Masujima M. Applied Mathematical Methods in Theoretical Physics

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568 References

problem in nonrelativistic quantum

mechanics.

9. Goldberger, M.L., and Watson, K.M.:

Collision Theory, John Wiley & Sons,

New York, 1964,Chapter10and

Appendix G.2.

16. DeAlfaro,V.,andRegge,T.:Po-

tential Scattering, North-Holland,

Amsterdam, 1965.

We remark that Appendix G.2 of the

book cited above, (R-9), discusses

Cauchy-type integral equations in the

scattering problem in nonrelativistic

quantum mechanics in terms of the

inhomogeneous Hilbert problems

with the complete solution.

We cite the following article for the

integro-differential equation arising

from Bose–Einstein condensation

in an external potential at zero

temperature.

17. Wu, T.T.: Phys. Rev. A58, 1465,

(1998).

We cite the following book for appli-

cations of the Wiener–Hopf method

in partial differential equations.

18. Noble, B.: Methods Based on the

Wiener–Hopf Technique for the Solu-

tion of Partial Differential Equations,

Pergamon Press, New York, 1959.

We note that Wiener–Hopf integral

equations originated from research

on the radiative equilibrium on

the surface of the star. We cite the

following articles for discussion of

Wiener–Hopf integral equations and

Wiener–Hopf sum equations.

19. Wiener, N., and Hopf, E.: S. B.

Preuss. Akad. Wiss. 696, (1931).

20. Hopf, E.: Mathematical Problems of

Radiative Equilibrium,Cambridge,

New York, 1934.

21. Krein, M.G.: Integral equation on

a half-line with the kernel depending

upon the difference of the arguments,

Am. Math. Soc. Transl.,(2)22, 163,

(1962).

22. Gohberg, I.C., and Krein, M.G.: Sys-

tems of integral equations on the

half-line with kernels depending on

the difference of the arguments, Am.

Math. Soc. Transl., (2) 14, 217, (1960).

We cite the following article for

discussion of the iterative solution

for a single Wiener–Hopf integral

equation and for a system of coupled

Wiener–Hopf integral equations.

23. Wu, T.T., and Wu, T.T.: Q. J. Appl.

Math., XX, 341, (1963).

We cite the following book for ap-

plication of Wiener–Hopf methods

to radiation from rectangular waveg-

uides and circular waveguides.

24. Weinstein, L.A.: The theory of diffrac-

tion and the factorization method,

Golem Press, Boulder, USA, 1969,

pp. 66–88, and pp. 120–156.

We cite the following articles

and books for application of the

Wiener–Hopf method to elastody-

namics of the crack motion.

25. Freund, L.B.: J. Mech. Phys. Solids,

20, 129, 141, (1972).

26. Freund, L.B.: Dynamic Fracture Me-

chanics, Cambridge University Press,

New York, 1990.

27. Broberg, K.B.: Cracks and Fracture,

Academic Press, New York, 1999.

We cite the following article and

the following book for application

of the Wiener–Hopf sum equation

to the phase transition of the two-

dimensional Ising model.

28. Wu, T.T.: Phys. Rev. 149, 380, (1966).

29. McCoy,B.,andWu,T.T.:The Two-

Dimensional Ising Model, Harvard

University Press, Cambridge, MA,

1971,ChapterIX.

We remark that Chapter IX of the

book cited above describes practical

methods to solve the Wiener–Hopf

sum equation with full mathemat-

ical details, including discussions

of Pollard’s theorem which is the

generalization of Cauchy’s theorem,

and the two special cases of the

Wiener–L

evy theorem.

We cite the following articles for

application of the Wiener–Hopf sum

equation to Yagi–Uda semi-inﬁnite

arrays .

30. Wasylkiwskyj, W.: IEEE Trans. Anten-

nas Propag., AP-21, 277, (1973).

31. Wasylkiwskyj, W., and

VanKoughnett, A.L.: IEEE Trans.

Antennas Propag., AP-24, 633,

(1974).

References 569

32. VanKoughnett, A.L.: Can. J. Phys.,

48, 659, (1970).

We cite the following book for the

historical development of the theory

of integral equations, the formal

theory of integral equations and a

variety of applications of the theory

of integral equations to scientiﬁc and

engineering problems.

33. Kondo, J.: Integral Equations,Kodan-

sha, Tokyo, 1991.

We cite the following book for the

pure-mathematically oriented reader.

