Quantum field theory i basics in mathematics and physics part 1

Quantum Field Theory I: Basics in Mathematics and Physics Eberhard Zeidler Quantum Field Theory I: Basics in Mathematics and Physics A Bridge between Mathematicians and Physicists With 94 Figures and 19 Tables 123 Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Inselstrasse 22 04103 Leipzig Germany e-mail: ezeidler@mis.mpg.de Library of Congress Control Number: 2006929535 Mathematics Subject Classification (2000): 35Qxx, 58-xx, 81Txx, 82-xx, 83Cxx ISBN-10 3-540-34762-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34762-0 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the author using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 46/3100/YL 543210 ă TO THE MEMORY OF JURGEN MOSER (1928–1999) Preface Daß ich erkenne, was die Welt im Innersten zusammenhă alt.1 Faust Concepts without intuition are empty, intuition without concepts is blind Immanuel Kant (1724–1804) The greatest mathematicians like Archimedes, Newton, and Gauss have always been able to combine theory and applications into one Felix Klein (1849–1925) The present comprehensive introduction to the mathematical and physical aspects of quantum field theory consists of the following six volumes: Volume Volume Volume Volume Volume Volume I: Basics in Mathematics and Physics II: Quantum Electrodynamics III: Gauge Theory IV: Quantum Mathematics V: The Physics of the Standard Model VI: Quantum Gravity and String Theory Since ancient times, both physicists and mathematicians have tried to understand the forces acting in nature Nowadays we know that there exist four fundamental forces in nature: • • • • Newton’s gravitational force, Maxwell’s electromagnetic force, the strong force between elementary particles, and the weak force between elementary particles (e.g., the force responsible for the radioactive decay of atoms) In the 20th century, physicists established two basic models, namely, • the Standard Model in cosmology based on Einstein’s theory of general relativity, and • the Standard Model in elementary particle physics based on gauge theory So that I may perceive whatever holds the world together in its inmost folds The alchemist Georg Faust (1480–1540) is the protagonist of Goethe’s drama Faust written in 1808 VIII Preface One of the greatest challenges of the human intellect is the discovery of a unified theory for the four fundamental forces in nature based on first principles in physics and rigorous mathematics For many years, I have been fascinated by this challenge When talking about this challenge to colleagues, I have noticed that many of my colleagues in mathematics complain about the fact that it is difficult to understand the thinking of physicists and to follow the pragmatic, but frequently non-rigorous arguments used by physicists On the other hand, my colleagues in physics complain about the abstract level of the modern mathematical literature and the lack of explicitly formulated connections to physics This has motivated me to write the present book and the volumes to follow It is my intention to build a bridge between mathematicians and physicists The main ideas of this treatise are described in the Prologue to this book The six volumes address a broad audience of readers, including both undergraduate students and graduate students as well as experienced scientists who want to become familiar with the mathematical and physical aspects of the fascinating field of quantum field theory In some sense, we will start from scratch: • For students of mathematics, I would like to show that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical questions • For students of physics, I would like to introduce fairly advanced mathematics which is beyond the usual curriculum in physics For historical reasons, there exists a gap between the language of mathematicians and the language of physicists I want to bridge this gap.2 I will try to minimize the preliminaries such that undergraduate students after two years of studies should be able to understand the main body of the text In writing this monograph, it was my goal to follow the advise given by the poet Johann Wolfgang von Goethe (1749–1832): Textbooks should be attractive by showing the beauty of the subject Ariadne’s thread In the author’s opinion, the most important prelude to learning a new subject is strong motivation Experience shows that highly motivated students are willing to take great effort to learn sophisticated subjects I would like to put the beginning of Ariadne’s thread into the hands of the reader On November 7th 1940, there was a famous accident in the U.S.A which was recorded on film The Tacoma Narrows Bridge broke down because of