Continued part 1, part 2 of ebook Quantum field theory III: Gauge theory provide readers with content about: applications of invariant theory to the rotation group; temperature fields on the euclidean manifold E3; velocity vector fields on the euclidean manifold E3; covector fields on the euclidean manifold E3 and cartan’s exterior differential – the beauty of differential forms; ariadne’s thread in gauge theory;...
9 Applications of Invariant Theory to the Rotation Group Geometry has to be independent of the choice of the observer Folklore 9.1 The Method of Orthonormal Frames on the Euclidean Manifold We want to use the method of orthonormal frames in order to define • the gradient grad Θ of a smooth temperature field Θ, and • both the divergence, div v, and the curl, curl v, of a smooth velocity vector field v on the Euclidean manifold E3 The physical meaning of grad Θ, div v, and curl v will be discussed in Sect 9.1.4 Einstein’s summation convention In this chapter, we sum over equal upper P and lower indices from to For example, xi ei = 3i=1 xi ei 9.1.1 Hamilton’s Quaternionic Analysis Consider a fixed right-handed Cartesian (x, y, z)-coordinate system of the Euclidean manifold E3 with the right-handed orthonormal basis i, j, k at the origin P0 Let iP , jP , kP be a right-handed orthonormal basis of the tangent space TP E3 at the point P , which is obtained from the basis vectors at the origin i, j, k by translation (Fig 9.1) In about 1850, Hamilton (1805–1865) introduced the differential operator D := ∂ ∂ ∂ ∂ + iP + jP + kP ∂t ∂x ∂y ∂z and applied it to the quaternionic function Q(t, x, y, z) := Θ(t, x, y, z) + u(t, x, y, z)iP + v(t, x, y, z)jP + w(t, x, y, z)kP The point P has the Cartesian coordinates (x, y, z) To simplify notation, we replace iP , jP , kP by i, j, k, respectively Furthermore, we set ∂ ∂ ∂ • ∇ := ∂x i + ∂y j + ∂x k (Hamilton’s nabla operator), and • v(P ) := u(P )i + v(P )j + w(P )k Finally, since the symbol ∇i denotes the covariant partial derivative in modern tensor analysis, we replace the vector symbol ∇ by ∂ := ∂ ∂ ∂ i+ j+ k ∂x ∂y ∂z E Zeidler, Quantum Field Theory III: Gauge Theory, DOI 10.1007/978-3-642-22421-8 10, © Springer-Verlag Berlin Heidelberg 2011 557 558 Applications of Invariant Theory to the Rotation Group Fig 9.1 Orthonormal basis of the tangent space TP E3 Setting ∂t := ∂ , ∂t we get D = ∂t + ∂ and Q(t, P ) = Θ(t, P ) + v(t, P ) Hamilton investigated the quaternionic product D · Q = (∂t + ∂) · (Θ + v) = ∂t Θ + ∂t v + ∂Θ − ∂v + ∂ × v This way, we get i + ∂Θ j + ∂Θ k (gradient of the temperature field Θ), • ∂Θ = grad Θ := ∂Θ ∂x ∂y ∂z ∂u ∂v ∂w • ∂v = div v := ∂x + ∂y + ∂z (divergence of the velocity vector eld v), ã ì v = curl v (curl of the velocity vector field v) Explicitly, ˛ ˛ ˛ i j k˛ ˛ ˛ ˛∂ ∂ ∂˛ curl v := ˛ ∂x (9.1) ∂y ∂z ˛ ˛ ˛ ˛ u v w˛ Hence „ « „ « „ « ∂w ∂v ∂u ∂w ∂v ∂u curl v = − i+ − j+ − k ∂y ∂z ∂z ∂x ∂x ∂y ” “ ∂ ∂ ∂ + v ∂y + w ∂z Θ Here, (v(P ) grad)Θ(P ) is called • (v∂)Θ = (v grad)Θ := u ∂x the directional derivative of the temperature field Θ at the point P in direction of the velocity vector v(P “ ) at the point P ” ∂ ∂ ∂ • (v∂)E := (v grad)E = u ∂x + v ∂y + w ∂z E Here, (v(P ) grad)E(P ) is called the directional derivative of the electric field E at the point P in direction of the velocity vector v(P )“ at the point P ” • ΔΘ = −∂ Θ := − ture field Θ).1 • ΔE = −∂ · E := − ∂2 ∂2 x “ + ∂2 ∂2x ∂2 ∂2y + + ∂2 ∂2y ∂2 ∂2 z + ∂2 ∂2 z Θ (Laplacian Δ applied to the tempera- ” E The definitions of grad Θ, div v, curl v, (v grad)Θ, (v grad)E, ΔΘ, ΔE given above depend on the choice of the right-handed Cartesian (x, y, z)-coordinate system Concerning our sign convention for the Laplacian, see page 471 9.1 The Method of Orthonormal Frames 559 However, we will show below that the definitions are indeed independent of the choice of the right-handed Cartesian coordinate system To this end, we will use the method of orthonormal frames which is the prototype for the use of invariant theory in geometry and analysis The idea of this method is to define quantities for a fixed right-handed Cartesian coordinate system Then we show next that the quantity under consideration is independent of the choice of the right-handed Cartesian coordinate system To this end, we set x1 := x, x2 := y, x3 := z, ∂i := ∂ ∂xi and e1 := i, e2 := j, e3 := k 9.1.2 Transformation of Orthonormal Frames To begin with, let us study the change of orthonormal systems Let e1 , e2 , e3 be a right-handed orthonormal system in the Euclidean Hilbert space E3 Furthermore choose three arbitrary vectors e1 , e2 , e3 in E3 such that 1 e1 e1 B C B C (9.2) @e2 A = G @e2 A e3 e3 where G is an invertible real (3 × 3)-matrix Proposition 9.1 The transformed vectors e1 , e2 , e3 form a right-handed orthonormal basis in the Euclidean space E3 iff the transformation matrix G is an element of the Lie group SO(3), that is, GGd = I and det G = Proof (I) Let e1 , e2 , e3 be a right-handed orthonormal system Then, we have the orthonormality condition, ei ej = δi j , i, j = 1, 2, 3, and the volume product satisfies the relation (e1 e2 e3 ) = because of the righthanded orientation Hence 0 1 e1 e1 e1 e2 e1 e3 e1 “ 100 ” B B C C B C @e2 A e1 , e2 , e3 = @e2 e1 e2 e2 e2 e3 A = @0 0A = I e3 e3 e1 e3 e2 e3 e3 001 This is equal to e1 “ ” B C G @e2 A e1 , e2 , e3 Gd = GIGd = GGd e3 Hence GGd = I Finally, by (9.2), we get = (e1 e2 e3 ) = det G · (e1 e2 e3 ) = det G (II) Conversely, if GGd = I and det G = 1, then the same argument shows that ✷ e1 , e2 , e3 is a right-handed orthonormal system 560 Applications of Invariant Theory to the Rotation Group Corollary 9.2 The vectors e1 , e2 , e3 form a left-handed orthonormal basis in the Euclidean space E3 iff the transformation matrix G is an element of the Lie group O(3) (that is, GGd = I) with det G = −1 Proof Note that (e1 e2 e3 ) = −1 if e1 , e2 , e3 is a left-handed orthonormal basis ✷ Set x = xi ei Here, x1 , x2 , x3 are the coordinates of the position vector x with respect to the basis e1 , e2 , e3 By (2.84) on page 164, it follows from xi ei = xi ei that 1 x1 x1 B 2C −1 d B C = (G (9.3) ) @x A @x A 3 x x If e1 , e2 , e3 is an orthonormal basis, then (G−1 )d = G This implies the following specific property of orthonormal frames (without taking orientation into account) Proposition 9.3 Under a change of orthonormal frames, the three basis vectors e1 , e2 , e3 and the corresponding Cartesian coordinates x1 , x2 , x3 transform themselves in the same way 9.1.3 The Coordinate-Dependent Approach (SO(3)-Tensor Calculus) We are now able to prove the main result of Hamilton’s vector analysis Theorem 9.4 The definitions of grad Θ, div v, curl v, (v grad)Θ, (v grad)E, ΔΘ, and ΔE not depend on the choice of the right-handed Cartesian coordinate system Proof The passage from a right-handed Cartesian coordinate system to another right-handed Cartesian coordinate system corresponds to an SO(3)-transformation Therefore, we will use the SO(3)-tensor calculus introduced on page 453 In particular, we have the form-invariant tensorial families δij , δ ij , δji , εijk , εijk (9.4) The basis vectors ei transform like a tensorial family Lifting and lowering of indices can be performed by means of δ ij and δij For example, ei := δ ij ej Furthermore, since the transformation formula for the coordinates xi is given by a matrix which does not depend on the position of the point on the Euclidean manifold E3 , the differential operator ∂i sends tensorial families again to tensorial families Note that • v := v i ei , E = E i ei , • grad Θ = ∂i Θ · ei , div v = ∂i v i , curl v = εijk ∂i vj · ek , • (v grad)Θ = v i ∂i Θ, (v grad)E = (v i ∂i )E j ej , • ΔΘ = −δ ij ∂i ∂j Θ = −∂ j ∂j Θ, , ΔE = −(δ ij ∂i ∂j )E k ek 9.1 The Method of Orthonormal Frames 561 Fig 9.2 Measuring velocity vector fields All the expressions not have any free indices Thus, the claim follows immediately from the principle of index killing ✷ If we allow the use of both right-handed and left-handed Cartesian coordinate systems, then we have to pass to the O(3)-tensor calculus Let us assign to righthanded (resp left-handed) coordinate systems the orientation number ι = (resp ι = −1) Then we have to use the O(3)-tensorial families δij , δ ij , δji , ι · εijk , ι · εijk , ei , xi In particular, we write curl v = ι · εijk ∂i vj ek All the other expressions considered above remain unchanged In addition, for the vector product we get v × w = ι · εijk v i wj ek Examples Let a be a real number, andplet a, ω be fixed vectors Furthermore, let x := xi + yj + zk, as well as r := |x| = x2 + y + z Then: • grad(ax) = a, • grad U (r) = U (r) xr , ` ´ • div a3 x = a, • div(U (r)x) = 3U (r) + rU (r), • curl( 12 ω × x) = ω 9.1.4 The Coordinate-Free Approach The physical interpretation of the temperature gradient grad Θ This will be discussed in Sect 10.1 on page 645 Roughly speaking, the vector grad Θ(P ) points to the direction of the maximal growth of the temperature Θ at the point P , and the length of the vector grad Θ(P ) measures the maximal growth rate of the temperature Θ at the point P The physical interpretation of div v and curl v Let v be a smooth velocity vector field defined in an open neighborhood of the point P0 in the Euclidean manifold E3 So far, we have defined div v and curl v by using a right-handed Cartesian coordinate system It follows from tensor analysis that this definition does not depend on the choice of the right-handed Cartesian coordinate system It is also possible to determine div v and curl v in an invariant way by the following limits (Fig 9.2) 562 Applications of Invariant Theory to the Rotation Group Fig 9.3 Special velocity vector fields Theorem 9.5 Consider a ball of radius R about the point P0 Contracting the ball to the point P0 , we get Z dS div v(P0 ) = lim R→0 4πR3 S2 (P ) R Here, n denotes the outer unit normal vector on the boundary of the ball Similarly, consider a disk of radius R about the point P0 which is perpendicular to the unit vector n Contracting the disk to the point P0 , we get Z vdx n curl v(P0 ) = lim R→0 πR S1 (P0 ) R Proof By the mean theorem for integrals, Z 4πR3 div v(P1 ) div v dxdydz = |x−x0 |≤R where P1 is a suitable point of the ball of radius R about the point P0 The Gauss– Ostrogradsky integral theorem on page 680 tells us that Z Z 3 div v dxdydz = nv dS 4πR3 |x−x0 |≤R 4πR3 S2 (P0 ) R Letting R → 0, we get div v(P0 ) Similarly, we obtain n curl v(P0 ) by using the Stokes integral theorem on page 680: Z Z n curl v dS = vdx B2 R (P0 ) S1 R (P0 ) ✷ Examples Choose the origin, P0 := O Consider the smooth map P → vP In terms of physics, this is a smooth velocity vector field on the Euclidean manifold E3 By definition, the streamline t → x(t) passing through the point P0 at time t0 is given by the solution of the differential equation ˙ x(t) = v(x(t)), t ∈ J, x(t0 ) = x0 , where J is an open interval on the real line which contains the point t0 Let us consider the prototypes of velocity vector fields (9.5) 9.1 The Method of Orthonormal Frames 563 Fig 9.4 Rotational velocity vector field • Source at the origin (Fig 9.3(a)): Choose the velocity vector field v(x) := with a > Then Z Z aR dS = · dS = a 4πR3 S2 (O) 4πR3 S2 (O) R a x R Letting R → 0, we get div v(O) = a, by Theorem 9.5 The origin is a source for the streamlines of the velocity vector field, and div v(O) measures the strength of this source • Sink at the origin (Fig 9.3(b)): Let a < Again we get div v(O) = a In this case, the origin is a sink for the streamlines of the velocity vector field • Circulation around the z-axis (Fig 9.4): Let us choose a right-handed Cartesian (x, y, z)-coordinate system with the right-handed orthonormal basis i, j, k at the origin O Let ω := ωk with ω > Consider the velocity vector field v(x) := 12 (ω × x) This corresponds to the counter-clockwise rotation of fluid particles about the z-axis with the angular velocity ω The streamlines are circles parallel to the (x, y)-plane centered at points of the z-axis Since the velocity vectors are tangent vectors to the streamlines, we get Z Z 1 ωR2 vdx = · ds = ω πR2 S1 (O) πR2 S1 (O) R R Letting R → 0, we get k curl v(O) = ω, by Theorem 9.5 Thus, the z-component of the vector curl v(O) measures the angular velocity of the fluid particles near the origin 9.1.5 Hamilton’s Nabla Calculus To begin with, let us summarize the key relations in classical vector calculus Let Θ, Υ : R3 → R and v, w : R3 → E3 be smooth temperature functions and smooth velocity vector fields, respectively Proposition 9.6 The following hold: (i) curl grad Θ = 0, (ii) div curl v = 0, (iii) grad(Θ + Υ ) = grad Θ + grad Υ , (iv) grad(ΘΥ ) = (grad Θ)Υ + Θ grad Υ, 564 Applications of Invariant Theory to the Rotation Group (v) grad(vw) = (v grad)w + (w grad)v + v × curl w + w × curl v, (vi) div(v + w) = div v + div w, (vii) div(Θv) = v(grad Θ) + Θ div v, (viii) div(v × w) = w curl v − v curl w, (ix) curl(v + w) = curl v + curl w, (x) curl(Θv) = (grad Θ) × v + Θ curl v, (xi) curl(v × w) = (w grad)v − (v grad)w + v div w − w div v, (xii) ΔΘ = − div grad Θ, (xiii) Δv = curl curl v − grad div v, (xiv) 2(v grad)w is equal to grad(vw) + v div w − w div v − curl(v × w) − v × curl w − w × curl v (xv) v(x + h) = v(x) + (h grad)v(x) + o(|h|), h → (Taylor expansion) The relations (xii)–(xiv) show that ΔΘ, Δv and (v grad)w can be reduced to ‘grad’, ’div’, and ‘curl’ All the relations (i)–(xiv) above can be verified by straightforward computations using a right-handed Cartesian coordinate system However, the nabla calculus works more effectively In this connection, we take into ∂ ∂ ∂ + j ∂y + k ∂z is both a differential operator account that the nabla operator ∂ = i ∂x and a vector Therefore, mnemonically, we will proceed as follows: • Step 1: Apply the Leibniz product rule by decorating the terms with dots • Step 2: Use algebraic vector operations in order to move all the dotted (resp undotted) terms to the right (resp left) of the nabla operator ∂ Proof Ad (i), (ii) It follows from a × Θa = and a(a × b) = that ∂ × ∂Θ = and ∂(∂ × v) = Hence curl grad Θ = and div curl v = Ad (iv) By the Leibniz product rule, ˙ ) + ∂(ΘΥ˙ ) ∂(ΘΥ ) = ∂(ΘΥ Moving the undotted quantities to the left of the nabla operator, we get ˙ + Θ(∂ Υ˙ ) ∂(ΘΥ ) = Υ (∂ Θ) Hence grad(ΘΥ ) = Υ grad Θ + Θ grad Υ Ad (xi) By the Leibniz rule, ˙ ∂ × (v × w) = ∂ × (v˙ × w) + ∂ × (v × w) Using the Grassmann expansion formula a × (b × c) = b(ac) − c(ab), we get ˙ ˙ + v(∂ w) ˙ − w(∂v) ˙ ∂ × (v × w) = v(∂w) − w(∂ v) Finally, moving the undotted terms to the left of the nabla operator ∂ by respecting the rules of vector algebra, we get ˙ + v(∂ w) ˙ − (v∂)w ˙ ∂ × (v × w) = (w∂)v˙ − w(∂ v) This is the claim (xi) Ad (v) Use the Grassmann expansion formula b(ac) = a(bc) + a × (b × c) The remaining proofs are recommended to the reader as an exercise ✷ 9.1 The Method of Orthonormal Frames 565 9.1.6 Rotations and Cauchy’s Invariant Functions Consider a right-handed Cartesian (x, y, z)-coordinate system with the right-handed orthonormal basis e1 , e2 , e3 Let x, y, z ∈ E3 , and let x = xi ei , y = y i ei , and z = z i ei Then the inner product xy = δij xi y j and the volume product (xyz) = εijk xi yj z k are invariants under the change of right-handed Cartesian coordinate systems If we consider the more general case of arbitrary Cartesian (x, y, z)-coordinate systems with an arbitrary orthonormal basis e1 , e2 , e3 , then the inner product xy remains an invariant However, this is not true anymore for the volume product (xyz) which changes sign under a change of orientation One of the main results of classic invariant theory tells us that these invariants are the only ones in Euclidean geometry Let us formulate this in precise terms The Cauchy theorem on isotropic functions The real-valued function f : E3 × · · · × E3 → R is called isotropic iff f (Gx1 , , Gxn ) = f (x1 , , xn ) (9.6) for all vectors x1 , , xn ∈ E3 and all unitary operators G ∈ U (E3 ) Moreover, the function f is called proper isotropic iff we have the relation (9.6) for all rotations G ∈ SU (E3 ) Note that a function is isotropic iff it is invariant under all rotations and reflections x → −x Theorem 9.7 (i) If the function f is isotropic, then it only depends on all the possible inner products xi xj , i, j = 1, , n (9.7) (ii) If the function f is proper isotropic, then it only depends on all the possible inner products (9.7), and all the possible volume products (xi xj xk ), i, j, k = 1, , n The polynomial ring of invariants The function f : E3 × · · · × E3 → R considered above is called a polynomial function iff it is a real polynomial with respect to the Cartesian coordinates of the vectors x1 , , xn Since the change of Cartesian coordinates is described by linear transformations, this definition does not depend on the choice of the Cartesian coordinate system Corollary 9.8 If the polynomial function f is proper isotropic, then it is a real polynomial of all the possible inner products xi xj , i, j = 1, , n, and all the possible volume products (xi xj xk ), i, j, k = 1, , n For the classic proofs of Theorem 9.7 and Corollary 9.8, we refer to the references given in Problem 9.5 Examples (a) Every proper isotropic, polynomial function f : E3 → R has the form for all x ∈ E3 f (x) = p(x2 ) where p is a polynomial of one variable with real coefficients Such a function is also isotropic (b) Every proper isotropic, polynomial function f : E3 × E3 → R has the form 566 Applications of Invariant Theory to the Rotation Group f (x, y) = p(x2 , y2 , xy), for all x, y ∈ E3 where p is a real polynomial of three variables Such a function is also isotropic (c) Every proper isotropic, polynomial function f : E3 × E3 × E3 → R has the form f (x, y, z) = p(x2 , y2 , z2 , xy, xz, yz, (xyz)) for all vectors x, y, z ∈ E3 Here, p is a real polynomial of seven variables (d) Set f (x, y, z) := (xyz)2 This polynomial function is isotropic By Theorem 9.7, we know that f only depends on all the possible inner products of the vectors x, y, z Explicitly, ˛ ˛ ˛ x2 xy xz˛ ˛ ˛ ˛ ˛ f (x, y, z) = ˛yx y2 yz˛ ˛ ˛ ˛ zx zy z ˛ This is the Gram determinant Let P(SU (E3 )) denote the set of all the real polynomials with respect to the variables xi xj , (xi xj xk ), i, j, k = 1, , n, n = 1, 2, This set is closed under addition and multiplication, hence it is a commutative ring The commutative ring P(SU (E3 )) is called the polynomial ring of invariants of the Lie group SU (E3 ) The Rivlin–Ericksen theorem on isotropic, symmetric tensor functions in elasticity theory Let Lsym (E3 ) denote the set of all linear self-adjoint operators A : E3 → E3 on the real Hilbert space E3 The linear operator T : Lsym (E3 ) → Lsym (E3 ) is called an isotropic tensor function iff we have R−1 T (A)R = T (R−1 AR) for all linear operators A ∈ Lsym (E3 ) and all rotations R ∈ SU (E3 ) Theorem 9.9 Let T be an isotropic tensor function Then there exist real functions a, b, c : R3 → R such that T (A) = aI + bA + cA2 for all A ∈ Lsym (E3 ) where a = a(tr(A), tr(A ), det A) together with analogous expressions for b and c Note the following: If λ1 , λ2 , λ3 are the eigenvalues of the operator A, then tr(A) = λ1 + λ2 + λ3 , tr(A2 ) = λ21 + λ22 + λ23 , det(A) = λ1 λ2 λ3 The proof of Theorem 9.9 together with applications to the formulation of general constitutive laws for elastic material (generalizing the classic Hooke’s law) can be found in Zeidler (1986), p 204, quoted on page 1089.3 Note that (xxy) = (yyx) = Therefore, the volume products disappear R Rivlin and J Ericksen, Stress-deformation relations for isotropic materials, J Rat Mech Anal (1955), 681–702 1112 Index Jordan (Pascal), 202, 293 algebra, 293 Jost (Jă urgen), XIII Joule, 953 – heat energy law, 944 Joyce, 283 Justitian, 588 K (kelvin), 983 Kă ahler, 473, 777 codierential, 474, 479 – differential, 474 – – ideal, 789 – duality, 474, 479 – form, 788 – interior differential calculus, 474, 790 – manifold, 788 – potential, 788 – star operator, 474, 479 Kassel, 291 Kastrup, XIII Kelvin (see also Thomson), 691, 777 – transformation, 692 Kendall, 896 Kepler, kernel, 146, 254, 1003 ket-vector, 171 Kijowski, XIII Killing, 179, 260, 284, 365, 440 – form, 234, 809 – – of slC (n, C), 248 – vector field – – metric, 732 – velocity vector field, 365 killing of indices, 443 Kirchhoff’s voltage rule, 1013 Kirsten, XIII Klein (Felix), 2, 20, 179, 623, 777 – Erlangen program, 20, 788, 899 Klein (Oskar), 2, 953 Klein–Fock–Gordon equation, 814, 815, 818 knot, 295 – invariant, 296 Kobayashi, 896 Kohn, 841 Kolmogorov, 348 – backward equation, 350 – forward equation, 350 Kreimer, XIII Kronecker, 179, 332 – product, 190 K-theory (see Vol II), 333 Kă unneth, 777, 1032 product formula, 1032 Kummer, 179 Kusch, 895 Lagrange, 21, 439, 777, 778 – identity, 84 – multiplier, 381 – multiplier rule, 601 Lagrangian, 404 Landau (Lev), 344 Landau (Lev9, 836 Lang, 1060 Langlands, 94 – program, 94 Laplace, 777, 780 – equation, 779 Laplace–Runge–Lenz vector, 280 Laplacian, 674, 779 – sign convention, 472 lattice, 310 Lebesgue, 329 – integral, 329, 330 Lederman, 896 Lee, 4, 895 Lefschetz, 1009 left translation, 357, 586, 804 left-invariant velocity vector field, 357 Legendre, 777 – manifold (see Vol II), 699 – polynomial, 780 – transformation, 409 – transformation (see Vol II), 699 Leibniz, 41, 74, 94, 348, 439, 666, 777 – rule, 328, 348, 612 – – classical, 42, 709 – – derivation, 532 – – differential forms on vector bundles, 864 – – for tensor fields, 49 – – generalized, 49 – – graded (supersymmetric), 37, 328, 465, 469, 701, 709 – – Lie derivative, 490 – – universal extension strategy on vector bundles, 858 Leipzig Acta Eruditorum, 328 length – contraction, 911 – of a curve, 627 – of a vector, 78 Lenz (Heinrich), 280 Lenz (Wilhelm), 280 Index Leutwyler, 243 lever principle, 375 Levi (Beppo), 265 Levi (Eugenio), 265 – decomposition theorem, 265 Levi-Civita, 7, 439 – connection on a sphere, 596 – duality, 460 – parallel transport – – prototype, 596, 846 – tensorial family, 460 lexicographic order – Young frames, 221 – Young tableaux, 221 LHC (Large Hadron Collider), 940 Li–York theorem on chaos, 345 Lie, 20, 41, 71, 179, 281, 358, 550, 659, 750, 777 – bracket, 488, 555 – derivative, 45, 523, 651 – – and the flow of a fluid, 651 – main trick of cancellation, 42 – product, 41 – – of vectors, 82 – – of vectors fields, 45 – super algebra, 288 Lie algebra – G2 , F4 , E6 , E7 , E8 , 260 – C ⊗ p(1, 3), 923 – gl(3, R),, 586 – gl(X), 193 – gl(n, R), gl(n, C), 109 – glC (n, R), gl(n, R), glC (n, C), 188 – o(1, 3), 919 – p(1, 3), 922 – sl(2, C), 246 – – and Pauli matrices, 99 – sl(2, R), 246 – slC (2, C), 246 – slC (3, C), 248 – so(3), 586 – su(2), 246 – – and Pauli matrices, 99 – – and the isospin of the proton, 437 – su(3), 230, 248 – su(3) so(3), 374 – su(N ), 50, 188 – sut(3, R) (Heisenberg algebra), 110, 263 – u(1) o(2) so(2), 355, 362 – u(E2 ) su(E2 ) o(2) so(2), 363, 368 – u(N ), 187 1113 – adjoint representation, 319, 583 – and crystal, 280 – basic properties (see Vol I), 110 – calculus, 802 – Cartan subalgebra of su(N ), 204 – classification, 259 – cohomology, 250 – complexification, 244, 245 – definition (see Vol I), 110 – Euclidean space, 82 – exceptional, 260 – – in physics, 260 – Gell-Mann matrices of su(3), 231 – Heisenberg algebra, 107 – isomorphism, 363 – linear representation, 193 – morphism, 363 – of velocity vector fields, 664 – Poisson algebra and quantization, 108 – prototype, 83 – realification, 244 – representation, 193 – semidirect product, 921 – semisimple, 262 – simple, 259 – – classification, 259 – solvable, 110, 261, 263 – velocity vector fields, 614 Lie group, 109, 804, 1083 – D3 , 210, 211 – GL(3, R), 586 – GL(n, C), GL(n, R), 188 – GL(n, R), GL(n, C), 109 – O(1, 1), 915 – O(1, 3), 916 – O(3), 372 – P (1, 3), 921 – SL(2, C), 300 – SO(3), 256, 586 – SO↑ (1, 3), 922 – SU (2) – – and the electron spin, 427 – – irreducible representations, 427 – SU (2) U (1, H), 433, 948 – SU (3), 230 – – and quarks, 225, 226 – SU (E2 ) SO(2) U (1), 366 – SU (E3 ) SO(3), 372 – SU (N ), 50, 102, 188, 226 – SU (X), 226 – SU T (3, R), 110 – Sp(2, C), 305 1114 Index Sp(2, R), 305 Sym(2), 214, 222 Sym(3), 215, 217, 223 Sym(n), 181, 224 U (E2 ) O(2), 362 U (E2 ) O(2) Z2 × SO(2), 366 U (E3 ) O(3), 372 U (N ), 102, 187, 226 Z2 (additive), 194, 367 Z2 (multiplicative), 367 Z2 Sym(2), 194 basic properties (see Vol I), 107 Birkhoff–Heisenberg quotient group, 110 – calculus, 802 – compact, 188, 355 – curvature, 806 – definition (see Vol I), 107 – epimorphism : SU (2) → SO(3), 433, 948 – global parallel transport, 804 – global structure, 1085 – Heisenberg group (see also Vol II), 107 – infinitesimal, 265 – introduction (see Vol I), 1083 – irreducible representations of SU (3), 233 – left-invariant velocity vector field, 805 – linear representation, 191 – locally compact, 188, 355 – maximal torus of SU (N ), 204 – prototype, 355 – the paradigm U (1), 187, 355 – universal covering group, 1085 Lie matrix algebra – basic definitions (see Vol I), 1084 Lie matrix group, 109, 1083, 1084 – basic definitions (see Vol I), 1084 – first main theorem, 1084 – second main theorem, 1084 – universal covering group, 1085 Lie–Cartan formula, 491 Lieb, XIII lifting of a curve in fiber bundles, 888 light – cone, 931 – ray, 969 – velocity in linear materials, 981 limit – inductive (direct) (see Vol II), 335 – projective (inverse) (see Vol II), 335 – – – – – – – – – – – – – linear – equivalence, 1003 – isomorphism (see Vol I), 105 – manifold, 150 – material, 981 – – electric, 980 – – magnetic, 980 – morphism (see Vol I), 78 – operator (Vol I), 1004 – quotient space, 1004 – space (see Vol I), 78 linearization – covariant directional derivative, 611, 821 – – connection, 877 – differential, 646 – directional derivative, 645 – Fr´echet derivative, 653 – functional derivative, 646 – global, 654 – Lie derivative, 647 – local, 645, 650, 653 – of a transformation, 739 – principle, 655, 659, 1079 – tangent bundle – – prototype, 654 – tangent map, 655 – variation, 652 linearly – dependent, 78 – independent, 78 link, 295 Liouville, 91 Lipschitz, 731 – constant, 731 – continuous, 731 Littlewood (Dudley), 274 Littlewood (John), 274 Littlewood–Richardson rules, 273 local – connection form, 582, 592 – equivalence principle, 13 – phase factor, 821, 827 – – prototype in geometry, 583 – – prototype in physics, 821 – symmetry, 817 local-global principle in – mathematics, 336 – quantum field theory (analyticity of the S-matrix), 336 locally – compact topological space, 370 – conformally flat, 518 Index – flat, 518 long-term development in mathematics and physics – differential geometry and Einstein’s theory of general relativity, – differential geometry and gauge theory, 891 – potential theory and differential forms, 777 – references, 792 Lorentz (Hendrik), 935, 953 – boost, 905, 922, 924 – force – – motivation, 971 – group, 917, 922, 924 – – orthochronous, 917, 922 – – proper, 917, 922 – matrix group, 917 – transformation, 906 – – orthochronous, 922, 924 – – proper, 922, 924 Lorenz (Ludvig), 953 – gauge condition, 400, 964, 966 Louis, XIII Lă ust, XIII MacLane, 1060 Maclaurin, 74 MacPherson, 294 magnetic – charge, 951, 984 – dipole, 947 – flow (weber), 985 – monopole, 951 – polarization (magnetization), 935 magnetic field B, 935 – derived (or effective) magnetic field H, 979 – of a magnetic dipole, 947 – of a magnetic monopole, 951 – strength, 935 magnetization (magnetic polarization), 585, 935, 979, 980 main theorem of calculus, 666, 671, 677, 729 Mainardi, 634 Mainardi–Codazzi equation, 631 manifold, 1069, 1072 – closed (compact and no boundary), 1072 – compact, 1070 – definition (see also Vol I), 1072 – global analysis, 526 1115 – isomorphism (diffeomorphism), 1075 – linear, 150 – morphism (smooth map), 1075 – orientable, 1075 – oriented, 1075 – paracompact, 1073 – prototype, 1071 – smooth, 1072 – terminology and conventions, 1072 – topological, 1072 – with boundary, 1071, 1072 – without boundary, 1072 Manin, 337 – quantum plane, 298 Marathe, XIII, 295 Marcolli, XIII Markov, 348 – process, 351 – property (causality), 351 Maskawa, 896 mathematical structures – statistics of, 1056, 1060 Mather, 896 matrix, 74 – and linear operator, 157 – orthogonal, 365 – product, 74 – unitary, 365 Maurer, 358 Maurer–Cartan – form, 357, 583, 585, 806, 849 – structural equation, 358, 769, 807 Maurin, XIII maximal torus, 204 Maxwell, 691, 731, 777, 908, 953 Maxwell equations in a vacuum, 983 – conservation laws, 957 – de Rham cohomology, 964, 1027, 1039 – electrostatics, 1045 – energy–momentum tensor, 971 – existence and uniqueness theorem, 1052 – four-potential, 964 – gauge invariance, 964, 977 – global form, 954 – historical background, 936 – Hodge duality, 962 – invariant formulation, 958 – language of – – differential forms, 962 – – fiber bundles, 967 – – tensor calculus, 960 1116 Index – – vector calculus, 958 – – Weyl derivative, 983 – local form, 957 – magnetostatics, 1048 – motion of charged particles, 970, 977 – physical units, 983 – principle of critical action, 976 – relativistic invariance, 935 Maxwell equations in materials – constitutive law, 980 – language of – – Cartan and Weyl derivative, 983 – – tensor calculus, 983 – – vector calculus, 980 – linear material, 981, 983 – physical units, 983 – Weyl duality, 979 Maxwell–Hodge–Yang–Mills equation, 472 Mayer (Robert), 972 mean curvature, 629 Menelaos, 599 Mercator, 15 meson, 816 messenger particle, 894 method of – moving frame, 580, 585, 621, 636 – – prototype, 585 – orthogonal projection, 694, 796 – orthonormal frame, 557 metric – Riemann curvature tensor, 30 – space (see Vol II), 333 – tensor, 169, 924 – – basic idea, 85 – – field, 715 – – prototype, 568 – tensorial family – – inner product, 512 – – Levi-Civita connection, 512 – – parallel transport, 513 – – pseudo-Riemannian, 469 – – Riemannian, 469 MeV (mega electron volt), 940 Millennium Prize Problems, 61, 728 Millikan, 948 Milnor, 623 Minerva, 21 minimal surface, 606 Minkowski, 71, 98, 905 – manifold, 71 – matrix, 924 – space, 71, 923 – symbol, 9, 924 Minkowski–Hasse theorem on Diophantine equations, 336 modularity theorem, 65 module, 310 – finitely generated, 311 – free, 311 – isomorphism, 311 – left R-module, 310 – main theorem on finitely generated modules, 310 – morphism, 311 – right R-module, 310 moduli, 18 – space, 64, 205, 1056 Molin, 285 moment, 780 – of inertia, 421 momentum, 69 – density vector, 975 – operator, 948 – vector, 927, 978 – – canonical, 978 Monge–Amp`ere equation, 789 monomial, 115 monomorphism, 254 monopole – electric, 937 – magnetic, 951 Moore (John), 59 morphism, 116 – classification, 254 Morse index, 202, 469 motion of a – charged relativistic particle, 970, 977 – free relativistic particle, 927 moving frame, 71, 557, 580, 621, 636, 788 – dual Cartan equation, 585 – Gauss’ equation, 580 MRI (magnetic resonance imaging), 948 Mă uller (Alexander), 836 multilinear functional, 75 multivector, 142 N (newton), 944, 984 nabla – calculus, 563 – operator, 104, 557, 774 – symbol, 557 Nambu, 896 natural Index – basis vectors, 609, 622 – frame, 570 – number, 1094 – – proper, 1094 Navier, 722 Navier–Stokes equations in hydrodynamics, 727 Ne’eman, 283 Neukirsch, 332 Neumann (Carl), 952 neutral current, 894 neutrino, 895 Newton, 69, 271, 348, 623, 652, 691, 777, 908, 952 – and Leibniz, 328 – law of motion, 324 Nobel prizes for discoveries in elementary particle physics, 895 Noether (Emmy), 179, 285, 413, 777, 1060 – principle – – symmetry and conservation laws, 737 – theorem (see also Vol II), 413, 737, 975 non-Abelian (noncommutative) group, 811 non-inertial system, 396 non-standard analysis, 330 non-standard analysis (see Vol II), 740 noncommutative geometry, 299 – and the Standard Model in particle physics, 345 noncommutative space-time, 347 nonholonomic constraints, 378, 382 nonlinear system, 1079 normal form strategy in mathematics, 1078 normal subgroup (see Vol II), 254, 368 Oberguggenberger, XIII observer, 442, 529 – admissible system, 449 – cocycle, 449 octonion, 176 odd element, 288 Oeckl, XIII Oerstedt, 952 Ohm, 944, 953 – law, 944, 980, 1012, 1013 ohm (electric resistance), 985 Okubo, 235 one-parameter group, 365 1117 – of diffeomorphisms, 739 operator module, 312 orbit, 205 order parameter, 839 orientable manifold, 1075 orientation, 452, 526, 926, 1075 – coherent, 1076 – function, 452 – number, 452 – positive, 926 – strictly positive, 926 orientation-preserving map, 442 orthogonal – complement, 146 – decomposition, 146 – matrix, 365 orthonormal frame, 559 Ortner, XIII Ostrogradsky, 777 Ostrowski, 91, 333 Pa (pascal/pressure), 985 p-adic number, 332 – basic properties, 333 – classification, 335 – construction, 333 – integer, 335 paracompact topological space, 1070 parallel transport, 31, 323, 581, 827, 846, 849, 882 – and curvature, 618 – basic ideas, 31 – global, 56, 576, 585 – of frames, 51 – of phase factor, 51 – of physical fields, 51 – on a sphere, 596 – on the Euclidean manifold, 581 parallelizable sphere, 56 parametrix, 265 parametrized linear algebra, 610 parity, 183, 195 – additive, 287 – group, 922 – multiplicative, 288 – number, 452 – of quantum states, 195 – permutation, 182 – transformation, 958 – violation, 196 Parseval, 80 – equation, 80, 82 – – generalized, 173 1118 Index partial derivative, 40 – commutativity relation, 42 – covariant, 43, 818, 844 – summary of classical identities, 40 partition – of unity, 1077 – Young frame, 217 Patras, XIII Pauli, 71, 98, 282, 950 – equation – – transformation law, 991 – exclusion principle, 197, 950 – – and the color of quarks, 241 – matrices, 100, 246 – spin equation, 948 – spin-statistics principle, 950 Penrose, 58 Penzias, 895 periods, 784 – Cauchy, 784 – Poincar´e, 787 Perl, 896 permutation, 73 – essential properties, 181 Peter–Weyl theory, 195 Pfaff, 278, 698, 777 – normal form – – second law of thermodynamics, 771 – problem, 698, 769 Pfaffian, 278 phase – factor, 51 – space – – extended, 752 – transition, 832 philosophical question – existence of – – a local-global principle in the universe, 336 – – adelic physics, 334, 338 photoelectric effect (see Vol I), 929 photon, 929 Planck – quantum of action, 297 – satellite, 699 Plateau, 18 – problem, 18 Plato, 1, 300, 588 – Academy, 588 – cave parable, 588 – realm of ideas, 588 Poincar´e, 113, 777, 787, 953 – algebra p(1, 3), 921, 922 – – complexified, 923 – cohomology rule, 465, 710 – – discrete, 1019, 1026 – cohomology theorem – – global, 1032 – – local, 763 – conjecture, 66 – duality, 1032 – group P (1, 3) P (M4 ), 259, 921, 924, 975 – – important subgroups, 922 – – infinitesimal transformations, 922 – homology, 1033 – no-go theorem for velocity vector fields, 623 – polynomial, 1030 Poincar´e–Cartan – contact 1-form, 747 – contact 2-form, 752 – integral invariant, 749 Poincar´e–Hopf theorem, 55 Poisson, 777, 952 – bracket, 109 – equation, 779 Poisson–Lie algebra, 108 polar coordinates, 571 polarization, 979, 980 – current, 955 – electric, 935, 936 – magnetic, 935, 936 Politzer, 244, 894 polynomial, 115 – algebra, 116 – – generalized, 298, 301 – antisymmetric, 269 – – Vandermonde polynomial, 272 – completely symmetric, 271 – elementary symmetric, 270 – generalized, 298 – power sum, 271 – ring of invariants, 565 – symmetric, 269 – – Schur polynomial, 272 Pontryagin duality, 369 Popov, 895 position space, 655 positive real number, 1094 potential, 778, 779, 939 – difference, 1041 – electrostatic potential and voltage, 939 – energy, 399 – four-potential, 964 Index – gauge transformation, 399, 964 – multi-valued, 687 – scalar potential, 399, 964 – single-valued, 685 – vector potential, 399, 964 Powell, 895 power sum, 271 Poynting vector, 1095 pre-Hilbert space (see Vol I), 694 principal – argument, 688 – axes of – – a self-adjoint operator, 200 – – inertia, 392 – axes theorem, 718 – – for a rigid body, 392 – – in geometry and quantum geometry, 200 – moments of inertia, 392 – – ball, 422 – – circular cylinder, 422 principal bundle – associated, 881 – cocycle strategy, 873, 881 – connection, 849, 881 – – prototype, 826 – curvature, 826, 851, 883 – – and the electromagnetic field, 823 – definition, 880 – general strategy, 875 – intuitive prototype, 880 – position space of a rigid body, 397 principle of – critical action – – Hamiltonian approach, 746 – exclusion (Pauli), 197, 950 – general relativity (Einstein), 9, 442 – indistinguishability of quantum particles, 197 – killing indices, 30 – local symmetry, 817 – – charged meson, 817 – special relativity (Einstein), 908 – spin-statistics (Pauli), 950 – the correct index picture, 236, 443, 444, 526 – – on bundles, 861 probability amplitude, 27 Proca equation, 835 product – alternating (Grassmann), 86 – contraction, 714 – direct, 255 1119 – exterior (Grassmann), 714 – interior (Clifford), 714, 790 – semidirect, 256 production – function, 695 – of heat energy, 695 projective – limit (see Vol II), 335 propagator, 827, 853 proper – map, 720 – time, 8, 9, 927, 970 pseudo-differential operator – prototyp, 360 – symbol, 360 pseudo-holomorphic curve, 61 pseudo-inner product, 512 pseudo-invariant, 452 pseudo-orthonormal basis, 926 – and inertial system, 925 pseudo-Riemannian manifold, 526 pseudo-tensorial – density, 462 – – family, 462 – family, 460 pull-back of – differential form, 476, 667, 673 – set, 667, 672, 673 – temperature field, 651, 657 – velocity vector field, 662 push-forward of – temperature field, 657 – velocity vector field, 661 QCD (quantum chromodynamics), 243, 854, 894 QED (quantum electrodynamics), see Vol II, 935 QFT (quantum field theory), see Vols I/II – open questions, VIII quadrupole moment, 780, 947 quantization, 108 quantum – algebra – – slq (2, C), 302, 305 – – Drinfeld–Jimbo, 318 – determinant, 301 – group – – duality, 370 – – further reading, 546 – – summary, 304 – group SLq (2, C), 301 1120 Index – plane, 298, 299 – – complex, 299 – – real, 299 – super plane, 299 – symmetry, 305, 414 quantum chromodynamics (see QCD), 243 quark, 225, 894 – color, 241 – – and the Pauli exclusion principle, 241 – confinement, 894 – flavor, 226 quasicrystal, 280 quaternion, 70, 97 – history, 94 quotient – algebra (see Vol II), 258, 261 – field (see Vol II), 333 – group (see Vol II), 111, 257, 258 – Lie algebra (see Vol II), 261, 264 – space (see Vol II), 150 – topology (see Vol II), 294 radical, 264 Ramanujan, 221 range of an operator, 146 rank of – linear operator, 146, 1006 – matrix, 161 – module, 307 rank theorem – linear, 1078 – nonlinear, 1079 rate-of-strain tensor, 715, 724 realification, 244, 245 reciprocal basis, 170 – in crystallography, 168 Reeb velocity vector field, 61 references, 1089 – complete list on the Internet, 1089 reflection, 195 – group, 195 – – representation, 195 regular k-cycle, 1035 – fundamental system, 1035 Rehren, XIII Reines, 896 Remmert, 90 renormalization – fixed-point theory, 345 – in chaotic dynamical systems, 340 – in classical physics, 981 – self-similarity, 345 – trick, 344 – universality, 345 representation – adjoint, 207, 234, 808 – basic notions, 189 – completely reducible (or semisimple), 190 – continuous, 191 – degree, 189 – dual, 450 – faithful, 189 – irreducible (or simple), 190 – linear, 189, 190 – of a group, 189 – of Lie algebras, 193 – regular, 209 – unitary, 190 resistance, 1013 resistor, 1013 Ricci – calculus, 45, 503 – – basic ideas, 45 – curvature tensor, 11, 515, 528 – curvature tensorial family, 515 – flat, 519 – flow, 66 – – and the Poincar´e conjecture, 66 – identity, 14, 45 – lemma, 44, 45, 513 – – prototype, 612 – scalar curvature, 515 Ricci–Weyl antisymmetry relations, 505 Ricci-Curbastro (see Ricci), 7, 22, 439, 504 Richter, 895 Riemann, IX, 7, 15, 179, 439, 634, 777 – curvature matrices, 35 – curvature operator, 33, 46, 615 – – prototype on a sphere, 615 – curvature tensor, 11, 12, 503, 528 – – and integrability condition, 581 – – and the theorema egregium, 14 – – classical, 12 – – generalized, 49 – – geometric meaning, 30 – – metric, 33 – – notation, 528 – – prototype on a sphere, 616 – – symmetry properties, 14 – – Yang’s matrix trick, 35 – inaugural lecture, 34, 640 – integral, 329 Index – prize problem of the Paris Academy and the Riemann curvature tensor, 17 – surface – – arg-function, 688 – – logarithmic function, 688 Riemann–Christoffel curvature tensor, 515 – prototype, 581 Riemann–Ricci curvature tensor, 517 Riemann–Weyl curvature tensor, 503 Riemannian – geometry – – prototype on a sphere, 593 – manifold, 358, 526 – metric tensor – – construction, 1077 Riesz (Fryges), 106 – duality, 106, 170 – – basic idea, 89 – – operator, 773 Riesz (Marcel), 106 right translation, 357, 586, 804 rigid body – continuous, 421 – discrete, 388 – equations of motion, 393 ring (see Vol II), 310, 331 Rivasseau, XIII Rivlin–Ericksen theorem, 566 R-module, 310 Ră omer, XIII root, 247, 248 – functional, 248, 250 – system, 252 Rosso, 291 rotation – active, 374, 431 – infinitesimal, 374 – passive, 374, 431 row rank, 161 Rubbia, 895 Rudolph, XIII Runge, 280 Salam, 893, 894 Salmhofer, XIII Salmon, 440 Sard’s theorem, 721 Savart, 952 scalar curvature, 11, 528 Scholz, XIII Schrieffer, 836 1121 Schur, 285 – lemma, 202 – polynomial, 272 Schwartz (Laurent), 330, 691, 953 Schwartz (Melvin), 896 Schwinger, 895, 935 section, 588, 876 – prototype, 876 section of – cotangent bundle, 325 – frame bundle, 327 – tangent bundle, 52, 325 – tensor bundle, 326 sectional curvature, 33 secular equation, 199 Segr`e, 895 Seiberg–Witten theory, 59 self-adjoint operator (see Vol I), 199 semidirect product, 256 semisimple – group, 253, 255 – Lie algebra, 262 – representation, 190 separated topological space (Hausdorff space), 1070 Serre, 283 short exact sequence, 1004 SI (Syst`eme International); see the Appendix of Vol I, 936 Sibold, XIII sigma model, 521 sign of – diffeomorphism, 445 – permutation, 183 signature, 469, 924 similarity – operator, 719 – transformation, 199 Simons (Jim), simple – group, 93, 253 – Lie algebra, 259 – representation, 190 Singer, 5, 777 singular chain, 684 skein relation, 296, 297 skew-adjoint operator (see Vol I), 199 skew-field (see Vol II), 117 skew-symmetric (antisymmetric), 457 Slavnov–Taylor identities, 416 slope function, 758, 759 small divisors, 340 Smoot, 896 1122 Index smooth – function, 1070 – map, 1075 solenoid, 340 solvability condition – global, 761 – local, 761 solvable – group, 255 – Lie algebra, 263 space – reflection, 958 – – group, 922 space-like, 930 sphere – as a paradigm in Riemannian geometry and gauge theory, 593 – locally Euclidean geometry, 621 spherical – geometry, 599 – pendulum, 600 – triangle, 599 spherical coordinates, 594 – Cartan – – connection form, 618 – – curvature form, 618 – Christoffel symbols, 604 – geodesics, 604 – Lagrangian of the spherical pendulum, 602 – regular, 572 – Riemann curvature – – operator, 618 – – tensor, 618 – singular, 572 spin – addition theorem, 434 – geometry, 59 – operator, 948 – state, 428, 429 spinning top, 395 spinor calculus (see Vol IV), 486 split exact sequence, 1004 spray of – a vector field generated by a connection, 520 – an affine connection, 520 stability, 377 stagnation point, 55 standard Cartan subalgebra of sl(3, C), 249 Standard Model in particle physics, 854, 894 – history, 894 star algebra (∗-algebra), 117 state, 172 – space, 655 Stein (William), 339 Steinacker, XIII stereographic projection, 15 Stern, 895, 948 Stern–Gerlach experiment and the electron spin, 948 Stiefel–Whitney class, 898 – historical remark, 898 Stirling, 181 – formula, 181 stochastic differential equation, 352 Stokes, 722, 777, 952 – integral theorem, 680, 1035 – – discrete, 1026 – – generalized, 729, 782 – – history, 782 strain tensor, 723 streamline, 562 stress tensor, 726 – inner friction, 726 strictly positively oriented inertial system, 926 string theory, 606 strong interaction, 242, 894 structure constants of a Lie algebra, 83, 231, 808 subalgebra, 116 submanifold, 1076 – with boundary, 1076 substitution rule, 667, 673, 678 subvelocity of light, 911 Sullivan, 339 super – algebra, 287, 288 – antisymmetry, 289 – charge, 288 – commutativity, 288, 464, 468, 555 – conductivity, 836, 837 – Jacobi identity, 289 – Lie algebra, 288 – plane, 299 – space, 287 – symmetry, 287 – – basic idea, 287 – – further reading, 543 – – general strategy, 287 – – graduation, 289 – – Leibniz rule, 709 – – product rule, 703 Index – vector product, 289 – velocity of light, 911 surface – differential, 679 – element, 679 – Gaussian curvature, 628 – integral, 678 – theory, 678 surjective (see Vol I), 253 susceptibility – electric, 980 – magnetic, 980 Sylvester, 20, 418, 439 Sym(2) – chirality, 196 – helicity, 196 – parity, 195, 196 – reflection, 195 – representations, 194 Sym(3) representations, 183 symmetric – group Sym(n), 181, 182 – polynomial, 269 symmetrization, 73, 457 symmetry – breaking, 831 – group of a set, 205 – in mathematics and physics, 181 – of tensorial families, 457 – prototype, 181 symplectic – geometry (see also Vol II), 306 – – basic ideas, 751 – group, 306 – isomorphism, 751 – matrix, 306 – morphism, 751 synchronization of clocks, 912 system of – differential forms, 789 – physical units – – Gauss, 985 – – Heaviside, 985 – – SI (Syst`eme International); see the Appendix of Vol I, 936, 983 syzygy, 418 T (tesla), 984 t’Hooft, 895 tangent – bundle, 52, 325, 654 – – prototype, 655 – – triviality, 56 1123 – map, 655 – – global, 655 – – local, 656 – space, 25, 322, 701 tangent vector (velocity vector), 322, 530, 533, 586 – extrinsic, 24 – intrinsic, 24 tangent vector field (see velocity vector field), 24, 659 Tannaka–Krein duality, 370 Tartaglia, 90 Taubes’ proof of the Weinstein conjecture, 61 Taylor (Brook), 652 – expansion, 652 Taylor (Richard), 896 temperature field – basic ideas, 645 – conservation law, 652 – directional derivative, 645 – first variation, 652 – Fr´echet derivative, 653 – gradient, 645 – higher variations and the Taylor expansion, 652 – Lie derivative, 651 – pull-back, 657 – push-forward, 657 tempered distribution, 1046 tensor – algebra, 120 – bundle, 326 – calculus, 439 – – basic ideas, 10, 611 – – on vector bundles, 859 – component, 138 – field, 326 – field of type (m, n), 30 – of inertia – – ball, 421 – – circular cylinder, 421 – of type (m, n), 137 tensor product – antisymmetrized (or alternating), 118 – Ariadne’s thread, 118 – in elementary particle physics, 89 – of multilinear functionals, 118 – symmetrized, 118 tensorial – density family, 462 – differential form, 482, 861 1124 Index – transformation law, tensorial family – m-fold contravariant, 11 – n-fold covariant, 11 – antisymmetric (skew-symmetric), 457 – Cartan, 464 – contravariant, 453 – covariant, 453 – form-invariant, 453 – in Gauss’ surface theory, 625 – of type (m, n), 11 – on vector bundles, 861 – scalar, 446 – symmetric, 457 – type (r, s), 453 Tesla, 953 tetrad formalism, 40 TeV (tera electron volt), 940 Thales of Miletus, 936 theorem of principal axes, 21 theorema egregium of Gauss, IX, 50, 517, 615 – and gauge theory, 896 – and modern differential geometry, 896 – and the Riemann curvature tensor, 16 – history, 16 theory of – general relativity – – basic ideas, – special relativity, 905 thermodynamics, 698 Thom, 1035 Thompson, 256 Thomson (Joseph John), 948, 1011 Thomson (Lord Kelvin), 691, 731, 777 time – dilation, 911 – reflection, 958 – – group, 922 Ting, 895 Tolksdorf, XIII Tomonaga, 895, 935 topological – charge, 784 – equivalence, 1032 – group, 370 topological manifold – definition, 1072 topological quantum field theory (see TQFT), 1056 topological space (see Vol I), 1033, 1069 – arcwise connected, 1069 – compact, 1069 – paracompact, 1069 – prototypes, 1069 – separated, 1069 Tor (torsion product), 309 torque, 69, 376 torsion – operator, 46, 661 – product, 309 – tensorial family, 499, 585 torsion-free group, 307 torus, 1032 – potential equation, 1038 totally disconnected, 340 TQFT (topological quantum field theory), 1055 – basic idea, 1056 – Jones polynomial in knot theory, 1056 trace invariance, 238 traceless matrix, 247 transition – map of – – a principal bundle, 881 – – a vector bundle, 876 – probability, 349 translation – left, 586, 804 – right, 586, 804 transport theorem, 735 – and Noether theorem, 737 transposition, 76 trivector, 143 Turaev, 291 twin paradox, 927 twistor, 58 type (m, n) of a tensor field, 11, 485, 500 – m-fold contravariant, 11 – n-fold covariant, 11 Uhlmann, XIII unimodular matrix, 454 unit circle – de Rham cohomology groups, 1031 unit matrix, 77 unital algebra, 116, 313 unitarian trick, 253 unitary, 371 – matrix, 365 Index – operator (see Vol I), 199, 362 universal – covering group, 70, 108, 356, 426, 1085 – object, 124 – property, 122, 174 universality – hypothesis, 344 – renormalization trick, 345 unknot, 295 upper half-plane, 1073 – open neighborhood, 1073 V (volt), 944, 984 vacuum polarization, 951 valuation, 334 – non-Archimedian, 334 van der Meer, 895 van der Waerden, 487, 1060 Vandermonde – function, 269 – polynomial, 272 variation, 405, 652 – first, 403 – language of – – mathematicians, 404 – – physicists, 407 – of a function, 403 – of the action functional, 404 – second, 403 vector – history, 69 – potential, 766, 947 – product, 82 vector bundle – associated, 827, 873, 881 – axioms, 875 – cocycle strategy, 872 – connection, 823, 875, 877 – – on a sphere, 608 – curvature, 823, 850, 877 – definition, 876 – general strategy, 875 – intuitive prototype, 876 – on a sphere, 609 – parametrized linear algebra, 610 vector field (see velocity vector field), 586, 659 velocity – field function, 756 – of light in linear material, 981 – potential, 726 velocity vector (tangent vector), 31, 69, 321, 533, 586 1125 – extrinsic, 530 – intrinsic, 531 velocity vector field – as a section of the tangent bundle, 325 – basic ideas, 659 – complete, 648 – conservation law, 663 – curl, 559 – divergence, 559 – dual covector field (differential form), 665 – flow, 647 – Lie algebra, 663 – Lie derivative, 663 – Lie parallel, 663 – main theorem in vector calculus, 767 – pull-back, 662 – push-forward, 662 – sink, 562 – source, 562 – special examples, 562, 648 – topological structure, 55 – transformations, 661 Veltman, 895 Verch, XIII vertical tangent vector, 882 – prototype, 850 Vi`ete (Vieta), 270, 1060 Virasoro algebra, 29 virtual – acceleration, 375 – degree of freedom, 399 – power, 377 – trajectory, 375, 377 – velocity, 375 – work, 378 viscous fluid, 727 Volta, 952 voltage, 691, 1009, 1041 Volterra, 777 volume – product, 82 – tensorial family, 463 volume form, 85, 359, 469, 762 – prototype, 569 volume-preserving operator, 718 von Neumann, 200, 440 vortex line, 735 vorticity – flow, 728, 735 – line, 728 – theorem due to 1126 Index – – Helmholtz, 735 – – Thomson (later Lord Kelvin), 735 W (watt), 944, 984 Wagner, XIII Ward–Takahashi identities, 414 wave operator, 966 Wb (weber/magnetic flow), 985 Weber, 952 wedge product, 118, 464, 467, 702 – of matrices with differential forms as entries, 509, 584 weight, 248, 252, 429 – diagram, 231 – functional, 252 – pseudo-tensorial density family, 462 – tensorial density family, 462 – tuple, 232, 252 Weinberg, 837, 893, 894 Weinstein conjecture, 61 Weitzenbă ock, 520 – formula, 519 well-posed equation, 154 Wess, 347 Westminster Abbey, 691 Weyl, 20, 99, 358, 439, 486, 487, 777, 875, 1087 – conformal curvature tensor, 517 – derivative, 524 – duality, 525 – – and the electromagnetic field in materials, 979 – family, 524 – group, 252 – his personality, 900 – his work, 900 Weyl–Schouten tensor, 519 Wiener, 348 – process, 350 Wilczek, 244, 894 Wiles, 94 Wilson (Kenneth), 344 – loop, 829 – – functional, 1058 Wilson (Robert), 895 Witt, 281 Witten, 777 WMAP (Wilkinson Microwave Anisotropy Probe), 699 work, 69 Woronowicz, 347 Wu, 895 Wulkenhaar, XIII Wußing, XIII Yang, 4, 777, 895, 1087 – matrix trick, 35 Yang–Baxter equation, 6, 291 Yang–Mills equation, 6, 857 – classical, 698, 766 – – prototype, 774 – discrete, 1020 Yau, 777 Yngvason, XIII Young (Alfred), 285 – frame, 218 – – and irreducible representations, 222, 227 – – and partitions, 217 – – and quarks, 227 – – dual, 219 – – regular filling, 229 – standard tableaux, 219 – symmetrizer, 220, 269 – tableaux, 220, 222 – – and irreducible representations, 222 Yukawa, 814, 895 – meson theory, 816 Zariski, 1070 – topology in algebraic geometry, 1070 Zhang, Zorn’s lemma, 314 ... ΔΘ = −∂ Θ := − ture field Θ).1 • ΔE = −∂ · E := − ? ?2 ? ?2 x “ + ? ?2 ∂2x ? ?2 ∂2y + + ? ?2 ∂2y ? ?2 ? ?2 z + ? ?2 ? ?2 z Θ (Laplacian Δ applied to the tempera- ” E The definitions of grad Θ, div v, curl v, (v... + r(t) r(t )2 cos2 ϑ(t) · ϕ(t) ˙ + r(t )2 ϑ(t) ˙ dt dt = dt t0 t0 The inverse matrix of (gij ) reads as 1 g 11 g 12 g 13 (r cos ϑ)? ?2 0 B 21 22 23 C B C r ? ?2 0A @g g g A = @ 0 g 31 g 32 g 33 The... g = dx ⊗ dx + dy ⊗ dy + dz ⊗ dz = r2 cos2 ϑ dϕ ⊗ dϕ + r dϑ ⊗ dϑ + dr ⊗ dr Moreover, g = det(gij ) = r4 cos2 ϑ Mnemonically, ds2 = r2 cos2 ϑ d? ?2 + r2 d? ?2 + dr2 This tells us that the length of