34. Kress, R.: Linear Integral Equations,

2nd edition, Springer, Heidelberg,

1999.

Calculus of Variations

We cite the following books for

an introduction to the calculus of

variations.

35. Courant, R., and Hilbert, D.: Methods

of Mathematical Physics,(Vols.1and

2), John Wiley & Sons, New York,

1966, Vol. 1, Chapter 4. Reprinted in

Wiley Classic Edition, 1989.

36. Akhiezer, N.I.: The Calculus of Vari-

ations, Blaisdell, Waltham, MA,

1962.

37. Gelfand, I.M., and Fomin, S.V.: Cal-

culus of Variations, Prentice-Hall,

Englewood Cliffs, NJ, 1963.

38. Mathews, J., and Walker, R.L.:

Mathematical Methods of Physics,

Benjamin, Reading, MA, 1970,

Chapter 12.

We cite the following book for the

variational principle in classical

mechanics, the canonical transforma-

tion theory, the Hamilton–Jacobi

equation and the semiclassical

approximation to nonrelativistic

quantum mechanics.

39. Fetter, A.L., and Walecka, J.D.: Theo-

retical Mechanics of Particles and Con-

tinua, McGraw-Hill, New York, 1980,

Chapter 6, Sections 34 and 35.

We cite the following books for ap-

plications of the variational principle

to nonrelativistic quantum mechanics

and quantum statistical mechan-

ics, which includes Hartree–Fock

program.

40. Feynman, R.P., and Hibbs, A.R.:

Quantum Mechanics and Path Inte-

grals, McGraw-Hill, New York, 1965,

Chapters 10 and 11.

41. Feynman, R.P.: Statistical Mechan-

ics,Benjamin,Reading,MA,1972,

Chapter 8.

42. Landau, L.D., and Lifshitz, E.M.:

Quantum Mechanics, 3rd edition,

Pergamon Press, New York, 1977,

Chapter III, Section 20.

43. Huang, K.: Statistical Mechanics,2nd

edition, John Wiley & Sons, New

York, (1983) Section 10.4.

We cite the following series of

articles for Schwinger’s action

principle.

44. Schwinger, J.: Phys. Rev. 82,(1951),

914. I. Phys. Rev. 91,(1953), 713.

II. Phys. Rev. 91,(1953), 728. III.

Phys. Rev. 92,(1953), 1283. IV. Phys.

Rev. 93,(1954), 615. V. Phys. Rev.

94,(1954), 1362. VI. Phys. Rev. 125,

(1962), 1043. VII. Phys. Rev. 127,

(1962), 324. VIII. Phys. Rev. 130,

(1963), 1253. IX. Phys. Rev. 132,

(1963), 1317.

The last four articles are concerned

with the non-Abelian gauge ﬁeld in

Coulomb gauge and the gravitational

ﬁeld.

We cite the following book and the

monograph for Noether’s theorem.

45. Treiman, S.B., Jackiw, R.,

Zumino, B., and Witten, E.: Cur-

rent Algebra and Anomalies, World

Scientiﬁc, Singapore, 1985, Lecture

by Jackiw, R.

46. Low, F.E.: Classical Field Theory,

John Wiley & Sons, 1997, New York,

Chapter 7.

We cite the following book as a gen-

eral reference for the theory of the

gravitational ﬁeld.

47. Weinberg, S.: Gravitation and Cos-

mology. Principles and Applications

of The General Theory of Relativ-

ity, John Wiley & Sons, New York,

1972.

We cite the following book for the

genesis of Weyl’s gauge principle

and the earlier attempt to unify the

electromagnetic force and the gravita-

tional force in the context of classical

570 References

ﬁeld theory before the birth of quan-

tum mechanics and quantum ﬁeld

theory.

48. Weyl, H.: Space–Time-Matter,Dover

Publications, New York, 1950.

Although H. Weyl failed to accom-

plish his goal of the uniﬁcation of

the electromagnetic force and the

gravitational force in the context of

classical ﬁeld theory, his spirit for the

uniﬁcation of all forces in nature sur-

vived even after the birth of quantum

mechanics.

We cite the following articles and

books for Weyl’s gauge principle

after the birth of quantum mechan-

ics for the Abelian electromagnetic

gauge group.

49. Weyl, H.: Proc. Nat. Acad. Sci. 15,

(1929), 323.

