Advertisement

New Philosophy of Renormalization: From the Renormalization Group Equations to Effective Field Theories

  • Tian Yu Cao

Abstract

The conceptual foundations of renormalization theory has undergone radical transformations during the last four decades. These changes are the result both of attempts to solve conceptual anomalies within the theory itself, and of fruitful interactions between quantum field theory (QFT) and statistical physics. Implicit assumptions concerning such concepts as regularization, cutoff, dimensionality, symmetry, and renormalizability have been clarified, and the original understanding of these concepts is being transformed. New concepts involve symmetry breaking, either spontaneous or anomalous, renormalization group transformations, decoupling of high-energy processes from low-energy phenomena, sensible nonrenormalizable theories, and effective field theories. These have been developed, drawing heavily on dramatic progress in statistical physics. The purpose of this essay is to examine these foundational transformations (Sec. 4.2), and to show that these advances have led to a new understanding of renormalization, a clarification of the theoretical structure of QFT and its ontological basis, and most importantly, a radical shift of outlook in fundamental physics (Sec. 4.3). For this purpose, however, a clarification of the conceptual background is indispensable (Sec. 4.1).

Keywords

Renormalization Group Scale Invariance Quantum Electrodynamic Effective Field Theory Renormalization Group Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adler, S. L. (1969), “Axial-vector vertex in spinor electrodynamics,” Phys. Rev. 177, pp. 24226–2438.CrossRefGoogle Scholar
  2. Aramaki, S. (1989), “Development of the renormalization theory in quantum electrodynamics. II,” Historia Sci. 37, pp. 91–113.MathSciNetGoogle Scholar
  3. Appelquist, T. and J. Carazzone (1975), “Infrared singularities and massive fields,” Phys. Rev. D 11, pp. 2856–2861.ADSCrossRefGoogle Scholar
  4. Becchi, C., A. Rouet, and R. Stora (1974), “The Abelian Higgs-Kibble model, unitarity of the S operator,” Phys. Lett. B 52, pp. 344–346.ADSCrossRefGoogle Scholar
  5. Bell, J. S. and R. Jackiw (1969), “A PCAC puzzle: tt0 yy in the a model,” Nuovo Cimento B 60, pp. 47–61.CrossRefGoogle Scholar
  6. Bethe, H. A. (1947), “The electromagnetic shift of energy levels,” Phys. Rev. 72, pp. 339–341.ADSCrossRefGoogle Scholar
  7. Bjorken, J. D. (1969), “Asymptotic sum rules at infinite momentum,” Phys. Rev. 179, pp. 1547–1553.ADSCrossRefGoogle Scholar
  8. Bopp, F. (1940), “Eine lineare theorie des elektrons,” Ann. Phys. 38, pp. 345–384.MathSciNetMATHCrossRefGoogle Scholar
  9. Born, M., W. Heisenberg, and P. Jordan (1926a), “Zur quantemmechnik. II,” Z. Phys. 35, pp. 557–615.ADSCrossRefGoogle Scholar
  10. Brown, L. M. and T. Y. Cao (1991), “Spontaneous breakdown of symmetry: Its rediscovery and integration into quantum field theory,” Historical Stud. Phys. Biol. Sci. 21, pp. 211–235.CrossRefGoogle Scholar
  11. Callan, C. G., Jr. (1970), “Bjorken scale invariance in scalar field theory,” Phys. Rev. D 2, pp. 1541–1547.ADSCrossRefGoogle Scholar
  12. Cao, T. Y. (1991), “The Reggeizaqtion program 1962–1982: Attempts at reconciling quantum field theory with S-matrix theory,” Arch. History Exact Sci. 41, pp. 239–283.MATHGoogle Scholar
