Mathematical analysis of quantum fields—Historical survey and a new asymptotic perturbation theory

Arai, Asao

doi:10.1090/suga/459

Mathematical analysis of quantum fields—Historical survey and a new asymptotic perturbation theory
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by Asao Arai
Translated by: the author

Sugaku Expositions 34 (2021), 93-121

DOI: https://doi.org/10.1090/suga/459

Published electronically: April 28, 2021

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Abstract:

A historical survey on mathematical analysis of quantum fields is presented and basic aspects of a new asymptotic perturbation theory recently developed by the present author are described.

References

Abdelmalek Abdesselam, The ground state energy of the massless spin-boson model, Ann. Henri Poincaré 12 (2011), no. 7, 1321–1347. MR 2846670, DOI 10.1007/s00023-011-0103-6
Abdelmalek Abdesselam and David Hasler, Analyticity of the ground state energy for massless Nelson models, Comm. Math. Phys. 310 (2012), no. 2, 511–536. MR 2890307, DOI 10.1007/s00220-011-1407-6
Michael Aizenman, Geometric analysis of $\varphi ^{4}$ fields and Ising models. I, II, Comm. Math. Phys. 86 (1982), no. 1, 1–48. MR 678000
Jean-Pierre Antoine, Atsushi Inoue, and Camillo Trapani, Partial $^*$-algebras and their operator realizations, Mathematics and its Applications, vol. 553, Kluwer Academic Publishers, Dordrecht, 2002. MR 1947892, DOI 10.1007/978-94-017-0065-8
Asao Arai, Rigorous theory of spectra and radiation for a model in quantum electrodynamics, J. Math. Phys. 24 (1983), no. 7, 1896–1910. MR 709529, DOI 10.1063/1.525922
Asao Arai, A note on scattering theory in nonrelativistic quantum electrodynamics, J. Phys. A 16 (1983), no. 1, 49–69. MR 700181
Asao Arai, Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons, J. Math. Anal. Appl. 140 (1989), no. 1, 270–288. MR 997857, DOI 10.1016/0022-247X(89)90108-X
Asao Arai, An asymptotic analysis and its application to the nonrelativistic limit of the Pauli-Fierz and a spin-boson model, J. Math. Phys. 31 (1990), no. 11, 2653–2663. MR 1075749, DOI 10.1063/1.528966
A. Arai, Fock Spaces and Quantum Fields I, in Japanese, Nippon-hyoron-sha, Tokyo, 2000.
A. Arai, Fock Spaces and Quantum Fields II, in Japanese, Nippon-hyoron-sha, Tokyo, 2000.
Asao Arai, Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians, J. Math. Anal. Appl. 246 (2000), no. 1, 189–216. MR 1761158, DOI 10.1006/jmaa.2000.6782
Asao Arai, Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation, Rev. Math. Phys. 13 (2001), no. 9, 1075–1094. MR 1853826, DOI 10.1142/S0129055X01000934
Asao Arai, Mathematical theory of quantum particles interacting with a quantum field, Non-commutativity, infinite-dimensionality and probability at the crossroads, QP–PQ: Quantum Probab. White Noise Anal., vol. 16, World Sci. Publ., River Edge, NJ, 2002, pp. 1–50. MR 2059856, DOI 10.1142/9789812705242_{0}001
Asao Arai, Mathematical aspects of quantum systems interacting with quantum fields, in Japanese, \cite[Chapter 4]NNW, 108–182.
Asao Arai, Mathematical Principles of Quantum Phenomena, in Japanese, Asakura-shoten, Tokyo. 2006.
A. Arai, Mathematics Principles of Quantum Statistical Mechanics, in Japanese, Kyoritsu-shuppan, Tokyo, 2008.
Asao Arai, Spectral analysis of an effective Hamiltonian in nonrelativistic quantum electrodynamics, Ann. Henri Poincaré 12 (2011), no. 1, 119–152. MR 2770092, DOI 10.1007/s00023-010-0071-2
Asao Arai. Mathematical Principles of Physics, from Newton mechanics to quantum mechanics, in Japanese, Maruzen-shuppan, Tokyo, 2012.
Asao Arai, A new asymptotic perturbation theory with applications to models of massless quantum fields, Ann. Henri Poincaré 15 (2014), no. 6, 1145–1170. MR 3205748, DOI 10.1007/s00023-013-0271-7
A. Arai and H. Ezawa. Mathematical Structures of Quantum Mechanics II. in Japanese, Asakura-shoten. Tokyo. 1999.
Asao Arai and Masao Hirokawa, On the existence and uniqueness of ground states of a generalized spin-boson model, J. Funct. Anal. 151 (1997), no. 2, 455–503. MR 1491549, DOI 10.1006/jfan.1997.3140
