Physics 525 (Methods of Theoretical Physics) Fall 2014 Theory of Distributions (v. 2) In this document I will summarize the definitions and statements of the simplest theorems of the theory of distributions. This gives a consistent framework for using things like the Dirac delta function. The concepts provide a very useful framework for many calculations in quantum mechanics (which was why Dirac invented the delta function). This document only has the mathematical definitions and the theorems, but leaves out the motivations etc., which I treated in class. A useful book on the subject is: Ian Richards and Heekyung Youn, \Theory of distributions, a non-technical introduction" (Cambridge University Press, 1990), call #: QA273.6.R53 1990 (in Physical Sciences Library). Another good treatment is in Sec. 4.1 of J.P. Keener, \Principles of Applied Mathematics" (Westview, 2000) ISBN 0{7382{0129{4, call #: QA601.K4 1999. Yet another is found in Ch. 2 of M. Stone and P. Goldbart, \Mathematics for Physics" (Cambridge University Press, 2009) ISBN 978{0{521{85403{0, call #: QC20.S76 2009. One annoyance is that there are a number of notations that are not, strictly speaking correct, but constitute an \abuse of notation". These notations are very useful for guiding a calculation. I like the term suggested by a student in one of my earlier courses: \bent notation". A lot of this document consists of definitions; there are not so many theorems. Think of this as being like a computer program with many subroutines; to use the subroutines, they have to be defined somewhere. Similarly to do mathematics with new objects, one has to define all the operations that one does with the new objects. 1 Distributions 1.1 Definition Definition: A distribution T is a continuous linear mapping from a space of test functions to (real or complex) numbers. We can write the value of T on a test function φ(x) as T [φ]. The term \generalized function" is a synonym for \distribution". The square-bracket notation emphasizes the aspect that the argument of a distribution is a whole function, not the value of a function at one point. A distribution is an example of what is called a functional. It is sometimes said that a functional is a \function of a function", as opposed to being a function of a real variable. In this document, the test functions will be functions of one real variable, and conse- quently we will talk about distributions defined on the real axis. Roughly speaking, what we mean by a test function is that A test function is any sufficiently differentiable function which vanishes suffi- ciently rapidly at infinity. However, there are certain complications concerning the nature of the test functions: • There are alternatives in making precise the space of test functions, which cause some complications that we will return to later. • As we will see with examples in Sec. 2, once a distribution is defined on a certain space of test functions, it can typically be naturally extended to a wider class of functions. While these details matter, it is also important that we obtain a physical intuition. A distribution is meant as a mathematical construct that corresponds, for example, to a distribution of electric charge in space, no matter whether we have a continuous distribution, defined by a continuous charge density, or something else, like a point charge or a point dipole. The operation of obtaining the value T [φ] corresponds to an (idealized) experiment that measures the charge weighted by the test function. 1.2 Spaces of test functions There are two standard spaces of test functions: Definition: A test function of compact support is a functions from the real line to the complex numbers that is infinitely differentiable and that is zero outside a bounded set. The space of compact-support test functions is called D. Definition: An open support test function φ(x) is a function from the real line to the complex numbers that is infinitely differentiable and that is of \rapid decrease", i.e., for every integer N, the product xN φ(x) remains bounded as x ! ±∞. The space of open support test functions is called S. Both of these spaces are vector spaces. The space S is sometimes called the Schwartz space, and its elements called Schwartz functions, after Laurent Schwartz, who was the leading pioneer of the mathematical theory of distributions. Note that the space D of compact- support test functions is a subspace of the Schwartz space S. Following Richards & Youn, I will define a general distribution as one on the space of compact-support test functions. A tempered distribution is one that is defined on Schwartz space. If T is a tempered distribution, it is also a general distribution; this is because T [φ] is defined for all φ 2 S, it is defined for all φ in the smaller space of compact-support test functions. Initially all our work will be with general distributions, and we can assume our test functions to have compact support. But when we treat Fourier transforms, it will be better to restrict to tempered distributions. 1.3 Extension of distributions beyond test functions Once we have defined a distribution on a space of test functions, its definition can typically be extended to a broader class of functions, as we will see. 2 1.4 Continuity of distributions It is required that a distribution is not just a linear functional of test functions but is a continuous linear functional. This will be true for all the distributions we normally encounter in physics. What this means is that if a sequence of test functions φn converges to a test function φ, then the values of a distribution on the φn must converge to its value on φ, i.e., if φn ! φ then T [φn] ! T [φ]. But it is important that there is a very strong definition of the meaning that φn ! φ. (As usual, n runs from 1 to 1.) In the following, for a function φ(x), its kth derivative is notated by φ(k)(x). Definition: For compact support test functions, by φn ! φ is meant that (k) (k) • For each k, φn (x) approaches φ (x) uniformly in x. I.e., for each k and (k) (k) > 0, there is an n0(k; ) such that jφn (x) − φ (x)j < for n > n0(k; ). Uniformity of the convergence means that the same n0 works for all x. • And there is a uniformly bounded support for all of φn, i.e., there is a number a such that φn(x) = 0 for all jxj > a. Uniformity here means that the same a works for all n. These restrictions ensure that the approach of φn to φ is not spoiled, for example, by oscilla- tions of decreasingly small amplitude but of rapidly increasing frequency; this would affect manipulations involving derivatives of the functions, such as we will encounter. Definition: For Schwartz functions, by φn ! φ is meant that N (k) • For each pair of non-negative integers k and N, jxj φn (x) approaches jxjN φ(k)(x) uniformly in x. 1.5 Generalization to higher dimensions The above definition, together with the ones we will encounter later, apply when the under- lying functions are of one real variable. But they generalize trivially to functions of more than one real variable. See Sec. 4 for one example. 2 Basic distributions There is standard correspondence between a distribution f¯ and an ordinary function f. It is defined by: Definition: f¯[φ] = R f(x) φ(x) dx. Comment: Note that a continuous function f(x) can be reconstructed if one knows f¯[φ] for all test functions. So one normally identifies the function and the corresponding distribution, and therefore drops the overbar on f¯. Also, the notation f¯, with the overbar, is my own notation and isn't standard. But there also distributions that do not correspond to functions, for example the well- known delta function: 3 Definition: The delta function at a point a is defined by δa[φ] = φ(a). (Strictly speaking, the use of the word \function" here is a misnomer, of course.) Notation: A notation that is often seen in the literature is that for any distribution T , hT; φi is defined to mean T [φ]. This is meant to be reminiscent of Dirac notation, but is not identical, because of the lack of a complex conjugate. I will not normally use this notation in this course, precisely because of the danger of confusion with Dirac bra and ket notation. Notation: We also use the notation R T (x) φ(x) dx to mean T [φ], even when T is not R obtained from a function. So we write, for example, δa[φ] = δ(x − a) φ(x) dx, even though δ(x − a) does not exist as a function. This bent notation is like the one for a derivative: df δf = lim ; (1) dx δx!0 δx where δf = f(x + δx) − f(x). It is useful to think of df=dx as df divided by dx even though neither df nor dx exists. It is useful when changing variables, for example: df(x(y))=dy = (df=dx) (dx=dy). Once we have defined a distribution with respect to a certain space of test functions, which for the moment we will take to be compact-support test functions, we can typically extend the definition in a natural way to more general functions. For example, the definitions of f¯[φ] can be extended to any function φ(x) for which the defining integral converges.