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Home , Electromagnetic tensor, Pseudotensor

Answers and Hints to Selected Exercises

EXERCISES 1.1 1 (ii).

EXERCISES 1.3

2 (i). Hint: Note that iij == (lj2)(iij + iji) + (lj2)(iij - i j ;).

EXERCISES 2.1

2 . .?li(t) = ± sin(t - ci).

EXERCISES 2.2 1. S(y) = 2.

EXERCISES 2.4

EXERCISES 3.2

2. Hint: Note that the 4 x 4 identity does not belong to this subset.

EXERCISES 3.4

3(') H" a a a a a a I. Illt: ax = au + au' at = au - au'

EXERCISES 4.1 1. Hint: Assuming xa = _xa prove that UPU t + p = 0, or U

202 Answers and Hints to Selected Exercises 203

EXERCISES 4.2 l(i). Hint: Compare the problem 2(ii) of Exercises 1.3.

EXERCISES 4.3

EXERCISES 5.1 1. H(8,Pe) = (pN2mt2) + mgt(l - cos 8).

EXERCISES 5.3

EXERCISES 5.4

. oA(x, u) OXk(X) oA(x, a) b0 2 Xk(X) 0 A(x, U) 2. Hmt: 8xi = ~ 8Xk + u oxboXi oak '

8A(x, u) 8Xk (x) 8A(x, a) ou i - -Oxi · 8a k •

EXERCISES 6.1

EXERCISES 6.2

. 81jJ 8h 01jJ [. oh Oh] 4. Hmt: 8x1 = (cosO() ow' 8x2 = (smO() 8w + o~ ,

81jJ .[8h . Oh] 81jJ 8h 8x3 = I aw + (sm O() o~ , ox4 = (cos ct) 8~'

EXERCISES 6.3 2. Hint: Use special Minkowski coordinates such that

F34(X) = F12 (x) == O. Index of Symbols

d(F) functional of a relativistic field Ak(X) potential a . b inner product between two vectors A x B Cartesian product of two sets M x N Kronecker product of two matrices BAi(X) gauge potential components riZ(D; IR); ri2 (D; S) the set of twice differentiable functions from D into IR, S respectively C A BD of a Lie algebra x(p) = x = (xl, X2, X3, x 4 ) local coordinates of a point p in the (X, U) chart in a differentiable manifold D = [diJ == diag[l, 1, 1, -1] Lorentz metric; a domain D c [Rn aD boundary of a domain D D d'Alembertian or wave operator, also denotes the completion of an example • Q.E.D. Dat/lK(x) gauge t5(k - k') Dirac delta function t5 a b Ad) perihelion shift per revolution 8(x l , x2 , x3 , x 4 )/8(x 1, x 2 , x 3, x 4 ) Jacobian of a coordinate transformation 1E3 three-dimensional Euclidean space {ei}i == {e 1,eZ,e3,e4} a set or tetrad ei ' ej = dij an M-orthonormal tetrad E(p) frequency associated with Dirac plane wave E belongs to Gijkl totally anti symmetric permutation symbol (Levi-Civita) '1ijkl totally antisymmetric pseudotensor (Levi-Civita) Fi(x, u) four-force components Fij electromagnetic components (field) Fi1 Hodge dual of electromagnetic tensor components FAij(X) gauge field components

204 Index of Symbols 205 t/JA (field) components G a group glj components gij(X, u) Finsler metric components 1': Xi = ,2"i(t) a parameterized curve on the manifold yk Dirac matrices yku v entries of Dirac matrices, also numerical bispinor-tensor components Yab(X) linearized gravitational ten-potential components H(t, x, p) (prerelativistic) Hamiltonian function 1= [c5 iJ the identity matrix J [I'] prerelativistic action functional Jh[f.] prerelativistic canonical action functional JL[y] relativistic action functional JH[y] relativistic canonical action functional ,1iab(X) relativistic lex) charge-current four-vector density Jab the total relativistic angular momentum tensor L, L == [[iJ a linear (or Lorentz) mapping and the corresponding matrix L T the transposed matrix L(t, x, v) prerelativistic Lagrangian L'(t, x, t', x') prerelativisitc Lagrangian in space and time ~(X,yA,yAi) the Lagrangian ofa field A. a scalar (a real or complex number) A(x, u) relativistic Lagrangian of a particle ~4 general Lorentz group ~4++ proper, orthochronous subgroup of ~4 M4 (flat) -time manifold .;v"o Null cone with vertex at Xo v(k) frequency associated with electromagnetic plane wave w(k) frequency associated with Klein-Gordon plane wave !lex, p) = 0 constraining mass hypersurface, Super-Hamiltonian P space reflection mapping Pa the total four-momentum vector Pk four-momentum components 9i'4 Poincare group f2 the total charge IR the set ofreal numbers IR" == IR x ... x IR (n factors) Cartesian product of the set IR S(y) separation functional on the curve I' SU(N) the set of unitary unimodular matrices u(v) separation number of the vector v u~ Pauli matrices U kAB numerical spinor-tensor "'(x) Klein-Gordon field "'", "'"(x) bispinor, Dirac field components 206 Index of Symbols