  13. Cao, T. Y. and S. S. Schweber (1993), “The conceptual foundations and philosophical aspects of renormalization theory.” Synthese (forthcoming: vol. 97, no. 1, Oct. 1993 ).Google Scholar
  14. Coleman, S. and E. Weinberg (1973), “Radiative corrections as the origin of spontaneous symmetry breaking,” Phys. Rev. D 7, pp. 1888–1910.ADSCrossRefGoogle Scholar
  15. Collins, J. C. (1984), Renormalization, Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  16. Cushing, J. T. (1990), Theory Construction and Selection in Modern Physics: The S-Matrix Theory, Cambridge: Cambridge University Press.MATHGoogle Scholar
  17. Dirac, P. A. M. (1927a), “The quantum theory of emission and absorbtion of radiation,” Proc. R. Soc. London Ser. A 114, pp. 243–265.ADSMATHCrossRefGoogle Scholar
  18. Dirac, P. A. M. (1927b), “The quantum theory of dispersion,” Proc. R. Soc. London Ser. A 114, pp. 716–728.ADSGoogle Scholar
  19. Dirac, P. A. M. (1930), “A theory of electrons and protons,” Proc. R. Soc. London Ser. A 126, pp. 360–365.ADSMATHCrossRefGoogle Scholar
  20. Dirac, P. A. M. (1933), “Theorie du positron,” in Rapport du le Conseil Solvay de Physique, Structure et Propriétés des noyaux Atomiques (22–29, Oct. 1933 ), Paris: Gauthier-Villars, pp. 203–212.Google Scholar
  21. Dirac, P. A. M. (1938), “Classical theory of radiating electrons,” Proc. R. Soc. London Ser. A 167, pp. 148–169.ADSCrossRefGoogle Scholar
  22. Dirac, P. A. M. (1942), “The physical interpretation of quantum mechanics,” Proc. R. Soc. London Ser. A 180, pp. 1–40.MathSciNetADSCrossRefGoogle Scholar
  23. Dirac, P. A. M. (1963), “The evolution of the physicist’s picture of nature,” Sci. Am. 208, pp. 45–53.CrossRefGoogle Scholar
  24. Dirac, P. A. M. (1969), “Methods in theoretical physics,” in Special Suppl. of IAEA Bulletin, Vienna: IAEA, pp. 21–28.Google Scholar
  25. Dirac, P. A. M. (1973a), “Relativity and quantum mechanics,” in The Past Decades in Particle Theory, C. G. Sudarshan and Y. Neéman, eds., New York: Gordon and Breach, pp. 741–772.Google Scholar
  26. Dirac, P. A. M. (1973b), “Development of the physicist’s conception of nature,” in The Physicist’s Conception of Nature, J. Mehra, ed., Dordrecht: D. Reidel, pp. 1–14.Google Scholar
  27. Dirac, P. A. M. (1983), “The origin of quantum field theory,” in The Birth of Particle Physics L. M. Brown and L. Hoddeson, eds., Cambridge: Cambridge University Press, pp. 39–55.Google Scholar
  28. Dyson, F. J. (1949), “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, pp. 486–502.MathSciNetADSMATHCrossRefGoogle Scholar
  29. Dyson, F. J. (1949), “The S matrix in quantum electrodynamics,” Phys. Rev. 75, pp. 1736–1755.MathSciNetADSMATHCrossRefGoogle Scholar
  30. Dyson, F. J. (1951), “The renormalization method in quantum electrodynamics,” Proc. R. Soc. London Ser. A 207, pp. 395–401.MathSciNetADSMATHCrossRefGoogle Scholar
  31. Dyson, F. J. (1952), “Divergence of perturbation theory in quantum electrodynamics,” Phys. Rev. 85, pp. 631–632.MathSciNetADSMATHCrossRefGoogle Scholar
  32. Essam, J. W. and M. E. Fisher (1963), “Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point,” J. Chem. Phys. 38, pp. 802– 812.ADSCrossRefGoogle Scholar
  33. Feldman, D. (1949), “On realistic field theories and the polarization of the vacuum,” Phys. Rev. 76, pp. 1369–1375.ADSMATHCrossRefGoogle Scholar
  34. Feynman, R. P. (1948a), “Space–time approach to nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20, pp. 367–387.MathSciNetADSCrossRefGoogle Scholar
