MIT 8 323 - DISTRIBUTIONS AND THE FOURIER TRANSFORM (6 pages)

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DISTRIBUTIONS AND THE FOURIER TRANSFORM



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DISTRIBUTIONS AND THE FOURIER TRANSFORM

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Pages:
6
School:
Massachusetts Institute of Technology
Course:
8 323 - Relativistic Quantum Field Theory I
Relativistic Quantum Field Theory I Documents

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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 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8 323 Relativistic Quantum Field Theory I Prof Alan Guth March 3 2008 INFORMAL NOTES DISTRIBUTIONS AND THE FOURIER TRANSFORM Basic idea In QFT it is common to encounter integrals that are not well de ned Peskin and Schroeder for example give the following formula p 27 after Eq 2 51 for the two point function 0 x y 0 for spacelike separations x y 2 r 2 i peipr D r dp 2 2 2 r p2 m2 If this integral is de ned in the usual way as lim peipr dp p2 m2 then it does not exist The integral can be de ned by putting in a convergence factor e p peipr e p lim dp 0 p2 m2 But how does one know whether a di erent convergence factor would get the same result One way to resolve these issues is to treat the ambiguous quantity as a distribution rather than a function All tempered distributions to be de ned below have Fourier transforms which are also tempered distributions Furthermore we can show that the prescription used above is equivalent to the tempered distribution de nition of the Fourier transform Distribution A distribution is a linear mapping from a space of test functions to real or complex numbers An operator valued distribution maps test functions into operators Test Functions The space of test functions t determines what type of distribution one is discussing The test functions for tempered distributions belong to Schwartz space the space of functions which are in nitely di erentiable and the func tion and each of its derivatives fall o faster than any power for large t The Gaussian is a good example of a Schwartz function Any function in Schwartz 8 323 LECTURE NOTES 3 SPRING 2008 Distributions and the Fourier Transform p 2 space has a Fourier transform in Schwartz space The Fourier transform of a Gaussian is a Gaussian Functions as



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