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**Arthur Michael Jaffe** (/ˈdʒæfi/; born December 22, 1937) is an American mathematical physicist at Harvard University, where in 1985 he succeeded George Mackey as the Landon T. Clay Professor of Mathematics and Theoretical Science.^{[1]}^{[2]}

Arthur M. Jaffe | |
---|---|

Born | December 22, 1937 |

Nationality | American |

Alma mater | Princeton University Clare College, Cambridge |

Scientific career | |

Fields | Mathematical physics |

Institutions | Harvard University |

Doctoral advisor | Arthur Wightman |

Doctoral students | Ezra Getzler Joel Feldman Clifford Taubes Eugene Wayne John Imbrie Christopher King Jonathan Weitsman |

After graduating from Pelham Memorial High School in 1955,^{[3]} Jaffe attended Princeton University as an undergraduate obtaining a degree in chemistry in 1959, and later Clare College, Cambridge, as a Marshall Scholar, obtaining a degree in mathematics in 1961. He then returned to Princeton, obtaining a doctorate in physics in 1966 with Arthur Wightman. His whole career has been spent teaching mathematical physics and pursuing research at Harvard University. Jaffe was appointed as Professor of Physics in 1970, and had his title changed to Professor of Mathematical Physics in 1974. As part of this transition, Jaffe became a member of the mathematics department. He served as chair from 1987 to 1990.^{[4]}

Arthur Jaffe's 30 doctoral students include Joel Feldman, Ezra Getzler, Clifford Taubes, Eugene Wayne, John Imbrie, Christopher King, and Jonathan Weitsman. In total, Jaffe has over 300 mathematical descendants. He has had many post-doctoral collaborators, including Robert Schrader, Konrad Osterwalder, Juerg Froehlich, Roland Sénéor , Thomas Spencer, Antti Kupiainen, Krzysztof Gawedzki, Tadeusz Balaban, Andrew Lesniewski, Slawomir Klimek, Zhengwei Liu, and Kaifeng Bu.

For several years Jaffe was president of the International Association of Mathematical Physics, and later of the American Mathematical Society. He chaired the Council of Scientific Society Presidents.^{[5]} He served as chair of the board of the Dublin Institute for Advanced Studies, School of Theoretical Physics, from 2005 to 2020.

Jaffe conceived the idea of the Clay Mathematics Institute and its programs, including the employment of research fellows and the Millennium Prizes in mathematics. He served as a founding member, a founding member of the board, and the founding president of that organization.

Arthur Jaffe began as chief editor of *Communications in Mathematical Physics* in 1979 and served for 21 years until 2001. He served as distinguished visiting professor at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences.

One of Arthur Jaffe's earliest contributions was his proof, joint with Henry Epstein and Vladimir Glaser, that energy densities in local quantum field theories are always nonpositive.^{[6]}

A large amount of Jaffe's work deals with the mathematical construction and proof of results in quantum field theory. Jaffe began his research on the topic in the late 1960s and early 1970s, at which point the only local quantum field theory which had been constructed mathematically was the free field model. In a series of landmark papers, Jaffe and collaborators made great progress in understanding the nature of quantum field theory.^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]} This culminated in the first ever mathematical local quantum field theory with non-linearity and non-trivial scattering.^{[13]} Thus it established the mathematical compatibility of special relativity, quantum theory, and interaction. For this work, Jaffe and James Glimm are acknowledged as the founders of the subject of constructive quantum field theory.

Another notable contribution of Jaffe's is his proof, joint with James Glimm and Thomas Spencer, that quantum field theories can have phase transitions.^{[14]}^{[15]} While physicists had conjectured for many years that this phenomenon took place, Jaffe-Glimm-Spencer's work gave the first mathematical proof. This work is also notable for using the formalism of reflection positivity to establish its results, which has since become common practice among researchers studying phase transitions in quantum field theory.^{[16]}

One recurring idea in Jaffe's works is the notion of reflection positivity, which was first introduced by Osterwalder and Schrader while they were Jaffe's post-doctoral fellows. The notion of reflection positivity has served since its inception as a key tool in the quantization of classical Euclidean field theories into relativistic quantum field theories. It also provides a basic tool to study phase transitions both in statistical physics as well as in quantum field theory. Jaffe has made major contributions to the development of this theory, by establishing key examples,^{[17]}^{[18]}^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}^{[24]}^{[25]} introducing important generalizations,^{[26]}^{[27]}^{[28]} and providing geometric interpretations.^{[29]}^{[30]}

