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.