A recently proposed statistical theory of the mean fields associated with the ground and excited collective states of a generic many-body system is extended by increasing the dimensions of the Pspace. In applying the new framework to nuclear matter, in addition to the mean field energies we obtain their fluctuations as well, together with those of the wavefunctions, in first order of the expansion in the complexity of the Q-space states. The physics described by the latter is assumed to be random. To extract numerical predictions from our scheme we develop a schematic version of the approach, which, while much simplified, yields results of significance on the size of the error affecting the mean fields, on the magnitude of the residual effective interaction, on the ground state spectroscopic factor, and on the mixing occurring between the vectors spanning the P-space. C 2002 Elsevier Science (USA) DE PACE AND MOLINARI lurking in a corner, not to mention the ambiguities existing in defining and constructing a potential model description of the NN interaction at short distances, where it is so hard to disentangle particle and nuclear physics from each other and where off-shell effects become larger.
It thus appears warranted to explore alternative routes in attacking the nuclear many-body problem. Hence, we have recently examined an approach to the theory of the ground state of atomic nuclei based on the concept of averaging rather than computing most of the physics related to the strong NN collisions at short distances. As a first step we have thus set up an energy averaging procedure suitable for constructing a mean field to be eventually identified with the shell model. Here, our procedure parallels the one successfully adopted in the development of the optical model of nuclear reactions [5], from which it is in fact directly inspired.
Clearly, for the mean field to have a meaning an assessment should be provided for the error associated with it. Indeed, if the error turns out to be large, then the concept of mean field is no longer tenable. Hence, we have derived an expression for the error through an expansion in the complexity of the states that the NN repulsion at short distances allows us to reach. At the basis of the derivation lies the hypothesis alluded to in the beginning of this Introduction, namely that the matrix elements of the NN repulsion are random, thus entailing the vanishing of the average value of the error, but, of course, not of its square. It is this one that our expansion (finite in finite nuclei and fastly convergent, as we shall prove) provides.
The above outlined framework basically corresponds to a statistical theory of the ground state (more generally, of the bound states) of the atomic nuclei. As a first example, we have attempted to implement it in the simplest among the latter, namely nuclear matter, assuming this system to represent, at least, partially, the physics of heavy nuclei. Already in performing this task, far from trivial technical problems have been encountered, notably the one of performing the sum over the complex nuclear excited states building up the error (square of ). We have been able to surmount this obstacle in Ref. [3].
It should be furthermore observed that, at the present stage, even in nuclear matter our statistical theory cannot be made parameter-free. As a consequence, its predictive power is somewhat limited: Indeed, what we actually do amounts to correlating a set of observables of nuclear matter (or heavy nuclei), either measured or computed (via, e.g., the BHF theory)-like the binding and the excited states energies and the level density-and predicting on this basis the mean field energies of the ground and of the collective nuclear states-together with the associated error-the residual effective interaction, which, while well defined, is presently hardly computable, and in addition the ground state spectroscopic factor.
Concerning the parameters entering our approach, one is needed to characterize our energy averaging procedure, whereas the others are required whenever the experimental values of the above referred to observables are lacking or their theoretical evaluation not available. We shall discuss in particular the significance of the former, which plays a central role in our approach.
This work is cast in the language introduced to build the unified theory of nuclear reactions [5], at the core of which lies the partition of the nuclear Hilbert space in the P and Q sectors. We implement this partition by inserting the random aspects of the physics of the nucleus in the latter. Hence the Q-space should not be viewed as specific of a given Hamiltonian, but rather it displays universal features and indeed it is in the Q-space that the energy average is performed. The P-space embodies instead the deterministic physics of the nucleus and its dimensions should be dovetailed for this purpose. This is why the chief scope of the present research is to expand the dimensions of the P sector of the nuclear Hilbert space, thus removing a major restriction of our past work, where the P-space was limited to one dimension only.
Specifically, the enlargement of the dimensions of the P-space allows us (i) to compute the fluctuations of the nuclear wave function and not only of the energy, thus assessing the error associated with both the mean field energy and its wave function; (ii) to show the fast convergence of the expansion for the error, essentially stemming from the rapid increase of the nuclear level density