Jaffe is also known for his mathematical proof of an aspect of the abelian Higgs mechanism. Namely, he showed that symmetry breaking in the abelian Higgs model induces a gap in the mass spectrum.^{[31]}^{[32]}^{[33]}

Within his work on supersymmetric quantum field theories Jaffe is most known for introducing the JLO cocycle, along with collaborators Andrzej Lesniewski and Konrad Osterwalder.^{[34]}^{[35]} The JLO construction takes as input a supersymmetric quantum field theory (mathematically, a θ-summable spectral triple) and outputs a cocycle in Alain Connes' cyclic cohomology.

In his later years Arthur Jaffe has made varied contributions to the theory of quantum information, along with postdoctoral researchers Zhengwei Liu, Kaifeng Bu, and students.^{[36]}^{[37]}^{[38]}^{[39]} Notable among these contributions are the introduction of quantum Fourier analysis,^{[40]}^{[41]} the study of quantum resources,^{[42]}^{[43]}^{[44]} quantum error correction,^{[45]} and the introduction of a 3D graphical language for quantum information.^{[46]}

Jaffe is the author of several essays on the philosophy of mathematics and physics, with a special emphasis on the role of proof and rigor in these subjects.^{[47]}^{[48]}^{[49]}^{[50]} The most influential of these works was his essay with Frank Quinn, which introduced the notion of "Theoretical Mathematics".^{[51]} An issue of the Bulletin of the American Mathematical Society was devoted to responses to this article, written by leading mathematicians.^{[52]}

Arthur Jaffe is the recipient of numerous awards and honors. In 1979 he was awarded the New York Academy of Science prize in Mathematics and Physics.^{[53]} In 1980 Arthur Jaffe was awarded the Dannie Heineman Prize for Mathematical Physics. In 1990 he was awarded the Medal Collège de France.^{[54]} In 2018 he was awarded the ICCM prize for best mathematical paper in the last five years.^{[55]} In 2020 he was awarded the Science China Mathematics Award for best editor.^{[53]} Jaffe has been an invited speak at many distinguished conferences, including the 1978 International Congress of Mathematicians at Helsinki.^{[56]}

Additionally, Jaffe is a fellow of many mathematical societies, including the Hagler Institute for Advanced Study, American Physical Society, Society of Industrial and Applied Mathematicians, American Mathematical Society, American Association for the Advancement of Science. He is a member of the American Academy of Arts and Sciences, US National Academy of Sciences, and an honorary member of the Royal Irish Academy.^{[53]}

Jaffe was married from 1971 to 1992 to Nora Frances Crow and they had one daughter, Margaret Collins, born in 1986. Jaffe was married to artist Sarah Robbins Warren from 1992 to 2002.

