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*              Quantum Field Theory Course Announcements            *
*                  Georgia Institute of Technology                  *

Domenico is making me do this:

in exercise 5.A.1

letter h is a number smaller than one, called hopping probability
letter s is a number smaller than one, called stopping probability

letter S is stepping matrix (in my messy lecture notes called
			   matrix h - hopping matrix)

probability conservation is
	1 = s + 2dh  (meaning you can hop left or right in each
		      of d dimensions separately, or give up and stop)

	m^2 = s / (h a^2)

combination is needed for the following reason:

if you halve the lattice spacing but keep the same h, then you are
effectively hopping much faster. In diffusion distance traveled in time t
is proportional to <x^2> (by symmetry average <x> = 0), that's roughly why
a^2 appears here. Or if it makes happier, Laplacian has dimmesnions of
1/(length)^2, and this is the only length available. So taking continuum
limit means keeping m^2 fixed, any other limit is either wrong or trivial.

When professionals do it, this is called the Theory of Critical Phenomena,
and exponent 1/2 in the diffusion relation |x(t)| --> t^{1/2} is called
the "critical exponent". This is a trivial exponent. The nontrivial ones
are sufficiently nontrival to award K. Wilson a single-winner Nobel Prize.

may the Force be with you
Predrag


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