The square of the delta function
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Unlike physicists and engineers, mathematicians insist that a function
that is zero almost everywhere has an integral of zero. This is indeed
a consequence of the definition of what mathematicians call a function
and an integral. It ensures that functional analysis is not plagued
by the same problems as quantum field theory.
Therefore mathematicians call the ''Dirac function'' delta(x),
which vanishes for nonzero x but integrates to 1, not a function
but a distribution. As a distribution (= a linear mapping that assigns
to nice functions a number), the delta distribution is well-defined:
delta(f) = f(0), (1)
and the notation
\int dx delta(x) f(x) = f(0)
is just considered to be a formal expansion of (1).
(The term ''linear functional'' is the generic term for linear mappings
from an arbitrary vector space to its defining field. One talks about
distributions when the space is the space of real- or complex valued
functions with the property that x^m d^n/ dx^n f(x) tends to zero as
|x| tends to infinity.
But no such distributional meaning (nor any other) can be attached
to the square of delta(x), whence mathematications call delta(x)^2
ill-defined.
Of course, one can give an ad hoc meaning to \int dx delta(x)^2;
onecan define it to be infinite, zero, or whatever else one finds
convenient.
But one cannot attach consistently a meaning to
\int dx delta(x)^2 f(x)
for arbitrary nice functions f, in a way that one can handle without
getting contradictions the integral according to the usual rules.
For example, the functions f_n defined for a given constant c by
f_n(x):=2n/(1+n^2x^2)-(2n-c\pi)/(1+(2n-c\pi)^2x^2)
converge to \pi delta(x) when n tends to infinity, so one would expect
that
\int dx delta(x)^2 = lim_{n to inf} \int dx f_n(x)/\pi \delta(x)
= lim_{n to inf} f_n(0)/\pi = c.
But c is an arbitrary finite value.
Therefore, delta(x)^2 is mathematically as ill-defined as
0/0 := lim a_n/b_n for a_n, b_n to 0,
though the latter is otherwise totally unrelated. But 0/0 can also be
defined to be 1 or infinity, or NaN, although any such definition is
inconsistent with the rules for handling numbers.
So talking about \int dx delta(x)^2 has no mathematical content.
It may convey some intuitive information, but this may turn out to
be correct or misleading, depending on the circumstances.
Mathematicians prefer arguments that are reliable. For them the
occurunce of a product of delta distributions is an indicator that
something went wrong with domains of definitions in an earlier step.
There is, however, an alternative mathematical theory in which one
can unrestrictedly multiply singular objects. This is called the theory
of generalized functions in the Colombeau sense. However, in the
Colombeau theory, infinitely many generalized functions correspond to
the same delta(x-x_0), so that already the latter is ambiguous.
On the other hand, this ambiguity can be fully controlled, making
generalized functions a useful tool so study some very singular
situations.
That generalized functions are not widely used has two reasons:
Getting started is difficult since a lot of nonstandard technical
notions have first to be swallowed. More importantly, all generalized
function theory in the sense of Colombeau can be equivalently restated
as theory about Young measures, and most functional analysts prefer the
latter formulation to study singular solutions of differential
equations.