Part 1 of ebook Quantum field theory II: Quantum electrodynamics provide readers with content about: mathematical principles of modern natural philosophy; the basic strategy of extracting finite information from infinities – ariadne’s thread in renormalization theory; the power of combinatorics; the strategy of equivalence classes in mathematics; basic ideas in classical mechanics;...
Quantum Field Theory II: Quantum Electrodynamics Eberhard Zeidler Quantum Field Theory II: Quantum Electrodynamics A Bridge between Mathematicians and Physicists Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Inselstr 22-26 04103 Leipzig Germany ISBN 978-3-540-85376-3 e-ISBN 978-3-540-85377-0 DOI 10.1007/978-3-540-85377-0 Library of Congress Control Number: 2006929535 Mathematics Subject Classification (2000): 35-XX, 47-XX, 49-XX, 51-XX, 55-XX, 81-XX, 82-XX c 2009 Springer-Verlag Berlin Heidelberg 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 to prosecution under the German Copyright Law 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 Cover design: WMXDesign GmbH Printed on acid-free paper springer.com TO FRIEDRICH HIRZEBRUCH IN GRATITUDE Preface And God said, Let there be light; and there was light Genesis 1,3 Light is not only the basis of our biological existence, but also an essential source of our knowledge about the physical laws of nature, ranging from the seventeenth century geometrical optics up to the twentieth century theory of general relativity and quantum electrodynamics Folklore Don’t give us numbers: give us insight! A contemporary natural scientist to a mathematician The present book is the second volume of a comprehensive introduction to the mathematical and physical aspects of modern quantum field theory which comprehends 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 Gravitation and String Theory It is our goal to build a bridge between mathematicians and physicists based on the challenging question about the fundamental forces in • macrocosmos (the universe) and • microcosmos (the world of elementary particles) The six volumes address a broad audience of readers, including both undergraduate and graduate students, as well as experienced scientists who want to become familiar with quantum field theory, which is a fascinating topic in modern mathematics and physics For students of mathematics, it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to VIII Preface discover interesting interrelationships between quite different mathematical topics For students of physics, fairly advanced mathematics are presented, which is beyond the usual curriculum in physics The strategies and the structure of the six volumes are thoroughly discussed in the Prologue to Volume I In particular, we will try to help the reader to understand the basic ideas behind the technicalities In this connection, the famous ancient story of Ariadne’s thread is discussed in the Preface to Volume I In terms of this story, we want to put the beginning of Ariadne’s thread in quantum field theory into the hands of the reader The present volume is devoted to the physics and mathematics of light It contains the following material: Part I: Introduction • Chapter 1: Mathematical Principles of Modern Natural Philosophy • Chapter 2: The Basic Strategy of Extracting Finite Information from Infinities – Ariadne’s Thread in Renormalization Theory • Chapter 3: The Power of Combinatorics • Chapter 4: The Strategy of Equivalence Classes in Mathematics Part II: Basic Ideas in Classical Mechanics • Chapter 5: Geometrical Optics • Chapter 6: The Principle of Critical Action and the Harmonic Oscillator as a Paradigm Part III: Basic Ideas in Quantum Mechanics • Chapter 7: Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization • Chapter 8: Quantum Particles on the Real Line – Ariadne’s Thread in Scattering Theory • Chapter 9: A Glance at General Scattering Theory Part IV: Quantum Electrodynamics (QED) • Chapter 10: Creation and Annihilation Operators • Chapter 11: The Basic Equations in Quantum Electrodynamics • Chapter 12: The Free Quantum Fields of Electrons, Positrons, and Photons • Chapter 13: The Interacting Quantum Field, and the Magic Dyson Series for the S-Matrix • Chapter 14: The Beauty of Feynman Diagrams in QED • Chapter 15: Applications to Physical Effects Part V: Renormalization • Chapter 16: The Continuum Limit • Chapter 17: Radiative Corrections of Lowest Order • Chapter 18: A Glance at Renormalization to all Orders of Perturbation Theory • Chapter 19: Perspectives Preface IX We try to find the right balance between the mathematical theory and its applications to interesting physical effects observed in experiments In particular, we not consider purely mathematical models in this volume It is our philosophy that the reader should learn quantum field theory by studying a realistic model, as given by quantum electrodynamics Let us discuss the main structure of the present volume In Chapters through 4, we consider topics from classical mathematics which are closely related to modern quantum field theory This should help the reader to understand the basic ideas behind quantum field theory to be considered in this volume and the volumes to follow In Chapter on the mathematical principles of modern natural philosophy, we discuss • the infinitesimal strategy due to Newton and Leibniz, • the optimality principle for processes in nature (the principle of critical action) and the calculus of variations due to Euler and Lagrange, which leads to the fundamental differential equations in classical field theory, • the propagation of physical effects and the method of the Green’s function, • harmonic analysis and the Fourier method for computing the Green’s functions, • Laurent Schwartz’s theory of generalized functions (distributions) which is related to the idea that the measurement of physical quantities by devices is based on averaging, • global symmetry and conservation laws, • local symmetry and the basic ideas of modern gauge field theory, and • the Planck quantum of action and the idea of quantizing classical field theories Gauge field theory is behind both • the Standard Model in elementary particle physics and • Einstein’s theory of gravitation (i.e., the theory of general relativity) In quantum field theory, a crucial role is played by renormalization In terms of physics, this is based on the following two steps: • the regularization of divergent integrals, and • the computation of effective physical parameters measured in experiments (e.g., the effective mass and the effective electric charge of the electron) Renormalization is a highly technical subject For example, the full proof on the renormalizability of the electroweak sector of the Standard Model in particle physics needs 100 pages This can be found in: E Kraus, Renormalization of the electroweak standard model to all orders, Annals of Physics 262 (1998), 155–259 Whoever wants to understand quantum field theory has to understand the procedure of renormalization Therefore, the different aspects of renormalization theory will be studied in all of the six volumes of this series of monographs This ranges from • resonance phenomena for the anharmonic oscillator (classical bifurcation theory), • the Poincar´e–Lindstedt series (including small divisors) in celestial mechanics, X Preface • and the Kolmogorov–Arnold–Moser (KAM) theory for perturbed quasi-periodic oscillations (e.g., in celestial mechanics) based on sophisticated iterative techniques (the hard implicit function theorem) to the following fairly advanced subjects: • • • • • • • • • • • • • • • the Feynman functional integral (the Faddeev–Popov approach), the Wiener functional integral (the Glimm–Jaffe approach), the theory of higher-dimensional Abelian integrals (algebraic Feynman integrals), Hopf algebras and Rota–Baxter algebras in combinatorics (the modern variant of the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach due to Kreimer), the Riemann–Hilbert problem and the Birkhoff decomposition (the Connes– Kreimer approach), Hopf superalgebras (the Brouder–Fauser–Frabetti–Oeckl (BFFO) approach), characterization of physical states by cohomology and algebraic renormalization (the Becchi–Rouet–Stora–Tyutin (BRST) approach), the Riesz–Gelfand theory of distribution-valued meromorphic functions (construction of the Greens functions), wave front sets and Hă ormanders multiplication of distributions (the Stueckelberg– Bogoliubov–Epstein–Glaser–Scharf approach), the Master Ward identity as a