1: 1 ,1:2 ,1:3 I-flat, 2-flat, 3-flat submanifolds T time reversal mapping Tij(x) canonical energy-momentum-stress tensor density rJ; :::~: Minkowski tensor components rj;:::~:(x) Minkowski components BiAx) symmetrized energy-momentum-stress tensor density u, Ui covariant vector (or covector) and components ui(s) four-velocity components Vex) the potential energy V 4 Minkowski v, Vi Minkowski vector, components z == Xl + ix2 a complex variable z == Xl - ix2 the complex conjugate variable 7L the set of integers 7L+ the set of positive integers Subject Index

abelian group (see group) bilinear mappings 14 acceleration four-vector 103 bispinors (Dirac) 85,86,152,153 action functional (or integral) 90, 93, bispinor-tensor 86 97,109,111 numerical bispinor-tensor 86 pre-relativistic mechanics 90 bispinor field 152, 153 canonical, prereIativistic mechanics Dirac equation 153 93 (see also and Dirac) canonical, prerelativistic mechanics in boost mapping 12,51 space and time 97 transformation 51 relativistic mechanics 109 (see also Lorentz) canonical, relativistic mechanics 111 relativistic fields 120, 171 algebra, Lie (see Lie algebra) canonical equations angle 3, 12, 64, 76 (see Hamilton) Eulerian 72 Cartesian coordinates 25, 48, 89 pseudoangle 64 characteristic coordinates 70 angular momentum 99, 130, 132 equation 2 relativistic angular momentum tensor (see also null coordinates) density 130 chart 20,21 spin and orbital tensor density 130 (see also differentiable manifold) total relativistic angular momentum charge (electric) 106, 135 tensor 132 charge- 130 arclength 27 Dirac field 158 (see also separation) Klein-Gordon field 135, 175 area 80, 117 charged particle (see particle) areal velocity 117 total charge 132 atlas (see differentiable manifold) chronometry, Minkowski 33, 34 axioms 1, 8, 27, 56 clock, apparent retardation 53 group 56 standard 28 linear mapping 8 commutation relations 147, 148 Minkowski vector space 1,2 complex numbers 72, 73, 123, nondegeneracy 2,27 147 relativistic equations of motion 103 conformal mappings 73 conjugation 72 holomorphic functions 74,147 basis set of Minkowski vectors (or tetrad) matrices 75, 77, 78 2,3,8 vector spaces 79, 85 spinor vectors (dyad) 79 wave fields (see wave fields)