50. Weyl, H.: Z. Physik. 56,(1929), 330.

51. Weyl, H.: Theory of Groups and Quan-

tum Mechanics, Leipzig, Zurich, 1928;

reprinted by Dover, 1950. Chapter 2,

Section 12, and Chapter 4, Section 5.

We cite the following article as a ﬁrst

attempt to unify the strong force and

the weak force in nuclear physics

in the context of quantum ﬁeld the-

ory without invoking Weyl’s gauge

principle.

52. Yukawa, H.: Proc. Phys. Math. Soc.

(Japan), 17, 48, (1935).

We cite the following articles for

the uniﬁcation of the electromag-

netic force and the weak force by

invoking Weyl’s gauge principle and

the Higgs–Kibble mechanism and

the proposal of the standard model

which uniﬁes the weak force, the

electromagnetic force, and the strong

force with quarks, leptons, the Higgs

scalar ﬁeld, the Abelian gauge ﬁeld,

and the non-Abelian gauge ﬁeld in

the context of quantum ﬁeld theory.

53. Weinberg, S.: Phys. Rev. Lett. 19,

1264, (1967); Phys. Rev. Lett. 27,

1688, (1971); Phys. Rev. D5, 1962,

(1971); Phys. Rev. D7, 1068, (1973);

Phys. Rev. D7, 2887, (1973); Phys.

Rev. D8, 4482, (1973); Phys. Rev. Lett.

31, 494, (1973); Phys. Rev. D9, 3357,

(1974).

We cite the following book for dis-

cussion of the standard model which

uniﬁes the weak force, the elec-

tromagnetic force, and the strong

force with quarks, leptons, the Higgs

scalar ﬁeld, the Abelian gauge ﬁeld

and the non-Abelian gauge ﬁeld by

invoking Weyl’s gauge principle, and

the Higgs–Kibble mechanism in the

context of quantum ﬁeld theory.

54. Huang, K.: Quarks, Leptons, and

Gauge Field, 2nd edition, World

Scientiﬁc, Singapore, 1992.

We cite the following article for LPZ

model and PZ II model.

55. Prentki, J., and Zumino, B.: Nucl.

Phys. B47, 99, (1972).

We cite the following article for the

O(3) model.

56. Georgi, H., and Glashow, S.L.: Phys.

Rev. Lett. 28, 1494, (1972).

We cite the following articles for the

instanton, the strong CP violation,

the Peccei–Quinn axion hypothesis

and the invisible axion scenario.

57. Belavin, A.A., Polyakov, A.M.,

Schwartz, A.S., and Tyupkin, Y.S.:

Phys. Lett. 59B,(1975), 85.

58. Peccei, R.D., and Quinn, H.R.: Phys.

Rev. Letters. 38,(1977), 1440; Phys.

Rev. D16,(1977), 1791.

59. Weinberg, S.: Phys. Rev. Letters. 40,

(1978), 223.

60. Wilczek, F.: Phys. Rev. Lett. 40,

(1978), 279.

61. Dine, M., Fishcler, W., and

Srednicki, M.: Phys. Lett. 104B,

(1981), 199.

We cite the following article for the

see-saw mechanism.

62. Yanagida, T.: Prog. Theor. Phys. 64,

(1980), 1103.

We cite the following article for the

identiﬁcation of two singlet Higgs

scalar ﬁelds, one for the ‘‘invisible’’

axion and the other for the see-saw

mechanism.

63. Langacker, P., Peccei, R.D.,

Yanagida, T.: Mod. Phys. Lett. A1,

(1986), 541.

We cite the following article for the

general structure of variant axion

models, their theoretical predictions

References 571

and the experimental bounds on

them.

64. Bardeen, W., Peccei, R.D., and

Yanagida, T.: Nucl. Phys. B279,

(1987), 401.

We cite the following book for the

grand uniﬁcation of the electromag-

netic, weak and strong interactions.

65. Ross, G.G.: GrandUniﬁedTheories,

Perseus Books Publishing, Mas-

sachusetts, 1984, Chapters 5 through

11.

We cite the following article for the

SU(5) grand uniﬁed model.

66. Georgi, H., and Glashow, S.L.: Phys.

Rev. Lett. 32, 438, (1974).

We cite the following article for dis-

cussion of the gauge principle in the

differential formalism originally due

to H. Weyl and the integral formal-

ism originally due to T.T. Wu and

C.N. Yang.

