Continued part 1, part 2 of ebook Quantum field theory II: Quantum electrodynamics provide readers with content about: basic ideas in quantum mechanics; quantization of the harmonic oscillator – ariadne’s thread in quantization; quantum particles on the real line – ariadne’s thread in scattering theory; quantum electrodynamics (QED); creation and annihilation operators; the basic equations in quantum electrodynamics;...
7 Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization Whoever understands the quantization of the harmonic oscillator can understand everything in quantum physics Folklore Almost all of physics now relies upon quantum physics This theory was discovered around the beginning of this century Since then, it has known a progress with no analogue in the history of science, finally reaching a status of universal applicability The radical novelty of quantum mechanics almost immediately brought a conflict with the previously admitted corpus of classical physics, and this went as far as rejecting the age-old representation of physical reality by visual intuition and common sense The abstract formalism of the theory had almost no direct counterpart in the ordinary features around us, as, for instance, nobody will ever see a wave function when looking at a car or a chair An ever-present randomness also came to contradict classical determinism.1 Roland Omn`es, 1994 Quantum mechanics deserves the interest of mathematicians not only because it is a very important physical theory, which governs all microphysics, that is, the physical phenomena at the microscopic scale of 10−10 m, but also because it turned out to be at the root of important developments of modern mathematics.2 Franco Strocchi, 2005 In this chapter, we will study the following quantization methods: • Heisenberg quantization (matrix mechanics; creation and annihilation operators), ã Schră odinger quantization (wave mechanics; the Schră odinger partial dierential equation), ã Feynman quantization (integral representation of the wave function by means of the propagator kernel, the formal Feynman path integral, the rigorous infinitedimensional Gaussian integral, and the rigorous Wiener path integral), • Weyl quantization (deformation of Poisson structures), From the Preface to R Omn`es, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 1994 Reprinted by permission of Princeton University Press We recommend this monograph as an introduction to the philosophical interpretation of quantum mechanics F Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for Mathematicians, Lecture Notes, Scuola Normale, Pisa (Italy) Reprinted by permission of World Scientific Publishing Co Pte Ltd Singapore, 2005 428 Quantization of the Harmonic Oscillator • Weyl quantization functor from symplectic linear spaces to C ∗ -algebras, • Bargmann quantization (holomorphic quantization), • supersymmetric quantization (fermions and bosons) We will choose the presentation of the material in such a way that the reader is well prepared for the generalizations to quantum field theory to be considered later on Formally self-adjoint operators The operator A : D(A) → X on the complex Hilbert space X is called formally self-adjoint iff the operator is linear, the domain of definition D(A) is a linear dense subspace of the Hilbert space X, and we have the symmetry condition χ|Aϕ = Aχ|ϕ for all χ, ψ ∈ D(A) Formally self-adjoint operators are also called symmetric operators The following two observations are crucial for quantum mechanics: • If the complex number λ is an eigenvalue of A, that is, there exists a nonzero element ϕ ∈ D(A) such that Aϕ = λϕ, then λ is a real number This follows from λ = ϕ|Aϕ = Aϕ|ϕ = λ† • If λ1 and λ2 are two different eigenvalues of the operator A with eigenvectors ϕ1 and ϕ2 , then ϕ1 is orthogonal to ϕ2 This follows from (λ1 − λ2 ) ϕ1 |ϕ2 = Aϕ1 |ϕ2 − ϕ1 |Aϕ2 = In quantum mechanics, formally self-adjoint operators represent formal observables For a deeper mathematical analysis, we need self-adjoint operators, which are called observables in quantum mechanics Each self-adjoint operator is formally self-adjoint But, the converse is not true For the convenience of the reader, on page 683 we summarize basic material from functional analysis which will be frequently encountered in this chapter This concerns the following notions: formally adjoint operator, adjoint operator, self-adjoint operator, essentially self-adjoint operator, closed operator, and the closure of a formally self-adjoint operator The reader, who is not familiar with this material, should have a look at page 683 Observe that, as a rule, in the physics literature one does not distinguish between formally self-adjoint operators and self-adjoint operators Peter Lax writes:3 The theory of self-adjoint operators was created by John von Neumann to fashion a framework for quantum mechanics The operators in Schră odingers theory from 1926 that are associated with atoms and molecules are partial differential operators whose coefficients are singular at certain points; these singularities correspond to the unbounded growth of the force between two electrons that approach each other I recall in the summer of 1951 the excitement and elation of von Neumann when he learned that Kato (born 1917) has proved the self-adjointness of the Schră odinger operator associated with the helium atom.4 P Lax, Functional Analysis, Wiley, New York, 2003 (reprinted with permission) This is the best modern textbook on functional analysis, written by a master of this field who works at the Courant Institute in New York City For his fundamental contributions to the theory of partial differential equations in mathematical physics (e.g., scattering theory, solitons, and shock waves), Peter Lax (born 1926) was awarded the Abel prize in 2005 J von Neumann, General spectral theory of Hermitean operators, Math Ann 102 (1929), 49–131 (in German) 429 And what the physicists think of these matters? In the 1960s Friedrichs5 met Heisenberg and used the occasion to express to him the deep gratitude of the community of mathematicians for having created quantum mechanics, which gave birth to the beautiful theory of operators in Hilbert space Heisenberg allowed that this was so; Friedrichs then added that the mathematicians have, in some measure, returned the favor Heisenberg looked noncommittal, so Friedrichs pointed out that it was a mathematician, von Neumann, who clarified the difference between a self-adjoint operator and one that is merely symmetric.