Huzihiro Araki, von Neumann algebras of local observables for free scalar field, J. Mathematical Phys. 5 (1964), 1–13. MR 160487, DOI 10.1063/1.1704063
H. Araki, Mathematics of Quantum Fields, in Japanese, Iwanami-shoten, Tokyo, 1993.
Volker Bach, Jürg Fröhlich, and Alessandro Pizzo, Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. 264 (2006), no. 1, 145–165. MR 2212219, DOI 10.1007/s00220-005-1478-3
Volker Bach, Jürg Fröhlich, and Alessandro Pizzo, Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field, Adv. Math. 220 (2009), no. 4, 1023–1074. MR 2483715, DOI 10.1016/j.aim.2008.10.006
Volker Bach, Jürg Fröhlich, and Israel Michael Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (1998), no. 2, 205–298. MR 1639709, DOI 10.1006/aima.1998.1733
Volker Bach, Jürg Fröhlich, and Israel Michael Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (1998), no. 2, 299–395. MR 1639713, DOI 10.1006/aima.1998.1734
Klaus Baumann, On canonical irreducible quantum field theories describing bosons and fermions, J. Math. Phys. 29 (1988), no. 5, 1225–1230. MR 941044, DOI 10.1063/1.527964
F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev., 52 (1937), 54–59.
N. N. Bogoliubov et al, Axiomatic Quantum Field Theory, Addison Wesley Publishing Company, 1973.
L. Brillouin, Champs self-consistents et electrons metalliques–III, J. de Phys. Radium, 4 (1933), 1–9.
Romeo Brunetti, Claudio Dappiaggi, Klaus Fredenhagen, and Jakob Yngvason (eds.), Advances in algebraic quantum field theory, Mathematical Physics Studies, Springer, Cham, 2015. MR 3381848, DOI 10.1007/978-3-319-21353-8
David C. Brydges, Jürg Fröhlich, and Alan D. Sokal, A new proof of the existence and nontriviality of the continuum $\varphi ^{4}_{2}$ and $\varphi ^{4}_{3}$ quantum field theories, Comm. Math. Phys. 91 (1983), no. 2, 141–186. MR 723546
J. M. Cook, The mathematics of second quantization, Trans. Amer. Math. Soc. 74 (1953), 222–245. MR 53784, DOI 10.1090/S0002-9947-1953-0053784-4
J. M. Cook, Asymptotic properties of a boson field with given source, J. Mathematical Phys. 2 (1961), 33–45. MR 121159, DOI 10.1063/1.1724210
J. Dereziński and C. Gérard, Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11 (1999), no. 4, 383–450. MR 1682684, DOI 10.1142/S0129055X99000155
Jan Dereziński and Christian Gérard, Mathematics of quantization and quantum fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2013. MR 3060648, DOI 10.1017/CBO9780511894541
P. A. M. Dirac, The quantum theory of the emission and absorption of radiation, Proceedings of the Royal Society of London, Series A, 114 (1927), 243–265.
H. Ezawa, Developments of mathematical sciences of fields, \cite[Chapter 6]NNW.
H. Ezawa and A. Arai, Quantum Field Theory and Statistical Mechanics, in Japanese, Nippon-hyoron-sha, Tokyo, 1988.
H. Ezawa and T. Tuneto (editors), Prospects of Quantum Physics I, in Japanese, Iwanami-shoton, Tokyo, 1977.
William G. Faris, Invariant cones and uniqueness of the ground state for fermion systems, J. Mathematical Phys. 13 (1972), 1285–1290. MR 321451, DOI 10.1063/1.1666133
J. Faupin, J. S. Møller, and E. Skibsted, Second order perturbation theory for embedded eigenvalues, Comm. Math. Phys. 306 (2011), no. 1, 193–228. MR 2819424, DOI 10.1007/s00220-011-1278-x
V. Fock, Konfigurationsraum und zweite Quantelung, Z. Physik, 75 (1932), 622–647.
K. O. Friedrichs, Mathematical aspects of the quantum theory of fields, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1953. MR 0056468
Jürg Fröhlich, On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. H. Poincaré Sect. A (N.S.) 19 (1973), 1–103 (English, with French summary). MR 368649
Jürg Fröhlich, On the triviality of $\lambda \varphi ^{4}_{d}$ theories and the approach to the critical point in $d{>atop (—)}4$ dimensions, Nuclear Phys. B 200 (1982), no. 2, 281–296. MR 643591, DOI 10.1016/0550-3213(82)90088-8
James Glimm and Arthur Jaffe, A $\lambda \phi ^{4}$ quantum field without cutoffs. I, Phys. Rev. (2) 176 (1968), 1945–1951. MR 247845
James Glimm and Arthur Jaffe, The $\lambda (\Pi ^{4})_{2}$ quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. of Math. (2) 91 (1970), 362–401. MR 256677, DOI 10.2307/1970582
