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The RPA was criticized in the late 1950s for overcounting the degrees of freedom and the call for justification led to intense work among theoretical physicists. In a seminal paper [[Murray Gell-Mann]] and [[Keith Brueckner]] showed that the RPA can be derived from a summation of leading-order chain [[Feynman diagram]]s in a dense electron gas.<ref name="Gell-Mann Brueckner pp. 364–368">{{cite journal | last1=Gell-Mann | first1=Murray | last2=Brueckner | first2=Keith A. | title=Correlation Energy of an Electron Gas at High Density | journal=Physical Review | publisher=American Physical Society (APS) | volume=106 | issue=2 | date=15 April 1957 | issn=0031-899X | doi=10.1103/physrev.106.364 | pages=364–368| bibcode=1957PhRv..106..364G | s2cid=120701027 | url=https://authors.library.caltech.edu/3713/1/GELpr57b.pdf }}</ref>
In [[superconductivity]], RPA can be used to derive [[Bardeen–Pines interaction]] between [[Phonon|phonons]] and electrons that leads to [[Cooper pair|Cooper pairing]].<ref>{{Cite book |last=Coleman |first=Piers |url=https://books.google.com/books?id=ESB0CwAAQBAJ&dq=bardeen+pines+interaction&pg=PA225 |title=Introduction to Many-Body Physics |date=2015-11-26 |publisher=Cambridge University Press |isbn=978-1-316-43202-0 |language=en}}</ref>
The consistency in these results became an important justification and motivated a very strong growth in theoretical physics in the late 50s and 60s.▼
▲The consistency in these results became an important justification and motivated a very strong growth in [[theoretical physics]] in the late 50s and 60s.
==Applications==
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:<math>\left|\mathrm{RPA}\right\rangle=\mathcal{N}\mathbf{e}^{Z_{ij}\mathbf{a}_{i}^{\dagger}\mathbf{a}_{j}^{\dagger}/2}\left|\mathrm{MFT}\right\rangle</math>
where ''Z'' is a [[symmetric matrix]] with <math>|Z|\leq 1</math> and
:<math>\mathcal{N}= \frac{\left\langle \mathrm{MFT}\right|\left.\mathrm{RPA}\right\rangle}{\left\langle \mathrm{MFT}\right|\left.\mathrm{MFT}\right\rangle}</math>
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