67. Yang, C.N.: Ann. N.Y. Acad. Sci. 294,

86, (1977).

We cite the following article for

WKB approximation to wavefunc-

tion in quantum mechanics, and the

connection formula.

68. Klein, O.: Z. Physik. 80,(1932), 792.

1. Bender,CarlM.andOrszag,Steven

A.: Advanced Mathematical Methods

For Scientists And Engineers, Springer,

New York, 1999.

We cite the following book for the

connection between Feynman’s

action principle in quantum mechan-

ics and the calculus of variations,

in particular, the second variation,

the Legendre test and the Jacobi

test. Also discussed in this book is

WKB approximation in path integral

formalism.

69. Schulman, L.S.: Techniques and Appli-

cation of Path Integration, John Wiley

& Sons, New York, 1981.

We cite the following seminal arti-

cle for the WKB approximation in

quantum ﬁeld theory in canonical

formalism by the method of third

quantization.

70. Nambu, Y.: Phys. Lett. 26B,(1968),

626.

We cite the following book for the

use of the calculus of variations in

the path integral quantization of

classical mechanics and classical

ﬁeld theory, Weyl’s gauge princi-

ple for the Abelian gauge group

and the semisimple non-Abelian

gauge group, the Schwinger–Dyson

equation in quantum ﬁeld theory and

quantum statistical mechanics, and

stochastic quantization of classical

mechanics and classical ﬁeld theory.

71. Masujima, M.: Path Integrals and

Stochastic Processes in Theoretical

Physics, Feshbach Publishing, Min-

neapolis, 2006, Chapter 1, Sections

1.1 and 1.2; Chapter 2, Sections 2.3,

2.4 and 2.5; Chapter 3, Sections 3.1,

3.2, and 3.3; Chapter 5, Sections 5.1

and 5.3; Chapter 7, Section 7.2.

We cite the following book for dis-

cussion of the Schwinger–Dyson

equation, and the Bethe–Salpeter

equation from the viewpoint of the

canonical formalism and the path

integral formalism of quantum ﬁeld

theory.

72. Huang, K.: Quantum Field Theory.

From Operators to Path Integrals,

John Wiley & Sons, New York, 1998,

Chapter 10, Sections 10.7 and 10.8;

Chapters 13 and 16.

54. Huang, K.: Quarks, Leptons, and

Gauge Field, 2nd edition, World Sci-

entiﬁc, Singapore, 1992,ChaptersIX

and X.

43. Huang, K.: Statistical Mechanics,2nd

edition, John Wiley & Sons, New

York, 1983, Chapter 18.

The Wick–Cutkosky model is the

only exactly solvable model for the

Bethe–Salpeter equation known

to this day. We cite the following

articles for this model.

73. Wick, G.C.: Phys. Rev. 96, 1124,

(1954).

74. Cutkosky, R.E.: Phys. Rev. 96, 1135,

(1954).

We cite the following articles for the

presentation of asymptotic disaster

of Abelian gauge ﬁeld with tri−

approximation in interaction with the

fermion ﬁeld.

75. Landau, L.D., Abrikosov, A.A., and

Khalatnikov, I.M.: Doklady. Akad.

572 References

Nauk. USSR. 95,(1954), 497, 773,

1177; 96,(1954), 261.

76. Abrikosov, A.A., Galanin, A.D., and

Khalatnikov, I.M.: Doklady. Akad.

Nauk. USSR. 97,(1954), 793.

77. Landau, L.D., Abrikosov, A.A., and

Khalatnikov, I.M.: Suppl. Nuovo

Cimento, 3,(1956), 80.

They attempted to smear out the

local interaction of QED to avoid

the asymptotic disaster of QED at

short distance with the use of the

θ -function with no avail.

We cite the following articles for the

presentation of asymptotic freedom

of non-Abelian gauge ﬁeld with tri-

approximation in interaction with the

fermion ﬁeld.

78. Fradkin, E.S., and Kalashnikov, O.K.:

Lett. Nuovo Cimento, 10,(1974), 771.

79. Fradkin, E.S., and Kalashnikov, O.K.:

Acta. Phys. Austriaca. 45,(1976), 81.

We cite the following monograph for

path integral renormalization.

72. Huang, K.: Quantum Field Theory.

From Operators to Path Integrals,

John Wiley & Sons, New York, 1998,

Chapter 16.

We cite the following standard

articles and the monographs for Gell-

Mann–Low renormalization group

equation and Callan–Symanzik

equation.

