Physical Mathematics Kevin Cahill Index More Information

Cambridge University Press 978-1-107-00521-1 - Physical Mathematics Kevin Cahill Index More information

Index

adjoint domain, 262 modified, 330–331, 341–343 analytic continuation, 179–180 Neumann functions, 341–343 dimensional regularization, 180 of the first kind, 143, 144, 325–341 analytic functions, 160–222 of the second kind, 341–345 branch of, 194 spherical, 145 definition of, 160 and partial waves, 338–340 entire, 161, 177 and quantum dots, 340–341 essential singularity, 177 Rayleigh’s formula for, 336 harmonic functions, 170–171 spherical Bessel functions of the second kind, holomorphic, 177 343–345 isolated singularity, 177 Bessel inequality, 284 meromorphic, 177 Bessel’s equation, 327 multivalued, 194 Bianchi identity, 413 pole, 177 binomial coefficient, 141 simple pole, 177 Bloch’s theorem, 105 angular momentum bodies falling in air, 247 lowering operators, 371 Boltzmann distribution, 54–55 raising operators, 371 Boltzmann’s constant, 149 spin, 371 boundary conditions annihilation and creation operators, 132, 233 Dirichlet, 262 arrays, 2–3 natural, 260, 268 associated Legendre functions, 317–323 Neumann, 260 Rodrigues’s formula for, 318 Bravais lattice, 104 associated Legendre polynomials, 317–323 Bromwich integral, 129 Rodrigues’s formula for, 318 asymptotic freedom, 237, 629–634 calculus of variations, 443–447 average value, 117 in nonrelativistic mechanics, 443–444 in relativistic electrodynamics, 445 basis, 7, 16 in relativistic mechanics, 444–445 beats, 77 particle in a gravitational field, 445–447 Bessel functions, 325–347 strings, 643–645 and charge near a membrane, 331–333 Callan–Symanzik equation, 630–634 and coaxial wave-guides, 342–343 canonical commutation relations, 132, 233 and cylindrical wave-guides, 333–335 Cartan subalgebras, 379 and scattering off a hard sphere, 344–345 Casimir effect, 214–217 exercises, 345–347 Cauchy’s principal value, 199 Hankel functions, 341–343 Feynman’s propagator, 201–205

656

INDEX

trick, 200 phase and group velocities, 208–210 chaos, 635–642 pole, 177 attractors, 639–642 residue, 176 limit cycle, 639 roots, 194 of fractal dimension, 642 simple pole, 177 strange, 642 singularities, 177–179 Bernoulli shift, 638 Taylor series, 171–173 chaotic threshold, 635 conformal mapping, 197–198 Duffing’s equation, 635 contractions, 416 dynamical system, 636 contravariant vector field, 401 autonomous, 636 contravariant vectors, 401 fractals, 639–642 convergence Cantor set, 640 of functions, 85 fractal dimension, 640 uniform, 85 Koch snowflake, 640 uniform self-similar dimension, 641 and term-by-term integration, 85 Hénon’s map, 638 convergence in the mean, 138–139 invertible map, 636, 637 convex function, 560 map, 636 convolutions, 121–124, 134 period-two sequence, 635 and Gauss’s law, 121–123 Poincaré surface of section, 636 and Green’s functions, 121–123 Rayleigh–Benard convection, 635 and translational invariance, 123 van der Pol’s equation, 639 coordinates, 400–401 characteristic function, 119 correlation functions and moments, 119 Glauber and Titulaer, 69–71 class Ck of functions, 85 cosmology, 289–291, 457–469 Clifford algebra, 393 , 462 commutators, 353 time evolution of, 461–463 compact, 351 comoving coordinates, 458 complex arithmetic, 2 cosmic microwave background radiation complex-variable theory, 160–222 (CMBR), 458, 469 Abel–Plana formula, 212–217 critical energy density, 458, 462 analytic continuation, 179–180 dark matter, 457 analyticity, 160–161 era of dark energy, 458 and string theory, 217–219 era of matter, 458 applications to string theory era of radiation, 457 radial order, 217 first three minutes, 457 argument principle, 178–179 homogeneous and isotropic calculus of residues, 180–182 line element, 458 Cauchy’s inequality, 173 Hubble constant, 458 Cauchy’s integral formula, 165–169 Hubble rate, 461 Cauchy’s integral theorem, 161–165 inflation, 457 and Stoke’s theorem, 169 models, 463–469 Cauchy’s principal value, 198–205 dark-energy dominated, 468 Cauchy–Riemann conditions, 169–170 equation of state, 464 conformal mapping, 197–198 inflation dominated, 465 contour integral with cut, 196 matter dominated, 466–468 cuts, 193–197 radiation dominated, 465–466 dispersion relations, 205–208 transparency, 468 essential singularity, 177 without acceleration, 464–465 and Picard’s theorem, 177 recombination, 468 exercises, 219–222 redshift, 463 fundamental theorem of algebra, 174 Robertson–Walker metric, 459–463 ghost contours, 182–191 energy–momentum tensor of, 461 harmonic functions, 170–171 Friedmann equations, 461 isolated singularity, 177 transparency, 458 Laurent series, 174–179 covariant derivatives Liouville’s theorem, 173–174 in Yang–Mills theory, 365 logarithms, 193–197 covariant vector field, 402 method of steepest descent, 210–212 covariant vectors, 402