“What’s the difference,” said Heisenberg As a rule of thumb, a formally self-adjoint (also called symmetric) differential operator can be extended to a self-adjoint operator if we add appropriate boundary conditions The situation is not dramatic for physicists, since physics dictates the ‘right’ boundary conditions in regular situations However, one has to be careful In Problem 7.19, we will consider a formally self-adjoint differential operator which cannot be extended to a self-adjoint operator The point is that self-adjoint operators possess a spectral family which allows us to construct both the probability measure for physical observables and the functions of observables (e.g., the propagator for the quantum dynamics) In general terms, this is not possible for merely formally self-adjoint operators The following proposition displays the difference between formally self-adjoint and self-adjoint operators Proposition 7.1 The linear, densely defined operator A : D(A) → X on the complex Hilbert space X is self-adjoint iff it is formally self-adjoint and it always follows from ψ|Aϕ = χ|ϕ for fixed ψ, χ ∈ X and all ϕ ∈ D(A) that ψ ∈ D(A) Therefore, the domain of definition D(A) of the operator A plays a critical role The proof will be given in Problem 7.7 Unitary operators As we will see later on, for the quantum dynamics, unitary operators play the decisive role Recall that the operator U : X → X is called unitary iff it is linear, bijective, and it preserves the inner product, that is, for all U χ|U ϕ = χ|ϕ χ, ϕ ∈ X This implies ||U ϕ|| = ||ϕ|| for all ϕ ∈ X Hence ||U || := sup ||U ϕ|| = ||ϕ||≤1 if we exclude the trivial case X = {0} The shortcoming of the language of matrices noticed by von Neumann Let A : D(A) → X and B : D(B) → X be linear, densely defined, formally J von Neumann, Mathematical Foundations of Quantum Mechanics (in German), Springer, Berlin, 1932 English edition: Princeton University Press, 1955 T Kato, Fundamental properties of the Hamiltonian operators of Schră odinger type, Trans Amer Math Soc 70 (1951), 195211 Schră odinger (18871961), Heisenberg (1901–1976), Friedrichs (1902–1982), von Neumann (1903–1957), Kato (born 1917) 430 Quantization of the Harmonic Oscillator self-adjoint operators on the infinite-dimensional Hilbert space X Let ϕ0 , ϕ1 , ϕ2 , be a complete orthonormal system in X with ϕk ∈ D(A) for all k Set ajk := ϕj |Aϕk j, k = 0, 1, 2, The way, we assign to the operator A the infinite matrix (ajk ) Similarly, for the operator B, we define bjk := ϕj |Bϕk j, k = 0, 1, 2, Suppose that the operator B is a proper extension of the operator A Then ajk = bjk for all j, k = 0, 1, 2, , but A = B Thus, the matrix (ajk ) does not completely reflect the properties of the operator A In particular, the matrix (ajk ) does not see the crucial domain of definition D(A) of the operator A Jean Dieudonn´e writes:6 Von Neumann took pains, in a special paper, to investigate how Hermitean (i.e., formally self-adjoint) operators might be represented by infinite matrices (to which many mathematicians and even more physicists were sentimentally attached) Von Neumann showed in great detail how the lack of “one-to-oneness” in the correspondence of matrices and operators led to their weirdest pathology, convincing once for all the analysts that matrices were a totally inadequate tool in spectral theory 7.1 Complete Orthonormal Systems A complete orthonormal system of eigenstates of an observable (e.g., the energy operator) cannot be extended to a larger orthonormal system of eigenstates Folklore Basic question Let H : D(H) → X be a formally self-adjoint operator on the infinite-dimensional separable complex Hilbert space X Physicists have invented algebraic methods for computing eigensolutions of the form Hϕn = En ϕn , n = 0, 1, 2, (7.1) The idea is to apply so-called ladder operators which are based on the use of commutation relations (related to Lie algebras or super Lie algebras) We will encounter this method several times In terms of physics, the operator H describes the energy of the quantum system under consideration Here, the real numbers E0 , E1 , E2 , are the energy values, and ϕ0 , ϕ1 , ϕ2 , are the corresponding energy eigenstates Suppose that ϕ0 , ϕ1 , ϕ2 , is an orthonormal system, that is, ϕk |ϕn = δkn , k, n = 0, 1, 2, There arises the following crucial question J Dieudonn´e, History of Functional Analysis, 1900–1975, North-Holland, Amsterdam, 1983 (reprinted with permission) J von Neumann, On the theory of unbounded matrices, J reine und angew Mathematik 161 (1929), 208–236 (in German) 7.1 Complete Orthonormal Systems 431 Is the system of the computed energy eigenvalues E0 , E1 , E2 complete? The following theorem gives us the answer in terms of analysis Theorem 7.2 If the orthonormal system ϕ0 , ϕ1 , is complete in the Hilbert space X, then there are no other energy eigenvalues than E0 , E1 , E2 , , and the system ϕ0 , ϕ1 , ϕ2 , cannot be extended to a larger orthonormal system of eigenstates Before giving the proof, we need some analytical tools Completeness By definition, the orthonormal system ϕ0 , ϕ1 , ϕ2 is complete iff, for any ϕ ∈ X, the Fourier series ϕ= ∞ X ϕn |ϕ ϕn n=0 P is convergent in X, that is, limN →∞ ||ϕ − N n=0 ϕn |ϕ ϕn || = The proof of the following proposition can be found in Zeidler (1995a), Chap (see the references on page 1049) Proposition 7.3 Let ϕ0 , ϕ1 , ϕ2 be an orthonormal system in the infinite-dimensional separable complex Hilbert space X Then the following statements are equivalent (i) The system ϕ0 , ϕ1 , ϕ2 , is complete (ii) For all ϕ, ψ ∈ X, we have the convergent series ψ|ϕ = ∞ X ψ|ϕn ϕn |ϕ , (7.2) n=0 which is called P the Parseval equation (iii) I = ∞ n=0 ϕn ⊗ ϕn (completeness relation) P (iv) For all ϕ ∈ X, we have the convergent series ||ϕ||2 = ∞ n=0 | ϕn |ϕ | (v) Let ϕ ∈ X If all the Fourier coefficients of ϕ vanish, that is, we have ϕn |ϕ = for all n, then ϕ = (vi) The linear hull of the set {ϕ0 , ϕ1 , ϕ2 , } is dense in the Hilbert space X Explicitly, for any ϕ ∈ X and any number ε > 0, there exist complex numbers a0 , , an such that ||ϕ − (a1 ϕ1 + + an ϕn )|| < ε Proof of Theorem 7.2 Suppose that Hϕ = Eϕ with ϕ = and that the eigenvalue E is different from E0 , E1 , E2 , Since the eigenvectors for different eigenvalues are orthogonal to each other, we get ϕn |ϕ = for all indices n By Prop 7.3(v), ϕ = This is a contradiction ✷ The Dirac calculus According to Dirac, we write equation (7.1) as H|En = En |En , n = 0, 1, 2, Moreover, the completeness relation from Prop 7.3(iii) reads as I= ∞ X |ϕn ϕn | (7.3) n=0 P This means that ϕ = limN →∞ N n=0 (ϕn ⊗ ϕn )ϕ for all ϕ ∈ X Here, we use the convention (ϕn ⊗ ϕn )ϕ := ϕn ϕn |ϕ 432 Quantization of the Harmonic Oscillator Mnemonically, from (7.3) we obtain |ϕ = P∞ n=0 ψ|ϕ = ψ| · |ϕ = ψ| · I|ϕ = |ϕn ϕn |ϕ and ∞ X ψ|ϕn ϕn |ϕ n=0 P