DOCSLIB.ORG

Introduction to the Second Quantization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to the Second Quantization Introduction to the Second Quantization Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 27, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 1 / 35 Identical particles Examples of quantum many body problems: I Photons in a block-body cavity I Electrons in a solid I Nucleons in a nucleus I Neutrons in a neutron star I Helium atoms in a superfluid The concept of identical particles: I Intrinsic: Those do not depend on the dynamical state of the particle, such as the rest mass, the charge, or the total spin. I Extrinsic: Those depend on the state of the particle or on its preparation: the position, and velocity, spin orientation and other internal parameters. Here two particles are considered identical if they possess the same intrinsic properties. More mathematically speaking, we say n particles are identical if all O(x1; p1; :::; ; xn; pn) are invariant under the exchange of their coordinates (xi ; pi ), i = 1;:::; n. n 2 X jpi j X H(n) = H (n) + V (n) = + V (jx − x j): 0 2m i j i=1 i<j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 2 / 35 One-particle system: states The collection of states of a single particle form a Hilbert space Z 2 3 2 H = L (R ) = f'(x); dxj'(x)j < 1g with the inner product Z h'j i = dx'∗(x) (x): If a particle posses a spin s, its wavefunction will have 2s + 1 components '(x; σ), σ = −s; −s + 1;:::; s − 1; s + 1. i.e. 0 '(x; −s) 1 B '(x; −s + 1) C B C B . C 2s+1 '(x) = B . C 2 H ⊗ C : B C @ '(x; s − 1) A '(x; s) Now, I will focus on the case s = 0. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 3 / 35 One-particle system: Observables Position q and momentum p are linear operators on H : (q')(x) = x'(x) @ @ @ (p')(x) = −i r'(x); r = ; ; ~ @x1 @x2 @x3 The canonical commutation relations α β [q ; p ] = i~δα,β; α; β = 1; 2; 3: In the Fourier representation 1 Z '(k) = dx e−ik·x'(x) e (2π)3=2 the position operator q and the momentum operator q becomes (q'e)(k) = −irk'e(k) (p'e)(k) = ~k'(k) Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 4 / 35 One-particle system: Evolution equations The evolution of the particle is governed by Schrodinger equation @ i '(x; t) = (H')(x; t) ~@t where H is the given Hamiltonian. The formal solution is given by '(x; t) = (U(t)')(x; 0); U(0) = I where U(t) is the evolution operator (semi-group on H). For example, if the external fields are independent of time, then itH U(t) = exp − ~ ∗ −1 U (t) = U (t); U(t1)U(t2) = U(t1 + t2): Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 5 / 35 n-particle system: States The collection of the states of n non-relativistic spinless particles form a Hilbert space Z n 2 3n 2 H = L (R ) = fΦ(x1;:::; xn); dx1 ::: dxnjΦ(x1;:::; xn)j < 1g: If the particles carry a spin s, the Hilbert space of states is n 2s+1 H ⊗ C . Now we are going to construct the n-particle states in Hn from the one-particle states. hx1;:::; xnj'1 ⊗ :::; ⊗'ni = '1(x1) :::'n(xn) Here j'i i, i = 1;::: n are the single-particle state in H. We interpret it as the joint probability of the state j'1 ⊗ :::; ⊗'ni to find particle 1 in x1, ::: , and particle n in xn. The inner product of j'1 ⊗ :::; ⊗'ni and j 1 ⊗ :::; ⊗ ni is h 1 ⊗ · · · ⊗ nj'1 ⊗ :::; ⊗'ni = h'1j 1ih'2j 2i ::: h'nj ni Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 6 / 35 n-particle system: Basis Suppose fj'i i; i = 1; 2;::: g form a basis in the Hilbert space of a single-particle system. We know fj'j1 ⊗ :::; ⊗'jn i; j1; j2; ··· = 1; 2;::: g form a basis of the space H⊗n = H ⊗ · · · ⊗ H. For all wavefunction Φ(x1;:::; xn), we can express as X Φ(x1;:::; xn) = cj1:::jn 'j1 (x1) :::'jn (xn): j1:::jn We could also write as X hx1;:::; xnjΦi = cj1:::jn hx1;:::; xnj'j1 ⊗ · · · ⊗ 'jn i: j1:::jn X jΦi = cj1:::jn j'j1 ⊗ · · · ⊗ 'jn i: j1:::jn This means we can identify n-particle space Hn with the tensor product H⊗n. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 7 / 35 n-particle system: One-body observables Observables of a n-identical-particle system can be constructed from observalbes in a single-particle system. If A is a single-particle observable in H, we define the action A on the jth factor of the tensor product H⊗n as (I ⊗· · ·⊗A⊗· · ·⊗I )j'1⊗· · ·⊗'j ⊗· · ·⊗'ni = j'1⊗· · ·⊗A'j ⊗· · ·⊗'ni and denote it as Aj . The corresponding extensive observable for the n-particle system is defined by n X A(n) = Aj : j=1 Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 8 / 35 n-particle system: Two-body observables Let V be an observable in two-particle space H ⊗ H, such as interaction potential. ⊗n ⊗n Define Vij on H as the operator acts on ith and jth factors of H and trivially on the others. The corresponding observable for a n-particle system is n X V (n) = Vij : i<j If V is acting on two identical particles, we have Vij = Vji . Therefore we have n 1 X V (n) = V : 2 ij i6=j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 9 / 35 Symmetrization: Permutation operator ⊗n Permutation operator Pπ actin on H is Pπj'1 ⊗ · · · ⊗ 'ni = j'π(1) ⊗ · · · ⊗ 'π(n)i 1 2 ::: n where π = . π(1) π(2) : : : π(n) Some properties: ∗ −1 Pπ1π2 = Pπ1 Pπ2 ; Pπ = Pπ Let A be an observable on a single-particle space H. We have [A(n); Pπ] = 0 for all permutation π Pn where A(n) = j=1 Aj . Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 10 / 35 Symmetrization: Exchange degeneracy Suppose jΦi is an eigenstate of a n-identical-particle system. i.e. H(n)jΦi = EjΦi We will find H(n)PπjΦi = EPπjΦi for all permutation π In a n-identical-particle system, PπjΦi and their linear combinations describe the same physical state. Principle of symmetrization: the state of n identical particles are either completely symmetric or completely antisymmetic. Bosons are particles in the symmetric case. Fermions are particles in the antisymmetic case. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 11 / 35 Symmetrization: Symmetric and Antisymmetric States For any jΦi in H⊗n, we can construct corresponding one which is totally symmetric or totally antisymmetric: jΦi+ = S+jΦi jΦi− = S−jΦi where two operators S+ and S− defined by 1 X 1 X S = P ; S = (−1)πP : + n! π − n! π π2Pn π2Pn Properties: 2 2 ∗ S+ = S+; S− = S− = S−; S+S− = S−S+ = 0: π Pπ = S+Pπ = S+; PπS− = S−Pπ = (−1) S−: According to the principle of symmetrization, the space of state of a system of n identical particle particles is not entire H⊗n. ⊗n ⊗n I Space for bosons (symmetric): H+ ≡ S+H . ⊗n ⊗n I Space for fermions (anti-symmetric): H− ≡ S−H . Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 12 / 35 Hilbert spaces for bosons or fermions ⊗n ⊗n Denote the symmetric or antisymmetric state of H+ or H− by j'1;'2;:::;'niν ≡ Sνj'1 ⊗ '2 ⊗ · · · ⊗ 'ni where ν = 1 for bosons and ν = −1 for fermions. Exchanging the ith and jth components, we find j'1;:::;'i ;:::;'j ;:::;'ni+ = j'1;:::;'j ;:::;'i ;:::;'ni+ j'1;:::;'i ;:::;'j ;:::;'ni− = −|'1;:::;'j ;:::;'i ;:::;'ni− The inner product is 1 X h' ;:::;' j ; : : : ; i = (ν)πh' j i ::: h' j i: ν 1 n 1 n ν n! 1 π(1) n π(n) π2Pn In particular for the fermions case, 1 h' ;:::;' j ; : : : ; i = detfh' j ig: − 1 n 1 n − n! i j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 13 / 35 Occupation Number Representation Let fj r ig be a basis of a single-particle space H. The occupation number normalized to 1 are defined by p n! jn1; n2;:::; nr ;::: i+ = p S+j r1 ⊗ · · · ⊗ rn i n1! ::: nr ! ::: p jn1; n2;:::; nr ;::: i− = n! S−j r1 ⊗ · · · ⊗ rn i where nr is the number of indices r1;:::; rn which are equal to r. For bosons, nr = 0; 1; 2; 3;::: . For fermions, nr = 0; 1. P The system has exactly n = r nr particles. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 14 / 35 Occupation Number Representation: Observables Let j r i are the eigenvector of a single-particle observable A such that Aj r i = αr j r i: Then jn ; n ;:::; n ;::: i are also eigenvectors of the one-body 1 2 t ν P observable A(n) with eigenvalue r nr αr . n ! X X A(n)j r1 ⊗ · · · ⊗ rn i = Aj j r1 ⊗ · · · ⊗ rn i = nr αr j r1 ⊗ · · · ⊗ rn i j=1 r Therefore if [Sν; A(n)] = 0, we have ! X A(n)jn1; n2;:::; nr ;::: iν = nr αr jn1; n2;:::; nr ;::: iν: r Orthogonality: νhm1; m2;:::; mr ;::: jn1; n2;:::; nr ;:::; iν = δm1;n1 δm2;n2 : : : δmr ;nr ::: ⊗n Basis: fjn1; n2;::: iνg form a basis of Hν .
Load more

Recommended publications
  • Introduction to the Many-Body Problem
    INTRODUCTION TO THE MANY-BODY PROBLEM UNIVERSITY OF FRIBOURG SPRING TERM 2010 Dionys Baeriswyl General literature A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall 1963. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill 1971. G. D. Mahan, Many-Particle Physics, Plenum Press 1981. J. W. Negele and H. Orland, Quantum Many Particle Systems, Perseus Books 1998. Ph. A. Martin and F. Rothen, Many-Body Problems and Quantum Field Theory, Springer-Verlag 2002. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press 2004. Contents 1 Second quantization 2 1.1 Many-particlestates ......................... 2 1.2 Fockspace............................... 3 1.3 Creationandannihilationoperators . 5 1.4 Quantumfields ............................ 7 1.5 Representationofobservables . 10 1.6 Wick’stheorem ............................ 16 2 Many-boson systems 19 2.1 Bose-Einstein condensation in a trap . 19 2.2 TheweaklyinteractingBosegas. 20 2.3 TheGross-Pitaevskiiequation . 23 3 Many-electron systems 26 3.1 Thejelliummodel........................... 26 3.2 Hartree-Fockapproximation . 27 3.3 TheWignercrystal .......................... 28 3.4 TheHubbardmodel ......................... 29 4 Magnetism 34 4.1 Exchange ............................... 34 4.2 Magnetic order in the Heisenberg model . 36 5 Electrons and phonons 39 5.1 Theharmoniccrystal.. .. .. 39 5.2 Electron-phononinteraction . 41 5.3 Phonon-inducedattraction
    [Show full text]
  • Field Formulation of Many-Body Quantum Physics
    A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools. Douglas Adams (1952-2001) 2 Field Formulation of Many-Body Quantum Physics A piece of matter composed of a large number of microscopic particles is called a many-body system. The microscopic particles may either be all identical or of dif- ferent species. Examples are crystal lattices, liquids, and gases, all of these being aggregates of molecules and atoms. Molecules are composed of atoms which, in turn, consist of an atomic nucleus and electrons, held together by electromagnetic forces, or more precisely their quanta, the photons. The mass of an atom is mostly due to the nucleus, only a small fraction being due to the electrons and an even smaller fraction due to the electromagnetic binding energy. Atomic nuclei are themselves bound states of nucleons, held together by mesonic forces, or more precisely their quanta, the mesons. The nucleons and mesons, finally, consist of the presently most fundamental objects of nuclear material, called quarks, held together by gluonic forces. Quarks and gluons are apparently as fundamental as electrons and photons. It is a wonderful miracle of nature that this deep hierarchy of increasingly funda- mental particles allows a common description with the help of a single theoretical structure called quantum field theory. As a first step towards developing this powerful theory we shall start from the well-founded Schr¨odinger theory of nonrelativistic spinless particles. We show that there exists a completely equivalent formulation of this theory in terms of quantum fields.
