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By Robert Geroch
The University of Chicago PressCopyright © 1985 The University of Chicago
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One sometimes hears expressed the view that some sort of uncertainty principle operates in the interaction between mathematics and physics: the greater the mathematical care used to formulate a concept, the less the physical insight to be gained from that formulation. It is not difficult to imagine how such a viewpoint could come to be popular. It is often the case that the essential physical ideas of a discussion are smothered by mathematics through excessive definitions, concern over irrelevant generality, etc. Nonetheless, one can make a case that mathematics as mathematics, if used thoughtfully, is almost always useful — and occasionally essential — to progress in theoretical physics.
What one often tries to do in mathematics is to isolate some given structure for concentrated, individual study: what constructions, what results, what definitions, what relationships are available in the presence of a certain mathematical structure — and only that structure? But this is exactly the sort of thing that can be useful in physics, for, in a given physical application, some particular mathematical structure becomes available naturally, namely, that which arises from the physics of the problem. Thus mathematics can serve to provide a framework within which one deals only with quantities of physical significance, ignoring other, irrelevant things. One becomes able to focus on the physics. The idea is to isolate mathematical structures, one at a time, to learn what they are and what they can do. Such a body of knowledge, once established, can then be called upon whenever it makes contact with the physics.
An everyday example of this point is the idea of a derivative. One could imagine physicists who do not understand, as mathematics, the notion of a derivative and the properties of derivatives. Such physicists could still formulate physical laws, for example, by speaking of the "rate of change of ... with ..." They could use their physical intuition to obtain, as needed in various applications, particular properties of these "rates of change." It would be more convenient, however, to isolate the notion "derivative" once and for all, without direct reference to later physical applications of this concept. One learns what a derivative is and what its properties are: the geometrical significance of a derivative, the rule for taking the derivative of a product, etc. This established body of knowledge then comes into play automatically when the physics requires the use of derivatives. Having mastered the abstract concept "rate of change" all by itself, the mind is freed for the important, that is, the physical, issues.
The only problem is that it takes a certain amount of effort to learn mathematics. Fortunately, two circumstances here intervene. First, the mathematics one needs for theoretical physics can often be mastered simply by making a sufficient effort. This activity is quite different from, and far more straightforward than, the originality and creativity needed in physics itself. Second, it seems to be the case in practice that the mathematics one needs in physics is not of a highly sophisticated sort. One hardly ever uses elaborate theorems or long strings of definitions. Rather, what one almost always uses, in various areas of mathematics, is the five or six basic definitions, some examples to give the definitions life, a few lemmas to relate various definitions to each other, and a couple of constructions. In short, what one needs from mathematics is a general idea of what areas of mathematics are available and, in each area, enough of the flavor of what is going on to feel comfortable. This broad and largely shallow coverage should in my view be the stuff of "mathematical physics."
There is, of course, a second, more familiar role of mathematics in physics: that of solving specific physical problems which have already been formulated mathematically. This role encompasses such topics as special functions and solutions of differential equations. This second role has come to dominate the first in the traditional undergraduate and graduate curricula. My purpose, in part, is to argue for redressing the balance.
We shall here take a brief walking tour through various areas of mathematics, providing, where appropriate and available, examples in which this mathematics provides a framework for the formulation of physical ideas.
By way of general organization, chapters 2–24 deal with things algebraic and chapters 25–42 with things topological. In chapters 43–50 we discuss some special topics: structures which combine algebra and topology, Lebesque integrals, Hilbert spaces. Lest the impression be left that no difficult mathematics can ever be useful in physics, we provide, in chapters 51–56, a counterexample: the spectral theorem. Strictly speaking, the only prerequisites are a little elementary set theory, algebra, and, in a few places, some elementary calculus. Yet some informal contact with such objects as groups, vector spaces, and topological spaces would be most helpful.