657

INDEX

decuplet of baryon resonances, 646–647 complete, 274, 277–283 degenerate eigenvalue, 40 orthonormal delta function, 97–101, 112–115, 120, 130, 131, Gram–Schmidt procedure, 274 281–283, 287–288 eigenvalues, 267–283 and Green’s functions, 122 algebraic multiplicity, 40 Dirac comb, 98–101, 114–115 degenerate, 39 eigenfunction expansion of, 281–283, 287–288 geometric multiplicity, 40 for continuous square-integrable functions, nondegenerate, 39 112–115 simple, 40 for real periodic functions, 101, 103 unbounded, 275–283 for twice differentiable functions on [−1, 1], 310 eigenvectors, 36–55 of a function, 113 Einstein’s summation convention, 404–405 density operators, 54–55 electric constant, 154, 413 determinants, 27–33 electric displacement, 413 and antisymmetry, 28, 29 electric susceptibility, 154 and Levi–Civita symbol, 28 electrodynamics, 411–414 and linear dependence, 29 electrostatic energy, 154 and linear independence, 29 electrostatic potential and permutations, 30 multipole expansion, 286 and the inverse of a matrix, 31 electrostatics cofactors, 28 dielectrics, 153–157 invariances of, 28 emission rate from a fluorophore, 247 Laplace expansion, 28 energy–momentum 4-vector, 410 minors, 28 entropy, 54–55 product rule for, 32 euclidean coordinates, 402–404 3 × 3, 27 euclidean space, 402–404 2 × 2, 27 Ewald summation, 115 dielectrics, 154 expected value, 117 differential equations, 223–295 exterior derivative, 417–419 exercises, 293–295 terminal velocity factorials, 141–145 of mice, men, falcons, and bullets, 247 double, 143, 145 diffusion, 133–134 Mermin’s approximation, 141 Fick’s law, 133 Mermin’s infinite-product formula, 141 dimensional regularization, 180 Ramanujan’s approximation, 141 Dirac mass term, 395 Stirling’s approximation, 141 Dirac notation, 19–27, 96–101 Faraday’s law, 154, 412 and change of basis, 22 Feynman’s propagator, 120–121, 201–205 and inner-product rules, 20 as a Green’s function, 201 and self-adjoint linear operators, 23 field of charge near a membrane, 155–157 and the adjoint of an operator, 22–23 field of charge near dielectric interface, 154–157 bra, 19 forms, 416–419, 427–431, 479–501 bracket, 19 closed, 496–498 examples, 22 differential forms, 416–419, 481–501 ket, 19 p-forms, 417 outer products, 21 1-forms, 416 tricks, 51 2-forms, 416 Dirac’s delta function, 285 and exterior derivative, 417 Dirac’s gamma matrices, 393 and gauge theory, 471 direct product, 377 and Stokes’s theorem, 418 adding spins, 68 closed, 418 and hydrogen atom, 68 complex, 498 dispersion relations, 205–208 curl, 436 and causality, 205 exact, 418 Kramers and Kronig, 206–208 Hodge star, 439–440, 442 divergence, 228–230 invariance of, 416 division algebra, 380, 384 wedge product, 416 double-factorials, 143 exact, 496–498 exercises, 500–501 eigenfunctions, 267–283 exterior derivatives, 486–491