  35. Feynman, R. P. (1948b), “A relativistic cutoff for classical electrodynamics,” Phys. Rev. 74, pp. 939–946.MathSciNetADSMATHCrossRefGoogle Scholar
  36. Feynman, R. P. (1948c), “Relativistic cutoff for quantum electrodynamics,” Phys. Rev. 74, pp. 1430–1438.MathSciNetADSMATHCrossRefGoogle Scholar
  37. Fisher, M. E. (1964), “Correlation functions and the critical region of simple fluids,” J. Math. Phys. 5, pp. 944–962.ADSCrossRefGoogle Scholar
  38. Frenkel, J. (1925), “Zur elektrodynamik punktfoermiger elektronen,” Z. Phys. 32, pp. 518–534.ADSCrossRefGoogle Scholar
  39. Furry, W. H. and J. R. Oppenheimer (1934), “On the theory of the electron and positron,” Phys. Rev. 45, pp. 245–262.ADSMATHCrossRefGoogle Scholar
  40. Gasiorowicz, S. G., P. R. Yennie, and H. Suura (1959), “Magnitude of renormalization constants,” Phys. Rev. Lett. 2, pp. 513–516.ADSMATHCrossRefGoogle Scholar
  41. Gell-Mann, M. (1987), “Particle theory from S matrix to quark,” in Symmetries in Physics (1600–1980), M. G. Doncel, A. Hermann, L. Michel, and A. Pais, eds., Barcelona: Bellaterra, pp. 474–497.Google Scholar
  42. Gell-Mann, M. (1989), “Progress in elementary particle theory, 1950–1964,” in Pions to Quarks, L. M. Brown, M. Dresdon, and L. Hoddeson, eds., Cambridge: Cambridge University Press, pp. 694–711.CrossRefGoogle Scholar
  43. Gell-Mann, M. and F. E. Low (1954), “Quantum electrodynamics at small distances,” Phys. Rev. 95, pp. 1300–1312.MathSciNetADSMATHCrossRefGoogle Scholar
  44. Georgi, H., H. Quinn, and S. Weinberg (1974), “Hierarchy of interactions in unified gauge theories,” Phys. Rev. Lett. 33, pp. 451–454.ADSCrossRefGoogle Scholar
  45. Georgi, H. (1989b), “Effective quantum field theories,” in The New Physics, Paul Davies, ed., Cambridge: Cambridge University Press, pp. 4446–4457.Google Scholar
  46. Green, M. B. and J. H. Schwarz (1984), “Anomaly cancellations in sypersymmetric D = 10 gauge theory and superstring theory,” Phys. Lett. B 149, pp. 117–122.MathSciNetADSCrossRefGoogle Scholar
  47. Green, M. B. (1985), “Unification of forces and particles in superstring theories,” Nature 314, pp. 409–414.ADSCrossRefGoogle Scholar
  48. Green, M. B., J. H. Schwarz, and E. Witten (1987), Superstring Theory, Cambridge: Cambridge University Press.MATHGoogle Scholar
  49. Gross, D. and R. Jackiw (1972), “Effect of anomalies on quasi-renormalizable theories,” Phys. Rev. D 6, pp. 477–493.ADSCrossRefGoogle Scholar
  50. Gross, D. and F. Wilczek (1973), “Ultraviolet behavior of non-abelian gauge theories,” Phys. Rev. Lett. 30, pp. 1343–1346.ADSCrossRefGoogle Scholar
  51. Gross, D. (1985), “Beyond quantum field theory,” in Recent Developments in Quantum Field Theory, J. Ambjorn, B. J. Durhuus, and J. L. Petersen, eds., Amsterdam: Elsevier, pp. 151–168.Google Scholar
  52. Hawking, S. (1980), Is the End in Sight for Theoretical Physicsl Cambridge: Cambridge University Press.Google Scholar
  53. Heisenberg, W. (1934), “Remerkung zur Diracschen theorie des positrons,” Z. Phys 90, pp. 209–231.ADSMATHCrossRefGoogle Scholar
  54. Heitier, W. (1961), in The Quantum Theory of Fields, R. Stoops, ed., New York: Interscience, p. 37.Google Scholar
  55. Hurst, C. A. (1952), “The enumeration of graphs in the Feynman-Dyson technique,” Proc. R. Soc. London Ser. A 214, pp. 44–61.MathSciNetADSMATHCrossRefGoogle Scholar
