Valery P. Gusynin
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine

Eduard V. Gorbar
Taras Shevchenko National University of Kyiv
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine


Year: 2023
Pages: 488
ISBN 978-966-360-487-9
Publication Language: Ukrainian
Publisher: PH “Akademperiodyka”
Place Published: Kyiv

Quantum theory of field is a base subject in elementary particle physics and condensing mediums and takes one of the leading places in academic programs of physics faculties in universities. The proposed monograph outlines the gauge fields theory basics, which are the core of modern quantum field theory. The presentation is based on the functional integration method application and introduces the quantization of free fields in the operator approach. The material of the monograph corresponds to a three-semester university course of lectures, and it includes the Standard Model of modern elementary particle physics and several areas of its expansion as well.
The book is for senior students, bachelors, masters, and post-graduate students of physical and mathematical specialties, as well as scientists who wish to familiarize themselves with the methods of the modern theory of gauge fields.




  1. A.I. Akhiezer and V.B. Berestetskii, Quantum Electrodynamics, John Wiley & Sons, 1965.
  2. Z. Baseyan, S.G. Matinyan, and G.K. Savvidi, Nonlinear plane waves in the massless Yang-Mills theory, Pis’ma v ZhETF/JETP Letters. 29, 587—589 (1980) [in Russian].
  3. I. Batsula and V.P. Gusynin, Plane-wave solutions in the framework of the SU(2) Yang–Mills theory, Ukrainian Journal of Physics. 26, 1233—1238 (1981) [in Russian].
  4. N. Bogolyubov, Kvazisredniye v zadachakh statisticheskoy mekhaniki [Quasi-averages in problems of statistical mechanics], JINR preprint D-781, Dubna, 1961, [in Russian].
  5. N. Bogolyubov and D.V. Shirkov, Introduction to the theory of quantized fields, Authorized English edition, revised and enlarged by the authors. Translated from the Russian by G.M. Volkoff. Interscience, New York, 1959.
  6. O. Vakarchuk, Kvantova mekhanika [Quantum mechanics], Lviv, 2012 [in Ukrainian].
  7. I. Holod and A.U. Klimyk, Matematychni osnovy teoriyi symetriy [Mathematical foundations of the theory of symmetries], Naukova Dumka, Kyiv, 1992 [in Ukrainian].
  8. E.V. Gorbar and V.P. Gusynin, Higgs Boson: Anticipation, Search, and Discovery, Nac. Akad. Nauk Ukr. (3), 31—41, (2014).
  9. P. Gusynin and V.A. Kushnir, Expansion of a one-loop effective action in powers of derivatives in a curved space, Yadernaya Fizika. 51, 587—597 (1990) [in Russian].
  10. D.D. Ivanenko (ed.), Noveysheye razvitiye kvantovoy elektrodinamiki [The latest development of quantum electrodynamics], Collection of articles, Moscow, 1954. [in Russian].
  11. Kvantovaya teoriya kalibrovochnykh poley [Quantum theory of gauge fields], Mir, Moscow, 1977 [in Russian].
  12. D. Landau, A.A. Khalatnikov and I.M. Khalatnikov, Asymptotic expression for the Green’s function of a photon in quantum electrodynamics, Doklady Akademii Nauk SSSR. 95, 1177—1180 (1954) [in Russian].
  13. D. Landau and I.M. Khalatnikov, The Gauge Transformation of the Green’s Function for Charged Particles, Soviet Physics JETP. 2(1), 69—72 (1956).
  14. A.V. Migdal and A.M. Polyakov, Spontaneous Breakdown of Strong Interaction Symmetry and the Absence of Massless Particles, Soviet Physics JETP. 24(1), 91—98 (1967).
  15. Polyakov A.M., Particle spectrum in quantum field theory, Pis’ma v ZhETF/JETP Letters. 20, 194—195 (1974).
  16. L. Rebenko, Osnovy suchasnoyi teoriyi vzayemodiyuchykh kvantovanykh poliv [Fundamentals of the modern theory of interacting quantized fields], Naukova Dumka, Kyiv, 2007 [in Ukrainian].