658

INDEX

exterior forms, 479–481 functional differential equation for ground Frobenius’s theorem, 498–500 state of a field theory, 583–585 integration, 491–496 functional differential equations, 583–585 Stokes’s theorem, 495 notation used in physics, 579–580 Fourier series, 75–107 of higher order, 581–582 and scalar fields, 132 Taylor series of, 582–583 better convergence of integrated series, 88 functionals, 578 complex, 75–79, 83–89, 96–107 functions for real functions, 76 analytic, 160–161 of nonperiodic functions, 78–89 differentiable, 160 convergence, 84–89 fundamental theorem of algebra, 174 convergence theorem, 85 exercises, 105–107 gamma function, 141–145, 179–180 Gibbs overshoot, 81–82 Gauss’s law, 154, 284, 412 nonrelativistic strings, 103 gaussian integrals, 586–588 of a C1 function, 87 Gell-Mann’s SU(3) matrices, 377 of nonperiodic functions, 78–89 general relativity, 289–291, 445–469 Parseval’s identity, 100 black holes, 456–457 periodic boundary conditions, 103–105 cosmological constant, 454 Born–von Karman, 105 cosmology, 457–469 poorer convergence of differentiated series, 89 dark energy, 454 quantum-mechanical examples, 89–96 Einstein’s equations, 453–456 real functions, 79–82 Einstein–Hilbert action, 454–455 Gibbs overshoot, 81 model cosmologies, 463–469 dark-energy dominated, 468 several variables, 84 equation of state, 464 stretched intervals, 83–84 inflation dominated, 465 the interval, 77 matter dominated, 466–468 where to put the 2πs, 77 radiation dominated, 465–466 Fourier transforms, 108–125, 129–134 transparency, 468 and Ampère’s law, 123 without acceleration, 464–465 and characteristic functions, 119 Schwarzschild’s solution, 456–457 and convolutions, 121–124 static and isotropic gravitational field, 455–457 and differential equations, 129–134 Schwarzschild’s solution, 456–457 and diffusion equation, 133–134 standard form, 455–456 and Fourier series, 110, 120 geometric series, 139–140 and Green’s functions, 121–125 gradient, 228–230 and momentum space, 116–119 grand unification, 377 and Parseval’s relation, 113 Grassmann numbers, 2, 6–7 and scalar wave equation, 131 Grassmann polynomials, 2 and the delta function, 112–115 Grassmann variables, 613–619 and the Feynman propagator, 120–121 Green’s function and the uncertainty principle, 117–119 for Helmholtz’s equation, 286 derivatives of, 115–119 for Helmholtz’s modified equation, 286 exercises, 134–135 for Laplacian, 123 in several dimensions, 119–121 for Poisson’s equation integrals of, 115–119 and Legendre functions, 287 inverse of, 109 Green’s functions, 284–289 of a gaussian, 110–111, 183 and eigenfunctions, 287–289 of real functions, 111–112 Feynman’s propagator, 287 Fourier–Legendre expansion, 310 for Helmholtz’s equation, 285–286 Fourier–Mellin integral, 129 for Poisson’s equation, 284–287 function of a self-adjoint operator, 288 f (x ± 0), 85 Poisson’s equation, 285 continuous, 85 group index of refraction, 209 piecewise continuous, 85 groups, 348–399 functional derivatives, 578–585 O(n), 349 and delta functions, 579–580 SO(3) and variational methods, 579–582 adjoint representation of, 366–367 exercises, 585 SO(n), 349

