Softcover ISBN:  9780821853672 
Product Code:  ULECT/59 
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Electronic ISBN:  9780821890356 
Product Code:  ULECT/59.E 
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Book DetailsUniversity Lecture SeriesVolume: 59; 2012; 190 ppMSC: Primary 11; 40; 47; 81; 65;
“Regularization techniques” is the common name for a variety of methods used to make sense of divergent series, divergent integrals, or traces of linear operators in infinitedimensional spaces. Such methods are often indispensable in problems of number theory, geometry, quantum field theory, and other areas of mathematics and theoretical physics. However arbitrary and noncanonical they might seem at first glance, regularized sums, integrals, and traces often contain canonical concepts, and the main purpose of this book is to illustrate and explain this.
This book provides a unified and selfcontained mathematical treatment of various regularization techniques. The author shows how to derive regularized sums, integrals, and traces from certain canonical building blocks of the original divergent object. In the process of putting together these “building blocks”, one encounters many problems and ambiguities caused by various socalled anomalies, which are investigated and explained in detail. Nevertheless, it turns out that the corresponding canonical sums, integrals, sums, and traces are well behaved, thus making the regularization procedure possible and manageable.
This new unified outlook on regularization techniques in various fields of mathematics and in quantum field theory can serve as an introduction for anyone from a beginning mathematician interested in the subject to an experienced physicist who wants to gain a unified outlook on techniques he/she uses on a daily basis.ReadershipResearch mathematicians interested in partial differential equations, asymptotic methods, and applications to quantum field theory.

Table of Contents

Chapters

Chapter 1. The Gamma function extended to nonpositive integer points

Chapter 2. The canonical integral and noncommutative residue on symbols

Chapter 3. The cutoff regularised integral

Chapter 4. The noncommutative residue as a complex residue

Chapter 5. The canonical sum on noninteger order classical symbols

Chapter 6. Traces on pseudodifferential operators

Chapter 7. Weighted traces

Chapter 8. Logarithmic residues

Chapter 9. Anomalies of regularised determinants


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“Regularization techniques” is the common name for a variety of methods used to make sense of divergent series, divergent integrals, or traces of linear operators in infinitedimensional spaces. Such methods are often indispensable in problems of number theory, geometry, quantum field theory, and other areas of mathematics and theoretical physics. However arbitrary and noncanonical they might seem at first glance, regularized sums, integrals, and traces often contain canonical concepts, and the main purpose of this book is to illustrate and explain this.
This book provides a unified and selfcontained mathematical treatment of various regularization techniques. The author shows how to derive regularized sums, integrals, and traces from certain canonical building blocks of the original divergent object. In the process of putting together these “building blocks”, one encounters many problems and ambiguities caused by various socalled anomalies, which are investigated and explained in detail. Nevertheless, it turns out that the corresponding canonical sums, integrals, sums, and traces are well behaved, thus making the regularization procedure possible and manageable.
This new unified outlook on regularization techniques in various fields of mathematics and in quantum field theory can serve as an introduction for anyone from a beginning mathematician interested in the subject to an experienced physicist who wants to gain a unified outlook on techniques he/she uses on a daily basis.
Research mathematicians interested in partial differential equations, asymptotic methods, and applications to quantum field theory.

Chapters

Chapter 1. The Gamma function extended to nonpositive integer points

Chapter 2. The canonical integral and noncommutative residue on symbols

Chapter 3. The cutoff regularised integral

Chapter 4. The noncommutative residue as a complex residue

Chapter 5. The canonical sum on noninteger order classical symbols

Chapter 6. Traces on pseudodifferential operators

Chapter 7. Weighted traces

Chapter 8. Logarithmic residues

Chapter 9. Anomalies of regularised determinants