  17. A. Rubakov, Superheavy magnetic monopoles and decay of the proton, Pis’ma v ZhETF/JETP Letters. 33, 644—646 (1981).
  18. B. Rumer and A.I. Fet, Teoriya unitarnoy simmetri [Theory of unitary symmetry], Nauka, Moscow, 1970 [in Russian].
  19. A. Slavnov, Ward identities in gauge theories, Theor. Math. Phys. 10, 153—161 (1972).
  20. A. Slavnov, Invariant regularization of gauge theories, Theor. Math. Phys. 13, 174—177 (1972).
  21. D. Faddeev, The Feynman integral for singular Lagrangians, Theor. Math. Phys. 1, 3—18 (1969).
  22. Abers and B.W. Lee, Gauge Theories, Phys. Rep. C. 9, 1—141 (1973).
  23. A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, London, 1975.
  24. Actor, Classical solutions of SU(2) Yang-Mills theories, Rev. Mod. Phys. 51, 461—525 (1979).
  25. L. Adler, Axial Vector Vertex In Spinor Electrodynamics, Phys. Rev. 177, 2426—2438 (1969).
  26. L. Adler and W.A. Bardeen, Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation, Phys. Rev. 182, 1517—1536 (1969).
  27. A. Albeverio, R.J. Høegh-Krohn, and S. Mazzucchi, Mathematical Theory of Feynman Path Integrals. An Introduction (Lecture Notes in Mathematics, Vol. 523), Springer-Verlag, Berlin, 2008.
  28. W. Anderson, Coherent Excited States in the Theory of Superconductivity: Gauge Invariance and the Meissner Effect, Phys. Rev. 110, 827—834 (1958).
  29. W. Anderson, Plasmons, Gauge Invariance, and Mass, Phys. Rev. 130, 439—442 (1963).
  30. W. Anderson, Basic Notions of Condensed Matter Physics, Benjamin- Cummings Publishing Company, Menlo Park, 1984.
  31. Aoyama, T. Kinoshita, and M. Nio, Revised and Improved Value of the QED Tenth-Order Electron Anomalous Magnetic Moment, Phys. Rev. D. 97, 036001 (2018).
  32. P. Armitage, E.J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  33. F. Ashmore, A Method of Gauge-Invariant Regularization, Nuovo Cimento Lett. 4, 289—290 (1972).
  34. Bagnuls and C. Bervillier, Exact Renormalization Group Equations: an Introductory Review, Phys. Rep. 348, 91—157 (2001).
  35. Bailin and A. Love, Introduction to Gauge Field Theory, Taylor & Francis Group, New York, 1993.
  36. S. Ball and T.-W. Chiu, Analytic Properties of the Vertex Function in Gauge Theories. I, Phys. Rev. D. 22, 2542—2549 (1980).
  37. S. Ball and T.-W. Chiu, Analytic Properties of the Vertex Function in Gauge Theories. II, Phys. Rev. D. 22, 2550—2557 (1980).
  38. A. Baikov and V.A. Il’in, Status of γ5 in Dimensional Regularization, Theor. Math. Phys. 88, 789—809 (1991).
  39. Bar-Gadda, Infrared Behavior of the Effective Coupling in Quantum Chromodynamics: A Non-perturbative Approach, Nucl. Phys. B. 163, 312—332 (1980).
  40. Baym and L.P. Kadanoff, Conservation Laws and Correlation Functions, Phys. Rev. 124, 287—299 (1961).
  41. Baym, Self-consistent approximations in Many-body Systems, Phys. Rev. 127, 1391—1401 (1962).
  42. A. Belavin, A.M. Polyakov, A.S. Schwartz, and Yu.S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B. 59, 85—87 (1975).
  43. S. Bell and R. Jackiw, A PCAC Puzzle: π0→γγ in the σ-model, Nuovo Cimento A. 60, 47—61 (1969).
  44. Berezin, The Method of Second Quantization, Academic Press, Orlando, 1966.
  45. B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Quantum Electrodynamics, Pergamon Press, Oxford, 1982.
  46. Berges, N. Tetradis, and C. Wetterich, Non-perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics, Phys. Rep. 363, 223—386 (2002).
  47. Berges, Introduction to Nonequilibrium Quantum Field Theory, AIP Conf. Proc. 739, 3—61 (2004).
  48. Bergmann and J. Brunings, Non–Linear Field Theories II. Canonical Equations and Quantization, Rev. Mod. Phys. 21, 480—487 (1949).