  56. Jaffe, A. (1965), “Divergence of perturbation theory for bosons,” Commun. Math. Phys. 1, pp. 127–149.MathSciNetADSMATHCrossRefGoogle Scholar
  57. Johnson, K. (1961), “Solution of the equations for the Green’s functions of a two-dimensional relativistic field theory,” Nuovo Cimento 20, pp. 773— 790.CrossRefGoogle Scholar
  58. Jost, R. (1965), The General Theory of Quantum Fields, Providence: American Mathematical Society.Google Scholar
  59. Kadanoff, L. P. (1966), “Scaling laws for Ising models near T c ,” Physics 2, pp. 263–272.Google Scholar
  60. Källen, G. (1953), “On the magnitude of the renormalization constants in quantum electrodynamics,” Dan. Mat.-Fys. Medd. 27, pp. 1–18.Google Scholar
  61. Källen, G. (1966), “Review of consistency problems in quantum electrodynamics,” Acta Phys. Austr. Suppl. II, pp. 133–161.Google Scholar
  62. Kamefuchi, S. (1951), “Note on the direct interaction between spinor fields,” Progr. Theor. Phys. 6, pp. 175–181.MathSciNetADSMATHCrossRefGoogle Scholar
  63. Kramers, H. (1938), Quantentheorie des Elektrons und der Strahlung, Leipzig: Akad. Verlag, trans. D. ter Haar, Amsterdam: North-Holland, 1957.Google Scholar
  64. Kramers, H. (1947), A review talk at the Shelter Island Conference (June 1947), Unpublished. For its content and significance in the development of renormalization theory, cf. S. S. Schweber: “A short history of Shelter Island I,” in Shelter Island //, R. Jackiw, N. N. Khuri, S. Weinberg, and E. Witten, eds., Cambridge MA: MIT Press, 1985.Google Scholar
  65. Lamb, W. E., Jr. and R. C. Retherford (1947), “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, pp. 241–143.ADSCrossRefGoogle Scholar
  66. Landau, L. D., A. A. Abrikosov, and I. M. Khalatnikov (1954a), “The removal of infinities in quantum electrodynamics,” Doki Akad. Nauka 95, pp. 497–499.MATHGoogle Scholar
  67. Landau, L. D., A. A. Abrikosov, and I. M. Khalatnikov (1954b), “An asymptotic expression for the electro Green function in quantum electrodynamics,” Dokl. Akad. Nauka 95, pp. 773–776.Google Scholar
  68. Landau, L. D., A. A. Abrikosov, and I. M. Khalatnikov (1954c), “An asymptotic expression for the photon Green function in quantum electrodynamics,” Dokl. Akad. Nauka 95, pp. 1117–1120.Google Scholar
  69. Landau, L. D., A. A. Abrikosov, and I. M. Khalatnikov (1954d), “The electron mass in quantum electrodynamics,” Dokl. Akad. Nauka 96, pp. 261–263.Google Scholar
  70. Landau, L. D. (1955a), “On the quantum theory of fields,” in Niels Bohr and the Development of Physics, W. Pauli, ed., London: Pergamon, pp. 52–69.Google Scholar
  71. Landau, L. D. and I. Pomeranchuck (1955b), “On point interactions in quantum electrodynamics,” Dokl. Akad. Nauka 102, pp. 489–491.MATHGoogle Scholar
  72. Landau, L. D., A. A. Abrikosov, and I. M. Kalatnikov (1956), “On the quantum theory of fields,” Nuovo Cimento Suppl. 3, pp. 80–104.CrossRefGoogle Scholar
  73. Lepage, G. P. (1989), “What is renormalization?” preprint, CLNS, 89/970. Newman Lab. of Nuclear Studies, Cornell University.Google Scholar
  74. Lewis, H. W. (1948), “On the reactive terms in quantum electrodynamics,” Phys. Rev. 73, pp. 173–176.ADSMATHCrossRefGoogle Scholar
  75. Lorentz, H. A. (1904a), “Maxwell’s elektromagnetische theorie,” in Encyc. Mat. Wiss. Vol. V /2, pp. 63–144.Google Scholar
  76. Lorentz, H. A. (1904b), “Weeiterbildung der Maxwellschen theorie: elektronen- theorie” in Encyc. Mat. Wiss. Vol. V/2, pp. 145–280.Google Scholar