**^**"Website of ACAP". Archived from the original on 13 July 2019. Retrieved 19 March 2018.**^**"Harvard University list of Faculty of Arts and Sciences". 1985.**^**"Oral History Interviews. Arthur Jaffe, interviewed by Katherine Sopka".*American Institute of Physics*. 15 February 1977.**^**"Harvard University list of Faculty of Arts and Sciences". 1987.**^**"CSSP Board History".*www.sciencepresidents.org*. Retrieved 24 April 2024.**^**Epstein, H.; Glaser, V.; Jaffe, A. (1 April 1965). "Nonpositivity of the energy density in quantized field theories".*Il Nuovo Cimento (1955-1965)*.**36**(3): 1016–1022. Bibcode:1965NCim...36.1016E. doi:10.1007/BF02749799. ISSN 1827-6121.**^**Jaffe, Arthur (1966). "Existence Theorems for a Cut-off λφ4 Field Theory".*Mathematical Theory of Elementary Particles*– via MIT Press.**^**Glimm, James; Jaffe, Arthur (25 December 1968). "A $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ Quantum Field Theory without Cutoffs. I".*Physical Review*.**176**(5): 1945–1951. doi:10.1103/PhysRev.176.1945.**^**Cannon, John T.; Jaffe, Arthur M. (1 December 1970). "Lorentz covariance of the λ(ϕ4)2 quantum field theory".*Communications in Mathematical Physics*.**17**(4): 261–321. doi:10.1007/BF01646027. ISSN 1432-0916.**^**"The $\lambda(\varphi^4)_2$ quantum field theory without cutoffs. II. The field operators and the approximate vacuum | Annals of Mathematics". Retrieved 19 April 2024.**^**Glimm, James; Jaffe, Arthur (1970). "The λ(φ4)2 quantum field theory without cutoffsquantum field theory without cutoffs: III. The physical vacuum".*Acta Mathematica*.**125**(none): 203–267. doi:10.1007/BF02392335. ISSN 0001-5962.**^**Jaffe, Arthur; Glimm, James (1973). "Positivity of the φ43 Hamiltonian".*Fortschritte der Physik*.**21**.**^**Glimm, James; Jaffe, Arthur; Spencer, Thomas (1974). "The Wightman Axioms and Particle Structure in the P(φ)2 Quantum Field Model".*Annals of Mathematics*.**100**(3): 585–632. doi:10.2307/1970959. ISSN 0003-486X. JSTOR 1970959.**^**Jaffe, Arthur; Glimm, James; Thomas, Spencer (1975). "Phase Transitions for φ42 Quantum Fields".*Communications in Mathematical Physics*(45): 203–216.**^**Jaffe, Arthur; Glimm, James; Spencer, Thomas (1976). "Existence of Phase Transitions for φ42 Quantum Fields".*Mathematical Methods of Quantum Field Theory*– via CNRS.**^**Fröhlich, Jürg; Israel, Robert; Lieb, Elliot H.; Simon, Barry (1 August 1978). "Phase transitions and reflection positivity. I. General theory and long range lattice models".*Communications in Mathematical Physics*.**62**(1): 1–34. Bibcode:1978CMaPh..62....1F. doi:10.1007/BF01940327. ISSN 1432-0916.**^**Glimm, James; Jaffe, Arthur (1 September 1979). "A note on reflection positivity".*Letters in Mathematical Physics*.**3**(5): 377–378. Bibcode:1979LMaPh...3..377G. doi:10.1007/BF00397210. ISSN 1573-0530.**^**Jaffe, Arthur; Klimek, Slawomir; Lesniewski, Andrzej (1 December 1989). "Representations of the Heisenberg algebra on a Riemann surface".*Communications in Mathematical Physics*.**126**(2): 421–431. Bibcode:1989CMaPh.126..421J. doi:10.1007/BF02125133. ISSN 1432-0916.**^**Jaffe, Arthur; Ritter, Gordon (1 May 2008). "Reflection Positivity and Monotonicity".*Journal of Mathematical Physics*.**49**(5): 052301. arXiv:0705.0712. Bibcode:2008JMP....49e2301J. doi:10.1063/1.2907660. ISSN 0022-2488.**^**Jaffe, Arthur; Jäkel, Christian D.; Martinez, Roberto E. (1 February 2014). "Complex classical fields: An example".*Journal of Functional Analysis*.**266**(3): 1833–1881. doi:10.1016/j.jfa.2013.08.033. ISSN 0022-1236.**^**Jaffe, Arthur; Pedrocchi, Fabio L. (1 February 2014). "Topological Order and Reflection Positivity".*EPL (Europhysics Letters)*.**105**(4): 40002. arXiv:1310.5370. Bibcode:2014EL....10540002J. doi:10.1209/0295-5075/105/40002. ISSN 0295-5075.**^**Jaffe, Arthur; Pedrocchi, Fabio L. (2015). "Reflection Positivity for Majoranas".*Annales Henri Poincaré*.