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.