide comments made by others relating to Lanczos's results around the time of their publication. Also, these introductions provide relevant historical information and connect Lanczos's results with more recent research which is discussed in the commentaries on selected papers provided in each section. Where appropriate, the introductions contain information and excerpts from the EinsteinLanczos correspondence.
Before 1935 Lanczos published most of his papers in German. After 1935 about 90 percent of the papers were published in English with the others being published in French, German and Hungarian. In the Collection, English translations of Lanczos's papers (fortyseven in all) are provided adjacent to each facsimile page of the originals. Accordingly, the translations have been carefully formatted page by page to reflect, as nearly as was practical, the original publications. Also, this arrangement facilitates checking mathematical equations in the translation with the original language paper.
Where necessary for clarity in the translations, the editors have included editorial notes. In these translations we have generally followed the principle of providing nearly literal translations, reflecting the syntax and sentence style of the traditional scholarly scientific language (German) in which Lanczos wrote.
These translations with each page facing its original language page could provide useful source material for graduate students in physics and mathematics to help them learn scientific foreign languages (German, French and Hungarian).
The Collection is divided into four major parts [Parts 14 should not to be confused with volume numbers IVI for this Collection]:
Part 1  Biographical and Professional Information is divided into five sections.
Section 2.1 is a general section covering the publications primarily relating to general relativity appearing during the period 19221934. Also, Section 2.1 considers the content of Lanczos's dissertation, a facsimile of which is contained in the Appendix. The subjects of particular interest dealt with in this section include the following: coordinate conditions and coordinate problems in cosmology, surface distributions of matter and junction conditions, general treatment of weak fields, Green functions, variational principles and methods, the problem of motion, integral conservation laws, and the Cauchy problem.
Section 2.2 is a specialized section dealing with Lanczos's contributions to nonrelativistic quantum mechanics. The subjects of particular interest in this section include Lanczos's fieldtype formulation of quantum mechanics late in 1925 that provides an integral formulation of Schrödinger's equations before Schrödinger's paper presenting his wave equation for quantum mechanics was published and before his paper that discussed the connection between matrix and wave mechanical formulations of quantum mechanics. Also, Section 2.2 deals with Lanczos's important work on the proper quantum mechanical treatment of the Stark effect in very strong electric fields and a number of important matters relating to the history of quantum mechanics.
Section 2.3 is a specialized section dealing with Lanczos's contributions to relativistic quantum mechanics. The subjects of particular interest included in this section are: a quaternion formulation and generalization of Dirac's equation that includes isospin, results connected with the particle doublets of Einstein and Mayer (1933), and results connected with the work of Proca (1932, 1936).
Section 2.4 is a specialized section dealing with Lanczos's new derivation of Dirac's equation for the electron in which he treats relativistic wave mechanics as the Hamiltonian dynamics of function space.
Section 2.5 deals with a survey paper written by Lanczos in 1931 which compares general relativity to the other physical theories and considers the problems of attempting to formulate unified field theories. Also, this section contains discussions connected with the modern notions of final theory, including matters concerning the recent HawkingPenrose debate and related philosophical issues.
Section 2.6 is a general section covering the published works of Lanczos in the period 19351955. The subjects of particular interest in physics in this period include: a curved spacetime elasticity type formulation, including general relativity; a characterization of the most general quadratic variational principle in Riemannian spacetime; a paper attempting to deal with gravitation, electricity, and matter waves; and writings on Einstein and the development of the theory of relativity. Also, the introduction to Section 2.6 contains some particularly interesting quotations from the EinsteinLanczos correspondence.
Section 2.7 is a general section covering the published works of Lanczos in the period 19561967. The subjects of particular interest in physics are: a unified theoretical formulation of gravitation and electromagnetism based on a variational principle quadratic in the components of the curvature tensor; Einstein and the role of contemporary physics; splitting the Riemann curvature tensor; propagation of signals in positive definite Riemannian metrics with a fourfold lattice structure; a microphysical oscillating tetrad lattice universe (lattice constant of the order of 10^{32} cm); and an appreciation of William Rowan Hamilton.
