International Lectures on Frontier Physics 1 Lecture 1 Exterior Product, Fermions Hirosi Ooguri - OPENCOURSEWARE The University of Tokyo Center for Research and Development of Higher Education

Lecture 1 In this course, we will learn methods of geometry in modern physics. Historically, mathe- matics and physics influenced each other in their developmen ts. We know, for example, that Newton invented calculus and classical mechanics simultan eously. However, in the mid 20th century, there was a period when interactions between physicists, in particular high energy physicists, and mathematicians almost stopped. There were two main reasons. One was on the mathematics side. The Bourbaki movement. Another was on the physics side. Quantum field theory was conc eived by Heisenberg and Pauli in 1929 and Feynman, Schwinger, and Tomonaga estab lished the renormalization procedure to make sense of QED. However, there was not mathem atical formulation of quantum field theory. In fact, there is still no rigorous mathematica l foundation for it, except for special cases, some of which we will see later in this course. Making sense of quantum Yang-Mills theory is posed as one of the seven Millennium problems by the Clay Mathematics Institute. The situtation has dramatically changed since the 1980’s. I t is because of the use of supersymmetry. Supersymmetry brought mathematics and phy sics closer since it is close to the language mathematicians use to describe geometric concepts.

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