This coincides with the Fourier series expansion ϕ = ∞ n=0 ϕn |ϕ ϕn and the Parseval equation (7.2) The following investigations serve as a preparation for the quantization of the harmonic oscillator in the sections to follow 7.2 Bosonic Creation and Annihilation Operators Whoever understands creation and annihilation operators can understand everything in quantum physics Folklore The Hilbert space L2 (R) We consider R ∞ the space L2 (R) of complex-valued (measurable) functions ψ : R → C with −∞ |ψ(x)|2 dx < ∞ This becomes a complex Hilbert space equipped with the inner product Z ∞ ϕ|ψ := ϕ(x)† ψ(x)dx for all ϕ, ψ ∈ L2 (R) −∞ p Moreover, ||ψ|| := ψ|ψ The precise definition of L2 (R) can be found in Vol I, Sect 10.2.4 Recall that the Hilbert space L2 (R) is infinite-dimensional and separable For example, the complex-valued function ψ on the real line is contained in L2 (R) if we have the growth restriction at infinity, |ψ(x)| ≤ const + |x| x ∈ R, for all and ψ is either continuous or discontinuous in a reasonable way (e.g., ψ is continuous up to a finite or a countable subset of the real line) Furthermore, we will use the space S(R) of smooth functions ψ : R → C which rapidly decrease at infinity (e.g., ψ(x) := e−x ) The space S(R) is a linear subspace of the Hilbert space L2 (R) Moreover, S(R) is dense in L2 (R) The precise definition of S(R) can be found in Vol I, Sect 2.7.4 The operators a and a† Fix the positive number x0 Let us study the operator a := √ „ x d + x0 x0 dx « More precisely, for each function ψ ∈ S(R), we define „ « xψ(x) dψ(x) (aψ)(x) := √ + x0 x0 dx for all x ∈ R (7.4) This way, we get the operator a : S(R) → S(R) We also define the operator a† : S(R) → S(R) by setting a† := √ „ x d − x0 x0 dx « (7.5) 7.2 Bosonic Creation and Annihilation Operators Explicitly, for each function ψ ∈ S(R), we set8 „ « xψ(x) dψ(x) (a† ψ)(x) := √ − x0 x0 dx for all 433 x ∈ R The operators a and a† have the following properties: (i) The operator a† : S(R) → S(R) is the formally adjoint operator to the operator a : S(R) → S(R) on the Hilbert space L2 (R).9 This means that ϕ|aψ = a† ϕ|ψ for all ϕ, ψ ∈ S(R) (ii) We have the commutation relation [a, a† ]− = I where I denotes the identity operator on the Hilbert space L2 (R) Recall that [A, B]− := AB − BA 2 (iii) Set ϕ0 (x) := c0 e−x /2x0 with the normalization constant c0 := √ √ Then x0 π aϕ0 = (iv) The operator N : S(R) → S(R) given by N := a† a is formally self-adjoint, and it has the eigensolutions N ϕn = nϕn , n = 0, 1, 2, where we set (a† )n ϕn := √ ϕ0 n! (v) For n = 0, 1, 2, , we have √ a† ϕn = n + ϕn+1 , (7.6) aϕn+1 = √ n + ϕn † Because of these relations, the operators a and a are called ladder operators.10 (vi) The functions ϕ0 , ϕ1 , form a complete orthonormal system of the complex Hilbert space L2 (R) This means that Z ∞ ϕn |ϕm = ϕn (x)† ϕm (x) dx = δnm , n, m = 0, 1, 2, −∞ 10 In applications to the harmonic oscillator later on, the quantity x has the physical dimension of length We introduce the typical length scale x0 in order to guarantee that the operators a and a† are dimensionless In functional analysis, one has to distinguish between the formally adjoint operator a† : S(R) → S(R) and the adjoint operator a∗ : D(a∗ ) → L2 (R) which is an extension of a† , that is, S(R) ⊆ D(a∗ ) ⊆ L2 (R) and a∗ ϕ = a† ϕ for all ϕ ∈ S(R) (see Problem 7.4) Ladder operators are frequently used in the theory of Lie algebras and in quantum physics in order to compute eigenvectors and eigenvalues Many examples can be found in H Green, Matrix Mechanics, Noordhoff, Groningen, 1965, and in ShiHai Dong, Factorization Method in Quantum Mechanics, Springer, Dordrecht, 2007 (including supersymmetry) We will encounter this several times later on 434 Quantization of the Harmonic Oscillator Moreover, for each function ψ in the complex Hilbert space L2 (R), the Fourier series ∞ X ϕn |ψ ϕn ψ= n=0 is convergent in L2 (R) Explicitly, lim ||ψ − k→∞ k X ϕn |ψ ϕn || = n=0 R∞ Recall that ||f ||2 = f |f = −∞ |f (x)|2 dx (vii) The matrix elements amn of the operator a with respect to the basis ϕ0 , ϕ1 , are defined by amn := ϕm |aϕn , m, n = 0, 1, 2, √ Explicitly, amn = n δm,n−1 Therefore, √ 1 0 B0 √2 0 C B C √ C (amn ) = B C B0 0 @ A Similarly, we introduce the matrix elements (a† )mn of the operator a† by setting (a† )mn := ϕm |a† ϕn , m, n = 0, 1, 2, Then (a† )mn = a†nm Thus, the matrix to the operator a† is the adjoint matrix to the matrix (amn ) Let us prove these statements To simplify notation, we set x0 := Ad (i) For all functions ϕ, ψ ∈ S(R), integration by parts yields „ « « Z ∞ Z ∞ „ d d † ϕ(x) x + ψ(x)dx = x− ϕ(x)† · ψ(x)dx dx dx −∞ −∞ Hence ϕ|aψ = a† ϕ|ψ d d )(x − dx )ψ = x2 ψ + ψ − ψ Similarly, Ad (ii) Obviously, 2aa† ψ = (x + dx „ «„ « d d 2a† aψ = x − x+ ψ = x2 ψ − ψ − ψ dx dx Hence (aa† − a† a)ψ = ψ √ Ad (iii) Note that ae−x /2 = (x + Ad (iv) For all ϕ, ψ ∈ S(R), d )e−x /2 dx = ϕ|a† aψ = aϕ|aψ = a† aϕ|ψ Hence ϕ|N ψ = N ϕ|ψ Thus, the operator N is formally self-adjoint We now proceed by induction Obviously, N ϕ0 = a† (aϕ0 ) = Suppose that N ϕn = nϕn Then, by (ii), N (a† ϕn ) = a† aa† ϕn = a† (a† a + I)ϕn This implies 7.2 Bosonic Creation and Annihilation Operators 435 N (a† ϕn ) = a† (N + I)ϕn = (n + 1)a† ϕn Thus, N ϕn+1 = (n + 1)ϕn+1 Ad (v) By definition of the state ϕn , a† ϕn = √ √ (a† )n+1 (a† )n+1 √ ϕ0 = n + p ϕ0 = n + ϕn+1 n! (n + 1)! Moreover, by (ii) and (iv), √ n + aϕn+1 = aa† ϕn = (a† a + I)ϕn = (n + 1)ϕn Ad (vi) We first show that the functions ϕ0 , ϕ1 , form an orthonormal system In fact, by the Gaussian integral, ϕ0 |ϕ0 = Z ∞ −∞ e−x √ dx = π We now proceed by induction Suppose that ϕn |ϕn = Then (n + 1) ϕn+1 |ϕn+1 = a† ϕn |a† ϕn = ϕn |aa† ϕn = ϕn |(N + I)ϕn By (iv), this is equal to (n + 1) ϕn |ϕn Hence ϕn+1 |ϕn+1 = Since the operator N is formally self-adjoint, eigenvectors of N to different eigenvalues are orthogonal to each other Explicitly, it follows from n ϕn |ϕm = N ϕn |ϕm = ϕn |N ϕm = m ϕn |ϕm that ϕn |ϕm = if n = m Finally, we will show below that the functions ϕ0 , ϕ1 , coincide with the Hermite functions which form a complete orthonormal system in L2 (R) Ad (vii) By (v), √ √ ϕm |aϕn = n ϕm |ϕn−1 = n δm,n−1 ✷ Moreover, (a† )mn = ϕm |a† ϕn = aϕm |ϕn = (anm )† Physical interpretation In quantum field theory, the results above allow the following physical interpretation • The function ϕn represents a normalized n-particle state • Since N ϕn = nϕn and the state ϕn consists of n particles, the operator N is called the particle number operator • Since N ϕ0 = 0, the state ϕ0 is called the (normalized) vacuum state; there are no particles in the state ϕ0 • By (v) above, the operator a† sends the n-particle state ϕn to the (n +1)-particle