= 0, this equation reads

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~=o . dx

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So the solution should be

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(F.43)

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f (x)

= {Cl

for x < 0 for x> 0,

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(F.44)

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where Cl and C2 are two constants. The question is: How do you match these two constants This is a first-order differential equation and there must be only one integration constant, so there must be a relation between Cl and C2. If we use (F.39) or (F.41), we conclude that this relation should be (F.45)

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DISCONTINUOUS FUNCTIONS

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If, on the other hand, we regard the delta function as the limit of a very thin and large barrier, we easily find that [252] (F.46)

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Which one is correct, (F.45) or (F.46) We can easily see that (F.46) is the correct relation because as Griffiths and Walborn [252] pointed out, (F.42) can be solved directly (F.47) where c is the single integration constant. Can we trace down the mistake in the steps that lead to (F.39) and (FA1) We notice that if we apply the steps that lead us to (FA1) to the product of four step functions instead of just two as in (FAO), we find that

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1 03(x)8(x) = 0(x)8(x) = 48(x),

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(F.48)

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which together with (F.41) implies the absurd conclusion that 1/4 = 1/2. In (F.39), the mistake is the assumption that as 8(x) is even, the total integrand would also be even. Depending on the microscopic behavior of both 8( x) and f (x) at x = 0, however, the delta function can sample the function more at one side of the discontinuity than at the other side, as we showed in the example above. Another way of writing (F.39) or (F.41) that often appears in the literature [610,620] is

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dx f(x) 8(x) =

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(1/2)f(0) f(O)

if ab = 0, if ab < O.

ff~>~

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(F.49)

This was even suggested in a paper [664] as an alternative, more precise definition of the delta function. It is embarrassing that even though the problem with the blind use of (F.39) or (F.41) or (F.49) has been known for a long time, papers misusing these relations keep reappearing in the literature. 6 Griffiths and Walborn [252] mention that in the physics literature, the problem was spotted at least as far back as 1981 [86, 126, 439,530, 585]. But well before that, in 1954, Schwartz [542] had already pointed out the problems with the multiplication of a distribution by another distribution. In that paper he shows that the associative property of multiplication and the concept of derivative are incompatible, within a mathematical framework where multiplication of distributions by each other is allowed. This incompatibility can be seen easily from our short demonstration about the flaw in our deduction of (FA1). There, the only logical steps are the use of the associative property of multiplication to infer that 03 (x) = O( x), and the rule for the derivative of a product. Together, these two simple steps lead to the absurd conclusion that 1/4 = 1/2. So they are incompatible within the context of multiplication of distributions.

6For some papers where this mistake was made, as well as their subsequent discussion and correction, see [41,86, 126, 127,253,393,425,426,570].

THE GOOD, THE BAD, AND THE UGLY

Schwartz's conclusion in that paper [542] was that the multiplication of distributions is impossible. But there is another solution to this problem. This solution, as we have already suggested, is to abandon the associative law for multiplication of distributions. There are several step functions and delta functions differing in their microscopic behavior. Then, even though the product of any number of step functions is also a step function, it is yet a step function of a different sort than the ones that were multiplied to originate it. Its microscopic behavior at its discontinuity point is different from that of its originators. This is basically the idea proposed by Colombeau in his theory of multiplication of distributions [110-112,114]. We do not discuss Colombeau's theory in detail here, but we recommend to the interested reader his paper [112], his book [113], and another book by Biagioni [521, all of which are quite accessible to physicists and engineers.

Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.