THE
DIRAC
6-FUNCTION
The Dirac 6-function is a strange, but useful function which has many applications in
science, engineering, and mathematics. The &function was proposed in
1930
by Paul
Dirac in the development of the mathematical formalism of quantum mechanics. He
required a function whlch was zero everywhere, except at a single point, where it was
discontinuous and behaved like
an
infinitely high, infinitely narrow spike of unit area.
Mathematicians were quick to point out that, strictly speaking, there is
no
function
which has these properties. But Dirac supposed there was, and proceeded to use it
so
successfully that a new branch of mathematics was developed in order to justify
its use.
This
area of mathematics is called the theory of
generalizedfunctions
and
develops, in complete detail, the foundation for the Dirac 6-function.
This
rigorous
treatment is necessary to justify the use of these discontinuous functions, but for the
physicist
the
simpler physical interpretations are just
as
important. We will take both
approaches
in
this chapter.
5.1
EXAMPLES OF
SINGULAR FUNCTIONS IN PHYSICS
Physical situations are usually described using equations and operations on contin-
uous functions. Sometimes, however, it is useful to consider discontinuous ided-
izations, such as the mass density of a point mass, or the force of an infinitely fast
mechanical impulse. The functions that describe these ideas are obviously extremely
discontinuous, because they and all their derivatives must diverge. For
this
reason
they are often called
singular
functions. The Dirac 6-function was developed
to
de-
scribe functions that involve these types of discontinuities and provide a method for
handling them in equations which normally involve only continuous functions.
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