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Action Functional 266, 304, 305, 311, 342, 345, 347, 348, 350 Action Index action functional 266, 304, 305, 311, 342, homogeneous 155 345, 347, 348, 350 inhomogeneous 153 action integral 267 integral equation of the second kind action principle 265 generalization 161 Feynman 266, 303–305, 308, 329, 370 inhomogeneous 157, 166 Hamilton 267, 272, 277 integral formula Schwinger 266 general decomposition problem 208, adjoint 211 boundary condition 9, 45 generalization 235 integral equation 282 review of complex analysis 21–24 matrix 9, 10, 12 Wiener–Hopf integral equation 196, operator 9–13, 17 218 problem 9, 11, 13, 15, 224–226, 233– Wiener–Hopf method 177 235 Wiener–Hopf sum equation 235 representation 343, 347 kernel Akhiezer, N.I. 368 Carleman integral equation 161 axion 356, 369 residue theorem Sturm–Liouville system 140 Bach, M. 365 singular integral equation 149 Bazant, M.Z. 366 causality 49, 50, 168 Belavin, A.A. 369 charge Bender, Carl M. 365 conserved matter 342 Bernoulli, Jacques and Jean 263 Bessel inequality 126 conserved Noether 344 Bethe–Salpeter equation 326, 370 group 345 bifurcation point 253–255 Collin, R.E. 366 Born approximation 312 complete 7 boundary terms 17 complete square 285 Brachistochrone 267, 270, 272–274 completeness of an orthonormal system of Broberg, K.B. 367 functions 8 concentrated load 20 Carleman conjugate point 284, 286–288, 295 integral equation conservation law homogeneous 161, 166 charge 339–342, 346 inhomogeneous 161, 162, 166 current 339–342, 345 Catenary 267, 276, 291 energy, momentum, and angular momen- Cauchy tum 348, 349, 352, 353 integral equation of the first kind Coulomb potential 312 374 Index Courant, R. 368 Fikioris, G. 107 covariant derivative Fishcler, W. 369 gravitational 350–355 Fomin, S.V. 32, 368 non-Abelian 343, 346 Fourier series 8 U(1) 339, 340 Fredholm current alternative 12–15 conserved Noether 344, 345 integral equation of the first kind 33, 37, gauged matter 344–346 115 ungauged matter 344 integral equation of the second kind 33, Cutkosky, R.E. 364, 371 79, 107, 125, 131, 281, 282 exactly solvable examples 93 De Alfaro, V. 366 homogeneous 34, 46, 47, 93 differential equation 54 inhomogeneous 33, 34, 43, 63, 75, differential operator 320, 325 95, 100, 114, 136 Dine, M. 369 system of 100 Dirac delta function 8 nonlinear integral equation 253 dispersion relations 166–168, 172 theory for a bounded kernel 86 distributed load 18 Fredholm Alternative 12, 36 disturbance solution 185 Fredholm solvability condition 12 dual integral equations 235, 239 Freund, L.B. 367 function space 7, 8, 10, 32 eigenfunction 11–16, 32 completeness 113, 129, 138, 142 Gamma Function 68 continuum 142, 143 gauge field discrete 142, 143 gravitational 339, 347, 353, 355, 368 eigenfunctions 11 non-Abelian 168, 339, 341, 343–347, eigenvalue problem 11, 45, 47, 129 353, 356, 369 infinite Toeplitz matrix 229 U(1) 339–341, 344–346, 369 eigenvalues 11 gauge group 341, 347, 356, 369–371 electrodynamics 166, 172, 339, 341 gauge principle energy integral 272 H. Weyl 339–341, 343, 347, 349, 352, equal time canonical commutator 306, 