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MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth March 13, 2008 INFORMAL NOTES DIRAC DELTA FUNCTION AS A DISTRIBUTION Why the Dirac Delta Function is not a Function: The Dirac delta function δ ( x ) is often described by considering a function that has a narrow peak at x = 0, with unit total area under the peak. In the limit as the peak becomes infinitely narrow, keeping fixed the area under the peak, the function is sometimes said to approach a Dirac delta function. One example of such a limit is g ( x ) lim g σ ( x ) , (4.1) σ 0 where g σ ( x ) ≡ √ 1 e 1 2 x 2 2 . (4.2) 2 π σ The area under g σ ( x ) is 1, for any value of σ > 0, and g σ ( x ) approaches 0 as σ 0 for any x other than x = 0. However, it was pointed out long ago that the delta function cannot be rigor- ously defined this way. The function g ( x ) is equal to zero for any x = 0, and is infinite at x = 0; it can be shown that any such function integrates to zero. To see this, define the integral as the area under the curve, and consider the construction: In this picture the vertical axis is entirely encased in rectangles, each of which has height 1. The width of the rectangles vary, with the lowest rectangle having width , for some > 0, and each successive rectangle has half the width of the rectangle below. Note that the outline of the boxes is everywhere above the curve g ( x ), so
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8.323 LECTURE NOTES 4, SPRING 2008: Dirac Delta Function as a Distribution p. 2 the area under g ( x ) must be less than the
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