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Some Physics for Mathematicians: Mathematics 7120, Spring 2011

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Some Physics for Mathematicians: Mathematics 7120, Spring 2011 Some Physics for Mathematicians: Mathematics 7120, Spring 2011 Len Gross with the assistance of Mihai Bailesteanu, Cristina Benea, Joe Chen, Nate Eldridge, Chi-Kwong (Alex) Fok, Igors Gorbovickis, Amanda Hood, John Hubbard, Yasemin Kara, Tom Kern, Janna Lierl, Yao Liu, Shisen Luo, Peter Luthy, Justin Moore, Maicol Ochoa Daza, Milena Pabiniak, Eyvindar Palsson, Ben Steinhurst, Baris Ugurcan, Hung Tran, Jim West, Daniel Wong, Tianyi Zheng,... June 23, 2011 These notes are based on a previous incarnation of this seminar: Physics for Mathematicians: Mathematics 712, Spring 2003 by Len Gross with the assistance of Treven Wall, Roland Roeder, Pavel Gyra, Dmitriy Leykekhman, Will Gryc, Fernando Schwartz, Todd Kemp, Nelia Charalambous, ... and the participation of Kai-Uwe Bux, Martin Dindos, Peter Kahn, Bent Orsted, Melanie Pivarski, Jos`eRamirez, Reyer Sjamaar, Sergey Slavnov, Brian Smith, Aaron Solo, Mike Stillman, ... 1 Contents 1 Introduction 5 2 Newtonian Mechanics 7 2.1 Work, energy and momentum for one particle . 7 2.2 Work, energy and momentum for N particles . 10 2.3 Angular Momentum . 12 2.4 Rigid bodies and SO(3) . 14 2.4.1 Angular velocity . 15 2.4.2 Moment of Inertia and Angular Momentum . 16 2.5 Configuration spaces and the Newtonian flow . 18 2.6 Lagrangian Mechanics . 21 2.6.1 Linear systems . 21 2.6.2 General configuration spaces. (Nonlinear systems.) . 24 2.7 Hamiltonian mechanics . 26 2.7.1 The Legendre transform . 28 2.7.2 From Lagrange to Hamilton via the Legendre transform 30 2.8 SUMMARY . 31 2.9 References . 31 2.10 Solutions to problems . 32 3 Electricity and Magnetism 34 3.1 Lodestones and amber from antiquity till 1600 . 34 3.2 Production, transfer and storage of electrostatic charge. 36 3.2.1 Production of electrostatic charge, 1704-1713. Hauksbee 36 3.2.2 Transfer of electrostatic charge, 1731. Grey and Dufay 37 3.2.3 Storage of electrostatic charge, 1746. Musschenbroek of Leyden. 37 3.3 Quantitative measurement begins: COULOMB . 42 3.3.1 How Coulomb did it. 42 3.3.2 Mathematical consequences . 43 3.4 The production of steady currents, 1780-1800 . 45 3.5 The connection between electricity and magnetism . 47 3.5.1 Oersted, Amp`ere,Biot and Savart . 47 3.5.2 FARADAY . 51 3.6 MAXWELL puts it all together, 1861 . 55 3.7 Maxwell's equations ala differential forms . 57 2 3.8 Electromagnetic forces ala Newton, Lagrange and Hamilton . 59 3.9 References . 63 4 Quantum Mechanics. 64 4.1 Spectroscope . 64 4.2 100 years of spectroscopy: radiation measurements that defied Newton [ Maxwell . 66 4.3 The rules of quantum mechanics . 69 4.4 The Heisenberg commutation relations and uncertainty principle 75 4.5 Dynamics and stationary states . 79 4.6 Hydrogen . 81 4.7 Combined systems, Bosons, Fermions . 85 4.8 Observables from group representations . 87 4.8.1 Example: Angular momentum . 88 4.9 Spin . 90 4.10 Pictures: Heisenberg vs Schr¨odinger. 92 4.11 Conceptual status of quantum mechanics . 95 4.12 References . 96 5 Quantum field theory 97 5.1 The harmonic oscillator . 98 5.2 A quantized field; heuristics. 99 5.3 The ground state transformation . 102 5.4 Back to the quantized field. 107 5.5 The time dependent field . 113 5.6 Many, many particles: Fock space . 117 5.6.1 Creation and annihilation operators . 118 5.6.2 The canonical commutation relations . 122 5.6.3 Occupation number bases . 125 5.6.4 Time evolution on Fb and Ff . 128 5.7 The Particle-Field isomorphism. 130 5.8 Pre-Feynman diagrams . 134 6 The electron-positron system 141 6.1 The Dirac equation . 141 6.2 Dirac hole theory . 145 6.3 Pair production . 151 3 7 The Road to Yang-Mills Fields 152 7.1 Quantization of a particle in an electromagnetic field . 153 7.2 Two classic debuts of connections in quantum mechanics . 155 7.2.1 The Aharonov-Bohm experiment . 156 7.2.2 The Dirac magnetic monopole . 157 8 The Standard Model of Elementary Particles 160 8.1 The neutron and non-commutative structure groups. Timeline. 163 9 Appendices 164 9.1 The Legendre transform for convex functions . 164 9.1.1 The Legendre transform for second degree polynomials 167 9.2 Poisson's equation . 168 9.3 Some matrix groups and their Lie algebras . 171 9.3.1 Connection between SU(2) and SO(3). 174 9.3.2 The Pauli spin matrices . 175 9.4 grad, curl, div and d . 176 9.5 Hermite polynomials . 178 9.6 Special relativity . 180 9.7 Other interesting topics, not yet experimentally confirmed . 180 9.7.1 The Higgs particle for the standard model . 180 9.7.2 Supersymmetry . 182 9.7.3 Conformal field theory . 182 9.7.4 String theory . 182 9.8 How to make your own theory . 