207 208 Subject Index conservation equations 130, 131, 132, Hodge dual tensor 18, 19, 80, 142, 176 146, 147 differential, angular momentum 129 dynamics (see mechanics) differential, charge-current 130 differential, energy-momentum-stress 128 eigenvalues 2, 108, 147 total, angular momentum 132, 176 metric tensor 2 total, charge 132, 176 energy-momentum-stress tensor 147 total, four-momentum 132, 176 Einstein ix, 71,113, 118 (see also Noether) 118 contraction, moving rod (see Lorentz• linearized gravity 113 Fitzgerald) philosophy ix (see tensors) summation convention 2, 3, 79, 85, contravariant vectors (see vectors) 147, 163, 169 coordinates 20,22,25, 115, 123 electric 106, 108, 112 Cartesian 25 charge (see charge, electric) charts (general) 20,21 field (see electromagnetic field) complex conjugate 72,123 electromagnetic field 18, 140, 141 ignorable 94,116 complex version 142, 146 Minkowski 25, 26, 33, 34 energy-momentum-stress tensor 19, transformation (see transformation) 145, 146, 147 constraints (or constraining hyper• 107, 113, 141 surface) 93, 97, 111 four-potential 113, 142, 143, 161, 162 (see also Lagrangian) 107, 108, 113, 141 curve, parameterized 22, 27, 28, 29, 30, Maxwell's equations 141 33,34 (see also Lorentz equations of motion, arclength 27 photon) separation 27, 28, 29, 30, 33, 34 electroweak theory 166 (see also motion curve, equations of elementary particles (see relativistic motion) fields) energy 89,92,96,105,132,174 hypersurface (or shell) 92,94, 100 diagonal 2,61,63 relative 105 Lorentz metric 2 rest 105 block diagonal matrix 61, 63 total 92, 132, 174 d'Alembertian operator 47, 123, 124, energy-momentum-stress tensor 128, 132,133 171,175 differentiable manifold 20,21,22,25, gauge field 150 115 Klein-Gordon field 135, 175 atlas 21 electromagnetic field 145, 146, 147 chart (or local ) 20, Dirac (bispinor) field 158 21,22,25, 115 equations of motion 89, 103, 106 Hausdorff topology 20 Newtonian (or prerelativistic) 89 projection mappings 20, 22 relativistic 103 subatlas 21 Lorentz 106 Dirac 85, 137, 151, 192 (see also Euler-Lagrange equations) delta function 137 Euclidean space (or geometry) 48, 72, equation, wave or field 151,153,160, 95 192 plane 7, 67, 73 matrices 86, 153, 154, 198 orthogonal 7 bispinors (see bispinors) Euler 63, 64, 72 dual 13,18,19,80,142,146,147 angles 63, 64, 72 vector (covariant vector) 13, theorem on homogeneous functions 80 96,109 Subject Index 209

Euler-Lagrange equations 90,93,95, (see also differentiable manifold) 97, 111, 123, 171 98, 113 events 25, 26 Newtonian constant 98, 115 future 27 linearized theory 113, 118 past 27 linearized Schwarzschild field 115 present 27 planetary orbits 98, 115 separation 27, 28 Greek indices 4 simultaneous 52 Green's theorem (see Gauss's theorem) group 55,56,63,66,147,163 Lie group 63, 66, 70 field 1, 120, 168, 191 Lorentz group (see Lorentz) bispinor (or Dirac field) 153,160,191 Poincare group 42, 56, 170 complex numbers 72, 73, 123 manifold 66, 147 gauge field 147, 163 representation 59, 61, 149, 163 real numbers I SU (N) 163 real scalar field 43, 47, 123 subgroup 56, 57, 58, 59 tensor field 43,47, 120, 142, 170 symmetric 56 (see also relativistic fields) unitary, unimodular 74, 77, 78, 148, Finsler metric 108, 109, 110 151,164,166 flat 20, 35, 41 (see also unitary, unimodular space-time (Minkowski) 20 conditions) submanifolds 35 I-flat 35 2-flat 35,36,37,38,39,41 Hamiltonian mechanics 91, 92, 93, 97, 3-flat 35, 37, 41 110, 112, 119 force 48, 89, 103 prerelativistic canonical equations four-force vector 103, 104, 106 92,98 monogenic 89 relativistic canonical equations 112 relative three-force vector 104 hermitian 74, 75,77,82, 153 three-force vector 89 conjugation 74, 153 frames of reference (see inertial frames) matrix 75, 77, 82 frequency (with wave) 135 history of a particle 26 functions (see mappings) Hodge dual 18, 19,80, 142 functionals 27,90,93,97, 109, 111, 120, pseudotensor 18, 19, 142, 146, 147 171 spinor 80 action (see action) homogeneous function 96, 97, 109, 110 separation 27 hypersurface (see constraints)

Galilean transformation 50 inertial observer (or frame) 26, 33,48, gauge fields (nonabelian) 147, 149, 163 52,53 energy-momentum-stress tensor 150, infinitesimal transformation 126, 128, 151 129,130,171,172 Yasskin class 151 inner product (or ) 1,2,3,5, gauge transformations 142, 143, 160, 7,8, 125,80, 148, 169 164 integral (Riemann) 45, 135, 138, 146, covariant derivative 161,166 200 global 130 Cauchy principal value 137, 138 local 142, 164 four-dimensional 45 Lorentz gauge condition 143, 147 improper 135,137,146,156,177, Gauss's theorem 46, 121, 131, 173 185,200 general relativity 118 conserved quantities (see geometry 20, 25, 108 conservation) Euclidean (see Euclidean) intersection of curves 30, 31, 35 210 Subject Index invariant (or scalar) 14,43, 108, 124, Lorentz equations of motion 106 127,171 Lorentz metric 2 invariance of action 108, 124, 171 gauge condition 143,147 inverse mapping 9,21,100,117 group 55,58 invertible linear mapping 9 general 58 isomorphic vector spaces 23 manifold 66 proper 58 proper and orthochronous 58, 59, Jacobi 21,48, 100, 148 63 identity 148 orthochronous 58 Jacobian 21,24,41,66, 125 representations 59, 61, 63, 84 Jacobi's principle matrix 10, 13, 42, 43, 49, 58, 65 mapping 8, 10 proper 10 Kepler problem (see planetary motion) improper 10 Klein-Gordonfield 133,175 orthochronous 10 Kronecker 9, 59, 60 transformation 43, 48, 49, 58 delta 9, 16, 17, 57, 74, 81, 125, 148, applications 51,52,53,55 171 boost 12,51 product of matrices 59,60,63 infinitesimal 128, 129