    [Show full text]
  • Second Quantization
    Appendix A Second Quantization An exact description of an interacting many-body system requires the solving of the corresponding many-body Schrodinger¨ equations. The formalism of second quantization leads to a substantial simplification in the description of such a many- body system, but it should be noted that it is only a reformulation of the original Schrodinger¨ equation but not yet a solution. The essential step in the second quanti- zation is the introduction of so-called creation and annihilation operators. By doing this, we eliminate the need for the laborious construction, respectively, of the sym- metrized or the anti-symmetrized N-particle wavefunctions from the single-particle wavefunctions. The entire statistics is contained in fundamental commutation rela- tions of these operators. Forces and interactions are expressed in terms of these “creation” and “annihilation” operators. How does one handle an N-particle system? In case the particles are distinguish- able, that is, if they are enumerable, then the method of description follows directly the general postulates of quantum mechanics: H(i) {|φ(i) } 1 : Hilbert space of the ith particle with the orthonormal basis α : (i) (i) φα |φβ =δαβ (A.1) HN : Hilbert space of the N-particle system H = H(1) ⊗ H(2) ⊗···⊗H(N) N 1 1 1 (A.2) with the basis {|φN }: |φ =|φ(1)φ(2) ···φ(N) =|φ(1) |φ(2) ···|φ(N) (A.3) N α1 α2 αN α1 α2 αN An arbitrary N-particle state |ψN , |ψ = c(α ···α )|φ(1)φ(2) ···φ(N) (A.4) N 1 N α1 α2 αN α1···αN underlies the same statistical interpretation as in the case of a 1-particle system.
    [Show full text]
  • Introduction to Quantum Field Theory and Quantum Statistics
    1 Introduction to Quantum Field Theory and Quantum Statistics Michael Bonitz Insitut f¨urTheoretische Physik und Astrophysik Kiel University February 9, 2015 preliminary lecture notes, not for distribution 2 Contents 1 Canonical Quantization 11 1.1 Minimal action principle . 11 1.1.1 Classical mechanics of a point particle . 16 1.1.2 Canonical momentum and Hamilton density of classical fields . 17 1.2 Conservation laws in classical field theory . 19 1.2.1 Translational invariance. Energy and momentum con- servation . 21 1.3 Field quantization . 24 1.4 Phonons . 26 1.4.1 Application of canonical field theory . 27 1.4.2 Expansion in terms of eigenfunctions . 28 1.4.3 Quantization of the displacement field . 30 1.5 Photons . 34 1.5.1 Maxwell's equations. Electromagnetic potentials. Field tensor . 34 1.5.2 Lagrange density of the free electromagnetic field . 38 1.5.3 Normal mode expansion of the electromagnetic field . 42 1.5.4 Quantization of the electromagnetic field . 44 1.6 EMF Quantization in Matter . 48 1.6.1 Lagrangian of a classical relativistic particle . 48 1.6.2 Relativistic particle coupled to the electromagnetic field . 50 1.6.3 Lagrangian of charged particles in an EM field . 51 1.6.4 Quantization of the electromagnetic field coupled to charges 54 1.6.5 Quantization of the EM field in a dielectric medium or plasma . 56 1.7 Quantization of the Schr¨odinger field . 58 1.8 Quantization of the Klein-Gordon field . 58 3 4 CONTENTS 1.9 Coupled equations for the Schr¨odingerand Maxwell fields .