state ϕn+1 Naturally enough, the operator a† is called the particle creation operator In particular, the n-particle state (a† )n ϕ0 ϕn = √ n! is obtained from the vacuum state ϕ0 by an n-fold application of the particle creation operator a.11 11 For the vacuum state ϕ0 , physicists also use the notation |0 436 Quantization of the Harmonic Oscillator • Similarly, by (v) above, the operator a sends the (n+1)-particle state ϕn+1 to the n-particle state ϕn Therefore, the operator a is called the particle annihilation operator The position operator Q and the momentum operator P We set x0 Q := √ (a† + a), P := i √ (a† − a) x0 This way, we obtain the two linear operators Q, P : S(R) → S(R) along with the commutation relation [Q, P ]− = i I This follows from [a, a† ]− = I In fact, [Q, P ]− = 12 [a† + a, i (a† − a)]− Hence 2[Q, P ]− = i [a, a† ]− − i [a† , a]− = 2i [a, a† ]− = 2i I Explicitly, for all functions ψ ∈ S(R) and all x ∈ R, (P ψ)(x) = −i (Qψ)(x) = xψ(x), Hence P = −i d dx dψ(x) dx The operators Q, P are formally self-adjoint, that is, ϕ|Qψ = Qϕ|ψ , ϕ|P ψ = P ϕ|ψ for all functions ϕ, ψ ∈ S(R) In fact, Z Z ∞ ϕ(x)† xψ(x) dx = ϕ|Qψ = −∞ ∞ (xϕ(x))† ψ(x) dx = Qϕ|ψ −∞ Furthermore, noting that (iϕ(x))† = −iϕ(x)† , integration by parts yields Z ∞ Z ∞ ϕ(x)† (−i ψ (x))dx = (−i ϕ (x))† ψ(x) dx = P ϕ|ψ ϕ|P ψ = −∞ −∞ The Hermite functions To simplify notation, we set x0 := We will show that the functions ϕ0 , ϕ1 , introduced above coincide with the classical Hermite functions.12 To this end, for n = 0, 1, 2, , we introduce the Hermite polynomials Hn (x) := (−1)n ex dn e−x dxn (7.7) along with the Hermite functions ψn (x) := e−x /2 Hn (x) p √ , 2n n! π x ∈ R (7.8) Explicitly, H0 (x) = 1, H1 (x) = 2x, and H2 (x) = 4x2 − For n = 0, 1, 2, , the following hold: 12 Hermite (1822–1901) Index Mandelstam variables, 912 manifold (see Vol I), 212, 283, 312, 399 – n-manifold, 343 Manin, 1056 Manoukian, 973 ∗-map, 627 mapping class, 219 Marcolli, XXII Mascheroni, 54, 55 Maslov, 673 – index, 377, 534, 576, 581 – phase factor, 538 mass renormalization, 957 material sciences, 424 mathemagics, 109 mathematical principles of natural philosophy, 11 matrix, 766 – function, 769 – group, 292 – mechanics, 444 Maupertuis, 359, 483 Maurer, 339 Maurer–Cartan – form, 339 – structural equation, 340 Maurin, 272, 279, 324, 390, 660, 1056 maximal element, 237 maximum principle – classical, 274 Maxwell, 13, 14, 263, 267, 507, 903, 1056 – equation – – in a vacuum, 934 – – in matter, 267 – theory of light, 267 Maxwell–Boltzmann statistics, 650 Mazur, 660 mean – fluctuation, 463 – value, 199, 463, 634, 638 measurable function (see Vol III), 772, 785 measure, 503 – finitely additive, 637 – integral, 587 – theory (see Vol I), 587, 763 – zero, 763 measurement, 462 – in quantum mechanics, 466 mechanics – Hamiltonian, 402 – Lagrangian, 372, 402 1087 – Newtonian, 366 – Poissonian, 406 Mehler formula, 537 meromorphic function, 56 – with values in the space of tempered distributions, 82 method of – least squares, 106 – residues (see also Vol I), 58 – the variation of the parameter, 421 metric space, 240, 317 – complete, 317 microstructures and material sciences, 424 mid-point approximation, 613 Miku´ nski field, 192 Mills, 44 Milnor, 1056 minimal, 237 – subtraction, 65 minimum – global, 360 – local, 360 – problem, 363 – – constrained, 363 – – quadratic, 361 Minkowski, 13, 297, 1056 – space (see Vol III), 826 Mittag-Leffler, 54 – theorem, 58 – – renormalization, 948 mnemonic approach, 284, 287 Mă obius, 166, 200, 317, 343, 691 inversion formula, 163, 166 – strip, 343 modular – automorphism, 660 – conjugation operator, 660 – operator, 659 moment, 562 – free, 565 – full, 567 – of a probability distribution (see Vol I), 564 – problem, 145 – reduced full, 569 momentum, 367, 368 – map of Lie, 419 – operator, 436, 521 momentum space – lattice, 801 monad, 659 Monastirsky, 1056 1088 Index monodromy, 1005, 1007 – in nature, 1007 monomorphism, 624 monster group, 185 Montesquieu, 175 moon landing ferry, 348 Moore–Penrose inverse, 107 Moore–Smith sequence, 240 morphism, 116 – theorem for – – algebras, 190 – – groups, 185 – – linear spaces, 190 – – rings, 184 Morse, 377, 1056 – index, 362, 363, 377, 379, 534, 581 Moser, 314, 1056 motives in mathematics, 250 moving frame (see Vol III), 43, 337 Moyal star product, 597 formal, 591, 603 rigorous, 607 Mă uller Stefan, 424 – Werner, 572 multi-loop computation in renormalization theory, 978 – automated, 978 multilinear functional, 120 multiplicative – operator, 155 – renormalization constant, 947 MWI (Master Ward Identity), 1027 Møller, 747 – wave operator, 750 Naimark, 660 Nash, 313 – embedding theorem, 314 – equilibrium, 314 natural – number, 240 – philosophy – – interaction principle, 483 – – mathematical principles, 11 neighborhood (see Vol I), 240 net, 240 Neumann – Carl, 744 – John (Janos) von (see von Neumann), 1058 Nevalinna–Sokal theorem, 99 Newton, 12, 200, 263, 367, 483, 729, 902, 1056 Newtonian – mechanics, 366 – motion, 87, 366 – – advanced, 88 – – calculus of variations, 372 – – Dirac delta distribution, 19 – – fundamental solution, 87 – – Green’s function, 17 – – initial-value problem, 17 – – kick force, 18 – – nonlinear, 20 – – principle of critical action, 15, 372 – – retarded, 88 – – retarded-advanced, 88 – – rigorous Fourier transform, 90 – – rigorous Laplace transform, 91 – – self-interaction, 20 Nirenberg, 508 Nobel prize – laureates, 1056 – lectures, 1056 Noether (Emmy), 13, 383, 1056 – symmetry principle, 12 – theorem, 36, 384, 417 – – general case, 386 – – special case, 385 Noether (Max) – genus of algebraic curves, 203 Nomizu, 43 non-causal, 499 non-contradiction, 110, 246 non-Euclidean geometry, 290 non-expansive semigroup, 507 non-resonance case, 383 – anharmonic oscillator, 50 non-set, 246 non-standard – analysis, 257 – mathematics, 257 – number, 249 noncommutative geometry, 660–662, 1047 nonlinear process, 13 nonlinearity principle, 13 norm (see Vol I), 462, 656 normal – operator, 692 – product, 438, 693, 818, 823, 1017 – – paired, 847 – – principle, 847 – subgroup, 184 normalization – momentum, 63 Index – volume, 801, 892 North Pole, 412 n-point – function (correlation function) – – free, 565 – – full, 568 – Green’s function (correlation function), 456 – Wightman function, 457 n-sphere, 343 nuclear – operator, 629 – space, 601 number – generalized, 249 – infinite, 249 – infinitesimal, 249 observable, 428, 462, 634, 639 – formal, 428, 446, 462 observers – cocycles of bundles, 208 – physical fields, 208 Odifreddi, 1056 Oeckl, 1016 off-shell, 63 Omn`es, 427, 637 on-shell, 63 one-parameter unitary group, 505 – strongly continuous, 505 one-particle irreducible (1PI) (see Feynman graph), 