  49. Bilal, Lectures on Anomalies, Preprint arXiv: 0802.06334 (2008).
  50. D. Bjorken and S. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1965.
  51. D. Bjorken and S. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965.
  52. N. Bogoljubov, V.V. Tolmachev, and D.V. Sirkov, A New Method in the Theory of Superconductivity, Fortschritte der Physik. 6, 605—682 (1958).
  53. G. Bollini and J.J. Giambiagi, Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter, Nuovo Cimento B. 12, 20—26 (1972).
  54. M. Brown and T.Y. Cao, Spontaneous Breakdown of Symmetry: Its Rediscovery and Integration Into Quantum Field Theory, Historical Studies in the Physical and Biological Sciences. 21, 211—235 (1991).
  55. S. Brown, Quantum Field Theory, Cambridge University Press, Cambridge, 1994.
  56. L. Buchbinder, S.D. Odintsov, and I.L. Shapiro, Effective Action in Quantum Gravity, IOP Publ. Ltd Techno House, Bristol, 1992.
  57. G. Callan, Disappearing dyons, Phys. Rev. D. 25, 2141—2146 (1982).
  58. Calzetta and B. Hu, Functional Methods in nonequilibrium QFT. In: Nonequilibrium Quantum Field Theory, Cambridge University Press, Cambridge, 2008. P. 170—208.
  59. Cawley, Determination of the Hamiltonian in the Presence of Constraints, Phys. Rev. Lett. 42, 413—416 (1979).
  60. Chanowitz, M. Furman, and I. Hinchliffe, The Axial Current in Dimensional Regularization, Nucl. Phys. B. 159, 225—243 (1979).
  61. Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, Oxford, 1988.
  62. Coleman, Non-Abelian Plane Waves, Phys. Lett. B. 70, 59—60 (1977).
  63. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge University Press, Cambridge, 1986.
  64. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective Action For Composite Operators, Phys. Rev. D. 10, 2428—2445 (1974).
  65. DeWitt, Dynamical theory of groups and fields, Gordon and Breach, New York, 1965.
  66. A.M. Dirac, Lectures on Quantum Mechanics, Dover Publications, Mineola, 2013.
  67. A.M. Dirac, Principles of Quantum Mechanics, Snowball Publishing, Lancaster, 2012.
  68. S. Dmytriiev and V.V. Skalozub, Low-energy Effective Lagrangian of The Two-Higgs-Doublet Model, Journal of Physics and Electronics. 29, 8—20 (2022).
  69. V. Dunne, New Strong-Field QED Effects at Extreme Light Infra-structure. Nonperturbative Vacuum Pair Production, Eur. Phys. J. D. 55, 327—340 (2009).
  70. V. Dunne, The Heisenberg–Euler Effective Action: 75 Years On, Int. J. Mod. Phys. A. 27, 1260004 (2012).
  71. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, and N. Wschebor, The Nonperturbative Functional Renormalization Group and Its Applications, Phys. Rep. 910, 1—114 (2021).
  72. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, 1999.
  73. Eichhorn, An Asymptotically Safe Guide to Quantum Gravity and Matter, Front. Astron. Space Sci.. 5, 47 (2019).
  74. Englert and R. Brout, Broken Symmetry and The Mass of Gauge Vector Bosons, Phys. Rev. Lett. 13, 321—323 (1964).
  75. Englert, The BEH Mechanism and Its Scalar Boson, Rev. Mod. Phys. 86, 843—850 (2014).
  76. D. Faddeev and A.A. Slavnov, Gauge Fields: An Introduction to Quantum Theory, CRC Press, Boca Raton, 2018.
  77. L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems, Dover Books on Physics, Courier Corporation, 2003.
  78. P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill College, New York, 1965.
  79. I. Fomin, V.P. Gusynin, and V.A. Miransky, Vacuum Instability of Massless Electrodynamics and the Gell-Mann-Low Eigenvalue Condition for the Bare Coupling Constant, Phys. Lett. B. 78, 136—139 (1978).
  80. I. Fomin, V.P. Gusynin, V.A. Miransky, and Yu.A. Sitenko, Dynamical Chiral Symmetry Breaking and Particle Mass Generation In Gauge Field Theories, Riv. Nuovo Cimento. 6, 1—90 (1983).
