Arthur_Jaffe

Arthur Jaffe

Arthur Jaffe

American mathematician

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]

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Education and career

After graduating from Pelham Memorial High School in 1955,[2] 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.

Arthur Jaffe's 30 doctoral students include Joel Feldman, Ezra Getzler, Clifford Taubes, Eugene Wayne, and John Imbrie. In total, Jaffe has over 300 mathematical descendants. He has had many post-doctoral collaborators, including Robert Schrader, Konrad Osterwalder, Juerg Froehlich, Roland Sénéor [fr], 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[3]. He presently 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.

Research

Nonpositivity of Energy Density

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[4].

Constructive Quantum Field Theory

A large amount of Jaffe's work deals with the rigorous 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 rigorously constructed 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[5][6][7][8][9][10]. This culminated in the first ever rigorous mathematical interacting local quantum field theory[11]. For this work, Jaffe and James Glimm are acknowledged as the founders of the subject of constructive quantum field theory.

Phase Transitions in 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[12][13].

Reflection Positivity

One recurring idea Jaffe's works is the notion of reflection positivity due to Osterwalder and Schrader. The notion of reflection positivity has served since its inception as a key tool in the quantization of classical Euclidian field theories into relativistic quantum field theories. Jaffe has made major contributions to the development of this theory, by establishing key examples[14][15][16][17][18][19][20][21][22], introducing important generalizations[23][24][25], and providing geometric interpretations[26][27].

Higgs Effect

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[28][29][30].

Supersymmetric Models

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[31][32]. 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.

Quantum Information

In his later years, Arthur Jaffe has made varied contributions to the theory of quantum information[33][34][35][36]. Notable among these contributions are the introduction of quantum Fourier analysis[37][38], the study of quantum resources[39][40][41], and the introduction of a 3D graphical language for quantum information[42].

Philosophy of Math and Physics

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[43][44][45][46]. The most influential of these works was his essay with Frank Quinn, which introduced the notion of "Theoretical Mathematics"[47]. An issue of the Bulletin of the American Mathematical Society was devoted to responses to this article, written by leading mathematical physicists[48].

Awards and honors

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[49]. In 1980 Arthur Jaffe was awarded the Dannie Heineman Prize for Mathematical Physics. In 1990 he was awarded the Medal Collège de France[50]. In 2018 he was awarded the ICCM prize for best mathematical paper in the last five years[51]. In 2020 he was awarded the Science China Mathematics Award for best editor [49]. Jaffe has been an invited speak at many distinguished conferences, including the 1978 international Congress of Mathematicians at Helsinki [52].

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 [49].

Personal life

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.

