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Some Problems with the Use of the Dirac Delta Function I: What Is the Value of ∫∫∫ ∞ Δ(X)Dx?

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Some Problems with the Use of the Dirac Delta Function I: What Is the Value of ∫∫∫ ∞ Δ(X)Dx? Revista Brasileira de Ensino de F´ısica, vol. 43, e20210132 (2021) Articles www.scielo.br/rbef c b DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0132 Licenc¸aCreative Commons Some problems with the use of the Dirac delta function I: R 1 What is the value of 0 δ(x)dx? M. Amaku1,2 , F.A.B. Coutinho1, O. Eboli´ *3 , E. Massad4 1Universidade de S˜aoPaulo, Faculdade de Medicina, 05405-000, S˜aoPaulo, SP, Brasil. 2Universidade de S˜aoPaulo, Faculdade de Medicina Veterin´aria e Zootecnia, 05508-970, S˜aoPaulo, SP, Brasil. 3Universidade de S˜aoPaulo, Instituto de F´ısica,05508-090, S˜aoPaulo, SP, Brasil. 4Escola de Matem´aticaAplicada, Funda¸c˜aoGet´ulioVargas, 22250-900, Rio de Janeiro, RJ, Brasil. Received on April 06, 2021. Revised on June 03, 2021. Accepted on June 05, 2021. This is the first of a series of articles dealing with the casual use of distributions made by physicists and engineers. This use is very economical but sometimes leads to embarrassing conclusions that are difficult to justify with rigorous theories of distributions. Keywords: Dirac delta function, distributions. 1. Motivation until the work of Schwartz [7]. The work of Dirac and Schwartz is praised in the book by Lighthill [8] as One of the authors received an e-mail from a famous follows: “To Paul Dirac who saw the it must be true; physicist about one of his articles [1]. The following Laurent Schwartz who proved it and to George Temple is a transcription of a relevant part that motivated who showed how simple it could be made.” The last this article: “As to your paper, I see it as an appeal reference is to the work George Temple [9]. It should that people stop using the now-standard notation for be mentioned the above mentioned symbolic methods the Dirac delta function, and switch to the notation of can also be justified by an almost equivalent theory by distributions. I don’t think this is practical, and I’m not J. G. Mikuzinski [10, 11]; see also [12, 13]. For further too supportive of papers that insist on this anyway.” information the history of distributions see [14, 15] and We have to respect this opinion because the physicists’ references therein. use of the Dirac delta function as if it were a function In spite of this some of the symbolic methods men- is very economical. However, in the mentioned paper, tioned above, the efforts to put them in a rigorous we were not “advocating stopping the now standard mathematical form still do not explain them clearly. notation of the Dirac function”. In fact we responded In fact, we believe that neither Schwartz [7] nor Tem- that “I disagree with you that our paper proposes a new ple [16] distribution theories can easily clarify the formal way of using the Dirac delta function. In fact we used methods commonly used by Physicists. In this series of methods that are becoming more and more common articles we try to show examples where this appears in physics and only mentioned in remarks that the to be so, comment on the possible relevance of the procedures could be done consistently by using Schwartz results, and show cases where the theory, when properly or Temple distribution theories. I am sure that you used, greatly illuminate the physics involved. The first of known examples of inconsistencies when using the now these problems presented in this article has no practical standard methods to handle blindly the Dirac function. consequences, but since it brings a lot of confusion to the Let us explain our position more clearly with a bit of literature we think it is a good idea to clarify it. More history. specifically, we focus on the value of the integral th th In the late 19 century and the beginning of the 20 Z ∞ century a number of special techniques called “improper δ(x)dx = X. (1) functions methods” or “symbolic methods” were being 0 used by engineers and mathematical physicists [2–6]. 