highly non-trivial renormalization condition and the generalized DysonSchwinger equation (the Dă utschFredenhagen approach), q-deformed quantum eld theory (the Wess–Majid–Wachter–Schmidt approach based on the q-deformed Poincar´e group, quantum groups, and the q-analysis on specific classes of q-deformed quantum spaces), deformation of bundles and quantization (the Weyl–Flato–Sternheimer–Fedosov– Kontsevich approach), microlocal analysis and renormalization on curved space-times (the Radzikowski BrunettiFredenhagenKă ohler approach), renormalized operator products on curved space-times (the Wilson–Hollands– Wald approach to quantum field theory), natural transformations of functors in category theory and covariant quantum field theory on curved space-time manifolds (the Brunetti–Fredenhagen–Verch approach), as well as • one-parameter Lie groups and the renormalization group, • attractors of dynamical systems in the space of physical theories (the Wilson– Polchinski–Kopper–Rivasseau approach to renormalization based on the renormalization group), • the Master Ward Identity and the StueckelbergPetermann renormalization group (the Dă utsch–Fredenhagen approach), • motives in number theory and algebraic geometry, the Tannakian category, and the cosmic Galois group as a universal (motivic) renormalization group (the Connes–Marcolli approach), • noncommutative geometry and renormalization (the Grosse–Wulkenhaar approach) The recent work of Alain Connes, Dirk Kreimer, and Matilde Marcolli shows convincingly that renormalization is rooted in highly nontrivial mathematical structures We also want to emphasize that the theory of many-particle systems (considered in statistical physics and quantum field theory) is deeply rooted in the theory of operator algebras This concerns Preface • • • • XI von Neumann algebras (the von Neumann approach), C ∗ -algebras (the Gelfand–Naimark–Segal approach), local nets of operator algebras (the Haag–Kastler approach) and, noncommutative geometry (the Connes approach) As a warmup, we show in Chapter that the regularization of divergent expressions represents a main subject in the history of mathematics starting with Euler in the eighteenth century In this connection, we will consider • the regularization of divergent series, and • the regularization of divergent integrals In particular, in Sect 2.1.3, we will discuss the classical Mittag–Leffler theorem on meromorphic functions f If the function f has merely a finite number of poles, then the method of partial fraction decomposition works well However, as a rule, this method fails if the function f has an infinite number of poles In this case, Mittag–Leffler showed in the late nineteenth century that one has to subtract special terms, which are called subtractions by physicists The subtractions force the convergence of the infinite series This is the prototype of the method of iteratively adding subtractions in the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach to renormalization theory The corresponding iterative algorithm (called the Bogoliubov R-operation) has to be constructed carefully (because of nasty overlapping divergences) This was done by Nikolai Bogoliubov in the 1950s An ingenious explicit solution formula for this iterative method was found by Wolfhart Zimmermann in 1969 This is the famous Zimmermann forest formula In the late 1990s, it was discovered by Dirk Kreimer that the sophisticated combinatorics of the Zimmermann forest formula can be understood best in terms of a Hopf algebra generated by Feynman diagrams By this discovery, the modern formulation of the BPHZ approach is based on both Hopf algebras and Rota–Baxter algebras As a warmup, in Chapter 3, we give an introduction to the modern combinatorial theory, which was founded by Gian-Carlo Rota (MIT, Cambridge, Massachusetts) in the 1960s This includes both Hopf algebras and Rota– Baxter algebras Surprisingly enough, it turns out that the Zimmermann forest formula is closely related to methods developed by Lagrange in the eighteenth century when studying the solution of the Kepler equation for the motion of planets in celestial mechanics In modern terminology, the Lagrange inversion formula for power series expansions is based on the so-called Fa`a