  77. Mack, G. (1968), “Partially conserved dilatation current,” Nucl. Phys. B 5, pp. 499–507.ADSCrossRefGoogle Scholar
  78. Mills, R. L. and C. N. Yang (1966), “Treatment of overlapping divergences in the photon self-energy function,” Suppl. Progr. Theor. Phys. 37–38, pp. 507–511.ADSCrossRefGoogle Scholar
  79. Nafe, J. E., E. B. Nelson, and I.I. Rabi (1947), “The hyperfine structure of atomic hydrogen and deuterium,” Phys. Rev. 71, pp. 914–915.ADSCrossRefGoogle Scholar
  80. Pais, A. (1945), “On the theory of the electron and of the nucléon,” Phys. Rev. 68, pp. 227–228.ADSCrossRefGoogle Scholar
  81. Pauli, W. and F. Villars (1949), “On the invariant regularization in relativistic quantum theory,” Rev. Mod. Phys. 21, pp. 434–444.MathSciNetADSMATHCrossRefGoogle Scholar
  82. Pauli, W. and V. Weisskopf (1934), “Uber die quantisierung der skalaren relativistischen wellengleichung,” Helv. Phys. Acta 7, pp. 709–731.Google Scholar
  83. Peterman, A. and E. C. G. Stueckelberg (1951), “Restriction of possible interactions in quantum electrodynamics,” Phys. Rev. 82, 548–549.ADSCrossRefGoogle Scholar
  84. Peterman, A. (1953), “Divergence of perturbation expression,” Phys. Rev. 89, pp. 1160–1161.ADSCrossRefGoogle Scholar
  85. Peterman, A. (1953), “Renormalisation dans les séries divergentes,” Helv. Phys. Acta 26, pp. 291–299.Google Scholar
  86. Poincaré, H. (1906), “Sur la dynamique de l’électron,” Read. Cire. Mat. Palermo 21, pp. 129–175.MATHCrossRefGoogle Scholar
  87. Polchinski, J. (1984), “Renormalization and effective Lagrangians,” Nucl. Phys. B 231, pp. 269–295.ADSCrossRefGoogle Scholar
  88. Politzer, H. (1973), “Reliable perturbative results for strong interactions?” Phys. Rev. Lett. 30, pp. 1346–1349.ADSCrossRefGoogle Scholar
  89. Rayski, J. (1948), “On simultaneous interaction of several fields and the self- energy problem,” Acta Phys. Polonica 9, pp. 129–140.MathSciNetGoogle Scholar
  90. Rivier, D. and E. C. G. Stueckelberg (1948), “A convergent expression for the magnetic moment of the neutron,” Phys. Rev. 74, p. 218.ADSMATHCrossRefGoogle Scholar
  91. Rohrlich, F. (1973), “The electron: Development of the first elementary particle theory,” in Physicist’s Conception of Nature, J. Mehra, ed., Dordrecht: Reidel, pp. 331–369.Google Scholar
  92. Sakata, S. and O. Hara (1947a), “The self-energy of the electron and the mass difference of nucléons,” Progr. Theor. Phys. 2, pp. 30–31.ADSCrossRefGoogle Scholar
  93. Sakata, S. (1947b), “The theory of the interaction of elementary particles,” Progr. Theor. Phys. 2, pp. 145–147.ADSCrossRefGoogle Scholar
  94. Sakata, S. (1950a), “On the direction of the theory of elementary particles” (English trans, in Suppl. Progr. Theor. Phys. 50 (1971): 155–158.ADSCrossRefGoogle Scholar
  95. Sakata, S. and H. Umezawa (1950b), “On the applicability of the method of mixed fields in the theory of the elementary particles,” Progr. Theor. Phys. 5, pp. 682– 691.MathSciNetADSCrossRefGoogle Scholar
  96. Sakata, S., H. Umezawa, and S. Kamefuchi (1952), “On the structure of the interaction of the elementary particles,” Progr. Theor. Phys. 7, pp. 377— 390.MathSciNetADSMATHCrossRefGoogle Scholar
  97. Sakata, S. (1956), “On a composite model for the new particles,” Progr. Theor. Phys. 16, pp. 686–688.ADSCrossRefGoogle Scholar
