Unification - The Standard Model

Transcription

1 Unification - The Standard Model Information on Physics level 4 Undergraduate Course PT.4.6 K.S. Stelle, Office Huxley 519 November 9, 2015 Rapid Feedback to be handed in to the UG office Level 3 (day t.b.a.) Rapid Feedback Presentation every 2nd week (location, dates & times t.b.a.) Course Background This course is a fourth year MSci undergraduate course and is also part of the Quantum Fields and Fundamental Forces MSc run by the Theoretical Physics Group at Imperial. However, it is aimed at the same level as other level four courses. The Masters students have a wide range of backgrounds including some who have completed three-year physics degrees at a 2.1 level or higher, identical to the requirements for an Imperial MSci student to take this course. Support material is provided as described below. Course Outline In the Unification course, we will study the implications of symmetry in field theory. Quantum field theory is used to describe the fundamental interactions as probed in the particle accelerators at CERN, Fermilab, DESY, SLAC etc. It is also the key to understanding phase transitions, whether they took place in the early universe or in modern superconductor physics, so these ideas are also of vital importance to cosmologists and to condensed matter physicists. The course will mainly be concerned with symmetries described by continuous groups (Lie groups) and the discussion will be largely classical, although it is of course motivated by the full quantum theory. Accordingly, this course should only be taken in parallel with the QFT course (the converse, however, is not necessarily true). The objective is to understand the role of symmetry in the Standard Model of elementary particle physics, including a discussion of the mass generating sector of the Standard Model, i.e. the Higgs particle(s) which are currently being intensively searched for at Fermilab and CERN. Key ideas encountered include conserved Noether currents, Goldstone s theorem, local (i.e. gauge) symmetry, and symmetry breaking including the Brout-Englert- Higgs-Kibble mechanism. The ideas have much wider uses, however: they form the backbone of string theory; similar symmetry principles are fundamental to General relativity and gravity; they also explain superfluids and superconductors the Cooper pair in superconductivity theory is just the counterpart of the Higgs particle in particle physics.

2 How the course is taught 26 lectures, 1 revision lecture. Problem sheets: There will be 6 or 7 question sheets plus another revision sheet for self-study. Students are expected to attempt all the questions on the question sheets unless they are indicated as optional. You must keep up with the non-optional questions, especially the questions discussed in the rapid feedback sessions, or you will not follow the later lectures. These questions are representative also of what might appear on the final exam. Answers to (almost) all questions will be handed out about a week after the question sheet. Past exam papers will be a good guide to future exam questions. (Note that the exam format has changed as of Autumn 2005, but previous exam questions can still be of use). For MSc students, there is an optional test a mock exam, in the first week of term in January. This is provided so that we can use positive results to support MSc students applying for PhDs (at any institution or in any topic). The test will be marked informally and will not contribute to the final course grade. It also provides feedback on students progress. Undergraduates are welcome to take the test but should ask me first. The January test will be put on the web, so it can be used for revision for the main summer exam. Office Hours. I will be available in my office to answer questions for two hours each week. I am also available for office hours in the summer term before the exam. My full lecture notes will be made available to students. Many quantum field theory books exist and have the important topics of the course scattered through their pages. As this is an advanced course, it is assumed that students will use books and other sources to supplement the lectures. The bibliography indicates which parts of the various books are relevant. Students may like to look at similar chapters of other quantum field theory books, especially any recommended for the level 4 MSci QFT (quantum field theory) course or suggested for any of the courses on the QFFF MSc. The departmental web site for this course will contain all the paperwork handed out. Any significant updates or corrections will be available only in these electronic versions.