unexpected nonlinear resonance effects I hope that my bridge between mathematicians and physicists is not of Tacoma type Preface IX Remember the following myth On the Greek island of Crete in ancient times, there lived the monster Minotaur, half human and half bull, in a labyrinth Every nine years, seven virgins and seven young men had to be sacrificed to the Minotaur Ariadne, the daughter of King Minos of Crete and Pasiphaăe fell in love with one of the seven young men – the Athenian Prince Theseus To save his life, Ariadne gave Theseus a thread of yarn, and he fixed the beginning of the thread at the entrance of the labyrinth After a hard fight, Theseus killed the Minotaur, and he escaped from the labyrinth by the help of Ariadne’s thread.3 For hard scientific work, it is nice to have a kind of Ariadne’s thread at hand The six volumes cover a fairly broad spectrum of mathematics and physics In particular, in the present first volume the reader gets information about • the physics of the Standard Model of particle physics and • the magic formulas in quantum field theory, and we touch the following mathematical subjects: • finite-dimensional Hilbert spaces and a rigorous approach to the basic ideas of quantum field theory, • elements of functional differentiation and functional integration, • elements of probability theory, • calculus of variations and the principle of critical action, • harmonic analysis and the Fourier transform, the Laplace transform, and the Mellin transform, • Green’s functions, partial differential equations, and distributions (generalized functions), • Green’s functions, the Fourier method, and functional integrals (path integrals), • the Lebesgue integral, general measure integrals, and Hilbert spaces, • elements of functional analysis and perturbation theory, • the Dirichlet principle as a paradigm for the modern Hilbert space approach to partial differential equations, • spectral theory and rigorous Dirac calculus, • analyticity, • calculus for Grassmann variables, • many-particle systems and number theory, • Lie groups and Lie algebras, • basic ideas of differential and algebraic topology (homology, cohomology, and homotopy; topological quantum numbers and quantum states) We want to show the reader that many mathematical methods used in quantum field theory can be traced back to classical mathematical problems In Unfortunately, Theseus was not grateful to Ariadne He deserted her on the Island of Naxos, and she became the bride of Dionysus Richard Strauss composed the opera Ariadne on Naxos in 1912 X Preface particular, we will thoroughly study the relation of the procedure of renormalization in physics to the following classical mathematical topics: • singular perturbations, resonances, and bifurcation in oscillating systems (renormalization in a nutshell on page 625), • the regularization of divergent infinite series, divergent infinite products, and divergent integrals, • divergent integrals and distributions (Hadamard’s finite part of divergent integrals), • the passage from a finite number of degrees of freedom to an infinite number of degrees of freedom and the method of counterterms in complex analysis (the Weierstrass theorem and the Mittag–Leffler theorem), • analytic continuation and the zeta function in number theory, • Poincar´e’s asymptotic series and the Ritt theorem in complex analysis, • the renormalization group and Lie’s theory of dynamical systems (oneparameter Lie groups), • rigorous theory of finite-dimensional functional integrals (path integrals) The following volumes will provide the reader with important additional material A summary can be found in the Prologue on pages 11 through 15 Additional material on the Internet The interested reader may find additional material on my homepage: Internet: www.mis.mpg.de/ezeidler/ This concerns a carefully structured panorama of important literature in mathematics, physics, history of the sciences and philosophy, along with a comprehensive bibliography One may also find a comprehensive list of mathematicians, physicists, and philosophers (from ancient until present time) mentioned in the six volumes My homepage also allows links to the leading centers in elementary particle physics: CERN (Geneva, Switzerland), DESY (Hamburg, Germany), FERMILAB (Batavia, Illinois, U.S.A.), KEK (Tsukuba, Japan), and SLAC (Stanford University, California, U.S.A.) One may also find links to the following Max Planck Institutes in Germany: Astronomy (Heidelberg), Astrophysics (Garching), Complex Systems in Physics (Dresden), Albert Einstein Institute for Gravitational Physics (Golm), Mathematics (Bonn), Nuclear Physics (Heidelberg), Werner Heisenberg Institute for Physics (Munich), and Plasmaphysics (Garching) Apology The author apologizes for his imperfect English style In the preface to his monograph The Classical Groups, Princeton University Press, 1946, Hermann Weyl writes the following: The gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle “Was das heissen will, weiss jeder, Der im Traum pferdlos geritten ist,”4 Everyone who has dreamt of riding free, without the need of a horse, will know what I mean Preface XI I am tempted to say with the Swiss poet Gottfried Keller (1819–1890) Nobody is more aware than myself of the attendant loss in vigor, ease and lucidity of expression Acknowledgements First of all I would like to thank the Max Planck Society in Germany for founding the Max Planck Institute for Mathematics in the Sciences (MIS) in Leipzig in 1996 and for creating a superb scientific environment here This treatise would have been impossible without the extensive contacts of the institute to mathematicians and physicists all over the world and without the excellent library of the institute My special thanks go to the intellectual fathers of the institute, Friedrich Hirzebruch (chairman of the Founder’s Committee) and Stefan Hildebrandt in Bonn, Karl-Heinz Homann and Julius Wess in Munich, and the late Jă urgen Moser in Zurich who was an external scientific member of the institute I would like to dedicate this volume to the memory of Jă urgen Moser who was a great mathematician and an amiable man Moreover, I would like to thank Don Zagier (Max Planck Institute for Mathematics in Bonn and Coll`ege de France in Paris), one of the greatest experts in number theory, for the kindness of writing a beautiful section on useful techniques of number theory in physics I am very grateful to numerous colleagues in mathematics and physics from all over the world for illuminating discussions It is not possible to mention the names of all of them, since the list is very long In particular, I would like to thank the professors from the Institute of Theoretical Physics at Leipzig University, Bodo Geyer, Wolfhard Janke, Gerd Rudolph, Manfred Salmhofer, Klaus Sibold, Armin Uhlmann, and Rainer Verch for nice cooperation For many stimulating discussions on a broad spectrum of mathematical problems, I would like to thank the co-directors of the MIS, Wolfgang Hackbusch, Jă urgen Jost, and Stefan Mă uller For getting information about new research topics, I am very grateful to my former and present collaborators: Gă unther Berger, Ludmilla Bordag, Friedemann Brandt, Friedemann Brock, Chand Devchand, Bertfried Fauser, Felix Finster, Christian Fleischhack, Jă org Frauendiener, Hans-Peter Gittel, Matthias Gă unther, Bruce Hunt, Konrad Kaltenbach, Satyanad Kichenassamy, Klaus Kirsten, Christian Klein, Andreas Knauf, Alexander Lange, Roland Matthes, Johannes Maul†, Erich Miersemann, Mario Paschke, Hoang Xuan Phu, Karin Quasthoff, Olaf Richter†, Alexander Schmidt, Rainer Schumann, Friedemann Schuricht, Peter Senf, Martin Speight, Jă urgen Tolksdorf, Dimitri Vassilevich, Hartmut Wachter, and Raimar Wulkenhaar For experienced assistance in preparing this book, I would like to thank Kerstin Făolting (graphics, tables, and a meticulous proof-reading of my entire latex-file together with Rainer Munck), Micaela Krieger–Hauwede (graphics, tables, and layout), and Jeffrey Ovall (checking and improving my English style) For supporting me kindly in various aspects and for helping me to save time, I am also very grateful to my secretary, Regine Lă ubke, and to the sta of the institute including the librarians directed by Ingo Bră uggemann, 396 Rigorous Finite-Dimensional Magic Formulas 7.20.1 Functional Derivatives Classical derivatives are generalized to functional derivatives; differentials are linear functionals in modern mathematics Folklore Let Z : X → C be a functional on the complex Hilbert space X We write Z = Z(J) That is, to each element J of X we assign the complex number Z(J) In quantum field theory, such functionals arise in a natural way Prototypes are the action, S(ψ), of a quantum field ψ and the generating functional Z for the correlation functions (see Chap 13) Then, Z(J) is the value of the generating functional at the point J Intuitively, the source function J describes an external force acting on the physical system The functional derivative Z  (J) tells us then the response of the physical system under a small change of the source It is our goal to investigate the following generalizations: • derivative ⇒ functional derivative; • partial derivative ⇒ partial functional derivative; • integral ⇒ functional integral Notation In mathematics, the following notions possess a precise meaning: • • • • • functional derivative and partial functional