The following texts are recommended for additional reading: A. H. Wallace, Algebraic Topology (Elmsford, NY: Pergamon, 1963), and C. Goffman and G. Pedrick, First Course in Functional Analysis (Englewood Cliffs, NJ: Prentice-Hall, 1965). Two examples of more advanced texts, to which the present text might be regarded as an introduction, are: M. Reed and B. Simon, Methods of Modern Mathematical Physics (New York: Academic, 1972), and Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics(Amsterdam: North-Holland, 1982).CHAPTER 2
In each area of mathematics (e.g., groups, topological spaces) there are available many definitions and constructions. It turns out, however, that there are a number of notions (e.g., that of a product) that occur naturally in various areas of mathematics, with only slight changes from one area to another. It is convenient to take advantage of this observation. Category theory can be described as that branch of mathematics in which one studies certain definitions in a broader context — without reference to the particular area to which the definition might be applied. It is the "mathematics of mathematics." Although this subject takes a little getting used to, it is, in my opinion, worth the effort. It provides a systematic framework that can help one to remember definitions in various areas of mathematics, to understand what many constructions mean and how they can be used, and even to invent useful definitions when needed. We here summarize a few facts from category theory.
A category consists of three things — i) a class O (whose elements will be called objects), ii) a set Mor(A, B) (whose elements will be called morphisms from A to B), where A and B are any two objects, and iii) a rule which assigns, given any objects A, B, and C and any morphism φ from A to B and morphism ψ from B to C, a morphism, written ψ º φ from A to C (this ψ º φ will be called the composition of φ with ψ) — subject to the following two conditions:
1. Composition is associative. If A, B, C, and D are any four objects, and φ, ψ and λ are morphisms from A to B, from B to C, and from C to D, respectively, then
(λ o ψ)o φ = λ o (ψ o φ).
(Note that each side of this equation is a morphism from A to D.)
2. Identities exist. For each object A, there is a morphism iA from A to A (called the identity morphism on A) with the following property: if φ is any morphism from A to B, then
φ o iA = φ;
if μ is any morphism from C to A, then
iA o μ = μ
That is the definition of a category. It all seems rather abstract. In order to see what is really going on with this definition — why it is what it is — one has to look at a few examples. We shall have abundant opportunity to do this: almost every mathematical structure we look at will turn out to be an example of a category. In order to fix ideas for the present, we consider just one example (the simplest, and probably the best).
To give an example of a category, one must say what the objects are, what the morphisms are, what composition of morphisms is — and one must verify that conditions 1 and 2 above are indeed satisfied. Let the objects be ordinary sets. For two objects (now, sets) A and B, let Mor(A, B) be the set of all mappings from the set A to the set B. (Recall that a mapping from set A to set B is a rule that assigns, to each element of A, some element of B.) Finally, let composition of morphisms, in this example, be ordinary composition of mappings. (That is, if φ is a mapping from set A to set B and ψ is a mapping from set B to set C,then ψ º φ is the mapping from set A to set C which sends the element a of A to the element ψ( φ(a)) of C.) We now have the objects, the morphisms, and the composition law. We must check that conditions 1 and 2 are satisfied. Condition 1 is indeed satisfied in this case: it is precisely the statement that composition of mappings on sets is associative. Condition 2 is also satisfied: for any set A, let iA be the identity mapping (i.e., for each element a of A, iA(a) = a) from A to A. Thus we have here an example of a category. It is called the category of sets.
This example is in some sense typical. It is helpful to think of the objects as being "really sets" (perhaps, as in later examples, with additional structure) and of the morphisms as "really mappings" (which, in these later examples, will be "structure preserving"). With this mental picture, it is easy to remember the definition of a category — and to follow the constructions we shall shortly introduce on categories.
This example suggests the introduction of the following notation for categories. We shall write A [??] B to mean "A and B are objects, and φ is a morphism from A to B."
We now wish to give a few examples of how one carries over notions from categories in general to specific categories.