659

INDEX

SU(2), 368–376 hermitian generators of, 363 defining representation of, 371–374 Jacobi identity, 374 spin and statistics, 370 noncompact, 361–364 tensor products of representations, 369 nonhermitian generators of, 363 SU(3), 377–379 of rotations, 366–374 SU(3) structure constants, 378 simple and semisimple, 376–377 Z2, 358 structure constants of, 363, 374–375 Z3, 358 SU(2) tensor product, 372 Zn, 358 SU(3) generators, 377 abelian, 349 symplectic group Sp(2n,R), 382 and symmetries in quantum mechanics, 351 symplectic groups, 381–383 and Yang–Mills gauge theory, 365 Lie groups have structure constants that are automorphism, 355 independent of the representation, 365 inner, 355 Lorentz, 349 outer, 355 Lorentz group, 386–396 block-diagonal representations of, 351 Dirac representation of, 393–395 centers of, 353 Lie algebra of, 386–389 characters, 356–357 two-dimensional representations of, 389–393 compact, 350–351 matrix, 349 compact Lie groups morphism, 354 real structure constants of, 364 noncompact, 350 totally antisymmetric structure constants of, nonabelian, 349 364 of matrices, 349–350 completely reducible representations of, 351 of orthogonal matrices, 349–350 conjugacy classes of, 353 of permutations, 360–361 continuous, 348–350 of rotations, 366–374 definition, 348 adjoint representation of, 366–367 direct sum of representations of, 351 explicit 3 × 3 representation of, 367 equivalent representations of, 351 generators of, 366–367 exercises, 396–399 spin and statistics, 370 factor groups of by subgroups, 354 tensor operators of, 376 finite, 350, 358–361 of transformations, 348–350 multiplication table, 358 Lorentz, 348 regular representation of, 359 Poincaré, 348 further reading, 396 rotations and reflections, 348 Gell-Mann matrices, 377–379 translations, 348 generators of adjoint representations of, of unitary matrices, 349–350 374–375 order of, 349, 358 invariant integration, 384–385 Poincaré, 349 irreducible representations of, 351 Poincaré group, 395–396 isomorphism, 354 Lie algebra of, 395–396 Lie algebras, 361–399 reducible representations of, 351 SU(2), 368–374 representations of, 350–352 Lie groups, 348–350, 361–399 dimensions of, 350 SO(3), 366–367 in Hilbert space, 351–352 SU(3), 377–379 rotations SU(3) structure constants, 378 representations of, 352 adjoint representations of, 374–375 Schur’s lemma, 355–356 antisymmetry of structure constants, 363 semisimple, 353 Cartan subalgebra of SU(3), 378 similarity transformations, 351 Cartan subalgebras of, 379 simple, 353 Cartan’s list of compact simple Lie groups, simple and semisimple Lie algebras, 376–377 383–384 subgroups, 353–355 Casimir operators of, 369, 375 cosets of, 354 compact, 361–385 invariant, 353 defining representation of SU(2), 371–374 left cosets of, 354 definition of, 361 normal, 353 exponential parametrization, 362 quotient coset space, 354 generators of, 362 right cosets of, 354 generators of adjoint representation, 374 trivial, 353

660

INDEX

symmetry Fourier series, 146 antilinear, antiunitary representations of, 352 geometric series, 139–140 linear, unitary representations of, 352 power series, 139–140, 146 tensor product Taylor series, 145–149 addition of angular momenta, 357 uniform convergence, 139 tensor products, 357–358 Riemann zeta function, 149 translation, 348 uniform convergence, 138 unitary representations of, 351 and term-by-term integration, 139 inner products, 3, 11–14 harmonic function, 170 and distance, 12 harmonic oscillator, 101–102, 272–273 and norm, 12 Heaviside step function, 106 degenerate, 12 helicity hermitian, 12 positive, 393 indefinite, 12, 14 right-handed, 393 Minkowski, 14 Helmholtz’s equation nondegenerate, 12 in cylindrical coordinates of functions, 13 and Bessel functions, 328–335 positive definite, 11 in spherical coordinates Schwarz, 11 and spherical Bessel functions, 335–341 inner-product spaces, 13–14 in three dimensions, 231–232 integral equations, 296–304 in two dimensions, 230–231 exercises, 304 rectangular coordinates, 231 Fredholm, 297–301 spherical coordinates, 232 eigenfunctions, 297–301 and associated Legendre functions, 317 eigenvalues, 297–301 and spherical Bessel functions, 317 first kind, 297 with azimuthal symmetry, 315–316 homogeneous, 297 and Legendre polynomials, 315–316 inhomogeneous, 297 and spherical Bessel functions, 315–316 second kind, 297 Hermite functions, 264 implications of linearity, 298–304 Hermite’s system, 264 integral transformations, 301–304 hermitian differential operators, 261 and Bessel functions, 302–304 Hilbert spaces, 13–14, 25–26 Fourier, Laplace, and Euler kernels, 302 homogeneous functions, 243–245 kernel, 297 Euler’s theorem, 243 numerical solutions, 299–301 virial theorem, 243–244 Volterra, 297–301 eigenfunctions, 297–301 index of refraction, 207 eigenvalues, 297–301 infinite products, 157–158 first kind, 297 infinite series, 136–159 homogeneous, 297 absolute convergence, 136 inhomogeneous, 297 asymptotic, 152–153 second kind, 297 WKB & Dyson, 153 integral transformations, 301–304 Bernoulli numbers and polynomials, 151–152 and Bessel functions, 302–304 binomial series, 148 Fourier, Laplace, and Euler kernels, 302 binomial theorem, 147, 148 invariant distance, 409 Cauchy’s criterion, 137 invariant subspace, 37 Cauchy’s root test, 137 comparison test, 137 Jacobi identity, 374 conditional convergence, 136 convergence, 136–140 kernel of a matrix, 355 d’Alembert’s ratio test, 138 Kramers–Kronig relations, 206–208 divergence of, 136 Kronecker delta, 5, 415 exercises, 158–159 Intel test, 138 Lagrange multipliers, 35–36, 54–55, 267–283 logarithmic series, 148–149 Lapack, 33, 66 of functions Laplace transforms, 125–134 convergence, 138–139 and convolutions, 134 convergence in the mean, 138–139 and differential equations, 128–134 Dirichlet series, 149–151 derivatives of, 127–128