**16**(1): 189–203. arXiv:1305.1792. Bibcode:2015AnHP...16..189J. doi:10.1007/s00023-014-0311-y. ISSN 1424-0637.**^**Jaffe, Arthur; Pedrocchi, Fabio L. (2015). "Reflection Positivity for Parafermions".*Communications in Mathematical Physics*.**337**(1): 455–472. arXiv:1406.1384. Bibcode:2015CMaPh.337..455J. doi:10.1007/s00220-015-2340-x. ISSN 0010-3616.**^**Chesi, Stefano; Jaffe, Arthur; Loss, Daniel; Pedrocchi, Fabio L. (27 May 2013). "Vortex loops and Majoranas".*Journal of Mathematical Physics*.**54**(11). arXiv:1305.6270v3. Bibcode:2013JMP....54k2203C. doi:10.1063/1.4829273.**^**Jaffe, Arthur; Janssens, Bas (12 June 2015). "Characterization of Reflection Positivity: Majoranas and Spins".*Communications in Mathematical Physics*.**346**(3): 1021–1050. arXiv:1506.04197v2. doi:10.1007/s00220-015-2545-z.**^**Jaffe, Arthur; Janssens, Bas (24 July 2016). "Reflection Positive Doubles". arXiv:1607.07126 [math-ph].**^**Jaffe, Arthur; Liu, Zhengwei (2017). "Planar Para Algebras, Reflection Positivity".*Communications in Mathematical Physics*.**352**(1): 95–133. arXiv:1602.02662. Bibcode:2017CMaPh.352...95J. doi:10.1007/s00220-016-2779-4. ISSN 0010-3616.**^**Jaffe, Arthur; Jäkel, Christian D.; Martinez II, Roberto E. (29 January 2012). "Complex Classical Fields: A Framework for Reflection Positivity". arXiv:1201.6003v2 [math-ph].**^**Jaffe, Arthur; Liu, Zhengwei (30 January 2019). "Reflection Positivity and Levin-Wen Models". arXiv:1901.10662v1 [math-ph].**^**Jaffe, Arthur; Liu, Zhengwei (6 June 2020). "A Mathematical Picture Language Project". arXiv:2006.03954v1 [math-ph].**^**Balaban, Tadeusz; Imbrie, John; Jaffe, Arthur (1985), Jaffe, Arthur; Lehmann, Harry; Mack, Gerhard (eds.), "Renormalization of the Higgs Model: Minimizers, Propagators and the Stability of Mean Field Theory",*Quantum Field Theory: A Selection of Papers in Memoriam Kurt Symanzik*, Berlin, Heidelberg: Springer, pp. 299–329, doi:10.1007/978-3-642-70307-2_17, hdl:2027.42/46529, ISBN 978-3-642-70307-2, retrieved 20 April 2024**^**Jaffe, Arthur; Imbrie, John; Balaban, Tadeusz (1988). "Effective Action and Cluster Properties of the Abelian Higgs Model" (PDF).*Communications in Mathematical Physics*.**114**(2): 257–315. Bibcode:1988CMaPh.114..257B. doi:10.1007/BF01225038.**^**Balaban, Tadeusz; Imbrie, John; Jaffe, Arthur; Brydges, David (1 December 1984). "The mass gap for Higgs models on a unit lattice".*Annals of Physics*.**158**(2): 281–319. Bibcode:1984AnPhy.158..281B. doi:10.1016/0003-4916(84)90121-0. ISSN 0003-4916.**^**Kastler, D. (1990). "KMS states, cyclic cohomology and supersymmetry". In Doebner, H. -D.; Hennig, J. -D. (eds.).*Quantum Groups*. Lecture Notes in Physics. Vol. 370. Berlin, Heidelberg: Springer. pp. 375–397. doi:10.1007/3-540-53503-9_55. ISBN 978-3-540-46647-5.**^**Jaffe, Arthur; Lesniewski, Andrzej; Osterwalder, Konrad (1988). "Quantum $K$-theory. I. The Chern character".*Communications in Mathematical Physics*.**118**(1): 1–14. Bibcode:1988CMaPh.118....1J. doi:10.1007/BF01218474. ISSN 0010-3616.**^**Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (1 May 2016). "Compressed Teleportation". arXiv:1605.00321v1 [quant-ph].**^**Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (19 November 2016). "Constructive simulation and topological design of protocols".*New Journal of Physics*.**19**(6). arXiv:1611.06447v2. doi:10.1088/1367-2630/aa5b57.**^**Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (30 April 2016). "Holographic software for quantum networks".*Science China Mathematics*.**61**(4): 593–626. arXiv:1605.00127v5. doi:10.1007/s11425-017-9207-3.**^**Li, Lu; Bu, Kaifeng; Koh, Dax Enshan; Jaffe, Arthur; Lloyd, Seth (12 August 2022). "Wasserstein Complexity of Quantum Circuits". arXiv:2208.06306v1 [quant-ph].**^**Jaffe, Arthur; Jiang, Chunlan; Liu, Zhengwei; Ren, Yunxiang; Wu, Jinsong (10 February 2020). "Quantum Fourier analysis".