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Diffusion, Quantum Theory, and Radically Elementary Mathematics. (MN-47)

Diffusion, Quantum Theory, and Radically Elementary Mathematics. (MN-47)

edited by William G. Faris
Copyright Date: 2006
Pages: 256
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  • Book Info
    Diffusion, Quantum Theory, and Radically Elementary Mathematics. (MN-47)
    Book Description:

    Diffusive motion--displacement due to the cumulative effect of irregular fluctuations--has been a fundamental concept in mathematics and physics since Einstein's work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. The book's inspiration is Princeton University mathematics professor Edward Nelson's influential work in probability, functional analysis, nonstandard analysis, stochastic mechanics, and logic. The book can be used as a tutorial or reference, or read for pleasure by anyone interested in the role of mathematics in science. Because of the application of diffusive motion to quantum theory, it will interest physicists as well as mathematicians.

    The introductory chapter describes the interrelationships between the various themes, many of which were first brought to light by Edward Nelson. In his writing and conversation, Nelson has always emphasized and relished the human aspect of mathematical endeavor. In his intellectual world, there is no sharp boundary between the mathematical, the cultural, and the spiritual. It is fitting that the final chapter provides a mathematical perspective on musical theory, one that reveals an unexpected connection with some of the book's main themes.

    eISBN: 978-1-4008-6525-3
    Subjects: Mathematics, Statistics, Physics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-x)
  4. Chapter One Introduction: Diffusive Motion and Where It Leads
    (pp. 1-44)
    William G. Faris

    The purpose of this introductory chapter is to point out the unity in the following chapters. At first this might seem a difficult enterprise. The authors of these chapters treat diffusion theory, quantum mechanics, and quantum field theory, as well as stochastic mechanics, a variant of quantum mechanics based on diffusion ideas. The contributions also include an infinitesimal approach to diffusion and related probability topics, an approach that is radically elementary in the sense that it relies only on simple logical principles. There is further discussion of foundational problems, and there is a final essay on the mathematics of music....

  5. Chapter Two Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys
    (pp. 45-74)
    Leonard Gross

    Hypercontractivity and logarithmic Sobolev inequalities have developed hand in hand over most of the past 38 years. Many of the developments and applications have already been discussed in books and surveys. After a brief sketch of the main ideas and of the equivalence of these two families of inequalities, a survey of these surveys will be given.

    The entire subject matter of this survey can be traced back to a short paper of Ed Nelson, published in 1966 [74]. Among the ideas in that paper, the one most relevant for this survey was developed over the next few years by...

  6. Chapter Three Ed Nelson’s Work in Quantum Theory
    (pp. 75-94)
    Barry Simon

    It is a pleasure to contribute to this celebration of Ed Nelson’s scientific work, not only because of the importance of that work but because it allows me an opportunity to express my gratitude and acknowledge my enormous debt to Ed. He and Arthur Wightman were the key formulative influences on my education, not only as a graduate student but during my early postdoctoral years. Thanks, Ed!

    I was initially asked to talk at the conference about Ed’s work in quantum field theory (QFT), but I decided to exceed my assignment by also discussing Ed’s impact on conventional nonrelativistic quantum...

  7. Chapter Four Symanzik, Nelson, and Self-Avoiding Walk
    (pp. 95-116)
    David C. Brydges

    Ed Nelson was not my thesis advisor, but his papers and books were continuous companions in my graduate student days and they still are. In particular, when I can round up some adventurous undergraduates, I would like to teach a course based on his bookRadically Elementary Probability Theory[46]. I learned about semigroups, the Trotter product formula, and the spectral theorem from his book [43]. His paper [42] was my first encounter with Brownian motion. His proof [41] of the lower bound on the energy of the$\phi^{4}_{2}$scalar quantum field theory was the critical discovery at the beginning...

  8. Chapter Five Stochastic Mechanics: A Look Back and a Look Ahead
    (pp. 117-140)
    Eric Carlen

    My introduction to Nelson’s stochastic mechanics came though his wonderful bookDynamical Theories of Brownian Motion[29], written in 1966. The first section is entitled “Apology” partly for the fact that the history of nineteenth century work on Brownian motion is considered in “unnecessary detail.” The reason offered for delving into the history nonetheless is the perspective it brings on scientific research:

    One realizes what an essentially comic activity scientific investigation is (good as well as bad).

    I will spend some time looking back over the history of this subject partly for this same reason, and partly because the story...

  9. Chapter Six Current Trends in Optimal Transportation: A Tribute to Ed Nelson
    (pp. 141-156)
    Cédric Villani

    The agitated history of optimal transportation meets Nelson’s research on several occasions, sometimes in an unexpected way. This theory has made its way into an impressive number of applications to problems in mathematical physics, or to science in general, just as Nelsonʼs work has done. This text, based on my lecture in Vancouver in June 2004, aims to collect, in a slightly impressionist way, some of the most striking elements in this long and rich, but not so complicated story.

    The story begins with some unusual variational principles. A few decades ago, while trying to revisit the fundamentals of quantum...

  10. Chapter Seven Internal Set Theory and Infinitesimal Random Walks
    (pp. 157-182)
    Gregory F. Lawler

    Let me start this chapter with a disclaimer: I am not an expert on logic or nonstandard analysis. I had the opportunity to learn Internal Set Theory (IST) as a graduate student and to use it in my research. It has been many years since I have used nonstandard analysis (at least in public), and there are others who are better equipped to discuss the fine points of Internal Set Theory and other flavors of nonstandard analysis. However, I happily accepted the opportunity to make some remarks for a number of reasons. First, Internal Set Theory is intrinsically a very...

  11. Chapter Eight Nelson’s Work on Logic and Foundations and Other Reflections on the Foundations of Mathematics
    (pp. 183-208)
    Samuel R. Buss

    This chapter was begun with the plan of discussing Nelson’s work in logic and foundations and his philosophy on mathematics. In particular, it is based on our talk at the Nelson meeting in Vancouver in June 2004. The main topics of this talk were Nelson’s predicative arithmetic and his unpublished work on automatic theorem proving. However, it proved impossible to stay within this plan. In writing the chapter, we were prompted to think carefully about the nature of mathematics and more fully formulate our own philosophy of mathematics. We present this below, along with some discussion about mathematics education.


  12. Chapter Nine Some Musical Groups: Selected Applications of Group Theory in Music
    (pp. 209-228)
    Julian Hook

    The groups mentioned in the title are not groups such as Metallica or the Chicago Symphony Orchestra, but groups in the algebraic sense. There are many ways in which elements of musical structure may be described by group-theoretic constructions; these constructions and related topics form the subject of this chapter.

    One important large class of musical groups may be calledinterval groups, formalizing and generalizing the notion of the musical interval between two notes. Ageneralized interval system, orGIS[15], consists of:

    a setS, called thespaceof the GIS;

    a groupG, called theinterval groupof...

  13. Chapter Ten Afterword
    (pp. 229-232)
    Edward Nelson

    It can be useful to pause in mid-career and reflect on work already done. But to do so in the company of friends, colleagues, and former students in an atmosphere of warm celebration was for me an astonishingly joyful experience. I am deeply grateful to David Brydges, Eric Carlen, Bill Faris, and Greg Lawler for all the hard work they did organizing the conference and editing this publication, to the speakers for uniformly interesting and stimulating talks, from which I learned some fascinating new mathematics and music theory, to the many people who came to Vancouver, some from a great...

  14. Appendix A. Publications by Edward Nelson
    (pp. 233-240)
  15. Index
    (pp. 241-245)
  16. Back Matter
    (pp. 246-246)