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{{Short
[[File:Random phase approximation ring diagrams.png|thumb|Bubble diagrams, which result in the RPA when summed up. Solid lines stand for interacting or non-interacting [[Green's function (many-body theory)|Green's functions]], dashed lines for two-particle interactions.]]
The '''random phase approximation''' ('''RPA''') is an approximation method in [[condensed matter physics]] and [[nuclear physics]]. It was first introduced by [[David Bohm]] and [[David Pines]] as an important result in a series of seminal papers of 1952 and 1953.<ref name="Bohm Pines pp. 625–634">{{cite journal | last1=Bohm | first1=David |author-link= David Bohm| last2=Pines | first2=David |author-link2=David Pines| title=A Collective Description of Electron Interactions. I. Magnetic Interactions | journal=Physical Review | publisher=American Physical Society (APS) | volume=82 | issue=5 | date=1 May 1951 | issn=0031-899X | doi=10.1103/physrev.82.625 | pages=625–634| bibcode=1951PhRv...82..625B }}</ref><ref name="Pines Bohm pp. 338–353">{{cite journal | last1=Pines | first1=David |author-link=David Pines| last2=Bohm | first2=David |author-link2=David Bohm| title=A Collective Description of Electron Interactions: II. CollectivevsIndividual Particle Aspects of the Interactions | journal=Physical Review | publisher=American Physical Society (APS) | volume=85 | issue=2 | date=15 January 1952 | issn=0031-899X | doi=10.1103/physrev.85.338 | pages=338–353| bibcode=1952PhRv...85..338P }}</ref><ref name="Bohm Pines pp. 609–625">{{cite journal | last1=Bohm | first1=David |author-link=David Bohm| last2=Pines | first2=David |author-link2=David Pines| title=A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas | journal=Physical Review | publisher=American Physical Society (APS) | volume=92 | issue=3 | date=1 October 1953 | issn=0031-899X | doi=10.1103/physrev.92.609 | pages=609–625| bibcode=1953PhRv...92..609B }}</ref> For decades physicists had been trying to incorporate the effect of microscopic [[Quantum mechanics|quantum mechanical]] interactions between [[electron]]s in the theory of matter. Bohm and Pines' RPA accounts for the weak screened Coulomb interaction and is commonly used for describing the dynamic linear electronic response of electron systems. It was further developed to the relativistic form (RRPA) by solving the [[Dirac equation]].<ref>{{Cite journal |last1=Deshmukh |first1=Pranawa C. |author-link=Pranawachandra Deshmukh |last2=Manson |first2=Steven T. |date=September 2022 |title=Photoionization of Atomic Systems Using the Random-Phase Approximation Including Relativistic Interactions |journal=Atoms |language=en |volume=10 |issue=3 |pages=71 |doi=10.3390/atoms10030071 |issn=2218-2004 |doi-access=free|bibcode=2022Atoms..10...71D }}</ref><ref>{{Cite journal |last1=Johnson |first1=W R |last2=Lin |first2=C D |last3=Cheng |first3=K T |last4=Lee |first4=C M |date=1980-01-01 |title=Relativistic Random-Phase Approximation |url=https://iopscience.iop.org/article/10.1088/0031-8949/21/3-4/029 |journal=Physica Scripta |volume=21 |issue=3–4 |pages=409–422 |doi=10.1088/0031-8949/21/3-4/029 |bibcode=1980PhyS...21..409J |s2cid=94058089 |issn=0031-8949|url-access=subscription }}</ref>
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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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