80. Gell–Mann, M., and Low, F.E.: Phys.

Rev. 95,(1954), 1300.

81. Callan, C.G.: Phys. Rev. D2,(1970),

1541.

82. Symanzik, K.: Comm. Math. Phys. 18,

(1970), 227.

83. Bogoliubov, N.N., and Shirkov, D.V.:

Introduction to the Theory of Quan-

tized Field, 3rd edition, John Wiley &

Sons, 1980,NewYork.Chapter9.

84. Nishijima, K.: Theory of Fields,

Kinokuniya-shoten, 1986,Tokyo,

Chapter 19. (In Japanese).

We cite the following article

for the relationship between

Gell–Mann–Low renormal-

ization group equation and

Callan–Symanzik equation.

85. Higashijima, K., and Nishijima, K.:

Prog. Theor. Phys. 64,(1980), 2179.

We cite the following monographs

for the proof the renormalizability

of the non-Abelian gauge ﬁeld the-

ory based on BRST invariance and

Zinn-Justin equation, electro-weak

uniﬁcation, standard model, grand

uniﬁcation of weak, electromagnetic

and strong interactions, and grand

uniﬁcation with the graded Lie gauge

group.

86. Weinberg, S.: Quantum Theory of

Fields I, Cambridge University Press,

New York, 1995.

87. Weinberg, S.: Quantum Theory of

Fields II, Cambridge University

Press, New York, 1996.

88. Weinberg, S.: Quantum Theory of

Fields III, Cambridge University

Press, New York, 2000.

We cite the following textbooks for

superstring theory.

89. Kaku, M.: Introduction to Superstrings

and M theory, Springer, New York,

1998.

90. Polchinski, J.: String Theory – An

Introduction to the Bosonic String,Vol-

ume 1, Cambridge University Press,

New York, 1998.

91. Polchinski, J.: String Theory-

Superstring Theory and Beyond,

Volume 2, Cambridge University

Press, New York, 1999.

573

Index

action functional, 312, 362, 365, 368, 369,

371, 428, 471, 473, 474, 488, 489, 507

– effective, 407, 469–471

– matter, 422, 424, 428, 429, 431, 432

– trial, 398–400, 406

action principle, 311

– Feynman, 311, 360–362, 364, 365, 368,

371

– Hamilton, 313, 318, 324, 353

– Schwinger, 312, 360, 361, 368, 369, 371

adiabatic switching hypothesis, 51, 58

adjoint,

– boundary condition, 5, 43

– integral equation, 330

–matrix, 6

– operator, 5–10, 14

– problem, 5, 8, 9, 11, 269–271, 279–281

– representation, 423, 427

asymptotic condition, 58

asymptotic disaster,

– QED, 475

asymptotic freedom,

– QCD, 213, 427, 479, 487

axion, 427, 507, 510–518

Bender, Carl M., 560

Bernoulli, Jacques and Jean, 309

Bethe–Salpeter equation, 384

bifurcation point, 299–302

Born approximation, 51

– distorted wave, 61, 398

– plane wave, 60, 398

Born series, 42

Born, M., 359

Brachistochrone, 313, 316, 317, 319–321

branch cut, 19, 182, 185, 193, 194, 198–200,

209, 232, 296

branch point, 182, 194, 195, 228, 255

Callan–Symanzik equation, 487, 493, 496,

498, 499

Carleman,

– integral equation,

– – homogeneous, 205, 211

– – inhomogeneous, 205, 206, 211

Catenary, 313, 323, 339

Cauchy,

– integral equation of the ﬁrst kind,

– – homogeneous, 199

– – inhomogeneous, 197

– integral equation of the second kind,

– – generalization, 205

– – inhomogeneous, 201, 211

– integral formula,

– – general decomposition problem, 253,

255

– – generalization, 280

– – review of complex analysis, 18–20, 22

– – Wiener–Hopf integral equation, 241,

263

– – Wiener–Hopf method, 223

– – Wiener–Hopf sum equation, 280

–kernel,

– – Carleman integral equation, 205

– residue theorem,

– – Sturm–Liouville system, 183

– singular integral equation, 193

causality, 48, 49, 51, 213

charge,

– conserved matter, 422

– conserved Noether, 412, 414, 415, 417, 424

Applied Mathematical Methods in Theoretical Physics, Second Edition. Michio Masujima