James Glimm and Arthur Jaffe, The $\lambda (\phi ^{4})_{2}$ quantum field theory without cutoffs. III. The physical vacuum, Acta Math. 125 (1970), 203–267. MR 269234, DOI 10.1007/BF02392335
James Glimm and Arthur Jaffe, The $\lambda \phi _{2}^{4}$ quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian, J. Mathematical Phys. 13 (1972), 1568–1584. MR 317683, DOI 10.1063/1.1665879
James Glimm and Arthur Jaffe, Collected papers. Vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1985. Constructive quantum field theory. Selected papers; Reprint of articles published 1968–1980. MR 947959
James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102, DOI 10.1007/978-1-4612-4728-9
Marcel Griesemer and David G. Hasler, Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation, Ann. Henri Poincaré 10 (2009), no. 3, 577–621. MR 2519822, DOI 10.1007/s00023-009-0417-9
Marcel Griesemer, Elliott H. Lieb, and Michael Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (2001), no. 3, 557–595. MR 1856401, DOI 10.1007/s002220100159
R. Haag, On quantum field theories, Danske Vid. Selsk. Mat.-Fys. Medd. 29 (1955), no. 12, 37. MR 71320
Rudolf Haag, Local quantum physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. Fields, particles, algebras. MR 1182152, DOI 10.1007/978-3-642-97306-2
Rudolf Haag and Daniel Kastler, An algebraic approach to quantum field theory, J. Mathematical Phys. 5 (1964), 848–861. MR 165864, DOI 10.1063/1.1704187
Christian Hainzl and Robert Seiringer, Mass renormalization and energy level shift in non-relativistic QED, Adv. Theor. Math. Phys. 6 (2002), no. 5, 847–871 (2003). MR 1974588, DOI 10.4310/ATMP.2002.v6.n5.a3
Y. Hara, T. Inami and K. Aoki, Elementary-particle Physics (Japanese), Asakura Shoten, Tokyo, 2000.
David Hasler and Ira Herbst, Ground states in the spin boson model, Ann. Henri Poincaré 12 (2011), no. 4, 621–677. MR 2787765, DOI 10.1007/s00023-011-0091-6
W. Heisenberg and W. Pauli, Zur Quantendynamik der Wellenfelder, Z. Physik, 56 (1929), 1–61.
Fumio Hiroshima, Analysis of ground states of atoms interacting with a quantized radiation field, Topics in the theory of Schrödinger operators, World Sci. Publ., River Edge, NJ, 2004, pp. 145–272. MR 2117109, DOI 10.1142/9789812562470_{0}005
Fumio Hiroshima, Perturbations of embedded eigenvalues in quantum field theory [translation of Sūgaku 57 (2005), no. 1, 70–92; MR2125197], Sugaku Expositions 21 (2008), no. 2, 177–207. Sugaku Expositions. MR 2493211
S. Hitotsumatsu et al, Seven Open Problems in Mathematics, in Japanese, Morikita-shuppan, Tokyo, 2002.
Raphael Høegh-Krohn, On the spectrum of the space cut-off $:P(\varphi ):$ Hamiltonian in two space-time dimensions, Comm. Math. Phys. 21 (1971), 256–260. MR 289074
A. Jaffe and E. Witten, Quantum Yang-Mills Theory, http: // www.claymath.org /sites /default /files/yangmills.pdf; http:// www.claymath.org /millenium-problems/yang-mills-and-mass-gap.
P. Jordan, $\ddot \textrm {U}$ber Wellen und K$\ddot \textrm {o}$rpuskeln in der Quantenmechanik, Z. Physik, 45 (1927), 766–775.
P. Jordan and O. Klein, Zum Mehrk$\ddot \textrm {o}$rperproblem der Quantentheorie, Z. Physik, 45 (1927), 751–765.
P. Jordan and E. P. Wigner, $\ddot \textrm {U}$ber das Paulische $\ddot \textrm {A}$quivalenzverbot, Z. Physik, 47 (1928), 631–651.
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Yusuke Kato, Some converging examples of the perturbation series in the quantum field theory, Progr. Theoret. Phys. 26 (1961), 99–122. MR 134225, DOI 10.1143/PTP.26.99
Yusuke Kato and Nobumichi Mugibayashi, Regular perturbation and asymptotic limits of operators in quantrm field theory, Progr. Theoret. Phys. 30 (1963), 103–133. MR 159586, DOI 10.1143/PTP.30.103
Yusuke Kato and Nobumichi Mugibayashi, Asymptotic fields in model field theories. I. $\lambda (\phi ^{4})_{2}$ with a space cutoff, Progr. Theoret. Phys. 45 (1971), 628–639. MR 303886, DOI 10.1143/PTP.45.628
T. Kinoshita, Present status of quantum electrodynamics, \cite[Chapter 12]ET.