derivative, directional derivative, variation, differential, infinitesimal transformation The confusion caused by infinitesimals The idea of infinitesimals was introduced by Newton and Leibniz in the 17th century They used the relation (δx)2 = (7.58) for the ‘infinitesimally small quantity’ δx Obviously, the only real number δx which satisfies the magic relation (7.58) is given by δx = 0, which does not fit the intention of Newton and Leibniz Thus, there is a lot of confusion concerning (7.58), which has survived in parts of the physical literature Nowadays, the notions are completely clarified in mathematics There exist two approaches, namely, • the standard approach, and • the non-standard approach 7.20 Functional Calculus 397 In the standard approach, one completely avoids the relation (7.58) This approach will be applied in the present first volume and in most parts of the further volumes In non-standard analysis, one also introduces the following notions: • infinitesimally small number, and • infinitely large number In the non-standard setting, the relation (δx)2 = takes on a precise meaning for an infinitesimally small non-standard number, δx.35 In the volumes of this treatise, we will show that relations of the form ∂ = 0, d2 = 0, δ2 = Q2 = play a fundamental role in modern mathematics and physics In particular, • the relation ∂ = for the boundary operator ∂ is responsible for Poincar´e’s homology theory in algebraic topology, • the Poincar´e lemma d2 = for differential forms (that is, d(dω) = 0) is the basis for de Rham’s cohomology theory in differential topology, • the relation δ = allows us to introduce the Hodge homology on Riemannian manifolds which is dual to the de Rham cohomology, and • the operator relation Q2 = is crucial for the BRST quantization (or cohomological quantization) of gauge theories (e.g., the Standard Model in particle physics and string theories) There exists a branch of mathematics called homological algebra which studies the far-reaching consequences of the relation Q2 = in terms of exact sequences and homology groups (resp the dual cohomology groups).36 In the 35 Non-standard analysis was rigorously founded in 1960 by the logician Abraham Robinson (1918–1974) This will be considered in Volume II, by using ultrafilters The elegant basic idea of non-standard analysis is to construct a field ∗R of mathematical objects called non-standard numbers such that ∗R represents an extension of the field R of real numbers, R ⊂ ∗R 36 Besides the real numbers, the field ∗R contains infinitesimally small numbers and infinitely large numbers For all the elements of ∗R, the operations of addition, multiplication, and division (by nonzero elements) are well defined In terms of algebra, the set ∗R is a field which extends the field R of classical real numbers For two positive elements x and y of R, there exists always a nonzero natural number n such that x < ny This so-called Archimedian property of the field R is not anymore valid for ∗R The classical book is H Cartan and S Eilenberg, Homological Algebra, Princeton University Press In terms of physics, we recommend M Henneaux and C Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1993 We will show in Volume IV on quantum mathematics that the physical origin of homology and cohomology is rooted in electric circuits and the Maxwell equations in electrodynamics 398 Rigorous Finite-Dimensional Magic Formulas early 1950s, Jean Leray and Henri Cartan showed that the theory of holomorphic functions of several variables can be reformulated elegantly in the language of sheaf cohomology37 The one-dimensional case As starting point, consider the Taylor expansion of the smooth function f : R → R,38 f (x + ∆x) = f (x) + f  (x)∆x + o(∆x), We define δx := ∆x, ∆x → δf := f  (x)δx, and df (x)(h) := f  (x)h, dx(h) := h for all h ∈ R With a view to generalizations to be considered below, note that • the variations δx and δf are real numbers, • whereas the differentials dx, df (x) : R → R are linear mappings (functionals) on the real line R Obviously, we have df (x)(h) = f  (x)dx(h) for all h ∈ R This is equivalent to the rigorous formula df (x) = f  (x)dx, in the sense of mappings The language of physicists In the physics literature, one proceeds frequently as follows By Taylor expansion, δf = f (x + δx) − f (x) = f  (x)δx + f  (x)δx2 + (7.59) Using (δx)2 = 0, we get δf = f  (x)δx (7.60) This is formally the same result as above In terms of mathematics, let us write f (x) = g(x) mod o(x), x→0 iff f (x) − g(x) = o(x) as x → In particular, (δx)2 = o(x) as x → means that (δx)2 = mod o(x), x → In standard mathematics, this replaces the magic relation (7.58) due to Newton and Leibniz The passage to the language of physicists