 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 978-3-527-40936-5

574 Index

– group, 425

conjugate point, 332, 334–336, 343

conservation law,

– charge, 418, 420–422, 426

– current, 418, 419, 421, 422, 425

– – energy, momentum, and angular

momentum, 429, 430, 433, 434

Coulomb potential, 398

covariant derivative,

– gravitational, 431, 432, 434, 436

– non-Abelian, 423, 426

– U(1), 419, 420

current,

– conserved Noether, 424, 425

– gauged matter, 424–426

– ungauged matter, 424

Cutkosky, R.E., 541

differential operator, 5, 7, 9, 10, 14, 16, 43,

181, 372, 379, 383, 430, 431, 475, 479, 493,

494

dispersion relations, 211–214, 216–218

dual integral equations, 281, 285

eigenfunction, 7, 8, 10, 12, 13, 29

– completeness, 157, 172, 181, 186

– continuum, 186

– discrete, 186

eigenvalue problem, 7, 43, 46, 172

– inﬁnite Toeplitz matrix, 274

electrodynamics, 211, 218, 418, 421

equal time canonical commutator, 365,

373–375, 387

Euler derivative, 363

Euler equation, 310, 312, 314–316, 319,

320, 322, 330, 331, 334–337, 347

Euler, Leonhard, 309, 310

Euler–Lagrange equation of motion,

363–365, 373, 375, 413, 414, 419, 422,

425, 429, 433

factorization,

– homogeneous Wiener–Hopf integral

equation of the second kind, 241, 246, 248,

254, 258, 260

– homogeneous Wiener–Hopf sum equation,

279

Fetter, A.L., 394

Feynman, R.P., 542

ﬁeld strength tensor,

– non-Abelian gauge ﬁeld, 423–425, 434

– U(1) gauge ﬁeld, 420, 421, 425

Fikioris, G., 150, 151

Fomin, S.V., 29

Fredholm,

– alternative, 9–12, 34

– integral equation of the ﬁrst kind, 31, 35,

158

– integral equation of the second kind, 31,

134, 168, 174, 329, 330

– – homogeneous, 32, 45, 46, 135, 150, 151

– – inhomogeneous, 31, 32, 42, 105, 117,

121, 143, 151, 158, 178

– – system of, 143

– – translation kernel, 136

– nonlinear integral equation, 299

– solvability condition, 9

– theory for a bounded kernel, 127, 133

function space, 3, 5, 6, 29

gauge ﬁeld,

– gravitational, 371, 418, 428, 434, 435, 437

– non-Abelian, 213, 371, 418, 421, 423,

425–427, 434, 435, 437–439, 441, 445, 447,

448, 450, 454, 457, 460, 461, 465, 468, 469,

479, 481, 507

– U(1), 371, 418, 420, 421, 424–426, 475,

476, 479

gauge principle,

– Kibble, T.W.B., 418, 428

– Utiyama, R., 428, 431

– Weyl, H., 418, 420–422, 428, 430, 431, 433

Gelfand, I.M., 29

generating function of j-fold mass-insertion

terms, 494, 499

generating functional, 373, 375, 471, 475,

479, 487–489

Georgi, H., 544

Glashow, S.L., 544

Goldberger, M.L., 218, 563

Goldstein, J., 538

Gottfried, K., 78

Green’s function, 1, 5, 14, 15, 18, 29, 31,

37–41, 44, 46, 47, 50, 52, 172, 181, 215, 371,

373, 375, 376, 378, 383, 389–392, 471,

475–477, 479–482, 484, 487–498

Hamilton’s canonical equation of motion,

311, 354, 359, 360

Index 575

Hamilton, William Rowan, 311

Hamilton–Jacobi equation, 29, 311, 353,

357, 358

Hamiltonian, 54, 58–61, 354, 358, 385,

395

Hammerstein nonlinear integral equation,

303

harmonic potential, 399

Hartree–Fock equation, 529

Heisenberg, W., 359

Hilbert problem,

– homogeneous, 194, 200, 207

– inhomogeneous, 194, 198, 202, 203, 206,

223

Hilbert–Schmidt,

– expansion, 159, 164, 174

– – generalization, 177

– – nonsymmetric kernel, 178

– theorem, 157, 161, 164, 173

– theory, 153

– – generalization, 174–176

index,

– Wiener–Hopf integral equation,

256–258, 260, 265, 267–269, 271

– Wiener–Hopf sum equation, 276–279,

281

inner product, 1–3, 5–12, 14, 15, 29, 43,

47, 173

instanton, 427, 507, 508, 516

invariance,

– BRST, 468, 470, 473, 474

– gauge, 418, 470, 471, 473, 474

–globalG phase, 422, 425

–globalU(1) phase, 419–421, 426

– inﬁnite-dimensional group, 371, 418,

428

–localG phase, 422–426

–localU(1) phase, 418–420, 426

– Lorentz, 473

– – local, 428

–Poincar

– – global, 429, 430, 432

– – local, 428, 433

– rotational, 413

– scale, 416–418

– translational, 52, 393, 413, 429

– U(1)

P.Q.