    [Show full text]
  • Identical Particles
    Y. D. Chong PH4401: Quantum Mechanics III Chapter 4: Identical Particles These our actors, As I foretold you, were all spirits and Are melted into air, into thin air: And, like the baseless fabric of this vision, The cloud-capp'd towers, the gorgeous palaces, The solemn temples, the great globe itself, Yea, all which it inherit, shall dissolve And, like this insubstantial pageant faded, Leave not a rack behind. William Shakespeare, The Tempest I. QUANTUM STATES OF IDENTICAL PARTICLES In the previous chapter, we discussed how the principles of quantum mechanics apply to systems of multiple particles. That discussion omitted an important feature of multi-particle systems, namely the fact that particles of the same type are fundamentally indistinguishable from each other. As it turns out, indistinguishability imposes a strong constraint on the form of the multi-particle quantum states, and looking into this will ultimately lead us to a fundamental re-interpretation of what \particles" are. A. Particle exchange symmetry Suppose we have two particles of the same type, e.g. two electrons. It is a fact of Nature that all electrons have identical physical properties: the same mass, same charge, same total spin, etc. As a consequence, the single-particle Hilbert spaces of the two electrons must be mathematically identical. Let us denote this space by H (1). For a two-electron system, the Hilbert space is a tensor product of two single-electron Hilbert spaces, denoted by H (2) = H (1) ⊗ H (1): (4.1) Moreover, any Hamiltonian must affect the two electrons in a symmetrical way.
    [Show full text]
  • Canonical Quantization
    Phys540.nb 27 2 Canonical quantization 2.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 2.1.1.Lagrangian Lagrangian mechanics is a reformulation of classical mechanics. L = T - V (2.1) Here, the quantity L is called the Lagrangian, where T is the kinetic energy and V is the potential energy. For a pendulum, the kinetic energy is 2 1 ⅆθ 1 9 2 T = m r2 = m r θ (2.2) 2 ⅆt 2 and the potential is V $ - m g r cos 8 (2.3) And thus 9 1 9 2 L:8, 8; $ m r2 8 1 m g r cos 8 (2.4) 2 9 In general, a Lagrangian depends on coordinates (e.g. 8) and their corresponding velocities (e.g. 8) 9 9 9 L(q1, q2 < qn; q1, q2 <qn) (2.5) 9 where qi represent all coordinates in the systems and qi are the velocities. 2.1.2. Equation(s) of motion The equation of motion in Lagrangian mechanics is - =L =L 9 $ (2.6) -t =qi =qi For the pendulum example discussed above, - =L =L 9 $ (2.7) -t =8 =8 So 9 1 2 2 - =L - =T - = m r 8 - 9 .. 2 2 2 (2.8) 9 $ 9 $ 9 $ :m r 8; $ m r 8 -t =8 -t =8 -t =8 -t 28 Phys540.nb and =L =V =m g r cos 8 $ - $ $ -m g r sin8 (2.9) =8 =8 =8 As a result, the EOM is .. m r2 8 $ -m g r sin 8 (2.10) .
