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Topology and Geometry for Physicists
Charles Nash, Siddhartha Sen
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eBook - ePub
Topology and Geometry for Physicists
Charles Nash, Siddhartha Sen
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About This Book
Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research.
`Thoroughly recommended` by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.
`Thoroughly recommended` by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.
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TopologiaCHAPTER 1
Basic Notions of Topology and the Value of Topological Reasoning
1.1 INTRODUCTION
Topology can be thought of as a kind of generalization of Euclidean geometry, and also as a natural framework for the study of continuity. Euclidean geometry is generalized by regarding triangles, circles, and squares as being the same basic object. Continuity enters because in saying this one has in mind a continuous deformation of a triangle into a square or a circle, or indeed any arbitrary shape. A disc with a hole in the centre is topologically different from a circle or a square because one cannot create or destroy holes by continuous deformations. Thus using topological methods one does not expect to be able to identify a geometrical figure as being a triangle or a square. However, one does expect to be able to detect the presence of gross features such as holes or the fact that the figure is made up of two disjoint pieces etc. This leads to the important point that topology produces theorems that are usually qualitative in nature—they may assert, for example, the existence or non-existence of an object. They will not in general, provide the means for its construction.
Let us begin by looking at some examples where topology plays a role.
Example 1. Cauchy’s residue theorem
Consider the contour integral for a meromorphic function f(z) along the path Γ1 which starts at a and finishes at b (c.f. Fig. 1.1). Let us write
Now deform the path Γ1 continuously into the path Γ2 shown in Fig. 1.1. Provided we cross no poles of f(z) in deforming Γ1 into Γ2, then Cauchy’s theorem for meromorphic functions allows us straightaway to deduce that
This is just the statement that
where C is the closed contour made up by joining Γ1 to Γ2 and reversing the arrow on Γ1 so as to give an anticlockwise direction to the contour C. The intuitive content of this result is that to integrate f(z) from a to b in the complex plane is independent of the path joining a to b (under the conditions stated). Even if we relax these conditions and allow that the deformation of Γ1 into Γ2 may entail the crossing of some poles of f(z), then we still have complete knowledge of the relationship of the two integrals. It is simply
where the sum is over the residues, if any, of the poles inside C. This simple example uncovers some topological properties underlying the familiar Cauchy theorem.
Example 2. The fundamental theorem of algebra
In this example we shall use the Cauchy theorem to give a proof of the fundamental theorem of algebra. First of all a simple consequence of Cauchy’s theorem is that for a meromorphic function f(z) we have
where n0 and np are the number of zeroes and the number of poles respectively of f(z) lying inside C Let f(z) be the polynomial Q(z) of degree q where
Then since Q(z) has no poles we have
It is clear that n0 is a continuous function of q of the q + 1 coefficients a0, . . ., aq. Let us select the coefficients a0, . . ., aq − 1 and write
However, n0 also only takes integer values. Now since it is impossible to continuously jump from one integer value to another it follows that n0 must be invariant under continuous change of the ai’s. This state of affairs permits the following argument: for large |z|, say |z| > R, |Q(z)|, grows as |aq||z|q which is a large number. Then if C is a circ...