  81. S. Fradkin, Concerning Some General Relations of Quantum Electrodynamics, Zh. Eksp. Teor. Fiz. 29, 258—261 (1955).
  82. Fritzsch and P. Minkowski, Unified Interactions of Leptons and Hadrons, Annals of Physics. 93, 193—266 (1975).
  83. Fujikawa, Path Integral for Gauge Theories with Fermions, Phys. Rev. D. 21, 2848—2857 (1980).
  84. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies, Clarendon Press, Oxford, 2004.
  85. K. Gaillard and B.W. Lee, Rare Decay Modes of The K Mesons In Gauge Theories, Phys. Rev. D. 10, 897—916 (1974).
  86. Gell-Mann and A. Pais, Behavior of Neutral Particles under Charge Conjugation, Phys. Rev. 97, 1387—1389 (1955).
  87. Gell-Mann, P. Ramond, and R. Slansky, Color Embeddings, Charge Assignments, and Proton Stability In Unified Gauge Theories, Rev. Mod. Phys. 50, 721—744 (1978).
  88. Georgi, Weak Interactions and Modern Particle Theory, Dover Publications, Mineola, N.Y., 2009.
  89. Georgi, Lie Algebras in Particle Physics: From Isospin to Unified Theories, Benjamin-Cummings, Reading, 1999.
  90. Georgi and S.L. Glashow, Unity of All Elementary-Particle Forces, Phys. Rev. Lett. 32, 438—441 (1974).
  91. Gies, Introduction to the Functional RG and Applications to Gauge Theories, In: A. Schwenk, J. Polonyi, (eds) Renormalization Group and Effective Field Theory Approaches to Many-Body Systems. Lecture Notes in Physics, Vol. 852. Springer, Berlin, Heidelberg, 2012.
  92. M. Gitman and I.V. Tyutin, Quantization of fields with constraints, Springer-Verlag, Berlin, 1990.
  93. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York, 1987.
  94. Goldstone, Field Theories with “Superconductor” Solutions, Nuovo Cimento. 19, 154—164 (1961).
  95. Goldstone, A. Salam, and S. Weinberg, Broken Symmetries, Phys. Rev. 127, 965—970 (1962).
  96. Greiner and J. Reinhardt, Field Quantization, Springer-Verlag, Berlin, 1996.
  97. Greiner and J. Reinhardt, Quantum Electrodynamics, Springer-Verlag, Berlin, 2003.
  98. Greiner, S. Schramm, and E. Stein, Quantum Chromodynamics, Springer-Verlag, Berlin, 2002.
  99. N. Gribov, Quantization of Non-Abelian Gauge Theories, Nucl. Phys. B. 139, 1—19 (1978).
  100. Grosche and R. Steiner, Handbook of Feynman path integrals, Springer-Verlag, Berlin, 1998.
  101. Gross and F. Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys. Rev. Lett. 30, 1343—1346 (1973).
  102. Gross and F. Wilczek, Asymptotically Free Gauge Theories. I, Phys. Rev. D. 8, 3633—3652 (1974).
  103. Grozin, Lectures on QED and QCD, preprint arXiv:hep-ph/0508242 (2005).
  104. F. Gunion and H.E. Haber, CP-Conserving Two-Higgs-Doublet Model: The Approach to the Decoupling Limit, Phys. Rev. D. 67, 075019 (2003).
  105. F. Gunion, H.E. Haber, G. Kane, and S. Dawson, The Higgs Hunter’s Guide, Frontiers in Physics, Perseus Publishing, Cambridge, 2000, V. 80, P. 1—404.
  106. P. Gusynin and V.A. Kushnir, On-Diagonal Heat Kernel Expansion in Covariant Derivatives In Curved Space, Class. Quantum Grav. 8, 279—285 (1991).
  107. P. Gusynin, V.A. Miransky, and I.A. Shovkovy, Catalysis of Dynamical Flavor Symmetry Breaking by a Magnetic Field in 2+1 Dimensions, Phys. Rev. Lett. 73, 3499—3502 (1994).
  108. P. Gusynin, V.A. Miransky, and I.A. Shovkovy, Dymensional Reduction and Catalysis of Dynamical Symmetry Breaking by a Magnetic Field, Nucl. Phys. B. 462, 249—290 (1996).