  98. Salam, A. (1951a), “Overlapping divergences and the S matrix,” Phys. Rev. 82, pp. 217–227.MathSciNetADSMATHCrossRefGoogle Scholar
  99. Salam, A. (1951b), “Divergent integrals in renormalizable field theories,” Phys. Rev. 84, pp. 426–431.MathSciNetADSMATHCrossRefGoogle Scholar
  100. Salam, A. (1968), “Weak and electromagnetic interactions,” in Proceedings of Nobel Conference VIII, pp. 367–377.Google Scholar
  101. Salam, A. and J. Strathdee (1970), “Quantum gravity and infinities in quantum electrodynamics,” Lett Nuovo Cimento. 4, pp. 101–108.CrossRefGoogle Scholar
  102. Salam, A. (1973), “Progress in renormalization theory since 1949,” in The Physicist’s Conception of Nature, J. Mehra, ed., Dordrecht: Reidel, pp. 430–446.Google Scholar
  103. Schweber, S. S., H. A. Bethe, and F. de Hoffmann (1955), Mesons and Fields, Vol. I, Row, Peterson, and Co.MATHGoogle Scholar
  104. Schwinger, J. (1948a), “On quantum electrodynamics and the megnetic moment of the electron,” Phys. Rev. 73, pp. 416–417.MathSciNetADSMATHCrossRefGoogle Scholar
  105. Schwinger, J. (1948b), “Quantum electrodynamics. I. A covariant formulation,” Phys. Rev. 74, pp. 1439–1461.MathSciNetADSMATHCrossRefGoogle Scholar
  106. Schwinger, J. (1951), “On the Green’s functions of quantized field. I,” Proc. Natl. Acad. Sei. U.S.A. 37, pp. 452–459.MathSciNetADSCrossRefGoogle Scholar
  107. Schwinger, J. (1970), Particles, Sources, and Fields, Reading MA: Addison-Wesley, Vol. I.Google Scholar
  108. Schwinger, J. (1973), “A report on quantum electrodynamics,” in The Physicist’s Conception of Nature, ( J. Meyra, ed., Dordrecht: Reidel, pp. 413–429.Google Scholar
  109. Schwinger, J. (1983), “Renormalization theory of quantum electrodynamics: An individual view,” in The Birth of Particle Physics, L. M. Brown and L. Hoddeson, eds., Cambridge: Cambridge University Press, pp. 329–353.Google Scholar
  110. Streater, R. F. and A. S. Wightman (1964), PTC, Spin and Statistics, and All That, Reading, MA: Benjamin.Google Scholar
  111. Stueckelberg, E. C. G. (1938), “Die Wechsel Wirkungskräfte in der elektrodynamik und in der feldtheorie der Kernkrafte,” Helv. Phys. Acta 11, pp. 225–244; 299–329.MATHGoogle Scholar
  112. Stueckelberg, E. C. G. and A. Peteerman (1953), “La normalisation des constantes dans la theorie des quanta,” Helv. Phys. Acta 26, pp. 499–520.MathSciNetMATHGoogle Scholar
  113. Symanzik, K. (1970), “Small distance behavior in field theory and power counting,” Commun. Math. Phys. 18, 227–246.MathSciNetADSMATHCrossRefGoogle Scholar
  114. Symanzik, K. (1973), “Infrared singularities and small-distance behavior analysis,” Commun. Math. Phys. 34, pp. 7–36.MathSciNetADSCrossRefGoogle Scholar
  115. Symanzik, K. (1983), “Continuum limit and improved action in lattice theories,” Nucl. Phys. B 226, pp. 187–227.MathSciNetADSCrossRefGoogle Scholar
  116. Takabayasi, T. (1983), “Some characteristic aspects of early elementary particle theory in Japan,” in The Birth of Particle Physics, L. M. Brown and L. Hoddeson, eds., Cambridge: Cambridge University Press, pp. 294–303.Google Scholar
  117. Thirring, W. (1953), “On the divergence of perturbation theory for quantum fields,” Helv. Phys. Acta 26, pp. 33–52.MathSciNetMATHGoogle Scholar
  118. Thompson, J. J. (1881), “On the electric and magnetic effects produced by the motion of electrified bodies,” Philos. Mag. 11, pp. 227–249.Google Scholar
  119. ’t Hooft, G. (1971a) “Renormalization of massless Yang-Mills fields,” Nucl. Phys. B 33, pp. 173–99.ADSCrossRefGoogle Scholar