di Bruno Hopf algebra.1 This will be studied in Sect 3.4.3 The Italian priest and mathematician Francesco Fa` a di Bruno (1825–1888) was beatified in 1988 6.11 The Spherical Pendulum 411 TP M P v✲ M Fig 6.16 Motion of a particle on a sphere M = S2R 6.11 The Spherical Pendulum Two-dimensional spheres are the simplest curved surfaces They serve as prototypes for the geometry and analysis of manifolds Folklore We want to study the motion of a point of mass m on a sphere S2R of radius R under the influence of the gravitational force This is called a spherical pendulum (Fig 6.16) In a Cartesian (x, y, z)-coordinate system, the position of a point of the sphere is described by the vector q = xi + yj + zk (6.78) with initial point at the center of the ball The equation of the sphere reads as x2 + y + z = R2 The point (0, 0, R) (resp (0, 0, −R)) is called the North Pole (resp South Pole) 6.11.1 The Gaussian Principle of Critical Constraint The Newtonian equation of motion for the spherical pendulum reads as mă q(t) = mgk + Fc (q(t), q(t)), t ∈ R (6.79) Here, the vector −gmk describes the gravitational force on the surface of earth This force acts in direction of the negative z-axis The additional constraining force Fc keeps the particle on the sphere In order to compute the constraining force, let us use the most general principle in classical mechanics, namely, the Gaussian principle of critical constraint (see Sect 7.28 of Vol I) We have to solve the following extremal principle: (mă q F)2 = critical! (6.80) with the gravitational force F = −gmk and the constraint q = R As we have shown on page 492 of Vol I, this leads to the constraining force 2 ˙ = λ(q, q) ˙ ·q Fc (q, q) which is a normal force depending on position and velocity Explicitly, ˙ := λ(q, q) mg · qk − q˙ R2 Using this, the equation of motion (6.79) looks rather complicated It is our goal to simplify the approach 412 Principle of Critical Action and the Harmonic Oscillator In order to eliminate the constraining force Fc , we will use the principle of critical action in terms of spherical coordinates The gravitational force on the surface of earth has the potential U (z) := mgz In fact, the gravitational force −mgk is equal to −U (z)k Therefore, the Lagrangian, L : = kinetic energy minus potential energy, reads as ˙ := 12 mq˙ − mgz L(q, q) (6.81) This Lagrangian is basic for the following approach 6.11.2 The Lagrangian Approach In mechanics it is important to use the appropriate coordinates Folklore Spherical coordinates Let ϕ be the geographic length of the sphere, and let ϑ be the geographic latitude with −π ≤ ϕ ≤ π and − π2 ≤ ϑ ≤ π2 The equator is described by the equation ϑ = (see Fig 5.16 on page 305) For the relation between Cartesian coordinates x, y, z and spherical coordinates, we get x = R cos ϕ cos ϑ, y = R sin ϕ cos ϑ, z = R sin ϑ (6.82) The coordinate line ϑ = const is a parallel line of latitude; it has the tangent vector b1 (P ) := ∂q (P ) = R(− sin ϕ cos ϑ i + cos ϕ cos ϑ j) ∂ϕ at the point P = (ϕ, ϑ), by (6.78) Similarly, the meridian ϕ = const has the tangent vector ∂q b2 (P ) := (P ) = R(− cos ϕ sin ϑ i − sin ϕ sin ϑ j + cos ϑ k) ∂ϑ at the point P The two vectors b1 , b2 form an orthogonal basis of the tangent plane TP S2R of the sphere at the point P 27 The exterior unit normal vector N of the sphere at the point P is given by N := e1 × e2 , where ej := bj (P ) , |bj (P )| j = 1, The three vectors e1 , e2 , N form a right-handed orthonormal system at the point P of the sphere S2R Velocity vector (tangent vector) The motion of a point on the sphere is described by the position vector q(t) = x(t)i + y(t)j + z(t)k, t ∈ R Differentiating the functions x(t) = R cos ϕ(t) sin ϑ(t), y(t) = R sin ϕ(t) cos ϑ(t), z(t) = R sin ϑ(t) with respect to time t, we obtain 27 The point P has to be different from the North Pole and the South Pole, since the spherical coordinates are singular at the two poles This can be cured by introducing new local coordinates near the two poles (e.g., the coordinates (x, y)) To simplify the approach, we restrict ourselves to spherical coordinates 6.11 The Spherical Pendulum 413 ˙ cos ϕ(t) sin ϑ(t) x(t) ˙ = −Rϕ(t) ˙ sin ϕ(t) cos ϑ(t) − Rϑ(t) ˙ cos ϑ(t), and Furthermore, z(t) ˙ = Rϑ(t) ˙ sin ϕ(t) sin ϑ(t) y(t) ˙ = Rϕ(t) ˙ cos ϕ(t) cos ϑ(t) − Rϑ(t) Since cos2 α + sin2 α = 1, we get ˙ ) ˙ = x(t) ˙ + y(t) ˙ + z(t) ˙ = R2 (ϕ(t) ˙ cos2 ϑ(t) + ϑ(t) |q(t)| The transformed Lagrangian In terms of spherical coordinates, the Lagrangian reads as ˙ = mR2 (ϕ˙ cos2 ϑ + ϑ˙ ) − mgR sin ϑ L(ϕ, ϑ, ϕ, ˙ ϑ) The principle of critical action postulates that Z t1 ˙ L(ϕ(t), ϑ(t), ϕ(t), ˙ ϑ(t)) dt = critical! (6.83) t0 with (ϕ(t0 ), ϑ(t0 )) = (ϕ0 , ϑ0 ) and (ϕ(t1 ), ϑ(t1 )) = (ϕ1 , ϑ1 ) (boundary condition) The Euler–Lagrange equation Each smooth solution of (6.83) satisfies d d L = Lϕ and dt Lϑ˙ = Lϑ Explicitly, the Euler–Lagrange equation for the motion dt ϕ˙ of the spherical pendulum looks like ´ d ` ϕ(t) ˙ cos2 ϑ(t) = 0, dt with ω := pg R ă + cos (t) = ϑ(t) (6.84) Setting ϑ = − π2 + , the second equation passes over to ă + ω sin α = This is the equation of a circular pendulum The first equation of (6.84) can be written as sin (t) = (t) ă − 2ϕ(t) ˙ ϑ(t) (6.85) Geodesics If the gravitational force vanishes (i.e., g = 0), then the spherical pendulum is called free The trajectories of the free pendulum are called geodesics of the sphere A piece of the equator (ϕ(t) = t, ϑ(t) = 0) is always a geodesic By definition, a great circle of the sphere is obtained by the intersection between the sphere and a plane passing through the center of the sphere After a rotation, if necessary, we can assume that the great circle is the equator Consequently, pieces of great circles are geodesics For the motion q = q(t) of the free pendulum, it follows from equation (6.79) ă (t) vanishes According that the tangential component of the acceleration vector q ă (t) is parallel along the trajectory As we to Levi-Civita, we say that the vector q will show in Vol III on gauge theory, the parallel transport of physical quantities is crucial for modern mathematics and physics This corresponds to the transport of physical information 414 Principle of Critical Action and the Harmonic Oscillator 6.11.3 The Hamiltonian Approach Introduce the generalized momenta pϕ := Lϕ˙ = mR2 ϕ˙ cos2 ϑ, ˙ pϑ := Lϑ˙ = mR2 ϑ, and the Hamiltonian, H := pϕ ϕ˙ + pϑ ϑ˙ − L That is, p2ϕ p2ϑ + mgR sin ϑ + 2 2mR cos ϑ 2mR2 Then the canonical equations read as H(ϕ, pϕ , ϑ, pϑ ) = p˙ ϕ = −Hϕ = 0, ϕ˙ = Hpϕ , p˙ ϑ = −Hϑ , ϑ˙ = Hpϑ In particular, pϕ (z-component of the angular momentum) and H (energy) are conserved Set z0 := R sin ϑ(0), pϕ := mR2 ϕ(0) ˙ cos2 ϑ(0), 2` ´ mR ˙ + mgR sin ϑ(0) , ϕ(0) ˙ cos2 ϑ(0) + ϑ(0) E := `E ´ ` p ´2 and V (z) := m − gz (R2 − z ) − mϕ ˙ Proposition 6.15 For given data ϕ(0), ϕ(0), ˙ ϑ(0), ϑ(0) with ϑ(0) ∈] − π2 , π2 [, there exists a unique motion ϕ = ϕ(t), ϑ = ϑ(t) of the spherical pendulum given by the following elliptic integrals: Z z Z dζ dζ pϕ R z p p t(z) = R , ϕ(z) = ϕ(0) + 2 m V (ζ) V (ζ)) z0 z0 (R − ζ ) Moreover, z(t) = R sin ϑ(t) Proof We have the two conservation laws ˙ cos2 ϑ(t) = pϕ , mR2 ϕ(t) 2mR2 ˙ p2ϕ mR2 ϑ(t) + + mgR sin ϑ(t) = E, cos ϑ(t) where pϕ and E (energy) are constants The substitution z = R sin ϑ yields ϕ˙ = pϕ , m(R2 − z ) dz dϑ dz dz ϑ˙ = · = = √ dz dt R cos ϑ dt R2 − z dt Therefore, by the energy conservation law, we obtain „ «2 p2ϕ dz mR2 + + 2mgz = 2E R2 − z dt m(R2 − z ) Hence R2 „ dz dt «2 = V (z) This implies dt R = p dz V (z) Moreover, dϕ pϕ dϕ dz dϕ p V (z) = = · = · dt dz dt dz R m(R2 − z ) Finally, integrating this over z, we obtain t = t(z) and ϕ = ϕ(z) ✷ 6.11 The Spherical Pendulum 415 6.11.4 Geodesics of Shortest Length Arc length Let q = q(t), t0 ≤ t ≤ t1 , be a curve on the sphere S2R By definition, the arc length of this curve between the points q0 and q(t) is given by Z t Z t q ˙ )2 dτ ˙ )| dτ = s(t) := |q(τ R ϕ(τ ˙ )2 cos2 ϑ(τ ) + ϑ(τ t0 t0 Differentiating this with respect to time t, we get q ˙ ˙ cos2 ϑ(t) + ϑ(t) s(t) ˙ = R ϕ(t) Hence „ „ «2 «2 „ «2 ds(t) dϕ(t) dϑ(t) = R2 cos2 ϑ(t) + R2 dt dt dt Mnemonically, we write ds2 = R2 cos2 ϑ · dϕ2 + R2 dϑ2 (6.86) Curves of shortest length Now we are looking for a smooth curve q = q(t), t ≤ t ≤ t1 S2R on the sphere which connects the two point q0 and q1 , and which has minimal length This is the optimal route for an aircraft which is flying from the city q0 to the city q1 We have to solve the variational problem Z t1 q ˙ dt = min! R ϕ(t) ˙ cos2 ϑ(t) + ϑ(t) (6.87) t0 with the side