consists then in 37 38 See K Maurin, Methods of Hilbert Spaces, PWN, Warsaw, 1972 The definition of the classical Landau symbols o(∆x) and O(∆x) can be found on page 932 In particular, we write r(χ) = o(χ) as χ → iff limχ→0 r(χ) = χ 7.20 Functional Calculus 399 dropping out the symbol mod o(x) for simplifying notation In this sense, equation (7.59) implies (7.60) The general case Consider now the functional Z : X → C on the complex Hilbert space X Our starting point is the definition of the functional derivatives δZ(J) δ Z(J) , δJ δJ Synonymously, we will write Z  (J) = δZ(J) , δJ δ Z(J) δJ Z  (J) = Naturally enough, we will formulate the corresponding definitions in such a way that, in the special case where Z = Z(J) is a complex-valued function of the real variable J (i.e., X = R), the functional derivative coincides with the classical derivative That is, Z  (J) = δZ(J) dZ(J) = , δJ dJ Z  (J) = δ Z(J) d2 Z(J) = δJ dJ Moreover, our notation will be chosen in such a way that in the classical case, X = R, we get δ Z(J) (h, k) = Z  (J)hk δJ δZ(J) (h) = Z  (h)h, δJ for all real numbers h, k In the volumes of this treatise, we will extensively use the calculus of differential forms In modern mathematics, differentials are not infinitesimally small quantities, but functionals Explicitly, dZ(J)(h) = Z  (J)(h), d2 Z(J)(h, k) = Z  (J)(h, k) In this sense, dZ(J) = Z  (J) and d2 Z(J) = Z  (J) In the calculus of variations, one writes δZ := δZ(J) (h) δJ This is called the first variation of the functional Z at the point J in direction of the vector h More precisely, one has to write δZ(J; h) := δZ(J) (h) δJ Similarly, δ Z := More precisely, δ Z(J) (h, k) δJ 400 Rigorous Finite-Dimensional Magic Formulas δ Z(J) (h, k) δJ Basic definition of the functional derivative Fix the point J ∈ X For given h ∈ X, define δ Z(J; h, k) = Z(J + th) − Z(J) δZ(J) (h) := lim t→0 δJ t Here, t is a real parameter If this limit exists, then it is called the directional derivative of the functional Z at the point J in direction of the vector h Equivalently, δZ(J) d (h) := (Z(J + th))|t=0 δJ dt This allows us the following physical interpretation Think of Z as temperature and regard J as a point in 3-dimensional Euclidean space Starting at the point J, we move along a straight line in direction of the vector h At time t we reach the point J + th and we observe the temperature Z(J + th) Differentiating this with respect to time t at the initial time, t = 0, we get the directional derivative of temperature Z at the point J in direction of the vector h This quantity is also called the temperature gradient at the point J in direction of h For the change of temperature, we get Z(J + th) = Z(J) + t In the general case, the map h → δZ(J) (h) + o(t), δJ δZ(J) δJ (h) t → represents an operator of the form δZ(J) : X → C δJ Parallel to classical calculus, this operator is also denoted by the symbol Z  (J) := δZ(J) δJ We call Z  (J) the functional derivative39 of the functional Z at the point J Higher-order functional derivatives Fix h, k ∈ X Naturally enough, we define δ Z(J) d δZ(J + tk) (h, k) := (h) δJ dt δJ |t=0 The map (h, k) → 39 δ Z(J) δJ (h, k) represents an operator of the form As a rule, the map Z  (J) : X → C is linear, but this is not always the case 7.20 Functional Calculus 401 δ Z(J) : X × X → C δJ We also introduce the notation Z  (J) := δ Z(J) δJ This operator is called the second functional derivative40 of the functional Z at the point J Summarizing, the first and second functional derivatives of the functional Z at the point J are operators of the form Z  (J) : X → C, Z  (J) : X × X → C Higher-order functional derivatives are defined analogously For example, fix h, k, l ∈ X We then define δ Z(J) d δ Z(J + tl) (h, k, l) := (h, k) δJ dt δJ |t=0 Example Define F (ψ) := ψ|ψ for all ψ ∈ X where X is a complex Hilbert space.41 Then, for all h, k ∈ X, F  (ψ)(h) = δF (ψ) (h) = ψ|h + ψ|h † δψ and F  (ψ)(h, k) = h|k + h|k † Proof Set χ(t) := F (ψ + th) for all t ∈ R Explicitly, χ(t) = ψ|ψ + t(ψ|h + h|ψ ) + t2 h|h This implies χ (0) = F  (ψ)(h) = ψ|h + h|ψ Moreover, let k ∈ X Set (t) := F  (ψ + kt)(h) = ψ + tk|h + h|ψ + tk , t ∈ R Hence  (0) = F  (ψ)(h, k) = k|h + h|k 7.20.2 Partial Functional Derivatives Here, one finds a method which requires only a simple use of the principles of differential and integral calculus; above all I must call attention to the fact that I have introduced in my calculations a new characteristic δ since this method requires that the same quantities vary in two different ways Comte de Joseph Louis Lagrange, 1762 40 41 As a rule, Z  (J)(h, k) is linear with respect to h and k, and we have the symmetry property Z  (J)(h, k) = Z  (J)(k, h), but this is not always the case In a real Hilbert space X, we have F  (ψ)(h) = 2