, 511, 513, 514

isoperimetric problem, 310, 321

Itzykson, C., 536, 541

Jackson, J.D., 293, 294

Jacobi test, 332, 334, 335

Jacobi, Carl Gustav Jacob, 309, 311

Jordan, P., 359

kernel, 5, 7, 31, 35, 37, 46, 108

– bounded, 127, 133

– general, 125

– inﬁnite translational, 108, 136

– iterated, 120, 121

– Pincherle–Goursat, 123

– resolvent, 120, 126, 134, 157, 185

– – Wiener–Hopf integral equation, 265,

267

– semi-inﬁnite translational, 237, 256, 269,

283

– square-integrable, 37, 107, 118, 158

– symmetric, 46, 153, 155, 330

– transposed, 35, 155, 179–181

Klein, O., 560

Kondo, J., 220, 294

Kress, R., 29

Lagrange equation of motion, 310, 311, 319,

325, 353, 363–365, 412, 413

Lagrange, Joseph Louis, 309, 310

Lagrangian, 310, 313, 319, 324, 353, 354,

368, 411–413, 513

– electron–phonon system, 400

Lagrangian density, 312, 368, 372, 385, 411,

413, 414, 416, 418, 469, 471

– axion, 512, 513, 516

– effective, 474, 475, 479

– gravitational ﬁeld, 428, 434, 435, 437

– Higgs scalar, 502, 503

– interaction, 375, 377, 419, 421, 424, 504,

505

– matter ﬁeld, 419, 420, 422, 423, 428,

431–433

– non-Abelian gauge ﬁeld, 423, 424

– QCD, 427, 504, 507, 509, 513

– QED, 498

– standard model, 499, 503, 505, 506, 510,

511

– U(1) gauge ﬁeld, 418, 420, 421

Lautrup, B., 537

Legendre test, 332

Legendre, Adrien Marie, 309, 310

576 Index

Liouville’s theorem, 19

– Cauchy-type integral equation, 199, 204,

209

– Wiener–Hopf method, 224, 244, 245,

250, 258, 259, 261, 268, 276–278, 280

Low, F.E., 82

Margetis, D., 150, 151

McCoy, B., 280

minimum, 319, 324, 327, 328, 331–336,

341, 344, 345, 348

Myers, J.M., 150

Nakanishi, N., 537

Newton, Isaac, 309

Nishijima, K., 152, 534–536, 541

norm, 1–3, 118, 130, 167, 174

Orszag, Steven A., 560

Pauli, W., 560

Prentki, J., 543

proper self-energy parts, 379, 476, 477,

481–483

Rayleigh quotient, 169, 171, 172

renormalization group, 487, 491–493, 495,

496, 498

– Callan–Symanzik equation, 487, 493,

494, 496, 499

––j-fold mass insertion terms, 493, 498

– Gell–Mann-Low equation, 487, 493,

494

resolvent,

– Fredholm integral equation, 120, 121,

125, 127

– Hilbert–Schmidt theory, 157, 185

– Wiener–Hopf integral equation, 265,

267

scattering problem,

– time-dependent, 47, 51

– time-independent, 40, 214

Schr

odinger equation,

– time-dependent, 47, 51, 395

– time-independent, 38, 40, 53, 215

Schwarz inequality, 2, 3, 106, 118, 119

Schwinger–Dyson equation, 371, 381, 383,

385, 392, 394, 475, 476, 479, 481, 483

self-adjoint, 7, 12, 13, 16, 43, 46, 47, 153, 169,

172, 174, 181, 185

singular point, 299

solvability condition,

– Wiener–Hopf integral equation of the

second kind, 271

– Wiener–Hopf sum equation, 280

square-integrable, 1, 4, 25–27, 29, 37, 105,

107, 117, 125, 135, 136, 143, 153, 156–158,

164, 168, 173–179, 216, 329, 330

Stueckelberg, E.C.G., 536

Sturm–Liouville,

– eigenvalue problem, 46, 172, 174, 326

– system, 43, 46, 174, 181

summary,

– behavior near the end points, 23

– Fredholm theory for a bounded kernel,

133

– Schwinger–Dyson equation, 381, 392,

394