Section 2.8 is a specialized section dealing with the discovery and properties of the third order Lanczos tensor, antisymmetric in a pair of components, which exists in all Riemannian spacetimes. In particular, the Lanczos tensor is discussed in connection with its possible fundamental role relating to gravitational radiation.
Section 2.9 is a general section covering Lanczos's publications in the period 19681975. The subjects of special interest in physics in this section include: improved method for dealing with a variational principle with scale invariant variational principle quadratic in the components of the curvature tensor; Einstein's path from special to general relativity; Einstein and the future; further elaboration of the signature of Minkowski spacetime derived in terms of an underlying submicroscopic highly agitated lattice metric substructure that is positive definite; speculations about a connection between the Minkowski signature and the quantum principle; further development of a scale invariant quadratic variational principle governing the fundamental lattice metric substructure; and solutions for the field equations of the scale invariant quadratic variational principle in the linear approximation.
Part 3  Mathematics is divided into four sections: Section 3.1Linear Algebra (19361958); Section 3.2Theory of Approximations (19381964); Section 3.3Fourier Analysis and Orthogonal Polynomials (19421975); and Section 3.4Differential Equations and Integration (19351968). Each section includes introductions with commentaries on selected papers.
Section 3.1 covers the following topics connected with Lanczos's work on problems in linear algebra: an iteration method for the solution of the eigenvalue problem of linear differential and integral operators; separation of close eigenvalues of a real symmetric matrix; solution of systems of linear equations by minimized iterations; spectroscopic eigenvalue analysis; and linear systems in selfadjoint form.
Section 3.2 deals with Lanczos's work on approximation methods: trigonometric interpolation of empirical and analytical functions; analytical and practical curve fitting of equidistant data; smoothing and evaluation of noisy data; and a precision approximation of the gamma function.
Section 3.3 deals with Lanczos's work in the area of Fourier series and orthogonal polynomials: some improvements in practical Fourier analysis; Chebyshev polynomials in the solution of largescale linear systems; boundary value problems and orthogonal expansions; Legendre versus Chebyshev polynomials; and Fourier analysis of random sequences.
Section 3.4 considers Lanczos's work on the following topics: extended boundary value problems; solution of ordinary differential equations by trigonometric interpolation; an integral approach to the calculus of variations; and boundary value problems and the algebraic method.
Part 4  Papers of General Interest Including Philosophy and Pedagogy presents in chronological order papers covering the following subjects: philosophic aspects of the theory of relativity; science and society; modern physics in perspective; the inspired guess in the history of physics; rationalism and the physical world; mathematical pedagogy; Judaism and science; and science and art. A few of the papers in this part relating to the philosophy of science also mention some of the ideas Lanczos developed in his physics research papers.
The first complete set of the Collection was received in Raleigh on 5 February 99. Volume I is opened to the Frontispiece (p. ii) which shows Cornelius Lanczos in 1946 and to the Collection Title Page (p. iii).
The four Parts (14) of this Collection and the Appendix are divided into six Volumes (IVI). Page xviii in each volume indicates in which volume each part, section and page is located. Each volume contains a copy of the pages ixlv for the complete Collection (including a list of the Contributors, Preface, Copyright Permissions, Contents, List of Photographs, List of Commentaries and Introductions and Foreword). In addition, each of the Volumes IIVI contains the complete list of the Published Papers and Books of Cornelius Lanczos and the location of the papers in the Collection on pp. xxixxxxvii. In Volume I this list is contained in Part 1, pp. 139147. Page i of each volume is a brief Volume Title Page which gives the location of (a) the complete contents for the Collection, (b) the list of the commentaries and introductions by author for the Collection, (c) a chronological list of Lanczos's papers, books and book translations, (d) a narrative description of the organization and contents of the Collection, and (e) a detailed guide to locating material in the Collection.
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