314– 368–370 316, 331 R. Utiyama 347, 349 Euler derivative 304 T.W.B. Kibble 339, 347 Euler equation 264, 267, 269, 270, 272–275, Wu-Yang formalism 370 283, 284, 287–289, 298 Gelfand, I.M. 32, 368 Euler, Leonhard 263, 264 generating functional 314, 316, 327, 328 Euler–Lagrange equation of motion 272, 313, Georgi, H. 369, 370 316, 340, 342, 345, 348, 351 Glashow, S.L. 369, 370 Gohberg, I.C. 367 factorization Goldberger, M.L. 172, 365, 366 Wiener–Hopf integral equation 196, Goldstein, J. 364 201, 203, 209, 214, 216 Green’s function 9, 16–18, 20, 21, 32, 39–43, Wiener–Hopf sum equation 234 45–49, 51, 52, 129, 139, 145, 169, Fetter, A.L. 339, 368 312, 314, 316, 317, 319, 320, 325, Feynman, R.P. 368 328, 333, 335, 336 field strength tensor non-Abelian gauge field 343–345, 353 Hadamard inequality 88 U(1) gauge field 341, 345 Hamilton, William Rowan 265 Index 375 Hamilton–Jacobi equation 32, 265, 368 Jacobi, Carl Gustav Jacob 263, 265 Hammerstein Jerri, A.J. 365 nonlinear integral equation 257 harmonic potential 143, 312 kernel 33 Hermitian 10 bounded 86, 92 Hermitian transpose 10 general 84 Hibbs, A.R. 368 infinite translational 67, 95 Higgs scalar field 369 iterated 78, 79 Higgs–Kibble mechanism 369 Pincherle–Goursat 81 Hilbert problem resolvent 78, 84, 92, 114, 142 homogeneous 150, 156, 162 Wiener–Hopf integral equation 220, inhomogeneous 150, 154, 158, 162, 177 222 Hilbert, D. 368 semi-infinite translational 191, 212, 224, Hilbert–Schmidt 238 expansion 116, 121, 131, 134, 136 square-integrable 39, 65, 76, 114, 115 theorem 113, 118, 121, 130 symmetric 48, 109, 111, 282 theory transposed 36, 37, 111, 137, 138 generalization 133 Kondo, J. 174, 247, 367 Hölder condition 24 Krein, M.G. 366, 367 homogeneous 33 Kress, R. 32, 367 Hopf, E. 366 Kronecker delta symbol 7 Huang, K. 368–370 Lagrange equation of motion 305 in-state 56 Lagrange, Joseph Louis 263, 264 incoming wave condition 54 Lagrangian 264, 267, 272, 277, 303, 307, 308, index 329 Wiener–Hopf integral equation 211, Lagrangian density 266, 308, 313, 329, 339 212, 214–216, 220, 222–226 Abelian U(1) gauge field 341 Wiener–Hopf sum equation 230–235 gravitational field 347, 353, 355 infinitesimal variation of the initial condition interaction 316, 318, 339, 341, 344 287 matter field 339, 340, 342, 343, 347, influence function 20 350–352 inhomogeneous 33 matter-gauge system 341, 344 inner product 5–7, 9–15, 17, 18, 32, 44, 45, non-Abelian G gauge field 344 48, 130 QCD 356 instanton 356, 369 Landau, L.D. 368 integral equation 54 Laplace transform 67 invariance 338, 339, 347 Lautrup, B. 361 gauge 339 Legendre test 284, 285 global G 343, 345 Legendre, Adrien Marie 263, 264 global U(1) 339–341, 346 Lifshitz, E.M. 368 local G 343–346 linear operator 8, 9, 11, 13, 16, 32 local U(1) 339–341, 346 linear vector space 6 scale 339 Liouville’s theorem 21 isoperimetric problem 264, 274 Carleman integral equation 164 iterative solution 75 Cauchy integral equation 154, 159 Wiener–Hopf method 178, 199, 200, Jackson, J.D. 246, 366 205, 213, 215, 216, 223, 231, 232, Jacobi test 284, 286, 287 235 376 Index Margetis, D. 107 Wiener–Hopf integral equation 220, 222 Masujima, M. 303, 