ψ|h and F  (ψ)(h, k) = 2
h|k for all h, k ∈ X 402 Rigorous Finite-Dimensional Magic Formulas By generalizing Euler’s 1744 method, Lagrange (1736–1813) got the idea for his remarkable formulas, where in a single line there is contained the solution of all problems of analytic mechanics Carl Gustav Jacob Jacobi (1804–1851) It is our goal to generalize the classical partial derivatives ∂ f (x, y) ∂x∂y ∂f (x, y) , ∂x to the partial functional derivatives δZ(J) , δJ(x) δ Z(J) , δJ(x)δJ(y) respectively In classical calculus, the problem f (x1 , x2 ) = critical! is equivalent to ∂f (x1 , x2 ) =0 for all indices j ∂xj In the calculus of variations, the solutions of the principle of critical action S(ψ) = critical!, ψ∈X satisfy the so-called variational equation δS(ψ) = δψ This implies δS(ψ) =0 δψ(x) for all indices x (7.61) which represents the desired equation of motion for the field ψ This equation is also called the Euler–Lagrange equation We will show in this treatise that all of the fundamental field equations in physics are of the type (7.61) For example, this concerns the electromagnetic field, non-relativistic and relativistic quantum mechanics, the Standard Model in particle physics, and the theory of general relativity The basic tool for introducing partial functional derivatives is the notion of the density of a given functional; this generalizes the classical mass density Density of a functional Consider a set M equipped with a measure µ Let X be an appropriate subspace of the space of all the functions h : M → C For given function ∈ X, define 7.20 Functional Calculus 403  F (h) := for all h ∈ X (x)h(x)dµ(x) C The function is called the density function of the functional F This definition is well formulated if the function is uniquely determined by the functional F This uniqueness has to be checked in each case Examples will has a be considered below Suppose now that the functional derivative δZ(J) δJ density , that is,  δZ(J) (h) = (x)h(x)dµ(x) for all h ∈ X δJ M We then define δZ(J) := (x) δJ(x) This is called the partial functional derivative of the functional Z at the point J with respect to the ‘index’ x Summarizing, for all h ∈ X, we get the suggestive formula δZ(J) (h) = δJ  M δZ(J) h(x)dµ(x) δJ(x) The variational lemma Let us now study a few examples which are prototypes for general situations arising in quantum field theory For checking the density property, we will use the following result Let −∞ < a < b < ∞, and let C[a, b] denote the space of all continuous functions f : [a, b] → R Proposition 7.22 Suppose that for given two functions f, g ∈ C[a, b], we have  b  b f (x)h(x)dx = g(x)h(x)dx for all h ∈ C[a, b] (7.62) a a Then f (x) = g(x) for all x ∈ [a, b] Proof Set F (x) := f (x) − g(x) Then h ∈ C[a, b] Choosing h = F ,  b a F (x)h(x)dx = for all functions b F (x)2 dx = a Hence F (x) = for all x ∈ [a, b] It is important for the calculus of variations that there exists a stronger variant of the preceding proposition To this end, let D(a, b) denote the set of all smooth functions h : [a, b] → R which have compact support in the open interval ]a, b[, that is, they vanish in some open neighborhoods of the boundary points a and b 404 Rigorous Finite-Dimensional Magic Formulas Proposition 7.23 Suppose that for given two functions f, g ∈ C[a, b], we have  b  b f (x)h(x)dx = g(x)h(x)dx for all h ∈ D(a, b) (7.63) a a Then f (x) = g(x) for all x ∈ [a, b] This is a special case of the variational lemma to be considered in Prop 10.15 on page 543 The idea of proof is to use a limiting process in order to get the identity (7.62) from (7.63) Example Set X := C[a, b] Fix the function ∈ X, and define  b Z(J) := (x)J(x)dx for all J ∈ X a This is a functional F : X → R on the real, linear function space X Proposition 7.24 For fixed J ∈ X, the functional derivative is given by δZ(J) (h) = δJ  b (x)h(x)dx for all h ∈ X a This functional has the function as density According to this fact, for each given point x ∈ [a, b] we define the partial functional derivative as δZ(J) := (x) δJ(x) Proof Fix h ∈ X Define  b χ(t) := Z(J + th) = (x){J(x) + th(x)}dx, t ∈ R a b Hence χ (0) = Z  (J)(h) = a (x)h(x)dx The uniqueness of the density function follows from Prop 7.22 Example Choose again X := C[a, b] We now set  Z(J) := (x, y)J(x)J(y)dxdy for all J ∈ X C Here, we are given the continuous function : M → R on the closed square M := [a, b] × [a, b] In addition, we assume that is symmetric, that is, we have (x, y) = (y, x) for all (x, y) ∈ M 7.20 Functional Calculus 405 Proposition 7.25 Fix J ∈ X For all h, k ∈ X, the first functional derivative and the second functional derivative are given by  δZ(J) (h) = (x, y)h(x)J(y)dxdy