661

INDEX

examples, 125 Matlab, 33, 61, 66 integrals of, 127 matrices, 4–7 inversion of, 129 adjoint, 4 Laplace’s equation and linear operators, 9 in two dimensions, 316–317 change of basis, 10 Laplacian, 228–230 characteristic equation of, 38, 41–42 Legendre functions, 263–264, 287, 305–324 CKM, 63 exercises, 323–324 and CP violation, 63 second kind, 266 congruency transformation, 49 Legendre polynomials, 305–313 defective, 40 addition theorem for, 321 density operator, 69 generating function, 307–309 diagonal form of square nondefective matrix, Helmholtz’s equation 41 with azimuthal symmetry, 315–316 functions of, 43–45 Legendre’s differential equation, 309–311 gamma, 393 normalization, 305 hermitian, 5, 45–49 recurrence relations, 311–312 and diagonalization by a unitary Rodrigues’s formula, 306–307 transformation, 48 Schlaefli’s integral for, 312–313 complete and orthonormal eigenvectors, 47 special values of, 312 degenerate eigenvalues, 46 Legendre’s system, 263–264 eigenvalues of, 45 Leibniz’s rule, 141 eigenvectors and eigenvalues of, 48 Lerch transcendent, 151 eigenvectors of, 46 Levi-Civita symbol, 366 identity, 5 Lie algebras imaginary and antisymmetric, 48 ranks of, 379 inverse, 5 roots of, 379 inverses of, 31 weight vector, 379 nonnegative, 6 weights of, 379 nonsingular, 40 light normal, 50–55 slow, fast, and backwards, 209–210 compatible, 52–55 linear algebra, 1–74 diagonalization by a unitary transformation, exercises, 71–74 50 linear dependence, 15–16, 224 orthogonal, 5, 25 and determinants, 29 Pauli, 5, 68, 371 linear independence, 15–16, 224 positive, 6 and completeness, 15 positive definite, 6 and determinants, 29 rank of, 65 linear least squares, 34–35 example, 65 linear operators, 9–11 rank-nullity theorem, 58 and matrices, 9 real and symmetric, 48 density operator, 69 similarity transformation, 10, 41 domain, 9 singular-value decomposition, 55–63 hermitian, 23 example, 62 range, 9 quark mass matrix, 62 real, symmetric, 23–24 square, 38–42 self-adjoint, 23 eigenvalues of, 38–42 unitary, 24–25 eigenvectors of, 38–42 Lorentz force, 414 trace, 4 Lorentz transformations, 405–411 cyclic, 4 boost, 406 unitary, 5 invariance under, 406 upper triangular, 33 Maxwell’s equations, 412 magnetic constant, 413 in vacuum, 413 magnetic field, 413 Maxwell–Ampère law, 412 magnetic induction, 411 mean value, 117 Majorana field, 235 method of steepest descent, 210–212 Majorana mass term, 391, 393 metric spaces, 13–14 Maple, 33, 66 Minkowski space, 405–407 Mathematica, 33, 66 Monte Carlo methods, 563–577