307, 370 resolvent kernel 220 Mathews, J. 368 retarded boundary condition 54 McCoy, B. 235, 367 Riccatti differential equation 286 mean value theorem 297 Riccatti substitution 286 Mikhlin, S.G. 366 Ross, G.G. 370 minimum 6, 128, 273, 277, 279, 280, 284– 288, 292, 296, 299 Sakurai, J.J. 365 momentum operator 305 scalar multiplication 5 Myers, J.M. 107 scattering problem 41 Schrödinger equation 169, 307, 308 Nakanishi, N. 361 boundary value problem 39 Newton, Isaac 263 time-dependent scattering problem 48 Nishijima, K. 365 time-independent scattering problem 41 Noble, B. 366 Schulman, L.S. 370 norm 5–7, 76, 89, 124, 131 Schwartz, A.S. 369 norm of a function 5 Schwarz inequality 6, 7, 64, 76, 77 normalized 7 Schwinger–Dyson equation 312, 322, 325, 329, 336, 338, 339, 370 operator 8 second variation 277, 283, 370 Orszag, Steven A. 365 self-adjoint 10, 15, 16, 19, 45, 48, 109, 126, orthogonal functions 7 129, 131, 138, 139, 142 orthonormal set of functions 7 self-adjoint operator 15 out-state 56 singular point 253 outgoing wave condition 54 solvability condition Wiener–Hopf integral equation of the Peccei, R.D. 369 second kind 226 Pincherle–Goursat type kernel 85 Wiener–Hopf sum equation 234 Pipkin, A.C. 365 square-integrability 101, 133, 171 Plemelj formula 27 square-integrable 5, 8, 28, 29, 31, 39, 63, 65, Polianin, A.D. 365 66, 75, 84, 93, 95, 100, 101, 109, Polyakov, A.M. 369 112–115, 121, 124, 130, 131, 133, positivity 5 134, 136, 171, 282 principle of superposition 303 Srednicki, M. 369 proper self-energy parts 320 Stakgold, I. 365 standard model 356, 369, 371 quantum mechanics 143, 168 step-discontinuous 31 Quinn, H.R. 369 strong minimum 296 strong variation 283 Rayleigh quotient 126–128 Sturm–Liouville reciprocity 48 eigenfunction 129 reflection coefficient 43 eigenvalue problem 47, 129, 131, 279 Regge, T. 366 operator 129 remainder 117 system 44, 47, 131, 138 resolution of identity 303 summary resolvent behavior near the endpoints 26 Fredholm integral equation 78–80, 83, Example 7.4 207 86 Fredholm theory for a bounded kernel Hilbert–Schmidt theory 114, 142 92 Index 377 Schwinger–Dyson equation 323, 336 Walecka, J.D. 339, 368 Wiener–Hopf integral equation 226 Walker, R.L. 368 Wiener–Hopf sum equation 235 Wasylkiwskyj, W. 234, 367 supergain antennas 105, 107 Watson, K.M. 172, 365, 366 symmetric kernel 109 wave function bound state 143 time-ordered product 305, 314, 315 scattering state 143 Toeplitz matrix 227 vacuum 303 infinite 227 Wazwaz, A.M. 365 semi-infinite 229 weak minimum 284 transformation weak variation 283 gauge 343, 346, 347 Weierstrass E function 296 G global 342, 345 Weierstrass–Erdmann corner relation 297, U(1) global 340, 345 299, 300 G local 343, 346 Weinberg, S. 368, 369, 371 local phase 339 Weinstein, L.A. 246, 367 U(1) local 339–341 Weyl, H. 339, 368–370 transformation function 303 Wick, G.C. 364, 370 transmission coefficient 44 Wick–Cutkosky model 364, 370 trial action functional 312 Wiener, N. 366 trial potential 312 Wiener–Hopf triangular inequality 5–7, 76 integral equation 177, 191, 226, 366, Tricomi, F.G. 365 367 Trout, B.L.
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