Lagrange multiplier 93,95,97, 111 magnetic field (see electromagnetic field) Lagrangian 89,95, 108, 120, 171 mappings 8,13,14,20,21,22,27, prerelativistic mechanics 89 59,90,93,97, 109, 111, 120, prerelativistic mechanics in space and 127 time 95 bilinear 14, 80 relativistic mechanics 108 conformal 74 relativistic fields 120, 123, 124, 134, conjugate linear 84 144,149,157,160,163,171,175, holomorphic 74,147 191 invertible 8, 9, 21, 100, 117 latin indices (see Roman indices) linear 8 Legendre transformation 91 Mobius 74 length 3,4, 7, 52 parameterized curve 22, 23, 25, 26, moving rod 52, 53 27,29,31,32,35,55,89,90,93, Levi-Civita 17,18 97,109 permutation symbol 17, 18, 79, 80, projection 20,21,40,41, 73, 121 81, 8~ 10~ 10~ 108 Lorentz 8, 10 pseudotensor 18,19,74,81,87,142 manifold, differentiable (see differentiable (see also permutation symbols) manifold) light cone (see null cone) group 66, 147 coordinates 70 mass 89,98, 103, 104, 105, 108, 109, Lie 63, 147, 148, 151, 163, 166 113, 115 group 63,66,67,70,147,151,163, proper (or rest) 102, 103, 105, 106 166 relative (or moving) 104, 105 semisimple 148, 163, 166 hypersurface (or shell) 111 algebra 147, 148, 163 matrix 8,9, 10, 11, 12, 13,42,43,49, 58, representation 148, 149, 163, 166 65,72,74,77,86,149,154,164, linear 8, 10, 13, 14,75,77,79 191,193,198 conjugate linear 84 Dirac matrices 86, 154, 198 independence 10, 14, 75, 77 hermitian 75, 77, 82 mappings 8, 13, 79 Lorentz matrices 10, 13,42,43,49, Lorentz 2, 53, 55, 58, 106, 143, 147 58,65 Lorentz-Fitzgerald contraction 53 Pauli matrices 74, 77, 78, 149, 166 Subject Index 211