condition (ϕ(t0 ), ϑ(t0 )) = (ϕ0 , ϑ0 ) and (ϕ(t1 ), ϑ(t1 )) = (ϕ1 , ϑ1 ) Theorem 6.16 If the arc length s is chosen as parameter, then every solution of the variational problem (6.87) satisfies the following system of equations: (s) ă 2(s) (s) sin 2(s) = 0, ă (s) = (6.88) Proof The variational problem (6.87) can be written as Z t1 L dt = min! t0 √ with L := L where L is the Lagrangian of the spherical pendulum (6.83) with mass m = and vanishing gravitational force (i.e., g = 0) The Euler–Lagrange equations d d Lϕ˙ = Lϕ = 0, L ˙ = Lϑ = ds ds ϑ “ ” “ L ” Lϕ˙ ˙ d d √ √ϑ = and ds = The variational integral is invariant read as ds L L under reparametrizations Therefore, we can choose the arc length as parameter (i.e., t = s) Then L = Hence d Lϕ˙ = 0, ds d L ˙ = ds ϑ 416 Principle of Critical Action and the Harmonic Oscillator By (6.84), we obtain the claim (6.88) ✷ Suppose we are given two points q0 and q1 on the sphere S2R After a rotation, if necessary, we always can assume that the two points lie on the equator of the sphere Then there are two arcs of the equator which connect the two given points The smaller arc is a geodesic of shortest length between q0 and q1 Observe that two points not always uniquely determine the connecting geodesic For example, if the given points q0 and q1 represent the North Pole and the South Pole, respectively, then each meridian is a connecting geodesic (see Fig 5.9 on page 277) 6.12 The Lie Group SU (E ) of Rotations Invariance under rotations leads to conservation of angular momentum Folklore For the motion q = q(t) of a particle in the 3-dimensional space, the time-dependent ˙ vector p(t) := mq(t) (mass times velocity) is called the momentum of the particle at time t The angular momentum at time t is defined by a(t) := q(t) × p(t) (6.89) In this section, we study the motion q = q(t) of a particle given by the Newtonian equation mă q(t) = −U (q(t)), t∈R (6.90) under the assumption that the potential U = U (q) only depends on the distance |q| In other words, the potential U is invariant under rotations about the origin 6.12.1 Conservation of Angular Momentum Proposition 6.17 The angular momentum is conserved for the motion (6.90) Proof The trick is that b × p b = holds for all vectors b Let us use Cartesian (x, y, z)-coordinates Set r := x2 + y + z Then U (q) = V (r) It follows from ∂r x ∂ V (r) = V (r) = V (r) ∂x ∂x r and analogous formulas for y and z that q U (q) = V (r) r Let q = q(t) be a solution of (6.90) By the product rule, d q(t) ˙ a(t) = q(t) ì mq(t) + q(t) ì mă q(t) = −q(t) × V (r(t)) = dt r(t) ✷ Next we want to discuss the relation between the symmetry group of rotations and angular momentum The Lie group SU (E ) of rotations In the 3-dimensional space, all the rotations about the origin form a Lie group denoted by SU (E ) Each such rotation q → q+ can be represented by the Euler formula 6.12 The Lie Group SU (E ) of Rotations ✛ ❪ q+ 417 ϕ=π ✣ n ✻ q O Fig 6.17 Rotation q+ = q cos ϕ + (n × q) sin ϕ + (qn)n(1 − cos ϕ) (6.91) This formula describes the counterclockwise rotation of the position vector q at the origin about the axis unit vector n with the rotation angle ϕ (Fig 6.17) We write q+ = Rn,ϕ q The Lie algebra su(3) of infinitesimal rotations Linearization of (6.91) with respect to the angle ϕ yields q+ = q + (ϕn × q) (6.92) We call the transformation Rb given by Rb q := b × q an infinitesimal rotation Using the the usual linear combination of maps and the Lie product [Ra , Rb ]− := Ra Rb − Rb Ra , the following hold: The set of infinitesimal rotations forms a real 3-dimensional Lie algebra denoted by su(E ) To prove this, we have to show that Ra , Rb ∈ su(E ) implies [Ra , Rb ]− ∈ su(E ) This follows from the general theory of Lie groups However, in order to display the geometric meaning behind the Lie algebra property of su(E ), let us use the following argument based on the well-known geometric properties of the vector product In fact, it follows from the cyclic Jacobi identity for the vector product a × (b × c) + b × (c × a) + c × (a × b) = and from a × b = −b × a that a × (b × c) − b × (a × c) = (a × b) × c Hence Ra Rb c − Rb Ra c = Ra×b c (6.93) In addition, we have the exponential formula Rn,ϕ = eϕRn Conservation of angular momentum and the Noether