From: chernoff@math.berkeley.edu (Paul R. Chernoff) Subject: Re: What is deep about Fourier!! Date: Fri, 26 Jan 2001 01:16:03 GMT Newsgroups: sci.physics.research Summary: Why does the Fourier transform arise naturally in physics? In article <94e27i$ukg$1@nnrp1.deja.com>, wrote: >well, we usually use fourier analysis to move from the momentum space >to the the position space and vice versa >my question now is , why specially fourier, i mean what is so special >about it? any deep reasons? does it has something to do with being p >and x physical quantities? >i will be very appreciated for any help for a better understanding > The position & momentum operators satisfy the canonical commutation relation [P,Q] = (1/sqrt(-1))I where I is the identity operator and we use units where h bar = 1. It is a theorem of Marshall Stone and John von Neumann (independently) that any irreducible pair of self-adjoint operators satisfying CCR is unitarily equivalent to the "Schrodinger representation" of P and Q, namely P = (1/sqrt(-1)) d/dx and Q = multiplication by the coordinate function x on the Hilbert space L^2(-oo,oo) of square-integrable complex valued functions f on the line (-oo,oo). Now note that if [P,Q] = (1/sqrt(-1))I then [-Q,P] also equals (1/sqrt(-1))I. Hence P' = -Q and Q' = P is another pair satisfying the CCR. Accordingly, by the Stone-von Neumann theorem, there is a unitary operator U : L^2 --> L^2 such that UPU* = -Q and UQU* = P. U is determined up to a phase factor, i.e. a complex number of absolute value 1, and it turns out that U is (up to a constant phase factor) the Fourier transform F. So it is *inevitable* that the Fourier transform must emerge when we "move from position space to momentum space", i.e. when we change from a representation of CCR in which Q is a multiplication operator ("position space") to a representation of the CCR in which P is a multiplication operator ("momentum space"). All the above generalizes from the 1-dimensional case to the n-dimensional case, of course. (But it is false when there are infinitely many independent canonical pairs, as in quantum field theory; this is just one of the serious mathematical problems that arise when one attempts to construct quantum fields rigorously.) -- # Paul R. Chernoff chernoff@math.berkeley.edu # # Department of Mathematics 3840 # # University of California "Against stupidity, the gods themselves # # Berkeley, CA 94720-3840 struggle in vain." -- Schiller #