József Lőrinczi, Fumio Hiroshima, and Volker Betz, Feynman-Kac-type theorems and Gibbs measures on path space, De Gruyter Studies in Mathematics, vol. 34, Walter de Gruyter & Co., Berlin, 2011. With applications to rigorous quantum field theory. MR 2848339, DOI 10.1515/9783110203738
Tadahiro Miyao, Nondegeneracy of ground states in nonrelativistic quantum field theory, J. Operator Theory 64 (2010), no. 1, 207–241. MR 2669436
Tadahiro Miyao, Self-dual cone analysis in condensed matter physics, Rev. Math. Phys. 23 (2011), no. 7, 749–822. MR 2826463, DOI 10.1142/S0129055X11004424
Osamu Miyatake, On the non-existence of solution of field equations in quantum mechanics, J. Inst. Polytech. Osaka City Univ. Ser. A 2 (1952), 89–99. MR 52990
Osamu Miyatake, On the singularity of the perturbation-term in the field quantum mechanics, J. Inst. Polytech. Osaka City Univ. Ser. A 3 (1952), 145–155. MR 58483
K. Nakamura, T. Nakamura and K. Watanabe (editors), Who saw quantum fields?, in Japanese, Nippon-hyoron-sha, Tokyo, 2004.
Edward Nelson, Interaction of nonrelativistic particles with a quantized scalar field, J. Mathematical Phys. 5 (1964), 1190–1197. MR 175537, DOI 10.1063/1.1704225
Edward Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis 12 (1973), 97–112. MR 0343815, DOI 10.1016/0022-1236(73)90091-8
Konrad Osterwalder and Robert Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31 (1973), 83–112. MR 329492
Konrad Osterwalder and Robert Schrader, Axioms for Euclidean Green’s functions. II, Comm. Math. Phys. 42 (1975), 281–305. With an appendix by Stephen Summers. MR 376002
W. Pauli and M. Fierz. (1938), Zur Theorie der Emission langwelliger Lichtquanten, Nuovo Cimento, 15 (1938), 167–188.
J. W. S. Rayleigh, Theory of Sound I (2nd ed.), London: Macmillan, 1894.
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Itaru Sasaki, Ground state of the massless Nelson model in a non-Fock representation, J. Math. Phys. 46 (2005), no. 10, 102107, 12. MR 2178579, DOI 10.1063/1.2050507
E. Schrödinger, Quantisierung als Eigenwertproblem, Ann. der Phys., 80 (1926), 437–490.
Selected papers on quantum electrodynamics, Dover Publications, Inc., New York, 1958. MR 0095032
I. E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106–134. MR 76317, DOI 10.1090/S0002-9947-1956-0076317-8
I. E. Segal, Tensor algebras over Hilbert spaces. II, Ann. of Math. (2) 63 (1956), 160–175. MR 77908, DOI 10.2307/1969994
Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974. MR 0489552
Herbert Spohn, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004. MR 2097788, DOI 10.1017/CBO9780511535178
R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, 2nd ed., Mathematical Physics Monograph Series, Benjamin/Cummings Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1978. MR 0468904
Toshimitsu Takaesu, Essential spectrum of a fermionic quantum field model, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), no. 4, 1450024, 11. MR 3281341, DOI 10.1142/S0219025714500246
T. Takabayashi, History of Developments of Quantum Theory, in Japanese, Chuo-Koron-sha, Tokyo, 1977.
S. Tomonaga, Concerning the difficulty of infinity, \cite[Chapter 11]ET.
Léon Van Hove, Les difficultés de divergences pour un modelle particulier de champ quantifié, Physica 18 (1952), 145–159 (French). MR 49093
A. S. Wightman, Quantum field theory in terms of vacuum expectation values, Phys. Rev. (2) 101 (1956), 860–866. MR 83914
A. S. Wightman and L. Gårding, Fields as operator-valued distributions in quantum field theory, Ark. Phys., 28 (1964), 129–184.
Eugene Paul Wigner, The collected works of Eugene Paul Wigner. Part A. The scientific papers. Vol. III, Springer-Verlag, Berlin, 1997. Part I. Particles and fields; Part II. Foundations of quantum mechanics; With a preface by Jagdish Mehra and Arthur S. Wightman; With annotations by Wightman and Abner Shimony; Edited by Wightman. MR 1635987
S. Yamada et al (editors), Handbook of Physics of Elementary Particles, in Japanese, Asakura-shoten, Tokyo, 2010.
J. M. Ziman, Elements of advanced quantum theory, Cambridge University Press, New York, 1969. MR 0251965
Yury M. Zinoviev, Equivalence of Euclidean and Wightman field theories, Comm. Math. Phys. 174 (1995), no. 1, 1–27. MR 1372797