– Wiener–Hopf integral equation, 271

– Wiener–Hopf sum equation, 280

time-ordered product, 364, 369, 373, 374

Toeplitz matrix, 272

– inﬁnite, 272

– semi-inﬁnite, 274

transformation, 5, 36, 37, 175

– canonical, 29, 311

– conformal, 416–418

– covariant derivative, 432

– ε

(x), 431, 432, 434, 435

– ξ

(x), 431, 433–435

– A

, 432

– e

, 432

– Fourier, 137, 140, 487

– function, 361, 406, 407

– gauge, 423, 426, 427, 431

– general coordinate, 431, 435

–globalG, 422, 425

–globalU(1), 419, 425

– internal, 418, 425, 428, 430

– invariant scalar density, 433

– Laplace, 108, 109, 111, 295, 407

– Legendre, 311, 312

–localG, 422, 423, 426

Index 577

–localU(1), 418–421

– Lorentz, 414–416, 418, 428, 431

–Poincar

e, 416, 428–431, 435

– renormalization, 497

– scale, 416–418, 488, 492

– space–time coordinate, 418, 428, 430

– symmetry, 418

triangular inequality, 2, 3, 118

uniﬁcation,

– electro-weak, 427

– grand, 512

unitarity, 51, 55–57, 213, 214

vacuum, 400

vacuum expectation value, 375, 376

VanKoughnett, A.L., 280

variation, 354

– calculus of, 29, 151, 309–314, 329, 353

– corner relation, 345, 346, 348

– inﬁnitesimal, 312, 314, 318, 320, 321,

331–333, 335, 362, 363, 365, 368, 369,

412, 413, 428, 430

– – initial condition, 334, 336

– regular, mathematical theory of, 487

– second, 324, 330

– strong, 331, 344

– weak, 331

variational principle,

– Euler–Lagrange, 310, 311

– integral equations, 329, 330

– invariant, 428

–Kohn,

– – atomic scattering, 563

– quantum statistical mechanics, 395,

397, 399, 400, 407

–Schwinger,

– – phase shift, 563

vertex operator, 380, 475, 477, 480, 483, 484

Volterra,

–Abelintegralequation, 111

– coupled integral equations of the second

kind,

– – solvable case, 111

– integral equation of the ﬁrst kind, 31

– – solvable case, 110

– integral equation of the second kind,

– – homogeneous, 107

– – inhomogeneous, 31, 36, 40, 105,

107

– – solvable case, 108

– nonlinear integral equation of convolution

type, 295

– nonlinear integral equations,

– – classiﬁcation, 298

Walecka, J.D., 394

Ward–Takahashi identity,

– QED, 476, 498, 499

Ward–Takahashi–Slavnov–Taylor identity,

– QCD, 482, 486

Wasylkiwskyj, W., 280

Watson, K.M., 218, 563

wavefunction, 358

– scattering problem, 51

– – asymptotic, 51, 57–59

– – ﬁnal state, 59

– – free, 57

––incoming, 54

– – initial state, 59

– – outgoing, 54, 59

– – rearrangement collision, 59

– – total, 52

Weierstrass E function, 344

Weierstrass–Erdmann corner relation, 345,

348, 349

weight, 43, 173, 397

Weinstein, L.A., 293

Weyl, H., 418

Wick, G.C., 541

Wiener–Hopf,

– integral equation, 223, 237, 271

– integral equation of the ﬁrst kind, 237,

281, 283, 284

– integral equation of the second kind, 237,

241

– – homogeneous, 237, 253

– – inhomogeneous, 237, 261, 280

– method, 223, 235, 252

– – factorization, 272

– – sum splitting, 224, 227, 228, 235

– sum equation, 272, 280

– – inhomogeneous, 274, 280

WKB approximation,

– Euclidean quantum ﬁeld theory, 526

– Euclidean quantum mechanics, 525

Wronskian, 45, 182, 335, 338

Wu, Tai Tsun, 150, 280