Re: a question about non - locality

From: C. M. Heard (heard_at_pobox.com)
Date: 03/26/05 

Date: Sat, 26 Mar 2005 08:42:20 +0000 (UTC)

Aaron Bergman wrote:
> "Robert C. Helling" wrote:
[ ... ]
> > You are right, any finite number of derivatives is local. However
> > powerseries in derivatives can be non-local. The standard example is
> > the translation operator exp(a d_x) (d_x is the derivative):
> >
> > (exp(a d_x) f)(x) = f(x+a)
> >
> > and that's genuinely non-local.
>
> This is said a lot, and I hadn't really thought about it much, but
> there's something funky here. The above formula only holds for analytic
> functions. Now, at least in whatever way we can make sense of it,
> analytic functions are going to be of measure zero in the path integral.

One way to make sense of this is to consider the map f -> (exp(a d_x) f)
as a linear operator in the Hilbert space L^2(R). For analytic functions
it is defined by a power-series in derivatives. In particular, it is
defined for all finite linear combinations of the Hermite-Gauss energy
eigenfunctions of a harmonic oscillator. Furthermore the norm is bounded
over this set. But this set is dense in L^2(R) since the Hermite-Gauss
functions are a basis. A bounded linear operator being continuous, the
definition can be extended to all of L^2(R).

The key to this is that a set of measure zero can be dense, and a
bounded linear operator that is densely defined extends to the whole
Hilbert space.

Mike

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