The Green's-function Monte Carlo (GFMC) technique [Kal62] has been successfully used to study ground-state properties in a variety of many-body systems. The GFMC employs a Monte Carlo technique to iterate the integral equation
where 
is the ground state energy, and 
is
the Green's function defined by
Here 
are the coordinates of the
N particles in the system, 
is the wave function after n
iterations, and 
is the ground state energy of that wave function.
The Hamiltonian, H, will be described by conventional
nuclear theory
where 
and 
are given by realistic models of nuclear
potentials such as the Argonne and Urbana models, respectively, as
described in [WSA84].
The initial wave function and energy, 
and 
are typically obtained by variational techniques [PaW79].
It is not usually feasible to solve equation 
directly.
However, all that is required is a technique for sampling
for random 
for a given 
.
Then Monte Carlo techniques can be applied, using
as a density function for 
conditional upon
. Each iteration of equation may be considered a
step in a continuous random walk [Kal62].
A form of the GFMC method is particularly suitable for treating the
complicated spin-isospin structure of few body systems [CaS92]. The
GFMC is used to project out the ground state from an initial trial function,
, using the imaginary time propagator [Car91]:
Generally, 
can not be directly computed, but by dividing the
propagation time, 
, into small steps:
The full propagator can be evaluated by choosing an accurate approximation for the short time propagator and using Monte Carlo techniques to sample the propagator.
The Euclidean response of a nucleus to some operator, 
, is the Laplace
transform of the nuclear response, 
, as a function of imaginary time
[Pan94]:
where A is the number of nucleons, q is momentum transfer, 
is
energy loss, and 
is a constant. The nuclear response is defined as
where 
and 
are the ground state and
eigenstates of the system with energies of 
and 
,
respectively. If we consider the longitudinal and transverse spin-isospin
operators 
:
then 
and 
.
If accurate descriptions of the ground state wave functions,
, obtained from variational Monte Carlo techniques
[Wir91], are used the Euclidean response can be calculated
[Pan94]:
where 
is the propagator:
To carry out the integrations over 
and 
the
propagator is considered to be the product of short-time propagators
where n goes from 1 to
N as defined in equation . These short-time propagators can
be approximated accurately enough as [Car91]
and are sampled by Monte Carlo techniques.
The Green's function Monte Carlo techniques have been used to calculate the
Euclidean proton response functions on A=3 and A=4 systems
[Car91] [CaS92]. These calculations show that the inclusion of
two-body exchange current operators can cause significant effects even at a
relatively low momentum transfer. A comparison of the calculations to
experimental data shown that
two-body mechanisms are necessary to reproduce the experimental results.
In particular, calculations of 
quasifree
scattering showing a large enhancement in the transverse channel due to the
exchange currents were well supported by experiment [CaS92].
Recently the GFMC techniques have been used [Pan94] to calculate
the spin isospin nuclear responses in light nuclei, 
and
, in light of the lack of expected enhancement in the nuclear
longitudinal to transverse response ratio. The calculations indicate that
an enhancement should exist, even in the lightest nuclei, but it may be
moved out to the high excitation energy tail of the quasifree distribution.
These new calculations provide an excellent motivation for
a measurement of spin isospin responses in deuterium.