662

INDEX

and evolution, 576–577 integration, 240–242 and lattice gauge theory, 574, 577 van der Waals’s equation, 239 detailed balance, 574 first-order, 235–248 exercises, 577 exact, 238–242 genetic algorithms, 577 separable, 235–238 in statistical mechanics, 572–575 self-adjoint, 266–267 Metropolis step, 572 Frobenius’s series solutions, 251–254 Metropolis’s algorithm, 572–575 Fuch’s theorem, 253–254 more general applications, 575–577 indicial equation, 252 of data analysis, 566–572 recurrence relations, 252 example, 566–572 Green’s formula, 260 of numerical integration, 563–566 hermitian operators, 267 partition function, 574 homogeneous first-order, 245 smart schemes, 575 homogeneous functions, 243–245 sweeps, 573 Lagrange’s identity, 260 thermalization, 573 linear, 223–225 Moore–Penrose pseudoinverse, 63–65 general solution, 224, 225 homogeneous, 224 natural units, 121 inhomogeneous, 224, 225 nearest-integer function, 108 order of, 223 nonlinear differential equations, 289–293 linear dependence of solutions, 224 general relativity, 289–291 linear independence of solutions, 224 solitons, 291–293 linear, first-order, 246–248 notation for derivatives, 226–228 exact, 246 null space of a matrix, 355 integrating factor, 246 numbers meaning of exactness, 240–242 complex, 1, 7 nonlinear, 225 atan2, 2 second-order phase of, 2 eigenfunctions, 273–275 irrational, 1 eigenvalues, 273–275 natural, 1 essential singularity of, 251 rational, 1 Green’s functions, 288 real, 1 irregular singular point of, 251 making operators self adjoint, 264–265 Octave, 33, 66 nonessential singular point of, 251 octet of baryons, 379 regular singular point of, 251 octet of pseudo-scalar mesons, 378 second solution, 255–257 octonians, 384 self-adjoint, 260–283 open, 351 self-adjoint form, 223 operators singular points at infinity, 251 adjoint of, 22–23 singular points of, 250–251 antilinear, 26–27 weight function, 273 antiunitary, 26–27 why not three solutions?, 257–258 compatible, 352 wronskians of self-adjoint operators, complete, 352 265–266 orthogonal, 25 self-adjoint, 260–283 real, symmetric, 23–24 self-adjoint operators, 265 unitary, 24–25 separable, 235–238 optical theorem, 207 general integral, 236 ordinary differential equations, 223–225 hidden separability, 238 and variational problems, 259–260 logistic equation, 236 boundary conditions, 258–260 Zipf’s law, 236 differential operators of definite parity, 255 separated, 235 even and odd differential operators, 254–255 singular points of Legendre’s equation, 251 exact, 238–242 Sturm–Liouville problem, 265, 267–283 Boyle’s law, 239 systems of, 248–250, 289–293 condition of integrability, 239 Friedmann’s equations, 289–291 Einstein’s law, 239 Lagrange’s equations, 248–250 human population growth, 239 Wronski’s determinant, 255–258 integrating factors, 242 orthogonal coordinates, 228–230