rank 35, 155 equations of motion 89 representation 8 mechanics 89, 92 group 59,61,63,84 Noether's theorem 126, 128, 130, 171 linear mappings 8 nonabelian group 147, 148, 163 Lie algebra 148, 163, 166 gauge fields 149,151, 160, 163, 165, 75, 165 166 tracefree 62, 75 nondegeneracy 2,27 transposition 10,60,63, 119 axiom 2 similarity transformation 154 curve 27 unitary 74,75,76,78, 148, 149, 163, null 26,27,49 198 cone (or light-cone) 26,27,49 unimodular 72,77,79,80,81,83,84, coordinates 70 85,87,88 2-flat 37,38, 39,41 unitary and unimodular 74, 75, 76, vectors 4,6,26, 102, 144 78,148, 149, 163 world-line 26, 55 measurement 33, 34 numerical 17, 19, 79, 80, 86 Minkowski coordinates 33, 34 bispinor (see bispinor) proper time 28, 29 pseudotensor (see tensors) spacelike separation 30, 32 spinor (see spinors) mechanics (see equations of motion, tensors (see tensors) Lagrangian, and Hamiltonian) metric tensor 2, 108, 169 Lorentz metric 2, 10, 16,25 observer 26,33,48 Finsler metric 108, 109, 11 0 inertial (see inertial) Michelson-Moreleyexperiment 49 open subset 20,21, 121 Minkowski 1,2,4, 10, 15,25,26,30,31, operational method 33, 34 33, 34, 41 orbit (see planetary) chronometry 33, 34 orientation 46, 131 coordinates 25, 26, 33, 34 orthogonality 2, 4, 6, 7 M-orthogonality 2,4,6,30,31,33 Euclidean 7 M -orthonormality 2,4, 10, 11,25, 34 Minkowski (see Minkowski) tensors (see tensors) solution vectors 156, 159 tensor fields (see tensor fields) orthonormality, Minkowski (see vector space 1, 23 Minkowski) momentum 91,102,110,132,160,174 M -orthonormal basis (or tetrad) 2,3, four-momentum 102, 110, Ill, 112, 4,9,10,25 113 outer product (see ) three-momentum 91,92 total four-momentum of fields 132, 160,174 parallel 7, 32, 33, 34 total three-momentum of fields 132, parallelogram law 7 139,147,174 world-lines 32,33,34 motion curve 26,89,92,96,98, 103, 111 particle, point 25,89, 100, 106 equations of motion (see equations of charged, massive 106, 107 motion), in extended phase space free, massive 26, 102 98,111,112,113,119 photon, massless 26, 102 in phase space 92, 94 (see also motion curve) in space 89 past, present, and future (see events) in space and time 96 Pauli matrices 74, 77, 78, 149, 166 in space-time 103 perihelion shift 118 permutation 17,18,56,79,80,81,85, 106, 107, 108 Newtonian 89,92,98, 115 symbol (see Levi-Civita) constant of gravitation 98, 115 group (symmetric) 56 212 Subject Index planetary motions (or orbits) 98, 113, electromagnetic field 15, 18, 19, 140, 115,118 141, 142, 143, 146, 147 phase space 91,92,94,97, 111, 119, 168 gauge fields 147, 149, 163 extended 97, 111, 119, 168 interacting fields 160, 161, 162, 163, photon (massless, electromagnetic field 164 quantum) 26, 30, 33, 102 Klein-Gordon field 133, 175 polar coordinates 22, 24, 25, 115, 181, wave equation 47, 70, 123, 124, 139 182, 183, 192 representation, matrix 8,59,61,63,84 spherical (see spherical polar) degree 59,61 Poincare transformation 42, 43, 45, 52, group 59,61,63,84 128,129,170,171 linear mapping 8 group 56, 57, 58, 170 irreducible 61,62,63 positive definite 2, 3 retardation, clock (see clock) potential 92,98, 110, 113, 142, 143, 145, Riemannian manifold 25 146, 147, 149, 150, 151, 160, 161, rod (see Lorentz-Fitzgerald) 162, 163, 164, 165 Roman indices 2,4, 79,85, 120, 133, electromagnetic four-potential 113, 147, 163, 169 142, 143, 145, 146, 147, 160, 161, lower case 2,4, 85, 169 162 capital 79, 120, 133, 147, 163, 169, gauge potentials 149,150,151,163, 168 164,165,166,167 rotation 9,11,64,67,72,76,78,87,168 gravitational ten-potential 113, 114, group 67 115,116 matrix 9,11,72 projection 20,21,40,41,70, 73, 121 plane 11,67,72, 168 mappings 20,21,40,41,73,121 pseudorotation 64, 78, 87 M -orthogonal 40, 41 spatial 72, 76 transformations 70 Routh's procedure 95 stereographic 73 proper 11,28,29,33,53,58,63,64,68, 77,83,88, 102, 103, 105, 106 scalar 1,8, 14,21,72,73, 79, 84,123, Lorentz mapping (or transformation) 133, 175 11,58,63,64,66,68,77,78,79,83, complex numbers 72, 73, 123 88 field (see Klein-Gordon) mass (or rest) 102, 103, 105, 106 multiplication 1, 79, 84 time 28,29, 33, 53, 100, 112, 113 product (see inner product) pseudoangle 64, 78 real numbers 1, 8, 14,21 