1 Two possible choices for X are 2 and 1, that are called, The Dirac delta function became very popular because respectively, the weak and strong definitions of the Dirac of the extensive use of them in Dirac’s work. However, delta. We discuss these two options in detail in Section 2, it was introduced before by Kirchhoff and widely used while we summarize the main points of the theories of by Heaviside. These methods were difficult to justify distributions in Section 3. Once a choice for the integral in equation (1), we must * Correspondence email address: [email protected] consistently work with its chosen value when we consider Copyright by Sociedade Brasileira de F´ısica.Printed in Brazil. e20210132-2 Some problems with the use of the Dirac delta function I other related problems as it is shown below. As an 2. The problem illustration, we present the relevance for physics of this choice in Section 4 of this paper where we describe and The value of the integral in equation (1) involving the deduce two routinely used formulas in electromagnetism: Dirac delta function is the cause of some perplexity in the literature. The following integral (1) The Laplacian of the Coulomb potential, namely Z ∞ 1 δ(x)dx = 1 (6) 4 = −4πδ3(~r), (2) −∞ r is well know, but the value of equation (1) is subject of where r stands for |~r|. some discussion in the literature. (2) The divergence of the Coulomb field of a unit point According to G. Barton [18] on page 33 the “strong charge, namely definition of the delta function requires X = 1,... but some books choose X = 1 ”. We quote below some books ~r 2 ∇ · = 4πδ3(~r). (3) and articles that choose this last value: r3 (1) The first reference is in page 29 of [19]. No These two formulas were not know before the theory of explanation for this is given, but we can think generalized functions. We show how the theory modifies that the reasoning is as follows. We can write our understanding of electromagnetism and clarify com- equation (1) as plex examples. A very good early paper on this is [17]. Z ∞ Z 0 Z ∞ The two formulas are well know by physics students, but 1 = δ(x)dx = δ(x)dx + δ(x)dx we show that their derivation appear to depends on the −∞ −∞ 0 value of the integral mentioned in the title. Z ∞ As mentioned above these formulas were unknown = 2 δ(x)dx (7) until the development of distribution theory and without 0 them some calculations are problematic. For instance, we since δ(x) is an even function. Here is a similar know that argument 1 1 ~r ∇ · ~r 1 Z ∞ 4 = ∇ · ∇ = −∇ · = + ~r · ∇ 1 ikx r r r3 r3 r3 δ(x) = e dk (8) 2π −∞ 3 ~r · ~r = − − 3 = 0 (4) then r3 r5 Z ∞ Z ∞ 1 ikx ~ ~ e dk dx = 1. (9) for ~r 6= 0 and that this laplacian is not defined for ~r = 0. −∞ 2π −∞ However, Since equation (8) is an even function of x we have Z Z 2 1 3 1 4πR ∇ · ∇ d ~r = ∇ · dS~ = − = −4π (5) ∞ ∞ 2 Z 1 Z 1 r≤R r S r R eikxdk dx = . (10) 0 2π −∞ 2 This was not considered a problem before the intro- duction of distributions. It was considered to be just (2) The second reference is given in page 791 of [20]. 1 the potential of a point charge, which, for the pioneers, The justification for using equation (1) with X = 2 didn’t exist anyway: it was an idealized case. Of course is that the n-dimensional delta function is given by if we consider equation (2) the above result becomes so natural that the above calculation is considered as a 1 δ(|~r|) δ(~r) = n−1 , (11) proof of this equation. Note, however, that since the right ωn |~r| side of this equations is the delta function, a distribution or generalized function as it is also called, the left where ωn is the surface area of the n-dimensional side must also be a generalized function. Therefore, we unit sphere. In order to the integral of δ(~r) over shall return to this in Sections 3 and 4 of this article, the whole space to be one we must use equation (1) with X = 1 . and explain how electromagnetic theory was before the 2 theory of distributions and the advantages gained after the electric field, magnetic field, electric potential, etc., Remark 1: Formula (14) is a simplified version of the are “promoted” to distributions so that both sides of argument given by Courant and Hilbert [20] described the equations are distributions. The end of Section 4 is above in equation (11). Take for instance n = 3, then especially important in this respect. ω3 = 4π. Revista Brasileira de Ensino de F´ısica, vol. 43, e20210132, 2021 DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0132 Amaku et al. e20210132-3 (3) The third reference is given by Blinder [21] in his by physicists. In fact, consider the delta function in equation (7). His justification for this is that “the spherical coordinates, which is given by [18] factor 1 reflects the fact that the delta function is 2 δ(r − r )δ(θ − θ )δ(φ − φ ) located in one of the limits, so that, only half of the δ3(~r − ~r ) = 0 0 0 (18) 0 r2 sin θ delta function is within the range of integration”.
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