References

  1. [1]
    "Website of ACAP". Archived from the original on 13 July 2019. Retrieved 19 March 2018.
  2. [2]
    "Oral History Interviews. Arthur Jaffe, interviewed by Katherine Sopka". American Institute of Physics. 15 February 1977.
  3. [3]
    "CSSP Board History". www.sciencepresidents.org. Retrieved 24 April 2024.
  4. [4]
    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. doi:10.1007/BF02749799. ISSN 1827-6121.
  5. [5]
    Jaffe, Arthur (1966). "Existence Theorems for a Cut-off λφ4 Field Theory". Mathematical Theory of Elementary Particles – via MIT Press.
  6. [6]
    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.
  7. [7]
    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.
  8. [8]
    "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.
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    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.
  10. [10]
    Jaffe, Arthur; Glimm, James (1973). "Positivity of the φ43 Hamiltonian". Fortschritte der Physik. 21.
  11. [11]
    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.
  12. [12]
    Jaffe, Arthur; Glimm, James; Thomas, Spencer (1975). "Phase Transitions for φ42 Quantum Fields". Communications in Mathematical Physics (45): 203–216.
  13. [13]
    Jaffe, Arthur; Glimm, James; Spencer, Thomas (1976). "Existence of Phase Transitions for φ42 Quantum Fields". Mathematical Methods of Quantum Field Theory – via CNRS.
  14. [14]
    Glimm, James; Jaffe, Arthur (1 September 1979). "A note on reflection positivity". Letters in Mathematical Physics. 3 (5): 377–378. doi:10.1007/BF00397210. ISSN 1573-0530.
  15. [15]
    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. doi:10.1007/BF02125133. ISSN 1432-0916.
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    Jaffe, Arthur; Ritter, Gordon (1 May 2008). "Reflection Positivity and Monotonicity". Journal of Mathematical Physics. 49 (5): 052301. doi:10.1063/1.2907660. ISSN 0022-2488.
  17. [17]
    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.
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    Jaffe, Arthur; Pedrocchi, Fabio L. (1 February 2014). "Topological Order and Reflection Positivity". EPL (Europhysics Letters). 105 (4): 40002. doi:10.1209/0295-5075/105/40002. ISSN 0295-5075.
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    Jaffe, Arthur; Pedrocchi, Fabio L. (2015). "Reflection Positivity for Majoranas". Annales Henri Poincaré. 16 (1): 189–203. doi:10.1007/s00023-014-0311-y. ISSN 1424-0637.
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    Jaffe, Arthur; Pedrocchi, Fabio L. (2015). "Reflection Positivity for Parafermions". Communications in Mathematical Physics. 337 (1): 455–472. doi:10.1007/s00220-015-2340-x. ISSN 0010-3616.
  21. [21]
    Chesi, Stefano; Jaffe, Arthur; Loss, Daniel; Pedrocchi, Fabio L. (27 May 2013). "Vortex Loops and Majoranas". arXiv.org. Retrieved 24 April 2024.
  22. [22]
    Jaffe, Arthur; Janssens, Bas (12 June 2015). "Characterization of Reflection Positivity: Majoranas and Spins". arXiv.org. Retrieved 24 April 2024.
  23. [23]
    Jaffe, Arthur; Janssens, Bas (24 July 2016), Reflection Positive Doubles, doi:10.48550/arXiv.1607.07126, retrieved 24 April 2024
  24. [24]
    Jaffe, Arthur; Liu, Zhengwei (2017). "Planar Para Algebras, Reflection Positivity". Communications in Mathematical Physics. 352 (1): 95–133. doi:10.1007/s00220-016-2779-4. ISSN 0010-3616.
  25. [25]
    Jaffe, Arthur; Jäkel, Christian D.; Martinez II, Roberto E. (29 January 2012). "Complex Classical Fields: A Framework for Reflection Positivity". arXiv.org. Retrieved 24 April 2024.
  26. [26]
    Jaffe, Arthur; Liu, Zhengwei (30 January 2019). "Reflection Positivity and Levin-Wen Models". arXiv.org. Retrieved 24 April 2024.
  27. [27]
    Jaffe, Arthur; Liu, Zhengwei (6 June 2020). "A Mathematical Picture Language Project". arXiv.org. Retrieved 24 April 2024.
  28. [28]
    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
  29. [29]
    Jaffe, Arthur; Imbrie, John; Balaban, Tadeusz (1988). "Effective Action and Cluster Properties of the Abelian Higgs Model" (PDF). Communications in Mathematical Physics: 257–315.
  30. [30]
    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: 281–319. doi:10.1016/0003-4916(84)90121-0. ISSN 0003-4916.
  31. [31]
    Kastler, D. (1990). Doebner, H. -D.; Hennig, J. -D. (eds.). "KMS states, cyclic cohomology and supersymmetry". Quantum Groups. Berlin, Heidelberg: Springer: 375–397. doi:10.1007/3-540-53503-9_55. ISBN 978-3-540-46647-5.
  32. [32]
    Jaffe, Arthur; Lesniewski, Andrzej; Osterwalder, Konrad (1988). "Quantum $K$-theory. I. The Chern character". Communications in Mathematical Physics. 118 (1): 1–14. ISSN 0010-3616.
  33. [33]
    Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (1 May 2016). "Compressed Teleportation". arXiv.org. Retrieved 20 April 2024.
  34. [34]
    Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (19 November 2016). "Constructive Simulation and Topological Design of Protocols". arXiv.org. Retrieved 20 April 2024.
  35. [35]
    Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex (30 April 2016). "Holographic Software for Quantum Networks". arXiv.org. Retrieved 20 April 2024.
  36. [36]
    Li, Lu; Bu, Kaifeng; Koh, Dax Enshan; Jaffe, Arthur; Lloyd, Seth (12 August 2022). "Wasserstein Complexity of Quantum Circuits". arXiv.org. Retrieved 20 April 2024.
  37. [37]
    Jaffe, Arthur; Jiang, Chunlan; Liu, Zhengwei; Ren, Yunxiang; Wu, Jinsong (10 February 2020). "Quantum Fourier Analysis". arXiv.org. Retrieved 20 April 2024.
  38. [38]
    Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (16 February 2023). "Discrete Quantum Gaussians and Central Limit Theorem". arXiv.org. Retrieved 20 April 2024.
  39. [39]
    Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (15 June 2023). "Stabilizer Testing and Magic Entropy". arXiv.org. Retrieved 20 April 2024.
  40. [40]
    Chen, Liyuan; Garcia, Roy J.; Bu, Kaifeng; Jaffe, Arthur (18 November 2022). "Magic of Random Matrix Product States". arXiv.org. Retrieved 20 April 2024.
  41. [41]
    Garcia, Roy J.; Bu, Kaifeng; Jaffe, Arthur (22 August 2022). "Resource theory of quantum scrambling". arXiv.org. Retrieved 20 April 2024.
  42. [42]
    Liu, Zhengwei; Wozniakowski, Alex; Jaffe, Arthur (8 December 2016). "Quons: A 3D Language for Quantum Information". arXiv.org. Retrieved 20 April 2024.
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    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.
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    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.
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    "Equations for universal truth". Times Higher Education (THE). 28 July 2000. Retrieved 24 April 2024.
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    Jaffe, Arthur (1997). "Proof and the Evolution of Mathematics". Synthese. 111 (2): 133–146. ISSN 0039-7857.
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    Jaffe, Arthur; Quinn, Frank (30 June 1993), Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics, doi:10.48550/arXiv.math/9307227, retrieved 24 April 2024
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    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, doi:10.48550/arXiv.math/9404229, retrieved 24 April 2024
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    Jaffe, Arthur (2021). "Arthur Jaffe's CV" (PDF).
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    "Arthur M. Jaffe – Hagler Institute for Advanced Study". hias.tamu.edu. Retrieved 20 April 2024.
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    "2018 annual meeting of International Congress of Chinese Mathematicians" (PDF). 2018.
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    "ICM Plenary and Invited Speakers | International Mathematical Union (IMU)". www.mathunion.org. Retrieved 20 April 2024.
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