Pythagoras, generalized theorem 31 Schwarz inequality 4, 5,6, 7, 8 Schwarzschild (see gravitation) separable solution 134, 139, 140 (QeD) 151, separation, along a curve 27,28,29, 30, 166 33, 34 functional 27 number 4, 5, 7, 8 rank 15,35,43, 155 signature 2, 168 matrix 35, 155 simultaneity 52 tensor 15, 43 similarity transformation 75, 87, 88 relative 48, 104, 105, 108 space 4,8,11,26,30,32,37,38,48,49, energy 105 59,67,89 force 104 coordinates 48,49, 89 kinetic energy 105 reflection 11, 59 mass 104, 105, 108 spacelike separation 30, 32, 33 relativistic fields 120, 168 spacelike 2-flat 37, 38 Dirac field 151, 153, 154, 157, 158, spacelike 3-flat 37 160, 161, 162, 163, 164, 191, 192 spacelike vectors 4, 8, 26, 104 Subject Index 213 space-time manifold 20, 25 timelike 4,5,6,8,26,37,38, 102 diagram 27,29,30,31,32,33,37,39, curve 26 40,46,50,52,53,100 2-flats 37,38 speed of light (c = 1) 27,49 momentum 102 spherical polar coordinates 22, 25, 115 velocity 101 spin angular momentum 130,171 vectors 4, 5, 6, 8, 26 spin-(1/2) particle (see Dirac equation) transformation 11,13,15,20,21,22,24, spin-O particle (see Klein-Gordon 40,41,43,67,72,74,76,79,84, equation) 121, 168 spinors 79, 80, 84, 85 coordinates 20,21,22,24 spin or fields 152, 159 Galilean 50 spinor-tensors 81, 82 infinitesimal (see infinitesimal) symmetric 84, 85 Legendre 91,92,97,110 straight lines in space-time 26, 29, 30, Lorentz (see Lorentz) 31, 32,40, 52, 53, 105, 110 Mobius 74 string 52, 133 Poincare (see Poincare) structure constants 148, 149, 150, 151, projective 20,21,40,41, 70, 73, 163, 167 121 summation convention 2,3,79,85, 147, rotation 11,67, 72, 76, 168 163, 169 spinor components 79, 84 symmetry (group) 55 tensor components 15,43 Young symmetries 63 vector components 13, 43 symmetric group 55 translation, coordinates 57, 58 symplectic 80, 119 infinitesimal 128, 171 Synge's theorems 5, 32, 39 trace 75, 165 standard clock (see clock) tracefree 62, 75 stationary (or critical) values 90, 122 twin paradox 28, 29 subspace, vector 3,4, 8 subatlas (see differentiable ) subgroup (see group) unimodular condition 74, 75, 76, 78, 148, 149, 163, 198 unitary unimodular condition 74,75, tangent 23, 147 76, 78, 148, 149, 163 vectors 23, 147 unit vector (see vectors) space 23, 147 units 27, 115, 134 tensors 4,16,17,18,19,43,44,47,106, speed of light c = 1 27 108,120,120,142,170 c = G = 1 115 Cartesian 4, 89, 106, 108 c=h=I134 contraction 16, 44 Minkowski (mixed) 14,15,16, 18, 19 fields 43,44,47,120,142 variational formalism (see Euler• numerical 17,19 Lagrange equations) permutation symbol 17,18 vectors, Minkowski 1, 23 pseudo tensor 18, 19,81,82,87, future and past pointing 5, 8 142 null 4,6,8,26,102,144 raising and lowering indices 17, 18, spacelike 4,8,26, 104 44 separation number 4, 5, 7, 8 tensor or outer product 16,44 tangent vectors 23, 147 tetrad (see basis set) timelike 4,5,6,8,26, 101, 102 time 11,28,29, 33, 53, 59, 88, 100, 112, un~ 4,5,6,4~ 101, 131, 173 113 vector space, inner product 1,23 proper 28,29,33, 53, 100, 112, 113 four-dimensional, complex 85 retardation (see clock) Minkowski 1, 2, 23 reversal 11,59,88 subspace 3, 4, 8 214 Subject Index vector space (cont.) wave equation 47, 70, 123, 124, 139 three-dimensional 3, 4 fields (see relativistic fields) two-dimensional, complex 79, frequency 135 80 operator or d'Alembertian (see velocity 50,51,89, 101, 113, 119 d' Alembertian) composition 55 plane wave 135, 143, 154, 177 four-vector 101, 106, 108, 109, 110, world-line 26, 28, 30, 32, 33, 100 112, 113, 114, 119 (see also motion curve) three-vector 89,91,93, 101, 104, 105, work 89,107 108, 112, 113 function 89, 91, 92 volume, integral 45, 46, 120, 131, 132, Weyl equation 152 135, 173 four-dimensional 45,46, 120, 135 three-dimensional 131, 132, 135 Yasskin class of gauge fields 151 Universitext (continued)

Sagan: Space-Filling Curves Samelson: Notes on Lie Algebras Schiff: Normal Families Shapiro: Composition Operators and Classical Function Theory Smith: Power Series From a Computational Point of View SmoryJiski: Self-Reference and Modal Logic Stillwell: Geometry of Surfaces Stroock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tondeur: Foliations on Riemannian Manifolds