954 OPE (operator product expansion), 1024, 1039 open, 241 operator – annihilation, 436 – anti-multiplicative, 155 – antilinear, 155 – antiself-dual, 516 – bounded, 690 – calculus, 499 – – prototype, 495 – closed, 684 – compact, 535 – continuous, 690 – creation, 429, 435 – densely defined, 681 – density, 543 – dual, 515 – essentially self-adjoint, 683, 686 – formally self-adjoint (or symmetric), 428, 683 1089 – Hilbert–Schmidt, 535, 629 – ideal, 629 – kernel, 490, 599 – – formal, 490 – – rigorous, 508 – – theorem, 536 – linear, 155, 681 – multiplicative, 155 – non-expansive, 497 – normal, 692 – nuclear, 535, 629 – product expansion (OPE), 1024 – – as fundamental quantity, 1026 – regularization, 156, 157 – Rota–Baxter, 156 – self-adjoint, 428, 683, 686 – self-dual, 515 – sequentially continuous, 690 – topology – – strong, 656 – – uniform, 656 – – weak, 656 – trace class, 535, 543, 629 – truncation, 156, 157 – unitary, 429 Oppenheimer, 4, 47 optical distance, 270 optimal control, 348 optimality principle, 12 optimization theory, 348 orbit space, 195 orthogonal complement, 635 orthonormal (see Vol I) – basis, 700 – system, 430 – – complete, 431 oscillation – damped, 382 oscillator – anharmonic, 370, 378 – – renormalization, 49 – harmonic (see harmonic oscillator), 369 overlapping – divergence, 65, 967, 972 – subgraphs, 970, 972 – – trouble, 983 Painlev´e, 998, 1056 – equations, 1003 – property, 1003 Pappus of Alexandria, 370 paracompact, 312 1090 Index parallel transport, 12, 39, 216 – covariant derivative, 334 – curvature, 335 – in gauge theory, 40 – in mechanics, 413 – of physical information, 12, 39, 334 – of tangent vectors of geodesics, 334 parallelizable sphere, 212 parameter integral, 763 parity, 186 Parseval des Ch´enes, 480, 719 Parseval equation, 431, 465, 489, 512, 719 – generalized, 517 partial isometry, 636 particle number – fluctuation, 640 – mean, 640 – operator, 435, 440, 774 partition function, 638, 647 Paschke, XXII path – component, 220 – integral (see also Feynman path integral), 547 – – Brownian motion, 671 – – Wiener integral, 671 path-connected, 307 Patras, 45 Paul, 959 Pauli, 47, 1056 – exclusion principle, 771, 783 – spin-statistics principle, 818 Pauli–Villars – integration trick, 73 – regularization, 76 Peierls bracket, 1037 pendulum, 389 – nonlinear, 392 – spherical, 411 Penrose, 45, 1056 Perelman, 346 permeability constant – in a vacuum, 267 – in matter, 267 permutation, 185 – even, 186 – group, 185 – odd, 186 – sign, 186 perturbation theory, 566, 568 phase – equation, 337 – – associated fiber bundle, 337 – – gauge transformation, 338 – – parallel transport of physical fields, 337 – – principal fiber bundle, 341 – function – – global gauge, 35 – – local gauge, 36 – of a unitary matrix, 767 – space – – flow, 404 – – in statistical physics, 555 – – Liouville measure, 555 – – of the pendulum, 395 – state, 398 – transition, 381, 654 – – Ginzburg–Landau potential, 380 – – Higgs potential, 380 – – material sciences, 424 – – prototype, 380 phonon, 360 photon, 360 – breaking radiation, 976 – longitudinal, 805, 812 – mill on earth, 654 – propagator, 858 – scalar, 805, 825 – transversal, 812, 833 – virtual, 812, 825, 834 physical field – language of – – bundles, 208 – – sheaves, 216 – section of a bundle – – prototype, 208, 401 Picard, 729 picture (see Vol I) – Dirac, 611 – Heisenberg, 611 interaction, 611 Schră odinger, 611 Pietsch, 601 Plancherel theorem, 514 Planck, 13, 903, 1056 – action quantum, 13, 449 – constant, 13, 449 – length, 14 – quantization principle, 12 – radiation law, 650 – scale hypothesis, 14 plasmon, 360 Plato, 448, 1057 Plemelj, 1006 Index Plă ucker, 200 Poincare, 13, 54, 200, 216, 227, 297, 314, 390, 400, 998, 1002, 1057 – conjecture, 346, 352 – – generalized, 347 – duality, 307 – identity, 40 – model of hyperbolic geometry, 297, 314 – Seminar, XXI, 1050 Poincar´e–Cartan integral invariant (or the Hilbert invariant integral), 423 Poincar´e–Stokes integral theorem (see Vol I), 304, 357 Poincar´e–Wirtinger calculus, 318, 616 point – measure, 503 – of infinity, 314 Poisson, 12, 406 – bracket, 407, 443, 591 Poisson–Peierls bracket, 1037 polar decomposition, 691 polarization, 805, 954 – longitudinal, 805 – scalar, 805 – transversal, 805 Polchinski, 1024 – renormalization group approach, 1023 Politzer, 63 Poncelet, 200 Pontryagin, 348, 1057 – duality, 152 – maximum principle, 348 poset, 237 position – operator, 436, 518 – space, 396 potential, 332, 368 – barrier, 701 – long-range, 757 – short-range, 756 – statistical, 640 – well, 701 power-counting theorem, 60, 974, 986 pQFT (perturbative quantum field theory), XVIII, 970, 994, 1018, 1027, 1036 – in curved space-time, 1025 pre-bundle, 208 pre-fiber, 209 pre-Hamiltonian, 755 pre-Hilbert space (see Vol I), 306 1091 pre-image (see Vol I), 241 pre-sheaf, 217 – of smooth functions, 218 pre-state, 637 precision tests of the Standard Model, 1019 pressure, 285, 640, 647 prestabilized harmony, 659 prime number, 166 primitive divergent Feynman graph, 954 principal – argument, 85 – fiber bundle, 42, 337 – part, 56 – value of, 32 – – an integral, 495 – – the logarithm, 480 – – the square root, 479 principle of – averaging due to Laurent Schwartz, 26 – coordinatization due to Descartes, 146 – critical action, 15, 360, 374, 379 – general relativity due to Einstein, 12 – geometrization in physics, 13 – harmonic analysis due to Fourier, 21 – Huygens, 270 – indistinguishable particles, 647, 771 – infinitesimals due to Newton and Leibniz, 12 – least action, 15, 379, 903 – least time due to Fermat, 268 – limiting absorption, 738 – linearity (superposition), 13 – linearization (from Lie groups to Lie algebras), 400 – locality due to Faraday, 13 – locality due to Huygens, 270 – nonlinearity, 13 – optimality, 12 – parallel transport of information, 334 – Pauli – – exclusion, 771, 783 – – spin-statistics, 818 – propagation of physical effects due to Einstein, 13 – quantization due to Planck, 12 – special relativity due to Einstein, 12 – symmetry – – global, 12 – – local, 36 1092 Index – the Green’s function, 17 – the Planck scale, 14 – unitarity in quantum physics due to Dirac, 12 principles of modern natural philosophy, 11 probability – amplitude, 480, 489 – density – – free, 565 – – full, 567 – distribution – – free, 565 – – full (under interaction), 567 – measure, 143 – of transition, 480 process – dissipative, 382 – irreversible, 653 – linear, 13 – nonlinear, 13 – quasi-stationary, 653 – reversible, 653 product – bundle, 210 – group, 223 – rule, 124 – – for the Feynman propagator, 491 – topology, 243 projection – of a fiber bundle, 208 – operator, 189 projective – space – – complex, 203 – – real, 200 – topology, 243 Prokhorov, 