*Proceedings of the National Academy of Sciences*.**117**(20): 10715–10720. arXiv:2002.03477v1. Bibcode:2020PNAS..11710715J. doi:10.1073/pnas.2002813117. PMC 7245120. PMID 32354991.**^**Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (16 February 2023). "Discrete Quantum Gaussians and Central Limit Theorem". arXiv:2302.08423v2 [quant-ph].**^**Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (15 June 2023). "Stabilizer Testing and Magic Entropy". arXiv:2306.09292v1 [quant-ph].**^**Chen, Liyuan; Garcia, Roy J.; Bu, Kaifeng; Jaffe, Arthur (2024). "Magic of random matrix product states".*Physical Review B*.**109**(17): 174207. arXiv:2211.10350v3. Bibcode:2024PhRvB.109q4207C. doi:10.1103/PhysRevB.109.174207.**^**Garcia, Roy J.; Bu, Kaifeng; Jaffe, Arthur (2023). "Resource theory of quantum scrambling".*Proceedings of the National Academy of Sciences*.**120**(17): e2217031120. arXiv:2208.10477v2. Bibcode:2023PNAS..12017031G. doi:10.1073/pnas.2217031120. PMC 10151511. PMID 37071685.**^**Cain, Madelyn; Zhao, Chen; Zhou, Hengyun; Meister, Nadine; Ataides, J. Pablo Bonilla; Jaffe, Arthur; Bluvstein, Dolev; Lukin, Mikhail D. (5 March 2024). "Correlated decoding of logical algorithms with transversal gates". arXiv:2403.03272 [quant-ph].**^**Liu, Zhengwei; Wozniakowski, Alex; Jaffe, Arthur (8 December 2016). "Quon 3D language for quantum information".*Proceedings of the National Academy of Sciences*.**114**(10): 2497–2502. arXiv:1612.02630v3. doi:10.1073/pnas.1621345114. PMC 5347593. PMID 28167790.**^**Jaffe, Arthur (2003). "The Role of Rigorous Proof in Modern Mathematical Thinking". In Hoff Kjeldsen, Tinne (ed.).*New Trends in the History and Philosophy of Mathematics*. University of Odense Press.**^**Jaffe, Arthur (2003). "Interactions between Mathematics and Theoretical Physics". In Hoff Kjeldsen, Tinne (ed.).*New Trends in the History and Philosophy of Mathematics*. University of Odense Press.**^**"Equations for universal truth".*Times Higher Education (THE)*. 28 July 2000. Retrieved 24 April 2024.**^**Jaffe, Arthur (1997). "Proof and the Evolution of Mathematics".*Synthese*.**111**(2): 133–146. doi:10.1023/A:1004903010713. ISSN 0039-7857. JSTOR 20117623.**^**Jaffe, Arthur; Quinn, Frank (30 June 1993),*Theoretical mathematics**: Toward a cultural synthesis of mathematics and theoretical physics*, arXiv:math/9307227, Bibcode:1993math......7227J**^**Atiyah, Michael; Borel, Armand; Chaitin, G. J.; Friedan, Daniel; Glimm, James; Gray, Jeremy J.; Hirsch, Morris W.; MacLane, Saunder; Mandelbrot, Benoit B. (31 March 1994),*Responses to*Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics*, by A. Jaffe and F. Quinn*, arXiv:math/9404229, Bibcode:1994math......4229A- ^
^{a}^{b}^{c}Jaffe, Arthur (2021). "Arthur Jaffe's CV" (PDF). **^**"Arthur M. Jaffe – Hagler Institute for Advanced Study".*hias.tamu.edu*. Retrieved 20 April 2024.**^**"2018 annual meeting of International Congress of Chinese Mathematicians" (PDF). 2018.**^**"ICM Plenary and Invited Speakers | International Mathematical Union (IMU)".*www.mathunion.org*. Retrieved 20 April 2024.

- Jaffe's website
- Mathematical Picture Language Project at Harvard University
- Arthur Jaffe at the Mathematics Genealogy Project
- Dublin Institute for Advanced Study: Governing Boards Archived 5 November 2020 at the Wayback Machine
- "School of Theoretical Physics Statutory Public Lecture "Beauty and Truth in Mathematics and Physics"".
*YouTube*. Oracle Pictures. 14 June 2016. Archived from the original on 19 December 2021. (lecture by Arthur Jaffe, 18 May 2016, Trinity College Dublin) - "Arthur Jaffe | Is relativity compatible with quantum theory?".
*YouTube*. Harvard CMSA. 2 December 2020. - "Arthur Jaffe (Harvard): Remembering the Future".
*YouTube*. IMBM Istanbul Matematiksel Bilimler Merkezi. 20 June 2021. Archived from the original on 19 December 2021. - List of Past AMS Presidents (Jaffe is the 54th.)