182 10 Yet More History 182 10.1 Timeline for electricity vs magnetism. 182 10.2 Timeline for radiation measurements . 184 10.3 Planck, Einstein and Bohr on radiation. Timeline. 185 10.4 List of elementary particles . 188 4 1 Introduction There have been a number of good books aimed at mathematicians, de- scribing the mathematical structures that arise in quantum mechanics and quantum field theory. Here are just a few. Most have the words \Quantum" and \Mathematicians" in the title. [21, 28, 3, 46, 57, 58, 59, 73, 74]. Some aim to explain how these mathematical structures build on those that that represent an earlier physical theory . Some aim to give a mathematically precise exposition of various topics for mathematicians who want to under- stand the meaning of terms and ideas developed by physicists. It is no secret that the writing styles of mathematicians and physicists are not conducive to cross-communication. This seminar is aimed at describing the experiments and observational data that led physicists to make up the successful theories that explained the observed phenomena. Of course we want to describe what these theories are and how they solved some of the experimentally produced problems. To this end we will make use of some of the common mathematical background of the participants (supplemented when necessary), and thereby save time which would otherwise have to be used in a physics course. The listed prerequisites for this seminar were a) highschool physics and b) two years of graduate mathematics courses. This tells you something. There is really no substitute for going into a laboratory and doing ex- periments with real objects. But our room is not well equipped for this. Instead I'm going to try to substitute some history of electricity, magnetism and radiation, where possible, and describe the experimental facts that de- manded explanation. There is, I think, some physical insight to be gained from understanding who tried to do what, who succeeded and who failed and how and why. As to whether or not a semi-historical exposition is re- ally in the least bit helpful for conveying the notions of physics, even to a mathematically sophisticated audience, remains to be seen. It would be easy and quick just to write down Maxwell's equations and give the definitions of the electric and magnetic fields. We will, however, repeat in class Faraday's great experiment of 1831, which put the final touch on the experimental facts that Maxwell used 30 years later to bring the theory of electricity and mag- netism into its final form. Fortunately it is possible to do this experiment with rather primitive equipment, which is all that we have. One failed goal of this seminar was to reach a description, however prim- 5 itive, of the current status of elementary particle physics. A list of omitted important topics on the straight road to this goal would very likely be longer than the table of contents. Maybe next time. Here is the progression of topics necessary to understand in order to get at least a little background for current elementary particle theory. [F = ma] ! Lagrangian Mechanics ! Hamiltonian Mechanics ! Quantum Mechanics ! Quantum Field Theory Parallel to this one needs to understand two other \classical" theories and their relation. Maxwell's equations ! Yang-Mills equations 6 2 Newtonian Mechanics 2.1 Work, energy and momentum for one particle The objective of this section is to review the notions of linear momentum, angular momentum, energy and the conservation laws that they satisfy. Recall Newton's equation: F = ma: We are going to elaborate on the various forms that this equation takes in gradually more sophisticated systems. To begin, consider a single particle moving in space, which we, perhaps understandably, will take to be R3. Denote the position of the particle at time t by x(t). Let m be a strictly positive constant. We will refer to m as the mass of the particle. The acceleration of the particle is by definition d2x=dt2. Newton's equation then reads F = m d2x(t)=dt2 where F is the force acting on the particle at time t. In the preceding paragraph several mathematically undefined words have been used, namely \particle", \time", \mass" and \force". These words acquire meaning only from experience with the physical world. I'm going to assume that the reader has some familiarity with their physical meaning and I'll use them without further ado. The force F is a vector in R3 and may depend on time, on the position, x(t), of the particle, on the velocity dx(t)=dt and could, in principle, depend on higher derivatives of the position. But the existence theory for solutions of 1) is greatly affected if F depends on higher derivatives than the first.
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