DIRAC DELTA FUNCTION AS A DISTRIBUTION

MASSACHUSETTS INSTITUTE OF TECHNOLOGYP hysics : Relativistic Quantum Field Theory IProf .Alan GuthMarch 13, 2008 INFORMAL NOTESDIRAC DELTA FUNCTION AS A DISTRIBUTIONWhy the DIRAC DELTA FUNCTION is not a FUNCTION :The DIRAC DELTA FUNCTION (x) is often described by considering a FUNCTION thathas a narrow peak atx= 0, with unit total area under the peak .In the limit as thepeak becomes infinitely narrow, keeping fixed the area under the peak, the functionis sometimes said to approach a DIRAC DELTA FUNCTION .One example of such a limitisg(x) lim 0g (x),( )whereg (x) 1 2 e 12x2/ 2.( )The area underg (x) is 1, for any value of >0, andg (x) approaches 0 as 0for anyxother thanx= , it was pointed out long ago that the DELTA FUNCTION cannot be rigor-ously defined this way.

The Fourier transform of this distribution is then defined by applying the same distribution to the Fourier transform of the test function, so ˜ T f [ϕ] ≡ T f [˜]= ∞ −∞ pe ipa ˜ ϕ (p). (4.11) But the inverse Fourier transform is given by ϕ (x)= 1 2 π ∞ −∞ d pe ipx ˜ ϕ (p), (4.12) so by comparing the two formulas above ...

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  Delta, Transform, Inverse, Carid, Dirac delta

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