δJ M and δ Z(J) (h, k) = δJ  (x, y)h(x)k(y)dxdy, M respectively The second functional derivative has the function density Therefore, for all x, y ∈ [a, b], we define δ Z(J) := (x, y) δJ(x)δJ(y) This is the second partial functional derivative of the functional Z Proof (I) First functional derivative Let t ∈ R From  χ(t) := Z(J + th) = (x, y){J(x) + th(x)}{J(y) + th(y)}dxdy M we get χ (0) = Z  (J)(h) =  { (x, y)h(x)J(y) + (x, y)J(x)h(y)}dxdy M  By symmetry of , Z  (J)(h) = M (x, y)h(x)J(y)dxdy (II) Second partial derivative Differentiating the function σ(t) := Z  (J + tk)(h) with respect to t at the point t = 0, we obtain   Z (J)(h, k) = (x, y)h(x)k(y)dxdy M (III) Uniqueness of the density Let ∗ : M → R be continuous, and suppose that   (x, y)h(x)k(y)dxdy = ∗ (x, y)h(x)k(y)dxdy (7.64) M M for all h, k ∈ X Let F : M → R be an arbitrary continuous function By the classical Weierstrass theorem, there exists a sequence pn : M → R of polynomials in two variables such that maxa≤x,y≤b |f (x, y) − pn (x, y)| → as   n → ∞ By (7.64), M (x, y)pn (x, y)dxdy = M ∗ (x, y)pn (x, y)dxdy for all indices n Letting n → ∞, we get 406 Rigorous Finite-Dimensional Magic Formulas   (x, y)F (x, y)dxdy = M ∗ (x, y)F (x, y)dxdy M for all continuous functions F : M → R The same argument as in the proof of Prop 7.24 tells us now that = ∗ on M Example In the calculus of variations, the density functions of functionals are obtained by using integration by parts As a prototype, consider the functional  t1 S[q] := q(t) ˙ dt for all q ∈ X t0 The dot denotes the time derivative By definition, the symbol X represents the space of all smooth functions q : [t0 , t1 ] → R on the compact time interval [t0 , t1 ] which vanish on the boundary, that is, q(t0 ) = q(t1 ) = Proposition 7.26 Let q ∈ X The functional derivative is given by  t1 δS[q] (h) = ă q(t)h(t)dt for all h X q t0 This functional has the density ă q Therefore, for each point t ∈ [t0 , t1 ], we define the partial functional derivative S[q] := ă q (t) q(t) Proof (I) Functional derivative Fix q, h ∈ X For each parameter τ ∈ R, define  t1 ˙ χ(τ ) := S[q + τ h] = (q(t) ˙ + τ h(t)) dt, τ ∈ R t0 t ˙ Hence χ (0) = S  [q](h) = t01 q(t) ˙ h(t)dt Observing the boundary condition h(t0 ) = h(t1 ) = 0, integration by parts yields  S [q](h) =  t1 qă(t)h(t)dt for all h ∈ X t0 This proves the claim for the functional derivative (II) Uniqueness of the density Let g : [t0 , t1 ] → R be a continuous function Suppose that  t1  t1 ă q (t)h(t)dt = g(t)h(t)dt for all h ∈ X t0 t0 In particular, this identity is true for all functions h ∈ D(t0 , t1 ) By the variational lemma (Prop 7.23), g = ă q on [t0 , t1 ] 7.20 Functional Calculus 407 The principle of least action for the classical harmonic oscillator Let us study the motion q = q(t) of a particle of mass m on the real line Fix the compact time interval [t0 , t1 ] The functional  t1 κq(t)2 mq(t) ˙ − + F (t)q(t) dt, q∈X S[q] := 2 t0 is called the action of the particle Here, κ is a positive number called coupling constant, and the given function F : [t0 , t1 ] → R is smooth By definition, the space X consists of all smooth functions q : [t0 , t1 ] → R which satisfy the following boundary condition q(t0 ) = 0, q(t1 ) = The principle of least action for the motion of the particle reads as S[q] = min!, q ∈ X (7.65) Let us first compute the functional derivative To this end, fix q, h ∈ X Set χ(τ ) := S[q + τ h], τ ∈ R Differentiating with respect to the parameter τ at the point τ = 0,  t1   ˙ mq(t) ˙ h(t) − κq(t)h(t) + F (t)h(t) dt χ (0) = S  [q](h) = t0 Observing the boundary condition h(t0 ) = h(t1 ) = 0, integration by parts yields  t1 S  [q](h) = (mă q(t) q(t) + F (t)) h(t)dt t0 Hence, for all t ∈ [t0 , t1 ], we obtain the partial functional derivative S[q] = mă q(t) − κq(t) + F (t) δq(t) Theorem 7.27 Each solution q = q(t) of the principle of least action (7.65) satises the EulerLagrange equation S[q] =0 q(t) Explicitly, mă q (t) = −κq(t) + F (t) for all t ∈ [t0 , t1 ] 408 Rigorous Finite-Dimensional Magic Formulas This is the equation of a so-called harmonic oscillator with the restoring force −κq and the external force F If the external force vanishes, F = 0, then we get the special solution q(t) = const · sin(ωt), t ∈ R This motion represents an oscillation on the real line with the positive angular frequency ω given by the relation ω = κ/m Proof Let q ∈ X be a solution of the minimum problem (7.65) Fix the function h ∈ D(t0 , t1 ) Introduce the function for all τ ∈ R χ(τ ) := S[q + τ h] Since h(t0 ) = h(t1 ) = 0, we get h ∈ X Consequently, the simplified problem χ(τ ) = min!, τ ∈R has the solution τ = By classical calculus, χ (0) = This yields S  [q](h) = for all h ∈ D(t0 , t1 ) Hence  t1 