Abstract
In recent years gauge theories have emerged as primary tools for research in elementary particle physics. Experimental as well as theoretical evidence of their utility has grown tremendously in the last two decades. The isospin gauge group SU(2) of Yang–Mills theory combined with the U(1) gauge group of electromagnentic theory has lead to a unified theory of weak interactions and electromagnetism. We give an account of this unified electroweak theory in Chapter 8. In this chapter we give a mathematical formulation of several important concepts and constructions used in classical field theories. We begin with a brief account of the physical background in Section 6.2. Gauge potential and gauge field on an arbitrary pseudo-Riemannian manifold are defined in Section 6.3. Three different ways of defining the group of gauge transformations and their natural equivalence is also considered there. The geometric structure of the space of gauge potentials is discussed in Section 6.4 and is then applied to the study of Gribov ambiguity in Section 6.5. A geometric formulation of matter fields is given in Section 6.6. Gravitational field equations and their generalization is discussed in Section 6.7. Finally, Section 6.8 gives a brief indication of Perelman’s work on the geometrization conjecture and its relation to gravity.
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Notes
- 1.
I had the opportunity to visit Prof. Heisenberg at the MPI in Munich in 1970 and cherish the autographed copy of his autobiography Der Teil und das Ganze, which he gave me at that time.
- 2.
I discussed this in my talk at the Geometry and Physics Workshop organized by Prof. Raoul Bott at MSRI, Berkeley in 1994. After my talk Bott remarked: “We teach Harvard students to think of functions as Lie algebra valued 0-forms so that they know the distinction between scale and phase.” When one of his students said he never learned this in his courses, Bott gave a heary laugh.
- 3.
We would like to thank Prof. Akira Asada of Shinshu University for introducing us to Prof. Miyachi and his work.
- 4.
This example was suggested by Stefan Wagner, a doctoral student of Prof. Neeb at TU Darmstadt.
- 5.
I first met Prof. Chern and his then newly arrived student S.-T. Yau in 1973 at the AMS summer workshop on differential geometry held at Stanford University. Chern was a gourmet and his conference dinners were always memorable. I attended the first one in 1973 and the last one in 2002 on the occassion of the ICM satellite conference at his institute in Tianjin. In spite of his advanced age and poor health he participated in the entire program and then continued with his duties as President of the ICM in Beijing.
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Marathe, K. (2010). Theory of Fields, I: Classical. In: Topics in Physical Mathematics. Springer, London. https://doi.org/10.1007/978-1-84882-939-8_6
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DOI: https://doi.org/10.1007/978-1-84882-939-8_6
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