903 propagator, 498, 506, 527, 528, 726, 845, 856 – advanced, 498, 726 – and the path integral, 611 – differential equation, 609 – equation, 507 – – irreversible, 507 – – reversible, 506 – Euclidean, 507 – Feynman (see Feynman propagator), 955 – hypothesis, 548, 555 – kernel – – global, 483 – – infinitesimal, 483 – retarded, 498, 506, 727 – theory – – formal, 488 – – rigorous, 505 protein synthesis, 642 pseudo-convergence, 104 pseudo-differential operator, 30, 490, 591 pseudo-resolvent, 51, 52 pseudo-Riemannian manifold, 312 pull-back – of differential form, 354 Pythagorean theorem, 635 – Euclidean, 303 – hyperbolic, 303 – spherical, 303 QED (see quantum electrodynamics), 789 QFT (see quantum field theory), 1036 q-integral, 158 quadratic – reciprocity law, 178, 181 – supplement, 560 quantization, 15, 676 – and action, 15 – Bargmann, 617 – deformation, 590 – Epstein–Glaser, 990 – Feynman, 479 – Fourier, 444 – free – – electron field, 814 – – particle, 465, 509 – – photon field, 812 – – positron field, 815 – general principle, 443 – harmonic oscillator, 427, 534 – Heisenberg, 440 – operator algebras, 633 – perturbed free particle, 699 – quantum electrodynamics, 811 – – Dyson series, 835 – – free quantum field, 811 – – Gupta–Bleuler, 831 – quantum field – – free, 811 – – interacting, 835 – Schră odinger, 459 von Neumann, 495 algebra, 635 – Weyl, 590 quantum Index – action principle, 979, 1037 – chaos, 673 – computer, 183 – dynamics, 466, 505, 526, 699, 722 – – Euclidean, 506 – fluctuation, 485, 577 – gravity, 675 – group, XII, 146, 151 – logic (see Vol IV), 637 – – Gleason’s theorem, 637 – mathematics, XIV – state, 199 quantum electrodynamics (QED), 789 – application to physical effects, 899 – continuum limit, 945 – Dyson series and S-matrix, 835 – Feynman – – diagrams, 875 – – rules, 895 – free quantum field, 811 – history, 2, 1043 – interacting quantum field, 835 – lattice strategy, 799 – main strategy, 788, 799 – radiative corrections, 953 – renormalization, 967 quantum field theory – axiomatic, 1039 – basic ideas, 359 – discrete, 564 – harmonic oscillator, 360 – hierachy of functions, 1038 quantum mechanics – energy and spectrum, 754 – Feynman, 479 – Heisenberg, 440 Schră odinger, 459 via deformation, 592 – von Neumann – – rigorous approach, 495 – Weyl, 590 quasi-stationary thermodynamical process, 287, 652 quaternion, 179 quintessence in cosmology, 204 quotient – algebra, 190 – group, 185 – ring, 184 – space, 178 – topology, 242 radioactive decay, 727 1093 Ramanujan, 7, 1057 random – variable, 143 – walk, 585 Ray, 572 ray, 199 real line – full-rigged, 251 – motion of a classical particle, 367 – motion of a quantum particle, 459 realification, 291, 295 reduced Compton wavelength, 901 reduction formula, 841, 842 references – biographies of mathematicians on the Internet, 1059 – Collected Works, 1051 – complete list on the Internet, 1059 reflection – amplitude, 706 – probability, 706 refraction index, 264, 927 Regis, 1057 regularization, 48, 93, 107, 561, 980 – adiabatic, 31, 93 – – divergent series, 94 – – oscillating integral, 94 – analytic, 77 – averaging, 96 – Borel, 98 – counterterm, 67 – dimensional, 73, 74 – divergent integrals, 73 – Hadamard’s finite part, 100 – method – – ambiguity, 63 – minimal subtraction, 65 – operator, 156 – overlapping divergence, 65 – Taylor subtraction, 64 – zeta function, 101 regularized – Feynman function, 862 – photon propagator, 862 Rehren, XXII Reid, 1057 relatively prime, 182 relativistic invariance, 912, 949 renormalizable quantum field theory, 980, 992 renormalization, 47, 48 – additive, 948 – algebraic, 1019 1094 Index – – advantage, 1020 – ambiguity of renormalization schemes, 977 – analytic, 1022 – automated multi-loop computation, 978 – – computer algebra system FORM, 978 – – Laporta algorithm, 978 – – software package Feynarts, 978 – axiomatic approach, 977 – basic ideas, 47, 48, 60, 979, 1018, 1021 – BPHZ (Bogoliubov, Parasiuk, Hepp, Zimmermann), 981 – BRST (Becchi, Rouet, Stora, Tyutin) symmetry, 1019, 1020 – compensation principle, 948, 957 – continuum limit, 945 – counterterms, 947, 957 – dimensional regularization, 74, 981, 1019 – electroweak Standard Model, 1019 – equivalent schemes, 977 – fundamental limits, 945 – group (see Vol I), 53, 63, 352, 977, 1022, 1023 – – equation, 1022, 1024 – hints for further reading, 1029 – Hopf algebra revolution, 990 – importance of Ward identities, 957, 1022 – multiplicative, 950, 957 – of partial differential equations, 353 – of quantum electrodynamics, 975 – OPE (operator product expansion), 1024 – parity violation, 1020 – perspectives, 1029 – Polchinski approach, 1024 – quantum electrodynamics, 996, 1026 – radiative correction – – in lowest order, 953 – vertex function, 1020 renormalized (or effective) Green’s function, 53 renormalon problem, 1029 representation of a group, 820 representative, 178 residue, 56 – method (see Vol I), 58 resolvent, 492, 498, 504 – equation, 492 – of a matrix, 766 – set, 492, 504 – – of a matrix, 766 resonance, 25, 383, 748, 1014 – case of the anharmonic oscillator, 51 – complicated phenomena, 383 – scattering state, 714 – small divisor, 383 retarded – function, 980, 1037, 1039 – fundamental solution, 84, 88 – – Fourier transform, 91 – Green’s function, 88, 731 – product, 1037 – propagator, 498, 727 – – distribution, 873 retract, 221 retraction, 221 return of a spaceship, 348 reversible, 287, 653 Ricci – calculus, 341 – flow, 346, 352 – tensor, 327 – – in general relativity, 330 Ricci-Curbastro, 325 Riemann, 13, 43, 54, 200, 297, 311, 324, 325, 392, 507, 998, 1000, 1057 – curvature tensor (see Vol III), 43, 327 – – essential components, 328 – hypergeometric equation, 1000 – legacy, 272, 279, 324 – monodromy, 1005 – sphere, 195 – surface, 195, 318, 480, 531, 714 – – energetic, 714 – symbol, 1001 – zeta function, 574 Riemann–Hilbert problem, 997, 1006 Riemann–Liouville integral, 92 Riemann–Roch–Hirzebruch theorem, 200 Riemannian – geometry, 324 – manifold, 312 Riesz – Fr´ed´eric (Fryges), 1057 – Marcel, 54, 92, 1057 ring – isomorphism, 179 – morphism, 179 Ritz, 448 Index – combination principle, 448 Rivasseau, XXI, 1023 RNA (ribonucleic acid), 642 rocket, 369 Rolle, 692 – theorem, 692 Rosen, 975 Rosenberg, Jonathan, 621 Rota, XI, 54, 1057 Rota–Baxter – algebra, XI, 48 – operator, 156, 950 rotation, 416, 820 – infinitesimal, 417 royal road to – geometrical optics, 271 – the calculus of variations, 419 Rudolph, XXII Runge–Kutta method, 147, 1042 running coupling constant, 52, 63 Russel, 12, 245, 246 Rutherford scattering formula, 915 saddle point, 362 Salam, 973, 1057 – criterion, 975 – criterion in renormalization theory, 975 Salmhofer, XXII Sato, 193 – hyperfunctions, 