theorem We want to show that the conservation of angular momentum is a consequence of the Noether theorem with respect to the rotation group To this end, suppose that U (q) = V (|q|) Then the Lagrangian ˙ = 12 mq˙ − U (q) L(q, q) (6.94) 418 Principle of Critical Action and the Harmonic Oscillator is invariant under rotations, and hence the action Z t1 ˙ L(q(t), q(t)) dt S := t0 is invariant under rotations Each rotation q+ = Rn,ϕ q about the origin can be represented by the Euler formula (6.91) Hence δq := ε d Rn,ε q|ε=0 = ε(n × q) dε Applying the Noether theorem (6.49) on page 387 to the three components of the vector function t → q(t), we get d ˙ (Lδt − (δq − qδt)L ˙) = q dτ (6.95) for each solution t → q(t) of (6.90) Here, δt = Therefore, δq · Lq˙ is conserved Consequently, δq(t) · p(t) = ε(n × q(t)) p(t) = εn(q(t) × p(t)) = const, t∈R for all parameters ε > and all unit vectors n Hence q(t) × p(t) = const Conservation of angular momentum and Poisson brackets Now we want to describe an alternative approach for proving the conservation of angular momentum This will be based on the use of Poisson brackets In fact, this approach will correspond to Lie’s momentum map to be defined in (6.97) below The Lagrangian L from (6.94) yields the momentum p = Lq˙ = mq˙ and the Hamiltonian p2 ˙ = H(q, p) := pq˙ − L(q, q) + V (|q|) 2m q This implies the partial derivatives Hp = p/m and Hq = V (|q|) |q| Let a(q, p) := q × p For fixed vector n, set A(q, p) := a(q, p)n, that is, A(q, p) := (q × p)n This way, the vector-valued function a = a(q, p) is replaced by the real-valued function A = A(q, p) To simplify notation, we not indicate that the function A depends on the vector n Our goal is to show that, for the Poisson bracket, we have {A, H} = (6.96) Then the function A is conserved Hence d (q(t) × p(t)) n = dt for all t This is true for all vectors n Consequently, we get d (q(t) × p(t)) = dt for all t This tells us that the angular momentum is conserved It remains to prove (6.96) Using cyclic permutation, we obtain the well-known vector identity (a × b)c = (b × c)a Hence A(q, p) = (p × n)q Then 6.13 Carath´eodory’s Royal Road to the Calculus of Variations 419 ∂A d (q, p)h = A(q + σh, p)|σ=0 = (p × n)h ∂q dσ (q, p)h = (n × q)h Finally, note that (a × b)c = iff the three Analogously, ∂A ∂p vectors a, b, c are not linearly independent Since Hp is parallel to p, and Hq is parallel to q, we get {A, H} = Aq Hp − Ap Hq = (p × n)Hp − (n × q)Hp = 6.12.2 Lie’s Momentum Map For all vectors q, p and all vectors n, define M (q, p)Rn := (q × p)n This is Lie’s momentum map related to the Lie algebra su(E ) More precisely, this is a map of the form M : (T E3 )d → su(E )d (6.97) Let us briefly discuss this If we fix the tuple (q, p), then the map Rn → M (q, p)Rn (6.98) assigns to each infinitesimal rotation Rn a real number This map is a linear functional on the Lie algebra su(E ) Hence the map (6.98) is an element of the dual space su(E )d to the Lie algebra su(E ) Consequently, the map (q, p) → M (q, p) assigns to each point (q, p) of the phase space an element of the dual space su(E )d Finally, the points (q, p) of the phase space form the cotangent bundle (T E3 )d of the Euclidean manifold E3 This yields the map (6.97) 6.13 Carath´ eodory’s Royal Road to the Calculus of Variations 6.13.1 The Fundamental Equation Field of trajectories Fix n = 1, 2, Caratheodory’s fundamental equation reads as St (q, t) = L(q, v(q, t), t) − v(q, t)Lq˙ (q, v(q, t), t), Sq (q, t) = Lq˙ (q, v(q, t), t), (q, t) ∈ Rn+1 (6.99) We are given the smooth Lagrangian L : R2n+1 → R, where L depends on the variables q, q˙ ∈ Rn and t ∈ R We set q = (q , , q n ), as well as Lq = (Lq1 , , Lqn ) (H) We assume that all the eigenvalues of the matrix Lq˙q˙ (q, q, ˙ t) of the second-order partial derivatives of L (with respect to q˙1 , , q˙n ) are positive for any fixed argument (q, q, ˙ t) ∈ R2n+1 420 Principle of Critical Action and the Harmonic Oscillator This means that the function q˙ → L(q, q, ˙ t) is strictly convex on Rn for any fixed (q, t) ∈ Rn+1 We are looking for smooth functions v : Rn+1 → R and S : Rn+1 → R, which are called velocity field and action function, respectively The smooth solutions q = q(t), t ∈ R of the so-called velocity equation q(t) ˙ = v(q(t), t), x∈R (6.100) are called trajectories of (6.99) The set of all the solutions of (6.100) is called a field of trajectories Note that different trajectories of the field not intersect, since the solution of the initial-value problem for the velocity equation (6.100) is unique This generalizes the situation pictured in Fig 5.8 on page 273.28 In addition, let us consider the principle of least action: Z t1 L(t, q(t), q(t)) ˙ dt = min!, q(t0 ) = q0 , q(t1 ) = q1 (6.101) t0 Theorem 6.18 Let v, S be a smooth solution of Carath´ eodory’s fundamental equation (6.99) Fix the points (t0 , q0 ) and (t1 , q1 ) in Rn+1 Then the following hold: (i) Let q = q∗ (t), t0 ≤ t ≤ t1 , be a trajectory of (6.99) which satisfies the boundary condition q∗ (t0 ) = q0 and q∗ (t1 ) = q1 Then q∗ is a solution of the minimum problem (6.101) (ii) The difference S(q1 , t1 ) − S(t0 , q0 ) is equal to the integral Z t1 L(q∗ (t), q˙∗ (t), t) dt t0 The proof proceeds analogously to the proof of Theorem 5.2 on page 272 Extremals By definition, precisely the solutions q = q(t) of the Euler– Lagrange equation d ˙ t) = Lq (q(t), q(t), ˙ t) Lq˙ (q(t), q(t), dt are called extremals of the Lagrangian L In particular, the solutions of the minimum problem (6.101) are extremals Theorem 6.18 shows that The trajectories of (6.99) are extremals This statement remains true if the convexity assumption (H) above is not satisfied Then, for the proof, one has to use the Carath´eodory equation (6.99) together with the integrability condition Stq = Sqt Therefore, the field of trajectories related to (6.100) is also called a field of extremals Legendre transformation We define the co-velocity (or momentum) ˙ t) p := Lq˙ (q, q, Because of assumption (H) above, this equation is locally invertible, by the implicit function theorem To simplify the formulation, we assume that the Legendre transformation (q, q, ˙ t) → (q, p, t) is a diffeomorphism from R2n+1 onto R2n+1 This yields the smooth Hamiltonian H : R2n+1 → R given by H(q, p, t) := pq(q, ˙ p, t) − L(q, q(q, ˙ p, t), t) for all (q, p, t) ∈ R2n+1 The proof of the following theorem proceeds as on page 274 28 We assume that the solutions of the velocity equation exist for all times t ∈ R Otherwise, the following results are only valid locally 6.13 Carath´eodory’s Royal Road to the Calculus of Variations 421 Theorem 6.19 If v, S is a smooth solution of Carth´ eodory’s fundamental equation (6.99), then the action function S is a solution of the Hamilton–Jacobi equation St (q, t) + H(q, Sq (q, t), t) = 6.13.2 Lagrangian Submanifolds in Symplectic Geometry In geometrical optics, one wants to construct wave fronts by means of families of light rays The point is that only special families of light rays allow this construction This is intimately related to the notion of Lagrange brackets, which were introduced by Lagrange in the 18th century in order to simplify computations in celestial mechanics in the framework of perturbation theory.29 The point is that the Lagrange brackets of a family of light rays are constant in time along the light rays (i.e., they are first integrals of the Hamilton canonical equations) In modern symplectic geometry, the Lagrange brackets are reformulated as Lagrangian submanifolds of a symplectic manifold Replacing light rays by trajectories of particles in classical mechanics, we will study • the construction of a solution of the Hamilton–Jacobi partial differential equation • by the help of suitable families of trajectories which satisfy the Hamilton canonical system of ordinary differential equations The following general approach contains geometrical optics in the 3-dimensional (x, y, z)-space as a special case To this end, we consider the light rays y = y(x), z = z(x), and we choose x := t, y := q , z := q , y := q˙1 , z := q˙2 , as well as L(x, y, z, y , z ) := n(x, y, z) p + y + z c The Lagrange brackets Consider the smooth map q = q(u), p = p(u), u∈U (6.102) from the nonempty open subset U of Rn into R2n The target space R2n is a symplectic manifold equipped with the symplectic 2-form ω := n X dq k ∧ dpk k=1 Pn P s s Since dq k = r=1 qukr dur and dpk = n s=1 (pk )u du , it follows from the antisymr s s metry relation du ∧ du = −du ∧ dur that X r s [u , u ] dur ∧ dus ω= r