Noncommutative Geometry, Quantum Fields, and Motives
By addebook • Jun 29th, 2008 • Category: Mathematics •
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Noncommutative Geometry, Quantum Fields, and Motives
The unifying theme, which the reader will encounter in di®erent guises throughout
the book, is the interplay between noncommutative geometry and number theory,
the latter especially in its manifestation through the theory of motives. For us, this
interwoven texture of noncommutative spaces and motives will become a tool in the
exploration of two spaces, whose role is central to many developments of modern
mathematics and physics:
² Space-time
² The set of prime numbers
One may be tempted to think that, looking from the vantage point of those who
sit atop the vast edi¯ce of our accumulated knowledge of such topics as space and
numbers, we ought to know a great deal about these two spaces. However, there
are two fundamental problems whose di±culty is a clear reminder of our limited
knowledge, and whose solution would require a more sophisticated understanding
than the one currently within our immediate grasp:
² The construction of a theory of quantum gravity (QG)
² The Riemann hypothesis (RH)
The purpose of this book is to explain the relevance of noncommutative geometry
(NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall
discover that there are deep analogies between these two problems which, if properly
exploited, are likely to enhance our grasp of both of them.
Although the book is perhaps more aimed at mathematicians than at physicists,
or perhaps precisely for that reason, we choose to begin our account in Chapter 1
squarely on the physics side. The chapter is dedicated to discussing two main topics:
² Renormalization
² The Standard Model of high energy physics
ftp://ftp.alainconnes.org/bookjuly.pdf
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