δS[q] δS[q] h(t)dt = (h) = δq(t) δq t0 for all h ∈ D(t0 , t1 ) This implies δS[q] =0 δq(t) for all t ∈ [t0 , t1 ], by the variational lemma (Prop 7.23) This finishes the classical proof invented by the young Lagrange in 1762 The same argument applies to all kinds of variational problems in mathematics and physics In the volumes of this treatise, we will encounter plenty of such variational problems The principle of critical action Consider first the smooth real function f : R → R By definition, the problem f (x) = critical!, x∈R is equivalent to f  (x) = The solutions are called the critical points of the function f This includes minimal points, maximal points, and horizontal inflection points Similarly, by definition, the following critical point problem S[q] = critical!, q∈X (7.66) 7.20 Functional Calculus 409 is equivalent to S  [q] = The same argument as above shows that each solution q of (7.66) satisfies the same Euler–Lagrange equation as obtained in Theorem 7.27 The principle of least action versus the principle of critical action Let us finish with the following remark Consider first the real function f : R → R given by f (x) := x3 The minimum problem x∈R f (x) = min!, has no solution, but the critical point problem f (x) = critical!, x∈R has the solution x = In fact, f  (x) = implies 3x2 = 0, and hence x = The same happens to more general variational problems Therefore, we will not use the principle of least action, but the more general principle of critical action The language of physicists In Sect 11.2.3 on page 591, we will consider a formal definition of partial functional derivatives based on the Dirac delta function This formal definition is used in most physics textbooks The experience shows that both our rigorous approach introduced above and the formal approach based on the Dirac delta function lead to the same results 7.20.3 Infinitesimal Transformations Infinitesimal symmetry transformations know much, but not all about global symmetry transformations Folklore In order to investigate the invariance of physical processes under symmetries, physicists simplify the considerations by using infinitesimal transformations This theory was created by Sophus Lie (1849–1899) in about 1870 Let us discuss some basic ideas in rigorous terms Roughly speaking, infinitesimal transformations are obtained by neglecting terms of higher order than one The prototype of infinitesimal transformations are infinitesimal rotations Let us study this first Infinitesimal rotations The transformation x = x cos θ − y sin θ, y  = x sin θ + y cos θ (7.67) represents a counterclockwise rotation about the origin in the plane, with rotation angle θ (Fig 7.2) For small rotation angle θ, we get x = x − yθ + o(θ), y  = y + xθ + o(θ), θ → 410 Rigorous Finite-Dimensional Magic Formulas K y y6  x* K θx Fig 7.2 Rotation Thus, the linearization of the rotation (7.67) reads as x = x − yθ, y  = y + xθ (7.68) where θ is a fixed real number This linear transformation is called an infinitesimal rotation with rotation angle θ Physicists set δθ := θ, δx := x − x, δy := y  − y Therefore, the infinitesimal rotation (7.68) reads as δx = −yδθ, δy = xδθ (7.69) Invariant functions The smooth function f : R2 → R is called invariant under rotations iff for all rotation angles θ and all x, y ∈ R, we have f (x , y  ) = f (x, y) Moreover, the function f is called invariant under infinitesimal rotations iff for all rotation angles δθ and all x, y ∈ R, we get f (x + δx, y + δy) = f (x, y) + o(δθ), δθ → where δx and δy are given by (7.69) As the prototype of the classical Lie theory on invariant functions, let us prove the following result The point is that global symmetry properties can be described by a local equation, namely, a partial differential equation Proposition 7.28 For each smooth function f : R2 → R, the following three conditions are equivalent (i) The function f is invariant under infinitesimal rotations (ii) The function f satisfies the Lie partial differential equation xfy (x, y) − yfx (x, y) = for all (x, y) ∈ R2 (iii) The function f is invariant under rotations (iv) In polar coordinates ϕ, r, the function f only depends on r (7.70) .. .Quantum Field Theory I: Basics in Mathematics and Physics Eberhard Zeidler Quantum Field Theory I: Basics in Mathematics and Physics A Bridge between Mathematicians and Physicists With 94 Figures... 515 515 516 516 517 518 519 519 10 In? ??nite-Dimensional Hilbert Spaces 10 .1 The Importance of In? ??nite Dimensions in Quantum Physics 10 .1. 1 The Uncertainty Relation... Related Fields, Springer, Berlin, 19 70 (reprinted with permission) 1. 1 The Revolution of Physics 23 To explain this basic principle for describing nature in terms of mathematics, consider our