  120. ’t Hooft, G. (1971b) “Renormalizable Lagrangians for massive Yang-Mils fields,” Nucl. Phys. B 35, pp. 167–188.ADSCrossRefGoogle Scholar
  121. ’t Hooft and M. Veltman (1972a), “Renormalization and regularization of gauge fields” Nucl. Phys. B 44, pp. 189–213.ADSCrossRefGoogle Scholar
  122. ’t Hooft and M. Veltman (1972b), “A Combinatorics of gauge fields,” Nucl. Phys. B 50, pp. 318–353.CrossRefGoogle Scholar
  123. ’t Hooft and M. Veltman (1972c), “Example of Gauge field theory,” Proc. Marseille Conf. June 1972, C. Korthals-Altes, ed.Google Scholar
  124. Tomonaga, S. (1946), “On a relativistically invariant formulation of the quantum theory of wave fields,” Progr. Theor. Phys. 1, pp. 27–42.MathSciNetADSMATHCrossRefGoogle Scholar
  125. Tomonaga, S. (1965), “Development of quantum electrodynamics,” in Noble Lectures (Physics): 1963–1970, Amsterdam: Elsevier, pp. 126–136.Google Scholar
  126. Umezawa, H., J. Yukawa, and E. Yamada (1948), “The problem of vacuum polarization,” Progr. Theor. Phys. 3, pp. 317–318.ADSCrossRefGoogle Scholar
  127. Umezawa, H. and R. Kawabe (1949a), “Some general formulae relating to vacuum polarization,” Progr. Theor. Phys. 4, pp. 423–442.MathSciNetADSCrossRefGoogle Scholar
  128. Umezawa, H. and R. Kawabe (1949b), “Vacuum polarization due to various charged particles,” Progr. Theor. Phys. 4, pp. 443–460.MathSciNetADSCrossRefGoogle Scholar
  129. Velo, G. and A. Wightman (eds.) (1973), Constructive Quantum Field Theory, Berlin: Springer-Verlag.MATHGoogle Scholar
  130. Veltman, M. (1968a), “Relation between the practical results of current algebra techniques and the originating quark model,” Copenhagen lectures, July 1968. Reprinted in Gauge Theory—Past and Future, R. Akhoury, B. De Wit, P. van Nieuwenhuizen, and H. Veltman, eds., Singapore, World Scientific, 1992.Google Scholar
  131. Veltman, M. (1968b), “Perturbation theory of massive Yang-Mills fields,” Nucl. Phys. B 7, pp. 637–650.ADSCrossRefGoogle Scholar
  132. Veltman, M. (1969a), Proc. Topical Conf. on Weak Interactions, CERN, Geneva, 14–17 Jan. 1969. CERN Yellow Report, 69–7, p. 391.Google Scholar
  133. Veltman, M. (with J. Reiff) (1969b), “Massive Yang-Mills fields,” Nucl. Phys. B 13, pp. 545–564.ADSCrossRefGoogle Scholar
  134. Veltman, M. (1970), “Generalized Ward identities and Yang-Mills fields,” Nucl. Phys. B 21, pp. 288–302.ADSGoogle Scholar
  135. Veltman, M. (1977), “Large Higgs mass andµ-e universality,” Phys. Lett. B 70, pp. 253–254.ADSCrossRefGoogle Scholar
  136. Waller, I. (1930), “Bemerkiingen iiber die Rolle der Eigenenergie des Elektrons in der Quantentheorie der Strahlung,” Z. Phys. 62, pp. 673–676.ADSCrossRefGoogle Scholar
  137. Ward, J. C. (1950), “An identity in quantum eleectrodynamics,” Phys. Rev. 78, p. 182.ADSMATHCrossRefGoogle Scholar
  138. Ward, J. C. (1951), “On the renormalization of quantum electrodynamics,” Proc. Soc. (London) Ser. A 64, pp. 54–56.ADSCrossRefGoogle Scholar
  139. Weinberg, S. (1960), “High energy behavior in quantum field theory,” Phys. Rev. 118, pp. 838–849.MathSciNetADSMATHCrossRefGoogle Scholar
  140. Weinberg, S. (1967), “A model of leptons,” Phys. Rev. Lett. 19, pp. 1264–1266.ADSCrossRefGoogle Scholar
  141. Weinberg, S. (1978), “Critical phenomena for field theorists,” in Understanding the Fundamental Constituents of Matter, A. Zichichi, ed. New York: Plenum, pp. 1–52.Google Scholar