663

INDEX

divergence in, 228–230 slow, fast, and backwards light, 209 gradient in, 228–230 Planck’s constant, 149 Laplacian in, 228–230 Planck’s distribution, 149–151 orthogonal polynomials, 313–315 points, 400–401 Hermite’s, 314 Poisson summation, 114–115 Jacobi’s, 313–314 power series, 139–140 Laguerre’s, 314–315 pre-Hilbert spaces, 13–14 outer products, 18–22 principle of least action, 259–260 example, 18 principle of stationary action, 248–250, 267–273, in Dirac’s notation, 21 443–447 in nonrelativistic mechanics, 328, 443–444 partial differential equations, 225–235 in quantum mechanics, 267–273 Dirac equation, 234–235 in relativistic electrodynamics, 445 general solution, 226 in relativistic mechanics, 444–445 homogeneous, 226 particle in a gravitational field, 445–447 inhomogeneous, 226 probability and statistics, 502–562 Klein–Gordon equation, 233 Bayes’s theorem, 502–505 linear, 225–235 Bernoulli’s distribution, 508 separable, 230–235 binomial distribution, 508–511 Helmholtz’s equation, 230–232 brownian motion, 520–527 wave equations, 233–235 Einstein–Nernst relation, 520–524 photon, 233–234 Langevin’s theory of, 520–527 spin-one-half fields, 234–235 Cauchy distributions, 532 spinless bosons, 233 central limit theorem, 532–543 path integrals, 586–625 illustrations of, 535–543 and gaussian integrals, 586–588 central moments, 505–508 and lattice gauge theories, 622 centroid method, 514 and nonabelian gauge theories, 619–624 characteristic functions, 527–530 ghosts, 622–624 chi-squared distribution, 531 the method of Faddeev and Popov, 620–624 chi-squared statistic, 551–554 and perturbative field theory, 605–624 convergence in probability, 535 and quantum electrodynamics, 609–613 correlation coefficient, 507 and Schrödinger’s equation, 592–593 covariance, 507 and the Bohm–Aharonov effect, 594–595 lower bound of Cramér and Rao, 546–550 and the principle of stationary action, 591–592 cumulants, 529 euclidean, 588–590 diffusion, 519–527 euclidean correlation functions, 599–600 diffusion constant, 523 exercises, 624–625 direct stochastic optical reconstruction fermionic, 613–619 microscopy, 515 finite temperature, 588–590 Einstein–Nernst relation, 520–524 for a free particle in imaginary time, 595 Einstein’s relation, 523 for a free particle in real time, 593–595 ensemble average, 521 for harmonic oscillator in imaginary time, error function, 515–518 597–598 estimators, 543–550 for harmonic oscillator in real time, 595–597 Bessel’s correction, 545 in field theory, 603–624 bias, 543 in finite-temperature field theory, 600–603 consistent, 543 in imaginary time, 588–590 standard deviation, 546 in real time, 590–593 standard error, 546 Minkowski, 590–593 exercises, 560–562 of fields, 600–624 expectation, 505–508 of fields in euclidean space, 600–603 expected value, 505–508 of fields in imaginary time, 600–603 exponential distribution, 531 ratios of and time-ordered products, 604–624 fat tails, 530–532 Pauli matrices, 371 Fisher’s information matrix, 546–550 permittivity, 154, 413 fluctuation and dissipation, 524–527 permutations, 360–361 gaussian distribution, 512–519 and determinants, 30 Gosset’s distribution, 530 cycles, 360 Heisenberg’s uncertainty principle, 506–507 phase and group velocities, 208–210 information, 546–550

664

INDEX

information matrix, 546–550 small oscillations, 250 Kolmogorov’s function, 556 solitons, 291–293 Kolmogorov’s test, 554–560 special relativity, 408–414 kurtosis, 530 4-vector force, 410 Lévy distributions, 532 and time dilation, 410–411 Lindeberg’s condition, 535 electrodynamics, 411–414 log-normal distribution, 531 energy–momentum 4-vector, 410 Lorentz distributions, 532 kinematics, 410–411 maximum likelihood, 550–551 spherical harmonics, 319–323 Maxwell–Boltzmann distribution, 518–519 spin and statistics, 370 mean, 505–508 standard model of particle physics, 377 moment-generating functions, 527–530 states moments, 505–508 coherent, 71 normal distribution, 514 Stefan’s constant, 150 Pearson’s distribution, 531 Stokes’s theorem, 170 Poisson distribution, 511–512 strings, 643–650 coherent states, 512 and infinities of quantum field theory, 643 power-law tails, 530 Dirichlet boundary condition, 645 probability density, 505–519 free-endpoint boundary condition, 645 probability distribution, 505–519 Nambu–Goto action, 643–647 random-number generators, 537–538 quantized, 647–650 skewness, 530 D-branes, 647–648 Student’s t-distribution, 530 Regge trajectories, 646–647 variance, 505–508 Riemann surfaces and moduli, 649–650 lower bound of Cramér and Rao, 546–550 scattering of, 648 viscous-friction coefficient, 523 Sturm–Liouville equation, 265, 267–283 proper time, 409 SU(3) and quarks, 378–379 and time dilation, 410–411 subspace, 351 pseudoinverse, 63–65, 551 invariant, 351 proper, 351 quantum mechanics, 267–283 summation convention, 404–405 quaternions, 379–383 symmetric differential operators, 261 and the Pauli matrices, 380 symmetry in quantum mechanics, 26 R-C circuit, 247 systems of linear equations, 34–35 regular and self-adjoint differential system, 262 relative permittivity, 154 Taylor series, 145–149 renormalization group, 237, 626–634 tensor products, 377 exercises, 634 adding angular momenta, 357–358, 371–374 in condensed-matter physics, 632–634 adding spins, 68, 371–374 in lattice field theory, 630–632 and hydrogen atom, 68 in quantum field theory, 626–634 tensors, 400–478 rotations, 366–374 affine connections, 431–433 and metric tensor, 436–437 scalar fields, 131, 401 and general relativity, 445–469 scalars, 401 antisymmetric, 415 Schwarz inequality, 14–15, 284 basic axiom of relativity, 422 examples, 14 basis vectors, 421 Schwarz inner products, 69–71 Bianchi identity, 436 seesaw mechanism, 49 Christoffel symbols, 431–433 self-adjoint differential operators, 260–283 connections, 431–433 self-adjoint differential systems, 262–283 contractions, 416 sequence of functions contravariant metric tensor, 422 convergence in the mean, 88 covariant curl, 434–436 simple and semisimple Lie algebras, 376–377 covariant derivatives, 431–434 simple and semisimple Lie groups, 376–377 and antisymmetry, 436 simple eigenvalue, 40 metric tensor, 437–438 simply connected, 164 covariant metric tensor, 422 slow, fast, and backwards light, 209 curvature, 451–453 and Kramers–Kronig relations, 210 curvature of a sphere, 451–453