  109. Halzen and A.D. Martin, Quarks and Leptons: Introductory Course in Modern Particle Physics, Wiley, Hoboken, 1991.
  110. Hanneke, S. Fogwell, and G. Gabrielse, New Measurement of the Electron Magnetic Moment and the Fine Structure Constant, Phys. Rev. Lett. 100, 120801 (2008).
  111. Hanneke, S.F. Hoogerheide, and G. Gabrielse, Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment, Phys. Rev. A. 83, 052122 (2011).
  112. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems, Accademia Nazionale dei Lincei, Roma, Italy, 1976.
  113. Heisenberg and H. Euler, Consequences of Dirac’s Theory of Positrons, Z. Physik. 98, 714—732 (1936).
  114. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, Princeton, 2020.
  115. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13, 508—509 (1964).
  116. W. Higgs, Spontaneous Symmetry Breakdown without Massless Bosons, Phys. Rev. 145, 1156—1162 (1966).
  117. W. Higgs, Nobel Lecture: Evading the Goldstone theorem, Rev. Mod. Phys. 86, 851—853 (2014).
  118. Brown, M. Riordan, M.Dresden, and L. Hoddeson, The Rise of the Standard Model: 1964–1979. In: L. Hoddeson, L. Brown, M. Riordan, and M. Dresden (Eds.), The Rise of the Standard Model: A History of Particle Physics from 1964 to 1979, Cambridge University Press, Cambridge, 1997. P. 3—35,
  119. Holdom, Raising Condensates Beyond the Ladder, Phys. Rev. Lett. 213, 365—369 (1988).
  120. Huang, Quarks, Leptons, and Gauge Fields, World Scientific, Singapore, 1982.
  121. Itzykson and J.-B. Zuber, Quantum Field Theory, Dover Publications, Mineola, 2006.
  122. Jackiw and C. Rebbi, Vacuum Periodicity in a Yang-Mills Quantum Theory, Phys. Rev. Lett. 37, 172—175 (1976).
  123. Jackiw, Introduction to the Yang-Mills quantum theory, Rev. Mod. Phys. 52, 661—673 (1980).
  124. L. Kataev and S.A. Larin, Analytical five-loop expressions for the renormalization group QED β-function in different renormalization schemes, JETP Lett. 96, 64—67 (2012).
  125. V. Keldysh, Ionization in the Field of a Strong Electromagnetic Wave, Soviet Physics JETP. 20(5), 1307—1314 (1965).
  126. B. Khriplovich, Green’s functions in theories with nonabelian gauge group, Yadernaya Fizika. 10, 409—424 (1969).
  127. K. Kim and M. Baker, Consequences of Gauge Invariance for the Interacting Vertices in Non-Abelian Gauge Theories, Nucl. Phys. B. 164, 152—170 (1980).
  128. F. King, Neutrino mass models, Reports on Progress in Physics. 67, 107—157 (2004).
  129. Kizilersu, T. Sizer, M.R. Pennington, A.G. Williams, and R. Williams, Dynamical Mass Generation in Unquenched QED Using the Dyson- Schwinger Equations, Phys. Rev. D. 91, 065015 (2015).
  130. Kleinert, Particles and Quantum Fields, World Scientific, Singapore, 2016.
  131. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore, 2009.
  132. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4Theories, World Scientific, Singapore, 2000.
  133. B. Kogut, E. Dagotto, and A. Kocic, New Phase of Quantum Electro- dynamics: A Non-perturbative Fixed Point in Four Dimensions, Phys. Rev. Lett. 60, 772—775 (1988).
  134. B. Kogut, E. Dagotto, and A. Kocic, On the Existence of Quantum Electrodynamics, Phys. Rev. Lett. 61, 2416—2419 (1988).
  135. B. Kogut, E. Dagotto, and A. Kocic, Strongly Coupled Quenched QED, Nucl. Phys. B. 317, 253—270 (1989).
  136. Kugo and I. Ojima, Manifestly Covariant Canonical Formulation for the Yang-Mills Field Theory. I. General Formalism, Prog. Theor. Phys. 60, 1869—1889 (1978).
  137. Kugo and I. Ojima, Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem, Prog. Theor. Phys. Suppl. 66, 1—130 (1979).
  138. D. Landau, A.A. Abrikosov, and I.M. Khalatnikov, On Quantum Field Theory, Nuovo Cimento. 3, Suppl. 1, 80—104 (1956).
  139. Langacker, The Standard Model and Beyond, CRC Press, Boca Raton, 2010.
  140. Laporta and E. Remiddi, The Analytical Value of the Electron (g − 2) at order α3 in QED, Phys. Lett. B. 379, 283—291 (1996).
  141. Laporta, High-Precision Calculation of the 4-loop Contribution to the Electron g − 2 in QED, Phys. Lett. B. 772, 232—238 (2017).
  142. Leibbrandt, Introduction to the Technique of Dimensional Regularization, Rev. Mod. Phys. 47, 849—876 (1975).
  143. N. Leung, S.T. Love, and W.A. Bardeen, Spontaneous Symmetry Breaking in Scale Invariant Quantum Electrodynamics, Nucl. Phys. B. 273, 649—662 (1986).
  144. D. Linde, Particle Physics and Inflationary Cosmology, Harwood, Chur, 1990.
  145. J. Long, E. Sabancilar, and T. Vachaspati, Leptogenesis and Primordial Magnetic Fields, Journal of Cosmology and Astroparticle Physics. 2014(2), 036 (2014).
  146. M. Luttinger and J.C. Ward, Ground-State Energy of a Many-Fermion System. II, Phys. Rev. 118, 1417—1427 (1960).
  147. S. Manton, Topology in the Weinberg-Salam Theory, Phys. Rev. D. 28, 2019—2026 (1983).
  148. Maskawa and T. Nakajima, Spontaneous Breaking of Chiral Symmetry in a vector-Gluon Model, Prog. Theor. Phys.. 52, 1326—1354 (1974).
  149. A. Migdal, Effective Low Energy Lagrangian for QCD, Phys. Lett. B. 81, 37—40 (1979).
  150. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories, World Scientific, Singapore, 1993.
  151. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I., Phys. Rev. 112, 345—358 (1961).
  152. B. Okun, Leptons and Quarks, North-Holland Publishing Company, Amsterdam, 1985.
  153. D. Peccei, New Phase for an Old Theory?, Nature. 332, 492—493 (1988).
  154. E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995.
  155. Petermann, Fourth Order Magnetic Moment of the Electron, Helv. Phys. Acta. 30, 407—408 (1957).
  156. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30, 1346—1349 (1973).
  157. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland Publishing Company, Amsterdam, 1982.
  158. Ramond, Field Theory: A Modern Primer, Westview Press; 2nd edition, Boulder, 2001.
  159. D. Roberts and A.G. Williams, Dyson-Schwinger Equations and their Application to Hadronic Physics, Prog. Part. Nucl. Phys. 33, 477—575 (1994).
  160. Rosenstein and B.J. Warr, Dynamical Symmetry Breaking in Four-Fetmion Interaction Models, Phys. Repts. 205, 59—108 (1991).
  161. A. Rubakov, Classical theory of gauge fields, Princeton University Press, Princeton, 2002.
  162. A. Rubakov and D.S. Gorbunov, Introduction to the Theory of the Early Universe: Hot Big Bang Theory, World Scientific, Singapore, 2011.
  163. H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 1996.
  164. Salisbury, K. Sundermeyer, Leon Rosenfeld’s general theory of constrained Hamiltonian dynamics, Eur. Phys. J. H. 42, 23—61 (2017).
  165. Savvidi, From Heisenberg–Euler Lagrangian to the Discovery of Chromomagnetic Gluon Condensation, Eur. Phys. J. C. 80, 65—184 (2020).
  166. K. Savvidy, Infrared Instability of the Vacuum State of Gauge Theories and Asymptotic Freedom, Phys. Lett. B. 71, 133—134 (1977).
  167. Schubert, On the γ5-Problem of Dimensional Renormalization, preprint HD-THEP-93-46 (1993).
  168. S. Schulman, Techniques and applications of path integration, John Wiley, New York, 1981.
  169. S. Schwarz, Quantum field theory and topology, Springer-Verlag, Berlin, 2014.
  170. S. Schweber, Relativistic quantum field theory, Dover Publications, Mineola, 1961.
  171. S. Schweber, QED and Men who did it: Dyson, Feynman, Schwinger, Tomonaga, Princenton University Press, Princeton, 1994.