  142. Weinberg, S. (1979), “Phenomenological Lagrangian,” Physica A 96, pp. 327–340.ADSCrossRefGoogle Scholar
  143. Weinberg, S. (1980a), “Conceptual foundations of the unified theory of weak and electromagnetic interactions,” Rev. Mod. Phys. 52, pp. 515–523.MathSciNetADSCrossRefGoogle Scholar
  144. Weinberg, S. (1980b), “Effective gauge theories,” Phys. Lett. B 91, pp. 51–55.ADSCrossRefGoogle Scholar
  145. Weinberg, S. (1983), “Why the renormalization group is a good thing,” in Asymptotic Realms of Physics: Essays in Honor of Francis. E. Low, A. H. Guth, K. Huang, and R. L. Jaffee, eds., Cambridge, MA: MIT Press.Google Scholar
  146. Weisskopf, V. F. (1936), “Uber die elektrodynamic des vakuums auf grund der quantentheorie des elektrons,” K. Danske Vidensk. Selsk., Math.-Fys. Medd. 14, pp. 1–39.Google Scholar
  147. Wentzel, G. (1943), Quantum Field Theory ( English trans., New York: Inter- science, 1949 ).Google Scholar
  148. Widom, B. (1965a), “Surface tension and molecular correlations near the critical point,” J. Chem. Phys. 43, pp. 3892–3897.ADSCrossRefGoogle Scholar
  149. Widom, B. (1965b), “Equation of state in the neighborhood of the critical point,” J. Chem. Phys. 43, pp. 3898–3905.ADSCrossRefGoogle Scholar
  150. Wightman, A. S. (1976), “Hilbert’s sixth problem: Mathematical treatment of the axioms of physics,” in Mathematical Developments Arising from Hilbert Problems, F. E. Browder, ed. pp. 147–240.Google Scholar
  151. Wightman, A. S. (1978), “Field theory, Axiomatic,” in the Encyclopedia of Physics, New York: McGraw-Hill, pp. 318–321.Google Scholar
  152. Wightman, A. S. (1986), “Some lessons of renoormalization theory,” in The Lesson of Quantum Theory, J. de Boer, E. Dal, and D. Ulfbeck, eds., Amsterdam: Elsevier, pp. 201–225.Google Scholar
  153. Wilson, K. G. (1965), “Model Hamiltonians for local quantum field theory,” Phys. Rev. B 140, pp. 445–457.ADSCrossRefGoogle Scholar
  154. Wilson, K. G. (1969), “Non-Lagrangian models of current algebra,” Phys. Rev. 179, pp. 1499–1512.MathSciNetADSCrossRefGoogle Scholar
  155. Wilson, K. G. (1970a), “Operator-product expansions and anomalous dimensions in the Thirring model,” Phys. Rev. D 2, pp. 1473–1477.ADSCrossRefGoogle Scholar
  156. Wilson, K. G. (1970b), “Anomalous dimmensions and the breakdown of scale invariance in perturbation theory,” Phys. Rev. D 2, pp. 1478–1493.ADSCrossRefGoogle Scholar
  157. Wilson, K. G. (1971), “Renormalization group and strong interactions,” Phys. Rev.D 3, pp. 1818–1846.MathSciNetADSCrossRefGoogle Scholar
  158. Wilson, K. G. and M. E. Fisher (1972), “Critical exponents in 3.99 dimensions,” Phys. Rev. Lett. 28, pp. 240–243.ADSCrossRefGoogle Scholar
  159. Wilson, K. G. (1975), “The renormalization group: Critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, pp. 773–840.ADSCrossRefGoogle Scholar
  160. Wilson, K. G. (1983), “The renormalization group and critical phenomena,” Rev. Mod. Phys. 55, pp. 583–600.ADSCrossRefGoogle Scholar
  161. Yang, C. N. and R. L. Mills (1954a), “Isotopic spin conservation and a generalized gauge invariance,” Phys. Rev. 95, p. 631.MathSciNetGoogle Scholar
  162. Yang, C. N. and R. L. Mills (1954b), “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, pp. 191–195.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Tian Yu Cao

There are no affiliations available

Personalised recommendations