665

INDEX

curvature scalar, 451 variance cylindrical coordinates, 425 of an operator, 118–119 divergence of a contravariant vector, 438–443 variational methods, 248–250, 259–260, 267–273, and Hodge star, 439–440 443–447 and Laplacian, 441–443 in nonrelativistic mechanics, 328, 443–444 Einstein’s equations, 453–456 in quantum mechanics, 267–273 exercises, 475–478 in relativistic electrodynamics, 445 gauge theory, 469–475 in relativistic mechanics, 444–445 geometry of, 474–475 particle in a gravitational field, 445–447 role of vectors, 471–473 strings, 643–645 standard model, 469–473 vector space, 16 gradient, 426–427 dimension of, 16 Hodge star, 428–431 vectors, 7–8 and divergence, 428 basis, 7, 16, 17 and Laplacian, 428 complete, 16 and Maxwell’s equations, 430–431 components, 7 Laplacian, 441–443 direct product, 66–68 and Hodge star, 442 example, 68 Levi-Civita’s symbol, 427–431 eigenvalues, 37–55 Levi-Civita’s tensor, 427–431 example, 37 metric of sphere, 421 eigenvectors, 37–55 metric tensor, 420–427 example, 37 moving frame, 421 notation for derivatives, 433 eigenvectors of square matrix, 40 orthogonal coordinates, 423–426 functions as, 8 parallel transport, 433 orthonormal, 16–17 particle in a weak, static gravitational field, Gram–Schmidt method, 16–17 448–451 partial derivatives as, 8 gravitational redshift, 450 span, 16 gravitational time dilation, 449–450 span a space, 16 perfect fluid, 453 states as, 8 polar coordinates, 424 tensor product, 66–68 principle of equivalence, 447–448 Virasoro’s algebra, 219, 222 geodesic equation, 448 virial theorem, 243–244 quotient theorem, 420 raising, lowering indices, 423 wedge product, 416 Ricci tensor, 451 Weyl spinor Riemann tensor, 451 left-handed, 390 second-rank, 414–416 right-handed, 393 spherical coordinates, 425–426 Wigner–Eckart theorem symmetric, 415 special case of, 355–356 tensor equations, 419–420 wronskian, 255–258 torsion tensor, 433 third-harmonic microscopy, 184 Yang–Mills theory, 469–475 time dependence of Heisenberg operators, 604 geometry of, 474–475 time dilation, 409–410 in muon decay, 409–410 role of vectors, 471–473 time-ordered product, 121, 604 standard model, 469–473 total cross-section, 207 Yukawa potential, 125, 286

uncertainty principle, 117–119, 244 zeta function, 149–151

666