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We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension$d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas ofinteracting bosons in the GP scaling, in the limit $N\rightarrow\infty$. Themain result of this paper is the proof of convergence of the correspondingBBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous workon the well-posedness of the Cauchy problem for GP hierarchies,\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based onspace-time norms introduced by Klainerman and Machedon in \cite{klma}. We notethat in $d=3$, this has been a well-known open problem in the field. While ourresults do not assume factorization of the solutions, consideration offactorized solutions yields a new derivation of the cubic, defocusing nonlinearSchr\"odinger equation (NLS) in $d=3$.

Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms