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8 Rigorous Finite-Dimensional Perturbation Theory Perturbation theory is the most important method in modern physics Folklore 8.1 Renormalization In quantum field theory, a crucial role is played by renormalization Let us now study this phenomenon in a very simplified manner • We want to show how mathematical difficulties arise if nonlinear equations are linearized in the incorrect place • Furthermore, we will discuss how to overcome these difficulties by using the methods of bifurcation theory The main trick is to replace the original problem by an equivalent one by introducing so-called regularizing terms We have to distinguish between • the non-resonance case (N) (or regular case), and • the resonance case (R) (or singular case) In celestial mechanics, it is well-known that resonance may cause highly complicated motions of asteroids.1 In rough terms, the complexity of phenomena in quantum field theory is caused by resonances In Sect 7.16, the non-resonance case and the resonance case were studied for linear operator equations We now want to generalize this to nonlinear problems 8.1.1 Non-Resonance Consider the nonlinear operator equation H0 ϕ + κ(v0 + V (ϕ)) = Eϕ, ϕ ∈ X (8.1) We make the following assumptions (A1) The complex Hilbert space X has the finite dimension N = 1, 2, This is described mathematically by KAM theory (Kolmogorov–Arnold–Moser theory) As an introduction, we recommend Scheck (2000), Vol 1, and Thirring (1997) 498 Rigorous Finite-Dimensional Perturbation Theory (A2) The operator H0 : X → X is linear and self-adjoint Furthermore, H0 |Ej0 = Ej0 |Ej0 , j = 1, , N Here, the energy eigenstates |E10 , , |EN form a complete orthonormal system of X (A3) We set V (ϕ) := W (ϕ, ϕ, ϕ) for all ϕ ∈ X, where the given operator W : X × X × X → X is linear in each argument For example, we may choose V (ϕ) := ϕ|ϕ ϕ (A4) We are given the complex constant κ called the coupling constant, and we are given the fixed element v0 of the space X We are looking for an element ϕ of X Theorem 8.1 Suppose that we are given the complex number E different from the Then, there exist positive numbers κ0 and r0 such that, energy values E10 , , EN for each given coupling constant κ with |κ| ≤ κ0 , equation (8.1) has precisely one solution ϕ ∈ X with ||ϕ|| ≤ r0 Proof Equation (8.1) is equivalent to ϕ = −κ(H0 − EI)−1 (v0 + V (ϕ)), ϕ ∈ X The statement follows now from the Banach fixed-point theorem in Sect 7.13 ✷ In particular, the solution ϕ can be computed by using the following iterative method ϕn+1 = −κ(H0 − EI)−1 (v0 + V (ϕn )), n = 0, 1, (8.2) with ϕ0 := This method converges to ϕ as n → ∞ in the Hilbert space X For the first approximation, we get ϕ1 = −κ(H0 − EI)−1 v0 = κ N X |Ej0 Ej0 |v0 E − Ej0 j=1 (8.3) Let us discuss this (N) The non-resonance case (regular case) The expression (8.3) makes sense, since we assume that the parameter E is different from the eigenvalues E10 , , EN We say that the value E is not in resonance with the eigenvalues E10 , , EN Then, the Green’s operator (H0 − EI)−1 is well-defined Explicitly, (H0 − EI)−1 = N X |Ej0 Ej0 | Ej0 − E j=1 (R) The resonance case (singular case) The situation changes completely if we choose E := E10 Here, we say that the value E is in resonance with the eigenvalue E10 Then, the Green’s operator (H0 − E10 I)−1 does not exist, and the iterative method (8.2) above fails completely As a rule, ϕ1 is an infinite quantity Furthermore, if we set ε = 0, E := E10 + ε, 8.1 Renormalization then we obtain (H0 − EI)−1 = N X j=1 499 |Ej0 Ej0 | Ej0 − E10 − ε Since 1 = − lim = −∞, ε→+0 ε E10 − E some of the expressions arising from perturbation theory become very large if the perturbation ε is very small lim ε→+0 Summarizing, it turns out that Naive perturbation theory fails completely in the resonance case This situation is typical for the naive use of perturbation theory in quantum field theory In what follows, we will show how to obtain a rigorous result To this end, we will replace the naive iterative method (8.2) above by the rigorous, more sophisticated iterative method (8.12) below 8.1.2 Resonance, Regularizing Term, and Bifurcation Set E := E10 + ε Consider the nonlinear operator equation H0 ϕ + κV (ϕ) = Eϕ, ϕ ∈ X (8.4) In addition to (A1) through (A4) above, we assume that the energy eigenvalue E10 is simple, that is, the eigenvectors to E10 have the form |E10 where is an arbitrary nonzero complex number We are looking for a solution (ϕ, E) of (8.4) with ϕ ∈ X and E ∈ C The proof of the following theorem will be based on the use of regularizing terms Theorem 8.2 There exist positive constants κ0 , s0 , η0 and r0 such that for given complex parameters κ and s with |κ| ≤ κ0 , < |s| ≤ s0 , equation (8.4) has precisely one solution ϕ, E which satisfies the normalization condition E10 |ϕ = s and the smallness conditions |E − E10 | ≤ η0 and ||ϕ|| ≤ r0 Before proving this, let us discuss the physical meaning of this result We will show below that the zeroth approximation of the solution looks like ϕ = s|E10 , E = E10 The first approximation of the energy is given by E = E10 + κs2 Greg V (ψ1 )|ψ1 where we set ψ1 := |E10 Observe the following point which is crucial for understanding the phenomenon of renormalization in physics From the mathematical point of view, we obtain a branch of solutions which depends on the parameter s The free parameter s has to be determined by physical experiments 500 Rigorous Finite-Dimensional Perturbation Theory Let us discuss this Suppose that we measure the • energy E and • the running coupling constant κ We then obtain the approximation κ= E − E10 s2 Greg V (ψ1 )|ψ1 This tells us the value of the parameter s This phenomenon is typical for renormalization in quantum field theory The energy E10 is called the bare energy However, this bare energy is not a relevant physical quantity In a physical experiment we not measure the bare energy E10 , but the energy E and the running coupling constant κ In elementary particle physics, this corresponds to the fact that the rest energy of an elementary particle (e.g., an electron) results from complex interaction processes Therefore, the rest energy E differs from the bare energy E10 In the present simple example, interactions are modelled by the nonlinear term κV (ϕ) Proof of Theorem 8.2 (I) The resonance condition To simplify notation, set ψj := |Ej0 , j = 1, , N For given χ ∈ X, consider the linear operator equation H0 ϕ − E10 ϕ = χ, ϕ ∈ X (8.5) By Theorem 7.15 on page 376, this problem has a solution iff the so-called resonance condition ψ1 |χ = is satisfied The general solution is then given by ϕ = sψ1 + N X j=2 ψj |χ ψj Ej0 − E10 (8.6) where s is an arbitrary complex parameter (II) The regularized Green’s operator Greg Set P ϕ := ψ1 |ϕ ψ1 The operator P : X → span(ψ1 ) projects the Hilbert space X orthogonally onto the 1-dimensional eigenvector space to the energy eigenvalue E10 We now consider the modified equation H0 ϕ + P ϕ − E10 ϕ = χ, ϕ ∈ X (8.7) Theorem 7.16 on page 377 tells us that, for each given χ ∈ X, equation (8.7) has the unique solution ϕ = (H0 + P − E10 I)−1 χ We define Greg := (H0 + P − E10 I)−1 Explicitly,2 ϕ = Greg χ = ψ1 |χ ψ1 + N X j=2 ψj |χ ψj Ej0 − E10 In particular, Greg ψ1 = ψ1 The term P ϕ = ψ1 |ϕ ψ1 in (8.7) is called regularizing term (III) The trick of regularizing term The original equation (8.4) on page 499 can be written equivalently as P In fact, (H0 + P − E10 I)ϕ is equal to ψ1 |χ ψ1 + N j=2 ψj |χ ψj = χ 8.1 Renormalization H0 ϕ − E10 ϕ + κV (ϕ) + ψ1 |ϕ ψ1 = sψ1 + εϕ, ϕ∈X 501 (8.8) along with the normalization condition ψ1 |ϕ = s (8.9) By (II), this is equivalent to the operator equation ϕ = Greg (sψ1 − κV (ϕ) + εϕ) along with (8.9) Finally, since Greg ψ1 = ψ1 , we obtain the equivalent operator equation ϕ = sψ1 − κGreg V (ϕ) + εGreg ϕ (8.10) along with (8.9) We have to solve the system (8.9), (8.10) To this end, we will use both a rescaling and the Banach fixed-point theorem (IV) Rescaling Set ϕ := s(1 + ε)ψ1 + sχ Equation (8.9) yields s ψ1 |ψ1 + εψ1 + χ = s Since s = and ψ1 |ψ1 = 1, we get ε = − ψ1 |χ Furthermore, it follows from (8.10) that s(1 + ε)ψ1 + sχ = sψ1 − κs3 Greg V ((1 + ε)ψ1 + χ) +sε(1 + ε)ψ1 + sεGreg χ Consequently, the system (8.9), (8.10) corresponds to the following equivalent system χ = A(χ, ε, κ, s), ε = − ψ1 |A(χ, ε, κ, s) , χ ∈ X, ε ∈ C (8.11) along with A(χ, ε, κ, s) := −κs2 Greg V ((1 + ε)ψ1 + χ) + εGreg χ + ε2 ψ1 (V) The Banach fixed-point theorem The system (8.11) represents an operator equation on the Banach space X × C with the norm ||(χ, ε)|| := ||χ|| + |ε| We are given the complex parameters s and κ with < |s| ≤ s0 and |κ| ≤ κ0 where s0 > and κ0 > are sufficiently small numbers By the Banach fixed-point theorem in Sect 7.13 on page 366, there exists a small closed ball B about the origin in the Banach space X × C such that the operator equation (8.11) has a unique solution in the closed ball B (V) Iterative method By the Banach fixed-point theorem, the solution (χ, ε) of (8.11) can be computed by using the following iterative method χn+1 = A(χn , εn , κ, s), εn+1 = − ψ1 |A(χn , εn , κ, s) , n = 0, 1, 2, (8.12) with χ0 := and ε0 := This method converges in the Banach space X × C In particular, we get χ1 = −κs2 Greg V (ψ1 ), ε1 = κs2 ψ1 |Greg V (ψ1 ) ✷ Bifurcation On the product space X × C, the original nonlinear problem (8.4) on page 499 has two different solution curves, namely, 502 Rigorous Finite-Dimensional Perturbation Theory • the trivial solution curve ϕ = 0, E = arbitrary complex number, • and the nontrivial solution curve (ϕ = ϕ(s, κ), E = E(s, κ)) given by Theorem 8.2 on page 499 The two curves intersect each other at the point ϕ = 0, E = E10 Therefore, we speak of bifurcation The nontrivial solution branch of equation (8.4) represents a perturbation of the curve ϕ = sψ1 , E = E10 , s∈C which corresponds to the linearized problem H0 ϕ = E10 ϕ Bifurcation theory is part of nonlinear functional analysis A detailed study of the methods of bifurcation theory along with many applications in mathematical physics and mathematical biology can be found in Zeidler (1986) 8.1.3 The Renormalization Group The method of renormalization group plays a crucial role in modern physics Roughly speaking, this method studies the behavior of physical effects under the rescaling of typical parameters We are going to study a very simplified model for this Let (ϕ(s, κ), E(s, κ)) be the solution of the original equation (8.4) on page 499, that is, H0 ϕ(s, κ) + κV [ϕ(s, κ)] = E(s, κ)ϕ(s, κ) along with ψ1 |ϕ(s, κ) = s Choose the fixed real number λ > Replacing s → λs and κ → λκ2 , we get “ “ κ” κ ”i κ h “ κ” “ κ” = E λs, ϕ λs, H0 ϕ λs, + V ϕ λs, λ λ λ λ λ ´ ` κ along with ψ1 |ϕ λs, λ2 = λs Define ψ(s, κ) := “ κ” · ϕ λs, λ λ Noting that V (λψ) = λ3 V (ψ), we obtain “ κ” H0 ψ(s, κ) + κV [ψ(s, κ)] = E λs, ψ(s, κ) λ along with ψ1 |ψ(s, κ) = s By the uniqueness statement from Theorem 8.2 on page 499, we get “ κ” · ϕ λs, = ϕ(s, κ) λ λ (8.13) “ κ” E λs, = E(s, κ) λ (8.14) along with for all nonzero complex parameters s and κ in a sufficiently small neighborhood of the origin 8.1 Renormalization 503 Summarizing, the homogeneity of the potential, V (λϕ) = λ3 V (ϕ), implies the symmetries (8.13), (8.14) of the solution branch Differentiating equation (8.13) with respect to the parameter λ, and setting λ = 1, we obtain ϕ(s, κ) − sϕs (s, κ) + 2κϕκ (s, κ) = (8.15) In our model, the differential equation (8.15) can be regarded as a simplified version of the Callan–Szymanzik equation in quantum field theory × Let R× + denote the set of all positive real numbers; that is, x ∈ R+ iff x > 2 , define the map T : C → C given by For each parameter λ ∈ R× λ + “ κ” Tλ (s, κ) := λs, λ For all parameters λ, µ Rì +, Tà = T Tà Therefore, the family {Tλ }λ∈R× of all operators Tλ forms a group This group is + called the renormalization group of the original operator equation (8.4) on page 499 8.1.4 The Main Bifurcation Theorem Let us now study the general case of the nonlinear equation H0 ϕ + κV (ϕ) = Eϕ, ϕ∈X (8.16) where the eigenvalue E10 of the linearized problem H0 ϕ = E10 ϕ is not simple as in Sect 8.1.2, but it has general multiplicity To this end, we will reduce the problem to the nonlinear system (8.17) below We make the following assumptions (A1) The complex Hilbert space X has the finite dimension N = 1, 2, (A2) Linear operator: The operator H0 : X → X is linear and self-adjoint Furthermore, j = 1, , N H0 |Ej0 = Ej0 |Ej0 , Here, the energy eigenstates |E10 , , |EN form a complete orthonormal system of X (A3) Multiplicity: The eigenvalue E10 has the multiplicity m, that is, the eigenvec0 form a basis of the eigenspace of H0 to the eigenvalue E10 tors |E10 , , |Em Let ≤ m < N To simplify notation, set ψj := |Ej0 Define the orthogonal projection operator P : X → X by setting P ϕ := m X ψj |ϕ ψj for all ϕ ∈ X j=1 (A4) Nonlinearity: We set V (ϕ) := W (ϕ, ϕ, ϕ) for all ϕ ∈ X, where the given operator W : X × X × X → X is linear in each argument For example, V (ϕ) := ϕ|ϕ ϕ 504 Rigorous Finite-Dimensional Perturbation Theory (A5) Resonance condition: The nonlinear equation3 σ ∈ P X, κ ∈ C σ = κP V (σ), (8.17) has a solution (σ0 , κ0 ) where σ0 = and κ0 = This solution is regular, that is, the linearized equation h = κ0 P · V (σ0 )h, h∈X (8.18) has only the trivial solution h = Theorem 8.3 There exists a number α0 > such that for each given complex number α with |α| ≤ α0 , the nonlinear problem (8.16) with the coupling constant κ0 has a solution ϕ = ασ0 + O(α2 ), α → E = E10 + α2 , Proof (I) The regularized Green’s operator For all χ ∈ X, define Greg χ := m X N X ψj |χ ψj + j=1 j=m+1 ψj |χ ψj Ej0 − E10 Suppose that we are given χ ∈ X with P χ = By Theorem 7.16 on page 377, the equation H0 − E10 = χ has precisely one solution ∈ X with P = This solution is given by N X = Greg χ = j=m+1 ψj |χ ψj Ej0 − E10 (II) Equivalent system Set E := E10 +ε, and introduce the orthogonal projection operator Q := I − P Explicitly, Qϕ = N X ψj |ϕ ψj for all ϕ ∈ X j=m+1 Then, the original nonlinear problem (8.16) is equivalent to Q(H0 ϕ − (E10 + ε)ϕ + κV (ϕ)) = 0, P (H0 ϕ − (E10 + ε)ϕ + κV (ϕ)) = (8.19) The idea of the following proof is Set σ := s1 ψ1 + + sm ψm Equation (8.17) is then equivalent to the system gj (s1 , , sm ; κ) = j = 1, , m, s1 , , sm ∈ C Here, gj (s1 , , sm ; κ) := sj − κ · ψj |V (s1 ψ1 + + sm ψm ) Condition (8.18) means that « „ ∂gj (s01 , , s0m ; κ0 ) |j,k=1, ,m = det ∂sk 8.1 Renormalization 505 (i) to solve the first equation from (8.19) by the Banach fixed-point theorem, (ii) to insert the solution from (i) into the second equation from (8.19), and (iii) to solve the resulting equation by using the implicit function theorem near the solution (σ0 , κ0 ) of equation (8.17) For each ϕ ∈ X, define χ := P ϕ and := Qϕ Then ϕ=χ+ , χ ∈ P X, ∈ QX Therefore, system (8.19) is equivalent to Q{H0 (χ + ) − (E10 + ε)(χ + ) + κV (χ + )} = 0, P {H0 (χ + ) − (E10 + ε)(χ + ) + κV (χ + )} = Observe that ψj |H0 ϕ − E10 ϕ = H0 ψj − E10 ψj |ϕ (8.20) = for j = 1, , m Hence P (H0 − E10 I) = Furthermore, P χ = χ, Qχ = and Q = , P = Thus, choosing the coupling constant κ := κ0 and recalling that Q := I − P , the system (8.20) is equivalent to H0 − E10 = ε − κ0 QV (χ + ), εχ = κ0 P V (χ + ) (8.21) Finally, using the regularized Green’s operator, this system is equivalent to the equation = εGreg − κ0 Greg QV (χ + ) (8.22) along with εχ = κ0 P V (χ + ) (8.23) (III) Rescaling We set χ := ασ and ε := α2 Equation (8.22) passes then over to = α2 Greg − κ0 Greg QV (ασ + ) (8.24) (IV) The Banach fixed-point theorem By Theorem 7.12 on page 367, there exist positive parameters α0 , β0 and r0 such that for given α ∈ C and σ ∈ P X with |α| ≤ α0 , ||σ|| ≤ β0 , equation (8.24) has precisely one solution ∈ QX with || || ≤ r0 This solution will be denoted by = (α, σ) By the analytic form of the implicit function theorem,4 the components of (α, σ) depend holomorphically on the complex parameter α The iterative method n+1 with = α2 Greg n − κ0 Greg QV (ασ + n ), := (or comparison of coefficients) shows that This can be found in Zeidler (1986), Vol I, Sect 8.3 n = 0, 1, 506 Rigorous Finite-Dimensional Perturbation Theory (α, σ) = −α3 κ0 Greg QV (σ) + O(α4 ), α → (V) The bifurcation equation Inserting (α, σ) into equation (8.23), we get α3 σ = κ0 P V (ασ + (α, σ)) Dividing this by α3 , we obtain the so-called bifurcation equation α → σ = κ0 P V (σ) + O(α), (8.25) For α = 0, this equation has the solution σ = σ0 , by assumption (A5) Choose h ∈ X Differentiating the equation σ0 + th = κ0 P V (σ0 + th) with respect to the real parameter t at t = 0, we get h = κ0 P · V (σ0 )h (8.26) This is the linearization of (8.25) at the point σ = σ0 , α = By assumption (A5), equation (8.26) has only the trivial solution h = By the implicit function theorem, the bifurcation equation (8.25) has a solution of the form σ = σ0 + O(α), α → ✷ Modification If the resonance condition (A5) above is satisfied for the modified equation σ = −κP V (σ), σ ∈ P X, κ ∈ C, then Theorem 8.3 remains true if we replace E = E10 + α2 by E = E10 − α2 8.2 The Rellich Theorem Let X be a complex Hilbert space of finite dimension N = 1, 2, Consider the eigenvalue equation Aϕ = λϕ, λ ∈ R, ϕ ∈ X \ {0} along with the perturbed problem A(ε)ϕ(ε) = λ(ε)ϕ(ε), λ(ε) ∈ R, ϕ(ε) ∈ X \ {0} where ε is a small real perturbation parameter, and A(0) = A We assume the following (H1) The linear operator A : X → X is self-adjoint (H2) There exists an open neighborhood U (0) of the origin of the real line such that for each ε ∈ U (0), the operator A(ε) : X → X is linear and self-adjoint, and it depends holomorphically on the parameter ε Explicitly, A(ε) = A + εA1 + ε2 A2 + This means that for each arbitrary, but fixed basis |1 , , |N of the space X, all of the matrix elements m|A(ε)|n are power series expansions which are convergent for all real parameters ε ∈ U (0) 1006 Index – energy, 759 – product, 336 Institute for Advanced Study in Princeton, 67 integral, 528 – equation, 386 – on Riemann surfaces, 220 integration – by parts, 544 – over orbit spaces, 886 – tricks, 640 interaction – four fundamental forces in nature, 127 – picture, 43, 394, 748 – – Haag’s theorem, 748 – see also gauge field theory, 882 International Congress of Mathematicians (ICM), 71 interplay between mathematics and physics, 913 intersection number, 230 introductory literature on quantum field theory, 907 invariant theory, 363 inverse – Laplace transform, 372 – map, 932 inversion with respect to the unit sphere, 562 irreducible vertex function, 751 irreversible, 165, 179 isolated pole, 510 isometric operator, 338 isomorphic Hilbert spaces, 338 isomorphism, 330, 343 isospin, 153 – number, 154 isotope, 150 iterative method, 366 Ito, 74 Ivanenko, 112, 128 Jacobi, 19, 30, 257, 309, 549, 721 – inverse problem, 549 Jacobian, 273 Jaffe, 20, 78, 871, 912 Janke, XI Jensen, 70 John, 213 Joliot, 69 Joliot-Curie, 69 Jones, 72 – polynomial, 264 Joos, 99 Jordan – Camille, 240 – curve, 240 – – theorem, 240 – Pascual, 29, 49, 64 Jordan–Wigner bracket, 56 Jorgenson, 257 Josephson, 70 Jost Jă urgen, XI, 189 Res, 173, 223, 958 Joule, 23 Joyce, 98 KacMoody algebra, 920 Kadano, 73 Kă ahler, 956 – manifold, 72, 275, 920 KAM (Kolmogorov, Arnold, Moser), 650 – theory, 288, 497, 650 Kammerlingh-Onnes, 70 Kant, VII Kapusta, 757 Kastler, 819, 867, 911 Keller – Gottfried, XI – Joseph, 74 Kelvin, 540, 545, 717 – transformation, 562 Kendall, 70 Kepler, 1, 180, 345, 945 kernel theorem, 681 ket symbol, 359 Ketterle, 70, 684 Killing form, 885 Kirby, 71 Kirchhoff, 24, 103, 724 Kirchhoff–Green representation formula, 724 Klein – Felix, VII, 19, 60, 241, 245, 363, 550 – Oscar, 116 Klein–Gordon equation, 459, 712, 752, 806, 808, 814, 865 Klein–Nishina formula, 116, 119 Kleppner, 73 Klima, 739 Kline, 244, 545 KMS (Kubo, Martin, Schwinger), 742 – state, 742 knot Index – classification, 262 – theory, 265 Kodaira, 71, 74 Koebe, 19 Kohn, 70, 153, 569 Kolmogorov, 74, 497, 528, 650 – law in turbulence, 945 Kontsevich, 72, 252, 904 Koshiba, 73 Kostyuchenko, 525 Kramers, 61 Kramers–Kronig dispersion relation, 701 Kreimer, 859 – Hopf algebra, 741, 859 Krein, 74 Kroemer, 71 Kronecker, 255 – integral, 255 – symbol, 56, 354, 933 – – generalized, 591 Kroto, 70, 245 Kummer, 958 Kusch, 70 Lafourge, 72 Lagrange, 28, 383, 546, 547, 650 Lagrangian, 30 – and the principle of critical action, 30 – approach to physics, 47 – density, 752, 774, 792, 797, 805 – multiplier, 487, 796, 877 Lamb, 69 Landau – Edmund, 67, 932 – Lev, 60, 70, 283 – symbol, 932 Landau–Ginzburg potential, 180 Lang, 257 Langlands, 74 Laplace, 89, 252, 257, 283, 372, 542, 555 – transform, 89, 92, 286, 290 – – discrete, 283, 285, 289 Laplace transform, 377 – discrete, 285 Laplacian, 258, 542, 555, 559, 940 laser, 126 lattice, 669 – approximation, 815 – gauge theory, 204, 576 Laue, 69 Laughlin, 70 1007 Lawrence, 69 laws of progress in theoretical physics, 79 Lax, 75 Le Verrier, 111 least-squares method, 352 Lebesgue, 526, 527 – integral, 530 – measure, 530 Lebowitz, 190 Lederman, 70, 73 Lee, 3, 70 left-handed neutrino, 145 left-invariant vector field, 901 Legendre, 292, 958 – transformation, 458, 483 Leggett, 70, 73 Lehmann, 439, 765 Leibniz, 1, 101, 252, 334, 395, 396, 545, 575, 576 Lenard, 1, 26, 69, 112 lepton, 130 – number, 154, 156 Leray, 74, 230, 398 Leucippus, 98 Lewis, 26, 112 Lewy, 74 LHC (Large Hadron Collider), 139 l’Huilier, 245 Libchaber, 73 Lichtenstein, 67 Lie, 24, 29, 60, 197, 345, 409, 721 – algebra, 24, 342 – – so(3), su(2), 267, 343 – – u(X), su(X), gl(X), sl(X), 343 – – basis of, 883 – – isomorphism, 343 – – morphism, 342 – – structure constants of, 883, 884 – bracket, 48, 56, 342 – – product, 267, 342 – functor, 14 – group, 347 – – SO(3), U (n), SU (n), Spin(3), 267, 342 – – U (X), SU (X), GL(X), SL(X), 342 – – basic ideas, 199 – – isomorphism, 348 – – morphism, 348 – – one-parameter, 199, 414 – linearization principle, 348 – subalgebra, 343 – theory for differential equations, 199 1008 Index Lieb, lifetime, 89, 378, 380 – of a black hole, 143 – of elementary particles, 134 light – cone, 713 – particle (photon), 26 – ray, 723 – wave, 85 LIGO (Laser Interferometer Gravitational-Wave Observatory), 137 limits in physics, 682 linear – functional, 332, 349 – hull, 329 – isomorphism, 330 – material, 697 – morphism, 330 – operator, 330 – response and causality, 701 – response theory, 701 – space, 329 – subspace, 329 link, 262 linking number, 252 – and magnetic fields, 251 Lions, 72 Liouville, 525 – theorem, 218 Lippmann, 29 Lippmann–Schwinger integral equation, 40, 724 Lipschitz, 554 – continuous, 554 – space, 554 Lipschitz-continuous boundary, 546 LISA (Laser Interferometer Space Antenna), 137 Listing, 252 Littlewood, 286 local – degree of homogeneity, 621 – functional derivative, 591, 750 – properties of the universe, 227 – symmetry, 174 local-global principle, 218 locality, 869 locally – holomorphic, 210, 219 – holomorphic at ∞, 217 logarithmic – function, 220 – matrix function, 346 Lojasewicz, 646 loop, 240 Lorentz, 69 – boost, 867 – condition, 795 – transformation, 110 Lovasz, 74 Low, 688, 765 low-energy limit, 850 lower – half-plane, 662 – semicontinuous, 569 LSZ (Lehmann, Szymanzik, Zimmermann), 439, 765 – axiom, 785 – reduction formula, 444, 449, 482, 487, 746, 765, 767, 784, 803 Luria, 71 Lyapunov–Schmidt method, 631 Maclaurin, 309 macrocosmos, 227 magic – Dyson S-matrix formula, 822 – Dyson series for the propagator, 388 – Faddeev–Popov formula, 888 – Feynman formula, 417, 762 – formulas for the Green’s operator, 372 – Gell-Mann–Low formula, 427, 845 – LSZ reduction formula, 449, 482, 487, 746, 765, 767, 784, 803 – quantum action formula, 448 – quantum action reduction formula, 746, 765, 767, 782, 802 – survey on magic formulas, 326, 765 – trace formula, 756 – Wick formula, 425 – zeta function formula, 434 magnetic – field, 696 – field constant, 718, 847 – – of a vacuum µ0 , 696, 936 – intensity, 696, 699 – moment, 150 – – anomalous, 148 – – of the electron, – – of the myon, – monopole, 699 – quantum number, 180 – susceptibility, 697 magnetism, 150 magnetization, 696, 699, 952 Index Maiman, 73 majorant criterion, 493, 529 Mandelbrot, 73 manifold, 234, 235 – complex, 235 – oriented, 235 – with boundary, 546 Manin, 916, 978 Mann, 178 mapping degree, 228 – and electric fields, 253 Marathe, 13, 252, 263 Marcolli, 860 Marczewski, 67 Margulis, 72, 74 Maslov, 703 – index, 431 mass – density, 591 – hyperboloid, 462 – of a relativistic particle, 25 – shell, 462, 634, 713 mathematical physics, 13 matrix – algebra, 340 – calculus, 334, 345, 346 – elements, 355 – – of an operator, 334 – group, 340 – mechanics, 64, 65 – rules, 340 Maupertius, 30 Maurin, 67, 567, 913 maximum principle, 259 Maxwell, 25, 100, 252, 282 – equations, 171, 718, 809, 939 – – for material media, 696 McMullen, 72 mean – energy, 38 – field approximation, 456, 751 – fluctuation, 350, 759 – inner energy, 759 – lifetime, 377, 378 – particle number, 759 – value, 34, 350, 528, 759 mean-square convergence, 531 measurable function, 528 measure, 528 – integral, 416, 528, 602 – zero, 531 measurement of an observable, 354, 758 mega, 935 1009 – electron volt (MeV), 944 Mellin, 290, 309 – transform, 261, 290, 305, 665 – – generalized, 305 – – normalized, 290 Mendeleev, 150 meridian, 248 meromorphic, 213 meson, 133, 156 – model, 459, 773, 865 messenger particle, 130, 131 method of – least squares, 352, 532 – orthogonal projection, 564, 567 – quantum fluctuations, 655 – second quantization, 52 – stationary phase, 32, 430, 435, 714 metric tensor, 248 Meyer, 150 Michel, 71 Michelson, 25, 69 micro, 935 microcosmos, 227 microlocal analysis, 703 microstructure, 189 Mikusi´ nski calculus, 288 millennium prize problems, 77 milli, 935 millibarn, 128 Millikan, 26, 69, 112 Mills, 181, 185, 249 Milnor, 71, 74 minimal surface, 811 Minkowski, 24, 769, 958 – metric, 768, 934 – space, 769 – symbol, 769, 933 Minkowskian versus Euclidean model, 864 mirror symmetry, 703, 914, 920 Mittag–Leffler theorem, 510 modular – curve, 19 – form, 284, 321 – function, 18 moduli space, 14, 223, 552, 920, 923 modulus, 209, 932 Mă obius, 241 Mă oòbauer, 70 eect, 113 moment of a probability distribution, 58 – problem, 751 1010 Index – trick, 432 momentum, 35, 145 – operator, 675 – – on the real line, 33 monomorphism, 343 monster group, 920 monstrous moonshine module, 920 Montesquieu, 730 moon landing, 488 Mori, 72 morphism, 330, 343 Morse, 251 – index, 250 – theorem, 250 – theory, 250 Moser, XI, 74, 497, 650 motivic Galois group, 860 Mă uller Karl, 70 Stefan, XI, 189 multi-grid method, 568 multi-index, 536 multilinear functional, 332, 333 multiplicity, 503 Mumford, 71 muon, 130 – lepton number, 154 Nambu, 73, 923 nano, 935 Napier, 180, 345 NASA, 80 natural – number, 932 – SI units, 937, 950 Navier, 948 Navier–Stokes equations, 78, 948 Ne’eman, 98 N´eel, 70 negative – energy, 378 – real number, 932 neighborhood, 236 – open, 236 net of local operator algebras, 742 Neumann – Carl, 541 – John von (see von Neumann), 21 neutrino, 128–130 – mass, 145 – oscillations, 145 neutron, 100 Nevanlinna prize in computer sciences, 72 Newman, 293 – adiabatic theorem, 293, 686 Newton, 28, 100, 101, 127, 395, 396, 545, 575, 945 – equation of motion, 35 – polygon, 649 – potential, 555 Nirenberg – Louis, 703 – Marshall, 71 Nishina, 116 Nobel prize – in chemistry, 69 – in physics, 69 Noether, 22, 29, 60, 67, 892 – theorem (see also Vol II), 22, 31 non-degenerate ground state (vacuum), 424 non-positive, 932 non-relativistic approximation, 690 non-resonance, 374, 498 non-standard analysis, 397 noncommutative geometry, 139, 874 nonnegative, 932 norm, 337, 366 normal product, 822, 824 – principle, 824 normalization volume, 669 normalized state, 350 normed space, 366 notation, 931 Novikov, 71, 74 Nozieres, 73 nucleon, 129 null Lagrangian, 797, 807 observable, 38, 350 observer, 354 Occhialini, 73 one-parameter Lie group, 199, 414 one-to-one, 932 Onsager, 70 open, 339 – neighborhood, 236 – set, 236 – upper half-plane, 662 operator – algebra, 332 – approach, 746, 813 – calculus, 37 – function, 357, 367 Oppenheimer, 60 orbifold, 922 Index orbit, 269 – space, 269 oriented manifold, 235 orthogonal matrix, 341 orthogonality, 354, 564 – relation, 669 orthonormal – basis, 337 – system, 355 – – complete, 355 orthonormality condition – continuous, 693 – discrete, 356 oscillating integral, 666, 715 Osterwalder, 866 Osterwalder–Schrader axioms, 866 Ostrogradski, 546 Ostwald’s classic library, 216 paired normal product, 825 Paley–Wiener–Schwartz theorem, 661 panorama of – literature, 907 – mathematics, 15 parallel – of latitude, 247 – transport, 184, 249 Parasiuk, 853, 854 parity, 154, 159 – transformation, 172 – violation, 165 Parseval – de Ch´enes, 356 – equation, 356, 359, 532, 535, 536 partial – derivative, 536 – functional derivative, 401, 403, 750 particle – density, 34 – stream, 34 partition – function, 15, 106, 279, 283, 757 – functional, 757 Pascal, 958 path integral, 32, 57, 416, 651 Paul, 70 Pauli, 64, 70, 128, 129 – exclusion principle, 147, 148, 151, 161, 176 – matrices, 790, 885 – spin-statistics principle, 147, 148 Pauli–Villars regularization, 637, 853 Pauling, 70 1011 pendulum, 944 Penrose, 73 Penzias, 69, 113 period, 81 periodic table of chemical elements, 150 Perl, 70, 73 Perrin, 69 perturbation theory, 45, 180, 390, 747, 754 – software systems, 929 peta, 935 P´eter, 955 phase – shift, 82 – transition, 683 – – in the early universe, 684 – velocity, 82 Phillips, 70 photoelectric effect, 25, 26 photon, 26, 112, 130, 133 physical – mathematics, 13, 75, 263 – states, 893 physics, basic laws in, 936 Picard–Lefschetz theorem, 644 pico, 935 picture – Feynman path integral, 31 – Feynman propagator kernel, 50 – Heisenberg particle, 42, 48 Schră odinger wave, 35, 49 von Neumann operator, 37 piecewise smooth boundary, 546 Piguet, 858 Planck, 22, 26, 70, 98, 101, 283, 739, 936 – charge, 937 – constant, 23, 108 – energy, 937 – function and number theory, 105 – length, 937 – mass, 937 – quantization rule, 23 – quantum of action h, 23, 108, 936 – radiation law, 98, 102 – reduced quantum of action, = h/2π, 936 – scale, 934 – system of units, 934, 938 – temperature, 937 – time, 937 plane wave, 719 Plato, 244 1012 Index Platonic solids, 244 Poincar´e, 19, 21, 29, 227, 541, 547, 724, 892 – conjecture, 78 – group, 867 – hairy ball theorem, 246 – lemma, 397, 895 Poincar´e–Friedrichs inequality, 571 Poincar´e–Hopf index, 246 Poincar´e–Hopf theorem, 245 Poincar´e–Lindstedt series, 864 Poisson, 28, 309, 542, 555 – approach, 48 – bracket, 47, 48 – equation, 541, 542, 555, 698, 711 – summation formula, 310 polarization, 85, 696, 698, 952 Polchinski equation, 512, 513, 873 pole, 213, 510 Politzer, 70, 201 Polyakov, 923 Pontryagin, 721 Pople, 70, 153 position operator, 675 – on the real line, 33 positive real number, 932 post-Newtonian approximation, 947 potential, 35, 894 Pound, 113 Powell, 69 power series expansion, 209 pre-Hilbert space, 337 pre-image, 931 pressure, 759 Prigogine, 70 prime number, 291 – theorem, 291, 292 principal – argument, 209 – axis theorem, 357 – branch, 220 – part of the square root, 83 – symbol, 710 – value, 88, 618, 702 principle of – critical action, 30, 402, 409, 445, 460, 752, 774, 792, 804 – – summary, 804 – – under constraints, 489 – critical constraint, 490 – general relativity, 111 – indistinguishable particles, 148, 161 – least action, 406, 409 – special relativity, 110 – the right index picture, 770 probability, 368 – distribution function, 369 Prochorov, 69, 126 propagation of singularities, 710 propagator, 383, 419, 577, 582 – advanced, 582 – equation, 384 – retarded, 582 pseudo-differential operator, 728 pseudo-limit, 12 pseudo-resolvent, 377, 629 Puiseux expansion, 649 punctured open neighborhood, 213 Pythagoras, 17, 567 Pythagorean theorem, 567 QA (quantum action reduction formula), 444, 765, 767 QA axiom, 784 QCD (quantum chromodynamics), 133, 880 QED (quantum electrodynamics; see also Vol II), 789, 792, 811, 846, 879, 886 quadrupole moment, 698 quantization, 32 – Batalin–Vilkovisky, 903 – in a nutshell, 26 – of phase space, 838 – second, 52 quantum – chemistry, 153 – computer, 126 – fluctuation, 32, 655, 792 – gravity, 139, 929 – information, 126 – of action, 98, 108, 140 – of action (reduced), 140 – particle, 29 – state, 269 – statistics, 283 – symmetry, 904 quantum action – axiom, 783, 803 – functional, 782, 803 – – extended, 486, 782, 802 – principle – – global, 447, 753, 787 – – local, 452, 787 – – prototype, 432 – reduction formula, 444, 746, 765, 767, 783 Index quantum chromodynamics, 576, 880 quantum electrodynamics (see also QED), 789 quantum field, 52 – Bogoliubov’s formula, 857 – classical, 858 – creation and annihilation operators (see also Vol II), 55 – free, 743 – full, 743, 748 – generalized, 858 – trouble with interacting quantum fields, 748 quantum field theory – algebraic, 866 – – Haag–Kastler approach, 742, 866, 911 – – Hadamard states, 742 – – Kubo–Martin–Schwinger (KMS) states, 742 – – survey on quantum gravity, 912 – – Tomita–Takesaki theory for von Neumann algebras, 742 – Ariadne’s thread, 326 – as a low-energy approximation of string theory, 742 – Ashtekar program, 742 – at finite temperature, 757 – axiomatic approach, 866 – – Epstein–Glaser, 749, 854 – – G˚ arding–Wightman, 866, 911 – – Glimm–Jaffe, 870, 912 – – Haag–Kastler, 911 – – Osterwalder–Schrader, 866 – – Segal, 911 – – Wightman, 866, 911 – basic formulas, 739 – – (QA) and (LSZ), 765 – – Dyson’s S-matrix formula, 822 – – magic formulas (see also magic), 326, 765 – basic strategies, 739, 813 – – Dyson’s operator approach, 813 – – Feynman’s functional integral approach, 57, 753, 804 – – Schwinger’s response approach, 765 – Batalin–Vilkovisky quantization, 903 – Becchi–Rouet–Stora–Tyutin (BRST) symmetry, 890 – conformal, 923 – constructive, 870 – Faddeev–Popov ghosts, 888 1013 – Faddeev–Popov–De Witt ghost approach, 886 – fascination of, – Haag theorem, 748 – in a nutshell, 27 – introductory literature, 907 – key formula – – for the cross section of scattering processes, 839 – – for the transition probability, 837 – lattice approximation, 815 – method of – – Fourier quantization (see also Volume II), 55 – – Heisenberg–Pauli canonical quantization, 52, 760 – – Lehmann–Szymanzik–Zimmermann (LSZ), 765 – – moments and correlation functions (Green’s functions), 742 – – quantum action (QA), 765 – – second quantization, 52 – model – – asymptotically free, 869 – – continuum, 773 – – discrete, 459 – – trivial, 869 – paradox of, – perturbation theory and – – Feynman diagrams (see also Vol II), 747 – – Feynman rules (see also Vol II), 843 – renormalization (see also Vol II), 848 – rigorous – – finite-dimensional approach, 325 – – perspectives, 862 – standard literature to, 910 – topological, 264 quantum number, 143, 154 – of leptons, 155 – of quarks, 155 quark, 130 – confinement, 134, 135, 204, 205 – hypothesis, 180 quark-gluon field, 881 quasi-crystal, 288 quaternion, 265 Quillen, 72 rabbit problem, 286 Rademacher, 283, 555 – theorem, 555 1014 Index radiation law, 102 Radzikowski, 704 Raman, 69 Ramanujan, 283 Ramsey, 70 randomness of quantum processes, 369 ray, 274 Rayleigh, 98, 545, 724 Rayleigh–Jeans radiation law, 103 Razborov, 72 real part, 209, 932 red shift, 143 reduced – correlation function, 750 – Planck’s quantum of action, = h/2π, 936 reduction formulas, 746 references (see also hints for further reading), 959 refractive index, 718 regular solution, 489 regularization of integrals, 511 regularized Green’s operator, 374, 500 regularizing term, 29, 500, 509, 512, 618 Reines, 70 relativistic electron, 810 Rellich theorem, 507 Remmert, 213, 861 renormalizability, 854 renormalization, 5, 375, 497, 509, 618, 624, 754, 768, 784, 846, 848, 856, 862 – algebraic, 858 – and bifurcation, 630 – and Hopf algebras, 860 – and tempered distributions, 622, 857 – basic ideas, 196, 624, 625, 768, 850 – of the anharmonic oscillator, 625 – see also Vol II, 768 renormalization group, 197, 503, 849, 872 – basic ideas, 196 – differential equation, 198 – equation, 197, 633 renormalized – electron charge, 194 – electron mass, 768 – Green’s function, 633 – integral, 624 Repka, 113 residue – method, 213, 640, 732 – theorem, 214 resolvent, 358, 365 – set, 365 resonance, 89, 374, 498, 625 – condition, 374, 503 response – and causality, 701 – approach, 746, 765 – – rigorous, 438 – equation, 446 – function, 94, 96, 446, 479, 485, 767 – – for electrons, 800 – – for gauge bosons, 885 – – for mesons, 775 – – for photons, 799 rest mass, 25 retarded – fundamental solution, 713 – propagator, 372, 373, 378, 384, 388, 419, 582 retract, 239 reversible, 165 Reynolds number, 948 Richter, 70 Riemann, 10, 19, 60, 220, 227, 257, 291, 549 – conjecture, 78 – curvature tensor, 248 – moduli space, 223 – – and string theory, 223 – sphere, 217 – surface, 19, 220, 235, 551 – zeta function, 277, 291, 321 Riemann–Hilbert problem, 662, 679, 860 Riemann–Roch–Hirzebruch theorem, 892 Riemannian geometry of the sphere, 247 Riesz – Fryges, 22, 531 – Marcel, 714 – representation theorem for functionals, 356 rigged Hilbert space, 35, 578, 675 Ritt theorem, 861 Ritz, 120 – method in quantum chemistry, 153 Rivasseau, 849 Roberval, 575 Robinson, 397 Ră ontgen, 69, 128 Rosanes, 64 Rosen, 848 Rossi, 73 Index Roth, 71 Rubbia, 70, 136, 182 Rudolph, XI running (renormalized) – coupling constant, 196, 202 – – prototype, 500 – fine structure constant, 196, 202 Rutherford, 69, 98, 100, 120, 129 Rydberg, 120 Rydberg–Ritz energy formula, 122 Ryle, 69 saddle, 246 Salam, 3, 60, 70, 79, 135 – criterion, 638, 848 Salmhofer, XI Sato, 74 Savart, 252 scattering – cross section, 839 – function, 444 – functional, 449, 785 – matrix (S-matrix), 38, 370, 391, 746, 785, 828 – – generalized, 854 – state, 525 – theory (see also Vol II), 828 Schauder, 230, 560 – theory, 560 Schechter, Schelling, 958 Scherk, 923 Schmidt, 22 Schoenflies, 162 Schrader, 866 Schrieffer, 70, 575 Schră odinger, 29, 37, 62, 65, 70, 128, 525 equation, 36, 392, 808, 939 – – stationary, 36 – operator picture, 392 – quantization, 36, 753 – wave picture, 35 Schră odingerMaxwell equation, 172, 174 Schwartz Laurent, 71, 325, 525, 575–577 – Melvin, 70 – space S(RN ), 536 Schwarz – Albert, 904 – Amandus, 541 – inequality, 338 – John, 923 1015 Schwarzschild radius, 141, 142 Schweber, 739, 861 Schwinger, 4, 28, 29, 66, 70, 283, 374, 395, 739, 765, 861 – function, 865 – integration trick, 643, 778 second law of – progress in theoretical physics, 79 – thermodynamics, 166 second quantization, 52 secular equation, 365 Segal, 911 Segr´e, 70, 131 Seiberg–Witten equation, 204, 811 Selberg, 71, 74 self-adjoint operator, 357, 677 self-similarity, 197 semicontinuous, 569 separable Hilbert space, 678 separated, 236 sequentially – closed, 339 – continuous, 537 Serre, 71, 74, 75 set – arcwise connected, 239 – bounded, 366 – closed, 236 – compact, 239 – neighborhood of a point, 236 – open, 236 – – neighborhood of a point, 236 – simply connected, 240 sharp state, 351 sheaf cohomology, 398 shell structure of atoms, 150 Shimura–Taniyama–Weil conjecture, 19 shock wave, 613 Shockley, 70 Shore, 72 short-wave asymptotics for light, 718 SI system of units, 934 – rescaled, 946 – tables, 950 Sibold, XI Siegel, 74 sigma – additivity (σ-additivity), 528 – algebra (σ-algebra), 528 similarity principle in physics, 946 simple group, 920 simply connected, 240 Sinai, 74 1016 Index Singer, 75, 257 singular – limits in physics, 689 – support, 704 sink, 246 skein relation, 264 skew-adjoint, 343 skew-symmetric, 344 SLAC (Stanford Linear Accelerator Center, California), 135 Smale, 71, 234 small divisor, 626 Smalley, 70, 245 S-matrix (see scattering matrix), 785 smooth, 234 – boundary, 546 – function, 31, 521 Sobolev, 542 – embedding theorem, 558 – space, 557 software systems in perturbation theory, 929 Sokhotski formula, 663 solid forward light cone, 665 solitons in mathematics, physics, and molecular biology, 700 Solvay Conference, 62 Sommerfeld, 120, 724 – radiation condition, 725 Sorella, 858 source, 246 – term, 374 – trick, 749 space reflection, 172 special – functions, 663 – linear group, 341 – unitary group, 341 specific volume, 683 spectral – family, 368, 369 – geometry, 260 – theorem, 678, 679 – theory in functional analysis (see Vol II), 369 spectrum, 358, 365 Sperber, 195 spherical – coordinates, 247 – wave, 719 spin, 144, 145 – operator, 161 – quantum number, 154, 268 spin-orbit coupling, 152 splitting of spectral lines, 179 spontaneous – emission, 124 – symmetry breaking, 180 square-integrable functions, 578 stability of matter, stable manifold, 206 Standard Model in particle physics – emergence of the, 70 – history, 135 – minimal supersymmetric, 138, 859 – renormalization, 859 – resource letter, 79 – see also Vols III–VI, 127 standing wave, 84 state, 269, 349, 350, 596 – generalized, 596 – of an elementary particle, 161 state of the art in – gravitation and cosmology, 929 – quantum field theory, 929 state, generalized, 680 stationary phase, 430, 435, 714 Schră odinger equation, 373 statistical – operator, 758 – potential, 280 Stein, 74 Steinberger, 70 Steinmann, 622 – extension, 622 – renormalization theorem, 622 step function, 528 stereographic projection, 217 Stern, 69 Stern–Gerlach effect, 148 Stevin, 286 stimulated – absorption, 124 emission, 124 stochastic process, 660 Stă ormer, 70 Stokes, 545, 546, 948 – integral theorem, 547 – – on manifolds, 547 Stone, 22, 29 Stone–von Neumann theorem, 819 strangeness, 154 string theory, 2, 76, 137, 222, 291, 742, 811 – history of, 923 Index strong – correlation, 353 – force, 127, 131 – hypercharge, 154 – isospin, 154 structure constants of a Lie algebra, 884 Sturm, 525 sub-velocity of light, 25 subgroup, 340 submanifold (see Vol III), 263 substitution trick, 360 Sudan, 72 suggested reading (see also hints for further reading), 907 sum rule for spins, 147 summation convention, 769, 933 superconductivity, 575 supernova, 129 supersymmetric Standard Model in particle physics, 138, 859 supersymmetry, 138, 811 support, 608 – of a distribution, 610 – of a measure, 602 – singular, 704 surjective, 932 Sylvester, 363 symbol of a differential operator, 710 symmetric, 333 symmetry, 162–178 – and special functions, 663 – breaking, 178 – – in atomic spectra, 152 – factor of a Feynman diagram, 832 symplectic, 47, 706 system of units – energetic system, 942 – Gauss, 941 – Heaviside, 941 – natural SI units, 937, 950 – Planck, 938 – SI (Syst`eme International), 81, 934 – – tables, 950 Szymanzik, 439, 765 Tacoma Narrows Bridge, VIII Tamm, 112 Tanaka, 71 tangent bundle, 249 Tannoudji, 70 Tarjan, 72 Tate, 74 1017 tau lepton number, 154 Tauber, 286 – theorem, 286, 687 tauon, 130 Taylor – Joseph, 69, 73, 136 – Richard, 70 Taylor–Slavnov identity, 853 T -duality, 703 Teichmă uller space, 14 Telegdi, 73 tensor product of distributions, 615 tera, 935 Tesla, 950 test function, 608 theorema egregium, 10, 248 theory of – general relativity, 111 – probability, 528 – special relativity, 109 thermal Green’s function, 575 thermodynamic limit, 682 third law of progress in theoretical physics, 79 Thirring, Thom, 71, 227, 234 Thompson, D’Arcy, 955 Thompson, John, 71, 74 Thomson – George, 691 – Joseph John, 69, 98, 100, 112, 117, 128 – series, 922 t’Hooft, 70, 73, 135 Thouless, 73 Thurston, 72 time – period, 81 – reversal, 172 time-ordered – contraction, 827 – paired normal product, 827 – product, 422, 827 time-ordering, 388 Ting, 70 Tits, 74 Tomonaga, 4, 28, 70, 739, 861 Tonelli, 542, 569 topness, 154 topological – charge, 214 – invariant, 239 – quantum field theory, 252, 264 1018 Index – quantum number, 214, 241 – space (see also Vol II), 236 – – arcwise connected, 239 – – compact, 239 – – separated, 236 – – simply connected, 240 topology, 227 total – cross section, 839 – pairing, 826 Townes, 69, 126 trace, 340, 363 – formula, 756 transformation theory, 325, 355 transition – amplitude, 39, 351, 755, 823 – maps, 234 – probability, 40, 351, 786, 823 transport equation, 721 triangle inequality, 366 Triebel–Lizorkin space, 560 trivial – linear space, 330 – model in quantum field theory, 869 Trotter product formula, 653 trouble – with divergent perturbation series, 861 – with interacting quantum fields, 748 – with scale changes, 187 truncated – damped wave, 89 – Dirac delta function, 815 – lattice in momentum space, 670 Tsu, 70 tube, 224 tunnelling of α-particles, 129 turbulence, 945 – problem, 78 Tycho Brahe, 180 Uehling, 195 Uhlenbeck, 73, 148 Uhlmann, XI, 60 ultrafilter, 397 ultraviolet limit, 850 uncertainty – inequality, 34, 142, 523 – relation, 62 uniformization theorem, 19, 221, 552 unit – ball, 268 – sphere, 268 – – surface measure, 561 unitarity of the S-matrix, 370, 890 unitary – equivalence, 338, 358 – group, 341 – matrix, 341 – operator, 338 universe – global structure, 227 – local structure, 227 unknot, 262 unphysical states, 893 unstable manifold, 206 upper half-plane, 284, 662 vacuum (ground state), 55, 181, 817 – energy, 300 – expectation value, 425 – polarization, 195 – state, 424 Valiant, 72 van der Meer, 70, 136 van der Waerden, 60 van Dyck, 241 Vandermonde, 252 vanishing measure, 531 variation of the parameter, 383 variational – lemma, 403 – – complex, 544 – – real, 543 – problem, 547 vector calculus, 171 velocity of light c, 696, 936 Veltman, 5, 70, 79, 135 Veneziano, 291, 923 – model, 923 Verch, XI vertex – algebra, 922 – distribution, 780 – function, 444, 458, 483, 751 – functional, 458 vibrating string, 805 Vilenkin, 527 Virasoro algebra, 920 virtual particle, 59, 836, 844 virus dynamics, 289 Voevodsky, 72 Volterra, 386, 395 – differential calculus, 750 – integral equation, 43, 386 volume – form, 248 Index – potential, 555 von Klitzing, 70 von Neumann, 21, 29, 35, 37, 38, 60, 67, 68, 281, 283, 368, 372, 525, 535, 536, 567, 819, 957 – spectral theorem, 678 von Waltershausen, 10 Ward–Takehashi identity, 853 warning to the reader, 739, 776, 784 Wattson, 71 wave, 81 – equation, 805 – front, 718, 723 – – equation, 718 – – set, 706, 708, 710 – length, 82 – – of matter waves, 142 – mechanics, 65 – number, 82 – operator, 460 – packet, 82 – vector, 706 weak – convergence, 570 – correlation, 353 – force, 128 – gauge bosons W ± , Z , 130 – hypercharge, 154 – isospin, 154 weakly lower semicontinuous, 570 Weber, 252 wedge product, 515 Weierstrass, 19, 510, 540, 549, 650 – counterexample, 549 – product theorem, 509 Weil, 74 Weinberg, 3, 70, 79, 98, 135, 201, 638, 957 Weinberg–Salam theory, 576 Weisskopf, 73 Wentzel, 66 Wess, XI, 915 Wess–Zumino model, 811 west coast convention, 934 Weyl, X, 3, 24, 67, 68, 257, 363, 521, 525, 542, 567, 955, 958 – asymptotics of the spectrum, 260 – lemma, 611 Wheeler, 39, 73, 228 Whitney, 74 Wick – moment trick, 58, 432 1019 – rotation, 425, 588, 635, 642 – theorem – – first, 825 – – main, 824 – – second, 826 Wick theorem – main, 828 Widgerson, 72 Wieman, 70, 684 Wien, 69 – radiation law, 102 Wiener, 21, 29, 395 – integral, 654, 655, 660 – measure, 654 Wightman, 148, 173, 638, 648, 704, 848, 866, 911 – axioms, 866 – functional, 858 Wigner, 49, 70, 163, 525, 955 Wilczek, 70, 201 Wiles, 18, 72, 74, 78 Wilson – Charles, 69 – Kenneth, 70, 73, 190 – loop, 576 – Robert, 69, 113 winding number, 214 Witten, 13, 72, 75, 78, 100, 924 – functor, 13 WKB (Wentzel, Kramers, Brillouin), 431 – approximation, 947 – method, 431 WMAP (Wilkinson Microwave Anisotropy Probe), 80, 113 Wolf prize – in mathematics, 73 – in physics, 73 wormhole, 228 Wright, 848 Wu, 73 Wă uthrich, 71 Yang, 3, 70, 181, 185, 249 Yang–Lee condensation, 683 Yang–Mills – action, 883 – equation, 811 Yau, 72 Yoccoz, 72 Yukawa, 69 – meson, 143, 712 – potential, 724 1020 Index Zagier, XI, 18, 293, 304, 305 Zamolodchikov, 923 Zariski, 74 Zeeman, 69, 180 – effect, 179 Zelmanov, 72 zero – measure, 530 – set, 528, 530 zeta function, 260, 291, 312, 434, 658 – determinant formula, 261, 658 – of a compact manifold, 260 – regularization, 301, 657 – trick, 434 Zimmermann, 439, 765, 853, 854 – forest formula, 741, 853, 860 Zumino, 915 Zustandssumme (partition function), 757 Zweig, 98, 135 ... )(? ?2 a 22 ? ?2 ) = a11 a 22 (? ?2 ? ?2 η1 ζ1 ) + 2 The same expression is obtained for (ζ a η )(ζ1 a11 η1 ) 2 22 Similarly, we get 1 (ζ1 a 12 ? ?2 )(? ?2 a21 η1 ) = − a 12 a21 (? ?2 ? ?2 η1 ζ1 ) + , 2 and. .. c(? ?2 ? ?2 η1 ζ1 ) + 1+ i,j=1 i,j=1 The dots denote the remaining terms It turns out that c = a11 a 22 − a 12 a21 = det(A) Let us show this Since ? ?2 ? ?2 = −? ?2 ? ?2 + γ, we get ζ1 η1 ? ?2 ? ?2 = −ζ1 η1 ? ?2. .. ? ?2 ? ?2 + γζ1 η1 Furthermore, it follows from η1 ? ?2 = −? ?2 η1 and ζ1 ? ?2 = −? ?2 ζ1 that ζ1 η1 ? ?2 ? ?2 = −? ?2 ζ1 η1 ? ?2 + = −? ?2 ? ?2 ζ1 η1 + = ? ?2 ? ?2 η1 ζ1 + The dots denote terms that contain less