Table 4-1: Monte Carlo concepts in mathematics and physics.
Concept Markov chain symbol Physics symbol
State Space S Q
State x q
Probability Distribution e- S/Z
rather than in discrete steps. The results are almost the same, although the mathematical
framework becomes a little bit more involved.
To continue the discussion, it is appropriate at this point to narrow the scope and
consider the physical context in which we will work.
4.1.2 Expectation Values of Operators
Let us consider a physical system with dynamical variables denoted collectively by
q and let q live in some space Q which will have any mathematical structure needed to
perform the operations that follow. If S is the euclidian action then the expectation value
of an operator is written:
(F) dqF(q)e-S(q), (46)
where Z = fQ dqe-S(q) and F is the operator in question. Clearly, from the form of (4-6)
the integral is supported by the regions in Q where e-s is large. Standard optimization
yield the classical equations of motion for q. In quantum field theory a traditional next
step would be normal perturbation techniques, to expand the non-gaussian part of e-s
in powers of the coupling constant. But in anticipation for the application of stochastic
methods we interpret (4-6) as the statistical average of F weighted with the probability
S
distribution -- which it of course is. With Q finite we are in an exact application of the
Monte Carlo methods discussed earlier, with a translation of notation summarized in table
4-1.
We are interested in the special case of a quantum field theory represented as
a Lightcone World Sheet. In this case the space Q is the set of all (allowed) field