193 SBEG (Stueckelberg, Bogoliubov, Epstein, Glaser) approach (see Epstein–Glaser approach), 988 scalar – curvature, 327 – photon, 805, 812 – polarization, 805 scattering – long-range, 751 – process, 699 – Rutherford formula, 915 – short-range, 751 – state, 700, 722 – theory – – basic ideas, 747 – – inverse, 1011 – – stationary, 753 – – time-dependent, 751 scattering matrix (S-matrix), 5, 699, 708, 718, 723, 747, 749 – and its Feynman path integral, 614 – bound states, 1022 1095 – element, 610, 750 – global, 725 – k-component with respect to the wave number k, 725 – magic Dyson series, 835 – unitarity, 949 scattering operator (see scattering matrix), 609 Schauder, 1057 Schmidt – Alexander, XXII, 159 Erhard, 51 Schră odinger, 13, 428, 429, 459, 485, 1057 – equation, 459, 489, 730 – – abstract, 505 – – stationary, 468, 703, 717 – picture, 473, 474, 611 Schreiber, 1057 Schur, 140, 637, 1057 – lemma, 637 – polynomial, 140, 144 Schwartz (Laurent), 13, 480, 508, 599, 729, 1057 – distributions, 26 – function space, 598 – kernel theorem, 601 Schwarz (Amandus), 1057 – inequality, 355, 356, 634 Schwarz (John) – superstring theory, 1057 Schweber, 6, 962, 973, 1028, 1057 Schwinger, 2, 5, 6, 13, 542, 729, 903, 1057 – integration trick, 73 Scriba, 1057 section, 402 – of a fiber bundle, 208 – – physical field, 209 – – prototype, 208 – of a principal fiber bundle, 340 – of the tangent bundle, 212 – – velocity vector field, 212 – physical fields – – prototype, 401 secular equation, 766 self-adjoint, 755 – essentially, 755 – matrix, 766 – operator, 465, 500, 683, 686 – – essentially, 465 – – formally, 433, 440, 465 self-dual operator, 513, 515 1096 Index self-energy – electron, 954 – photon, 954 semi-axis, 132 semi-bounded from below, 755 semi-norm, 656 semi-ring, 229 – commutative, 229 – isomorphism, 229 – morphism, 229 – regular, 229 semiclassical – statistical physics, 544 – WKB method, 580 semigroup, 507 – non-expansive, 507 S´eminaire – Bourbaki, XXI – Bourbaphy, XXI separable Hilbert space (see Vol I), 432 separated topological space, 240 sequentially continuous (see Vol I), 515, 599 Serre, 218, 226, 227, 1057 – finiteness theorem, 227 set, 246 – completely ordered, 256 – of measure zero, 763 – partially ordered, 237 – totally ordered, 237 – well-ordered, 237 Severi, 1057 Shafarevich, 1057 Shannon, 641 sheaf, 217 – of holomorphic functions, 217 Shor, 183 short-range force, 271 SI (international system of units) (see Vol I), XXI, 360 Sibold, XXII – lectures given at the Max Planck Institute for Physics, Werner Heisenberg, Munich, 1036 Sigal, 751, 763 Simons, Jim, 332 simple – group, 185 – ring, 184 simply connected (see Vol I), 222, 307, 343 sin-Gordon equation, 391 Singer, Isadore, 572 singular – support (see Vol I), 746 – value, 106, 111, 629 singularity, 349 – strong, 1001 – weak, 1001 sinus – amplitudinis function, 392 – function – – classical, 391, 392 – – lemniscatic, 394 skew-adjoint matrix, 766 skew-field, 179 – isomorphism, 179 – morphism, 179 slash – matrix, 905 – symbols, 1066 Slavnov–Taylor identities, 6, 980, 997, 1020 slope, 265 – function, 272 Smale, 347, 396, 1057 small divisor and resonance, 383 S-matrix (see scattering matrix), Sobolev, 514, 1057 – space, 510, 514 Sokhotski formula, 86 soliton, 1007 Sommerfeld, 732, 1058 – radiation condition, 732, 734 South Pole, 412 spaces of – distributions (see Vol I), 116 – functions (see Vol I), 116 spectral – family, 429, 501 – – and measurements, 518, 522 – hypothesis, 548, 579 – line – – forbidden, 449 – – intensity, 447 – – wave length, 447 – measure, 501 – theorem, 500 – – proof, 502 – transform (inverse scattering theory), 1012 – triplet, 661 spectrum, 492, 504, 535, 755 – absolutely continuous, 504, 756 – discrete, 504 – essential, 504, 756 Index – of a matrix, 766 – pure point, 504, 756 – singular, 504 Speiser, 109 sphere, 343 – topological properties, 344 spherical – coordinates, 412 – pendulum – – free, 413 spin, 294, 959 – and infinitesimal rotations, 820 – of a photon, 820 – of an electron, 820 spin-down state, 808 spin-statistics principle of Pauli, 819 spin-up state, 808 Spitzer exponential formula, 161 splitting of physical states, 122 square-well potential, 704 Srivasta, 663 St Andrews, 1059 stability, 382 stabilizer, 197 standard model in – gauge theory, 34 – particle physics (see Vols I, III–VI), 34 – – historical remarks, 43, 331 – – renormalization, IX, 1020 – quantum mechanics on the real line, 459 – scattering theory on the real line, 699 – statistical physics, 638 – – semiclassical, 644 state, 446, 462, 513, 634, 639 – bound, 503 – classification, 699 – eigenstate, 503 – in Gibbs statistics, 646 – mixed, 634, 638 – pure, 634, 638 – scattering, 503 – singular, 503 – space, 397 stationary phase method, 561, 564 statistical – physics – – finite standard model, 638 – – language of C ∗ -algebras, 634, 638 – – semiclassical, 645 – weight, 641 statistics 1097 – Bose–Einstein, 649 – Fermi–Dirac, 650 – Maxwell–Boltzmann, 650 Steenrod, 332 Steinmann, 980 – extension theorem (see Vol I), 59 Stieltjes, 719 Stirling, 100 – asymptotic series, 100 – formula, 69, 648 Stone, 506 – theorem, 506 Stone–von Neumann uniqueness theorem, 621, 626 strip, 282 Strocchi, 427 strongly – closed, 656 – open, 656 – singular, 1001 structural equation, 326, 328, 341 – local, 329 Stueckelberg, 809, 968, 1037 Sturm, 277 Sturm–Liouville problem, 509 – regular, 277 – singular, 277 subgraph, 970 subring, 179 subtraction terms, XI – Laurent series method, 56 – Mittag-Leffler theorem, 57 – Taylor series method, 64 Suijlekom, 997 summation of a divergent series, 93 Sunder, 655 super-renormalizable, 980 superfunction, 192 superposition principle, 13 supersymmetry, 663, 666, 679 – harmonic oscillator, 663 – in genetics, 642 surface – classification theorem, 343 Sweedler notation, 125, 129 symbol of an operator, 28, 598, 740, 745 symbolic method, 28, 110 symmetric operator, 683 symmetrization, 772 symmetry, XII, 384, 988 – and conservation laws – – Noether theorem, 383 – and geometry 1098 Index – – Erlangen program due to Felix Klein, 194 – and group theory, 399 – and Hopf algebras, 146 – and theory of invariants, 399 – local principle, 36 symplectic, 623 – form, 283, 291, 624, 630 – geometry, 403, 409 – isomorphism, 291 – linear space, 630 – morphism, 291 – space, 291 – transformation, 409 symplectomorphism, 630 syzygies – gauge theory, 980 – Hilbert’s theory, 980 Tamm, 926 tangent – bundle, 337, 397, 398, 403 – space, 318, 396 – vector, 318 Tannaka–Krein duality, 152 Tauber, 98 – theorem, 98 Taylor – expansion, 378, 381 – subtraction method, 64, 985 temperature, 639 – absolute, 285 tempered distribution (see Vol I), 510, 598, 731 – derivative, 516 tensor product – algebras, 122, 261 – antisymmetrization, 772 – Hilbert spaces, 773 – linear – – differential operators, 124 – – functionals, 120 – – operators, 773 – – spaces, 121, 260 – multilinear functionals, 120 – particle states, 785 – physical fields, 772 – symmetrization, 772 test function, 373, 778 TeV (tera electron volt), 1067 theorema egregium, 311, 328, 329, 341 theory of invariants, 399 thermodynamic – equilibrium, 635, 640 – limit, 653 – potential, 287 – process, 652 thermodynamics, 285 – first law, 653 – second law, 653 – third law, 653 theta function, 393 Thikonov regularization, 107 Thirring, Walter, 1058 Thom, 1058 Thomson, Joseph John, 903 – scattering formula, 914 ’t Hooft, XVIII, 959, 993, 1058 thread, 245 time-ordered – contraction, 850, 854 – product, 1017, 1038 Titchmarsh, 191 – theorem, 192 Tolksdorf, XXI, 1050 Tomita–Takesaki theory, 659 Tomonaga, 4–6, 13, 903, 1058 topological – charge, 152, 220 – space (see Vol I), 205 – – separated, 240 – type, 221 topologically equivalent, 221 topology, 205, 241 – coproduct, 242 – inductive, 242 – product, 243 – projective, 243 – quotient, 242 – stronger, 241 – weaker, 241 totally ordered set, 237 Townes, 903 trace, 535, 543, 629, 766, 903 – class operator, 543, 629 – method in quantum electrodynamics, 903 – of a matrix, 904 – of an infinite-dimensional operator, 571 – rules, 904 transfer matrix, 708 transition – amplitude, 749, 842 – map, 213 Index – probability, 449, 473, 480, 610, 749, 893 transmission – amplitude, 706 – – analytic continuation, 715 – probability, 706 transport equation, 268 transport of – energy in nature, 1007 – information in nature, 1007 transposed matrix, 766 transversal – photon, 805, 812 – polarization, 805 trouble with – divergent – – integrals, 73, 100 – – series, 94 – formal perturbation theory, 109 – gauge condition in QED, 832 – ill-posed problems, 105 – infinitely many degrees of freedom, 101 – interchanging limits, 102 – oscillating integrals, 95 – pseudo-convergence, 104 – the ambiguity of regularization methods, 104 – virtual photons, 825 tunnelling, 702, 727 Turing, 1058 Uehling potential, 959 ultra-cofinite subset, 251 ultrafilter, 250 uncertainty – inequality for – – energy and time, 477 – – position and momentum, 446 – relation, 446 – – classical, 442 uniformization theorem, 400 uniformly – closed, 656 – open, 656 unit ball, 69 – measure, 69 unit matrix, 766 unit sphere, 69 – surface measure, 69, 111 unital, 116, 179, 627 unitality, 128 – map, 128 1099 unitarity of the S-matrix, 949, 988 unitary, 294 – extension, 720 – group, 505 – invariance, 687 – matrix, 767 – operator, 429 unity of mathematics, 484 universal – covering group, 294, 400 – covering space, 399 – enveloping algebra, 623 universe (in set theory), 246 unreasonable effectiveness of mathematics in the natural sciences, 295 unstable quantum states, 476 upper – bound, 237 – half-plane – – closed, 764 – – hyperbolic geometry, 314 – – open, 297, 764 – – Poincar´e model, 314 vacuum (ground state), 2, 435, 469, 618, 774, 780 – energy, 818 – nontrivial, 1018 – polarization, 954 van der Waerden, 440, 1058 Varadarajan, 1058 variation, 362 – first, 362, 374, 378 – of nth order, 378 – second, 362, 374, 378 variational lemma (see Vol I), 270, 374, 934 – complex, 933 variety, 579 V´ arilly, 661 vector – bundle, 211 – – associated, 337 – – smooth, 211 – product, 365 velocity, 368 Veltman, 959, 961, 993, 1058 Verch, XXII vertex function, 142, 1020 Virasoro algebra, 626 virtual – electron, 881 1100 Index – particle, 970 – photon, 805, 812, 832 – – mass limit, 894 Vladimirov, 987 Volterra, 1058 – exponential formula, 160 – integral equation, 160 volume form, 283 von Neumann, 13, 179, 246, 428–430, 480, 508, 543, 654, 1058 – algebra, 654 – – type I, II, III, 635 – spectral theorem, 500 – theory of classes, 246 Wachter, XXII, 159 Waerden (see van der Waerden), 1058 Wallis, 392 Ward, – identity, 6, 957, 958 Ward–Takahashi identities, 6, 958, 980, 997 water waves, 1014 wave – front, 265 – – set (see Vol I), 746 – number, 511, 708 – – operator, 522 – operator, 723, 750, 753 – – completeness, 752 – packet, 477, 724 weak – convergence, 655, 750 – limit, 501 weakly – closed, 656 – open, 656 – singular, 1001 wedge product, 120 Weierstrass, 264, 392, 1058 – excess function, 273 Weil, 200, 1058 Weinberg, 13, 47, 1058 – power-counting theorem (see Vol I), 61, 974, 986 Weisskopf, 955, 963 well-ordered set, 238 well-ordering principle, 247 well-posed problem, 105 Wentzel, Wess, XXII, 1050 Wess–Zumino model, 679 Weyl, 13, 297, 325, 331, 399, 504, 508, 509, 532, 719, 1058 – algebra, 628 – calculus – – formal, 602 – – rigorous, 596, 606 – group, 624 – map, 630 – ordering, 596 – quantization, 590 – – functor, 632 – relation, 621 – sequence, 504 – system, 623 Weyl–Kodaira theory (see Vol III), XVI Wheeler, 747, 1058 white – dwarf, 651 – noise, 95 Wick – differentiation trick, 73 – operator, 619 – rotation trick, 70 – symbol, 619 – theorem, 560, 564, 566, 848 – – first, 848 – – Gaussian integrals, 566 – – main, 846 – – prototype, 842 – – second, 851 – – vacuum expectation values and contractions, 846 – trick, 542, 562 Wielandt theorem, 69 Wiener, 154, 485, 1058 – measure, 587 – path integral, 587 – pre-measure, 587 Wiener–Hopf – integral equation, 154 – operator, 154 Wightman, XVIII, 218, 454, 967 – function, 451, 845, 860, 1038, 1039 Wigner, 295, 499, 988, 1058 Wilczek, 63 Wiles, 179 Wilson, 1023 – loop, 225 – operator product expansion (OPE), 1024 – renormalization group, 1023 winding number, 152, 220, 1005 WKB (Wentzel, Kramers, Brioullin) – method, 380, 484, 575, 576, 580 Index Wolf, 263 Worbs, 311 work, 368 Wright, 975 Wronskian, 703 Wu, 332 Wulkenhaar, XXII Wußing, 1058 Yandell, 1058 Yang, 13, 44, 325, 331, 1058 Yang–Mills theory (see Vol III), 44, 306, 331 Yosida, 1059 Yukawa, – potential, 915 Zee, 45, 360 Zeemann, 797 Zeldovich, 1059 Zermelo, 245, 247 zero divisor, 186 zeta function, 54, 162 – of an operator, 572, 574 – regularization, 101, 572 – Riemann, 166 Zhang, 331 Zimmermann, XI, 115 – convergence result, 986 – forest formula, XI, 140, 984 – – generalized, 115 Zorn’s lemma, 248 1101 ... = For example, Q2 = 12 (a + a† )(a + a† ) is equal to 12 (a2 + aa† + a† a + (a† )2 ) Hence : Q2 := 12 a2 + a† a + 12 (a† )2 This implies : Q2 : ψ = (x2 − 12 )ψ Hence : Q2 := x2 − Qn = xn + ... t2 , t3 , t4 ) = W2 (t1 , t2 )W2 (t3 , t4 ) + 2W2 (t1 , t3 )W2 (t2 , t4 ) for all time points t1 , t2 , t3 , t4 ∈ R (iv) Wn (t1 , t2 , , tn )† = W (tn , , t2 , t1 ) for all times t1 , t2... state a1 a? ?2 ϕ0 gives a non-vanishing contribution to the Wightman function W2 By (7.33), W2 (t1 , t2 ) is equal to ϕ0 |a1 a? ?2 ϕ0 = x20 −iωt1 iωt2 x2 e ϕ0 |aa† ϕ0 = · e−iω(t1 −t2 ) ·e 2 Ad (ii)