  172. S. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664—679 (1951).
  173. S. Schwinger, Gauge Invariance and Mass, Phys. Rev. 125, 397—398 (1962).
  174. S. Schwinger, Gauge Invariance and Mass. II, Phys. Rev. 128, 2425—2429 (1962).
  175. Senjanovic, Path Integral Quantization of Field Theories with Second-Class Constraints, Ann. Phys. 100, 227—261 (1976).
  176. Shifman, Historical curiosity: How asymptotic freedom of the Yang-Mills theory could have been discovered three times before Gross, Wilczek, and Politzer, but was not, in At the Frontier of Particle Physics, Handbook of QCD (World Scientific, Singapore), Vol. 1, 2001.
  177. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987.
  178. M. Sommerfield, The Magnetic Moment of the Electron, Annals of Physics. 6, 26—57 (1958).
  179. Steinberger, On the Use of Subtraction Fields and the Lifetimes of Some Types of Meson Decay, Phys. Rev. 76, 1180—1186 (1949).
  180. F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, Princeton, 1989.
  181. Sundermeyer, Constrained Dynamics, Springer-Verlag, Berlin, 1982.
  182. G. Sutherland, Current Algebra and Some Non-Strong Mesonic Decays, Nucl. Phys. B. 2, 433—440 (1967).
  183. C. Taylor, Ward identities and charge renormalization of the Yang-Mills field, Nucl. Phys. B. 33, 436—444 (1971).
  184. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B. 44, 189—213 (1972).
  185. ’t Hooft, Dimensional Regularization and the Renormalization Group, Nucl. Phys. B. 61, 455—468 (1973).
  186. ’t Hooft, Symmetry Breaking through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37, 8—11 (1976).
  187. ’t Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B. 79, 276—284 (1974).
  188. ’t Hooft, The Evolution of Quantum Field Theory: From QED to Grand Unification. In: The Standard Theory of Particle Physics, Advanced Series on Directions in High Energy Physics, Vol. 26, World Scientific, Singapore, 2016.
  189. B. Treiman, R. Jackiw, and D.J. Gross, Lectures on Current Algebra and Its Applications, Princeton University Press, Princeton, 2018.
  190. S. Vanyashin and M.V. Terentyev, The Vacuum Polarization of a Charged Vector Field, Zh. Eksp. Teor. Phys. 48, 565—573 (1965).
  191. N. Vasil’ev and A.K. Kazanskii, Legendre Transforms of the Generating Functionals in Quantum Field Theory, Theor. Math. Phys. 12, 875—887 (1972).
  192. N. Vasil‘ev, Functional Methods in Quantum Field Theory and Statistical Physics, CRC Press, Boca Raton, 1998.
  193. Veltman, Theoretical aspects of high energy neutrino interactions, Proc. R. Soc. A. 301, 107—112 (1967).
  194. Weinberg, New Approach to the Renormalization Group, Phys. Rev. D. 8, 3497—3509 (1973).
  195. Weinberg, Critical Phenomena for Field Theorists, In: A. Zichichi (ed.), Understanding the Fundamental Constituents of Matter. Springer New York, NY, 1978.
  196. Weinberg, The Quantum Theory of Fields. Vol. 1. Foundations, Cambridge University Press, Cambridge, 1995.
  197. Weinberg, The Quantum Theory of Fields. Vol. 2. Modern Applications, Cambridge University Press, Cambridge, 1996.
  198. Wetterich, Exact Evolution Equation for the Effective Potential, Phys. Lett. B. 301, 90—94 (1993).
  199. T. Wu, C.N. Yang, T.T. Wu, and C.N. Yang, Some Solutions of the Classical Isotopic Gauge Field Equations, In: H. Mark, S. Fernbach (Eds), Properties of Matter Under Unusual Conditions, Wiley-Interscience, New York, 1969. P. 344—354.
  200. Zener, A theory of the Electric Breakdown of Solid Dielectrics, Proc. R. Soc. A. 145, 523—529 (1934).
  201. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, Oxford, 2021.
  202. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, Oxford, 2005.
  203. Zumino, Gauge Properties of Propagators in Quantum Electrodynamics, J. Math. Phys. 1, 1—7 (1960).