Let φ be a morphism from A to B. This φ is said to be a monomorphism if the following property is satisfied: given any object X and any two morphisms, a and a', from X to A such that φ º a = φ º a', it follows that a = a' (figure 1). This φ is said to be an epimorphism if the following property is satisfied: given any object X and any two morphisms, β and β', from B to X such that β º φ = β' º φ, it follows that β = β' (figure 2). (That is, monomorphisms are the things that can be "canceled out of morphism equations on the left"; epimorphisms can be "canceled out of morphism equations on the right.")
As usual, one makes sense out of these definitions by appealing to our example, the category of sets.
THEOREM 1. In the category of sets, a morphism is a monomorphism if and only if it is one-to-one.
(Recall that a mapping from set A to set B is said to be one-to-one if no two distinct elements of A are mapped to the same element of B.)
Proof. Let φ be a mapping from set A to set B, which is one-to-one. We show that this φ is a monomorphism. Let X be any set, and let a and a' be mappings from X to A such that φ º a = φ º a'. We must show that a = a'. If a and a' were different, they would differ on some element of X; that is, there would be an x in X such that a(x) would be different from a'(x). Then, since f is one-to-one, we would have φ(a(x)) different from φ (a'(x)). But this contradicts φ º a = φ º a'. Hence φ is a monomorphism.
Let φ be a mapping from set A to set B which is a monomorphism. We show that this φ is one-to-one. Let a and a' be elements of A such that φ (a) = φ (a'). We must show that a = a'. Let X be the set having only one element, x. Let a be the mapping from X to A with a(x) = a, and let a' be the mapping from X to A with a'(x) = a'. Then, since φ (a) = φ (a'), φ º a(x) = φ º a'(x). That is, φ º a = f º a'. But φ is supposed to be a monomorphism; hence a =a'. In particular, we must have a(x) = a'(x); that is, we must have a = a'. Hence, φ is one-to-one. 
THEOREM 2. In the category of sets, a morphism is an epimorphism if and only if it is onto.
(Recall that a mapping from set A to set B is said to be onto if every element of B is the image, under the mapping, of some element of A.)
Proof. Let φ be a mapping from set A to set B, which is onto. We show that this φ is an epimorphism. Let X be any set, and let βand β' be mappings from B to X such that β º φ = β' º φ. We must show that β = β'. If β and β' were different, they would differ on some element of β; that is, there would be a b in B such that β (b) would be different from β'(b). But, since φ is onto, there is an a in A such that φ = b. Hence β º φ(a) would be different from β' º φ(a). This contradicts β º φ = β' º φ. Hence φ is an epimorphism.
Excerpted from Mathematical Physics by Robert Geroch. Copyright © 1985 The University of Chicago. Excerpted by permission of The University of Chicago Press.
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Table of Contents1. Introduction
3. The Category of Groups
5. Normal Subgroups
7. Direct Products and Sums of Groups
9. The Category of Vector Spaces
11. Linear Mappings; Direct Products and Sums
12. From Real to Complex Vector Spaces and Back
14. Multilinear Mappings; Tensor Products
15. Example: Minkowski Vector Space
16. Example: The Lorentz Group
18. The Category of Associative Algebras
19. The Category of Lie Algebras
20. Example: The Algebra of Observables
21. Example: Fock Vector Space
22. Representations: General Theory
23. Representations on Vector Spaces
24. The Algebraic Categories: Summary
25. Subsets and Mappings
26. Topological Spaces
27. Continuous Mappings
28. The Category of Topological Spaces
31. The Compact-Open Topology
33. Example: Dynamical Systems
36. Homology: Relation to Homotopy
37. The Homology Functors
38. Uniform Spaces
39. The Completion of a Uniform Space
40. Topological Groups
41. Topological Vector Spaces
42. Categories: Summary
43. Measure Spaces
44. Constructing Measure Spaces
45. Measurable Functions
48. Hilbert Spaces
49. Bounded Operators
50. The Spectrum of a Bounded Operator
51. The Spectral Theorem: Finite-dimensional Case
52. Continuous Functions of a Hermitian Operator
53. Other Functions of a Hermitian Operator
54. The Spectral Theorem
55. Operators (Not Necessarily Bounded)
56. Self-Adjoint Operators
Index of Defined Terms