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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Massachusetts Institute of Technology
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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 Distributions Distributions are sometimes called generalized functions which suggests that a function is also a distribution This is not quite true but a wide range of functions can also be thought of as distributions Given any function f t which is piecewise continuous and bounded by some power of t for large t one can de ne a corresponding distribution Tf by Tf dt f t t Since t falls o faster than any power this integral will converge Note that because the class of t s is very restricted the class of possible f t s is very large Fourier Transform For any function f t which is integrable meaning that dt f t converges de ne f dt e i t f t Fourier Transform of a Distribution To motivate the de nition suppose f t is integrable and consider f d Tf dt e i t f t d dt f t t Tf Note that these integrals are absolutely convergent so there is no problem about interchanging the order of integration So for any distribution T de ne its Fourier transform by T T 8 323 LECTURE NOTES 3 SPRING 2008 Distributions and the Fourier Transform p 3 Note that any function f t which is piecewise continuous and bounded by some power of t for large t can de ne a distribution and can therefore be Fourier transformed as a distribution Relation to convergence factor Suppose f t is not integrable and so does not have a Fourier transform Sup pose however that there exists a continuous sequence of regulated functions f t which are integrable for 0 which satisfy f t f t and which for each t satisfy lim f t f t 0 Example f t f t e t Note that the regulator that we used for the two point function at spacelike separations has this property To show if we Fourier transform f t and take the limit 0 at the end it is the same as the distribution theory de nition of the Fourier transform Proof The distribution theory de nition of the Fourier transform is T f Tf dt f t t The prescription is to use Tf lim Tf 0 We need to show these are equivalent Use d f Tf lim 0 lim 0 d lim 0 dt e i t f t dt f t t 8 323 LECTURE NOTES 3 SPRING 2008 Distributions and the Fourier Transform p 4 If we can take the limit inside the integral we are done Last step is proven with Lebesgue s Dominated Convergence Theorem If h t is a sequence of functions for which lim h t h t for all t 0 and if there exists a function g t for which dt g t converges and for which g t h t for all t and all then lim 0 dt h t dt h t Note by the way that the existence of the integrable bounding function g x is absolutely necessary A simple example of a function h t for which one CANNOT bring the limit through the integral sign would be a function that looks something like Analytically this function can be written as 1 if 1 t 1 1 h t 0 otherwise Note that the square well moves in nitely far to the right as 0 so h t 0 for any t But the integral of the curve is 1 for any and hence it is 1 in the limit The Lebesgue Dominated Convergence theorem excludes functions like this because any bounding function g t must be 1 for all t so g t cannot be integrable The theorem does apply however to lim dt f t t 0 8 323 LECTURE NOTES 3 SPRING 2008 Distributions and the Fourier Transform p 5 Take h t f t t h t f t t and g t f t t Bottom Line The prescription used by physicists is equivalent to the unambiguous de ni tion of the Fourier transform in tempered distribution theory That is if the function to be Fourier transformed f t is not integrable one can proceed as long as one can nd an integrable regulator f t such that f t f t and for each t lim f t f t 0 One can then Fourier transform f t instead In the general case one cannot take the limit 0 immediately but one must leave in the expression for the distribution Only after the distribution is evaluated for a particular test function can the limit 0 be taken Remember for example that we wrote the Fourier transform of the Feynman propagator as p2 i m2 i With the in place one can carry out integrals involving the propagator and then one can take the limit 0 at the end If one tried to set the term to zero immediately then the poles in the propagator would lead to ill de ned integrations


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