    [Show full text]
  • 7. Quantization of the Harmonic Oscillator – Ariadne's Thread In
    7. Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization Whoever understands the quantization of the harmonic oscillator can un- derstand 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 be- cause it is a very important physical theory, which governs all microphysics, that is, the physical phenomena at the microscopic scale of 10−10m, 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), • Schr¨odinger quantization (wave mechanics; the Schr¨odinger partial differential equation), • Feynman quantization (integral representation of the wave function by means of the propagator kernel, the formal Feynman path integral, the rigorous infinite- dimensional Gaussian integral, and the rigorous Wiener path integral), • Weyl quantization (deformation of Poisson structures), 1 From the Preface to R.
    [Show full text]
  • Theory of Many-Particle Systems
    Theory of Many-Particle Systems Lecture notes for P654, Cornell University, spring 2005. c Piet Brouwer, 2005. Permission is granted to print and copy these notes, if kept together with the title page and this copyright notice. Chapter 1 Preliminaries 1.1 Thermal average In the literature, two types of theories are constructed to describe a system of many particles: Theories of the many-particle ground state, and theories that address a thermal average at temperature T . In this course, we'll limit ourselves to finite-temperature thermal equilibria and to nonequilibrium situations that arise from a finite-temperature equilibrium by switch- ing on a time-dependent perturbation in the Hamiltonian. In all cases of interest to us, results at zero temperature can be obtained as the zero temperature limit of the finite-temperature theory. Below, we briefly recall the basic results of equilibrium statistical mechanics. The thermal average of an observable A is defined as A = tr A^ρ,^ (1.1) h i where ρ^ is the density matrix and A^ is the operator corresponding to the observable A. The density matrix ρ^ describes the thermal distribution over the different eigenstates of the system. The symbol tr denotes the \trace" of an operator, the summation over the expectation values of the operator over an orthonormal basis set, tr Aρ = n Aρ n : (1.2) h j j i n X The basis set n can be the collection of many-particle eigenstates of the Hamiltonian H^ , fj ig or any other orthonormal basis set. In the canonical ensemble of statistical mechanics, the trace is taken over all states with N particles and one has 1 ρ^ = e−H^ =T ; Z = tr e−H^ =T ; (1.3) Z 1 2 CHAPTER 1.
    [Show full text]
  • Elements of Quantum Field Theory
    Elements of Quantum Field Theory Janos Polonyi Strasbourg University, Strasbourg, France (Dated: September 30, 2020) Contents I. Introduction 3 A. Why Classical Field Theory? 3 B. Why Quantum Field Theory? 3 C. Basic idea of Quantum Field Theory 4 D. Fock space 5 II. Canonical quantization 7 A. Single particle 7 B. Quantum field 7 C. Explicit solution for free neutral, scalar particle 9 D. Free charged scalar particle 11 E. Neutral vector field 12 F. Charged fermion 13 III. Scattering processes 14 A. Asymptotic states 14 B. Cross section 15 IV. Perturbation expansion 16 A. Green functions 16 B. Propagator 17 C. Weak form of Wick theorem 19 D. Examples 21 E. Feynman rules 25 V. Gauge theories 26 A. Local symmetries 26 B. Geometry and dynamics 27 C. Covariant derivative 27 D. Parallel transport 28 E. Wilson-loop 30 F. Curvature or field strength tensor 32 G. Gauge field dynamics 33 H. Fundamental interactions 33 VI. Spontaneous symmetry breaking 35 A. Discrete symmetries 35 B. Adiabatic approximation 38 C. Continuous symmetries 39 D. Higgs mechanism 40 E. Dual superconductor model for quark confinement 42 VII. Renormalization group and asymptotic freedom 44 A. Regularization 44 B. Renormalization 44 C. Renormalization group 46 VIII. Standard Model 50 2 A. Fermi theory of the weak interaction 50 1. β-decay 50 2. Cabibbo angle 51 3. Universality 51 4. Neutral current 51 5. Generalization of the Fermi-model 52 B. Unified electro-weak theory 53 1. Vector bosons 53 2. Higgs sector 53 3. Quarks and leptons 54 4. Mixing 56 C.
    [Show full text]