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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

A new sum rule for the two-body distribution function of an infinite saturated fermion system

E. Guardiola, A. Polls

Published Dec 1, 1979• 2 citations
Publisher
DOI
10.1007/bf02750246

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where e is the dens i ty of the system, xi represents the set of space, spin and isospin co-ordinates , xi = {ri , ai , ~i} and the in tegrals include t h e summat ion ove r discrete var iables as wel l as the in tegra l on the space co-ordinates. Fu r the rmore , eq. (1) mus t be unders tood in the A + ~ l imit . The funct ion ~(Xl, x2) is an opera tor in the spin and isospin space, and is t rans la t iona l ly invar ian t , i.e. i t depends only upon the r x r 2 var iable . W i t h t he a id of the d is t r ibut ion func t ion one m a y compute the expect a t ion va lue of any local, two-body opera tor by means of the fo rmula

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where e is the dens i ty of the system, xi represents the set of space, spin and isospin co-ordinates , xi = {ri , ai , ~i} and the in tegrals include t h e summat ion ove r discrete var iables as wel l as the in tegra l on the space co-ordinates. Fu r the rmore , eq. (1) mus t be unders tood in the A + ~ l imit . The funct ion ~(Xl, x2) is an opera tor in the spin and isospin space, and is t rans la t iona l ly invar ian t , i.e. i t depends only upon the r x r 2 var iable . W i t h t he a id of the d is t r ibut ion func t ion one m a y compute the expect a t ion va lue of any local, two-body opera tor by means of the fo rmula
We use a dispersion relation in conjunction with the operator product expansion (OPE) to derive model independent sum rules for the dynamic structure functions of systems with large scattering lengths. We present an explicit sum rule for the structure functions that control the density and spin response of the many-body ground state. Our methods are general and apply to either fermions or bosons which interact through two-body contact interactions with large scattering lengths. By employing a Borel transform of the OPE, the relevant integrals are weighted toward infrared frequencies, thus allowing for greater overlap low energy data. Similar sum rules can be derived for other response functions. The sum rules can be used to extract the contact parameter introduced by Tan, including universality violating corrections at finite scattering lengths.
A formalism for treating the many-body problem of composite particles is presented for the case of composite particles consisting of two fermions. Commutation relations for composite-particle operators are derived, as well as a sum rule satisfied by the composite-particle Green's function. In an approximation that shuts off the interactions between composite particles in a consistent manner, the dynamical equation for the one-composite-particle Green's function is solved and the distribution function for the composite particles obtained. Possible applications to real systems are discussed.