3 Requirements Quantum Field Theory QFT (Quantum Field Theory) is used to describe the interactions of fundamental particles (electrons, positrons, photons, quarks, neutrinos, etc.). It is also the fundamental description required for the description of phase transitions in systems such as superconducting and superfluid transitions in condensed matter theory. The Unification course is expressed in the language of QFT and therefore the Unification course only makes sense when taken in conjunction with a QFT course. The Unification course itself uses classical analysis but will make frequent reference to the quantum generalizations provided by the QFT course. So QFT is not a prerequisite but it should at least be taken in parallel with Unification. The two courses complement each other: many ideas appear in both courses and seeing them twice usually enhances the understanding of both courses. Lagrangians and actions Lagrangians and actions are a main way of summarizing all the information about a classical system. They form the usual starting point for QFT as used in particle physics and, as such, the entire Unification course will be expressed in this language. It is therefore essential that students be familiar with the use of Lagrangians and actions. In the Unification course, we will use the Lagrangian/action formulation for fields (rather than for the coordinates of particles) in order to derive the Euler-Lagrange equations as the equations of motion. The necessary level of knowledge should be provided by the undergraduate course on Advanced Classical Physics, although elements also appear in the second year Mathematical Methods course. Almost any book on QFT will provide a suitable summary and introduction in the context of fields; see the Unification or QFT course bibliographies. Special Relativity and Index Notation The course will work throughout in the context of relativistic particle physics. This means that knowledge of standard relativistic notation will be assumed from the start, so knowledge of 4-index notation and of the Einstein summation convention in four space-time dimensions will be needed. The required level of knowledge should be provided by the undergraduate course on Advanced Classical Physics. Group Theory At the core of the Unification course is the link between symmetries of field-theory Lagrangians and the properties of the particles arising as excitations of such theories. In particular we will look at symmetries which can be described mathematically by continuous groups known as Lie groups, and by the closely related structures known as Lie algebras. In practice it will be sufficient to understand these simply in the context of matrix representations, either for general unitary matrices (i.e. the group U(d) and its unimodular subgroup SU(d)), or general

4 orthogonal matrices (i.e. the group O(d) and its unimodular subgroup SO(d)). Specific examples will mostly be limited to 2x2 or 3x3 matrices and one-dimensional phase factors. A general knowledge of group theory and group representations is a prerequisite, including the group axioms and a basic understanding of matrix representations. Lie Groups and Lie Algebras are also prerequisites. However, the course will not assume any high degree of fluency in these topics and I expect that for most students this will be the first time they have applied these ideas in practice. We will use the simplest examples, namely the trivial, fundamental and adjoint matrix representations for the groups mentioned above (U(d), SU(d), O(d), SO(d)). The third year undergraduate group theory course should be sufficient to provide a necessary coverage of these topics. A detailed understanding of finite groups and associated topics such as the use of characters, Schur's lemma or orthogonality theorems will not be required, A fluency with vector spaces and matrix algebra is essential but this should have been encountered in the second year Mathematical Methods course. Some QFT books provide a short introduction at the level required - see the Unification or QFT course bibliographies. Particle Physics The course will assume a rough familiarity with the Standard Model of particle physics. The basic properties of the four fundamental forces of nature (electromagnetism, strong nuclear, weak nuclear and gravity) and the associated particles (photon, gluons, W and Z bosons and the graviton) should be known at a basic level. Similarly, the fundamental fermions (electrons, neutrinos, quarks and their bound states the baryons (neutron, proton etc.) and the mesons (pions etc.), and the scalar(s) (Higgs) should all be familiar. You should be aware that these come in three generations. The particle part of the third year undergraduate nuclear and particle physics course should be sufficient to provide the needed background. The fourth year particle physics course contains elements of the unification course without the mathematical development. Both undergraduate particle physics courses generally have good web-based materials (see the physics department web site).

5 Bibliography Background H.F. Jones, Groups, Representation and Physics" (Institute of Physics Publishing, Bristol, 2nd edition 1998, ISBN ). [Chapter 1, section 2.2, chapter 3 (not section 3.2), sections 6.1, 6.2, 8.1 and 8.2 cover all the group theory needed for Unification. Chapters 6, 8, 9, 10 and 11 cover Lie groups in sufficient depth for the full MSc course including a good introduction to symmetries of spacetime in chapter 9 on the Poincaré group.] L.H.Ryder, Quantum Field Theory" (Cambridge University Press, Cambridge, 1985). [General QFT book at an appropriate level for the QFT and Unification courses.] F.Mandl and G.Shaw, Quantum Field Theory" (Wiley, Chichester, revised edition 1996). [General QFT book recommended for the QFT course and for the MSc course in general. Sections 2.1 and 2.2 provide a summary of the required knowledge of special relativity and Lagrangian mechanics.] Overview E.S. Abers and B.W. Lee, Gauge Theories (Physics Reports 9C, No. 1, November 1973). [The first part of this classic Physics Reports review covers many of the main topics of the Unification course. Even if it is hard to read at the beginning of the course, it is hoped that students will be able to understand it by the end of the course.] Cliff Burgess and Guy Moore, The Standard Model a Primer (Cambridge University Press, 2007). [A brand-new graduate level course, goes beyond the Unfication course level. Looks to be a complete and up-to-date treatment of the subject.] Further resources T-P. Cheng and L-F. Li, Gauge Theory of Elementary Particle Physics" (Oxford Univ. Press, 1984). [General QFT book. Starts with good overview of classical Lagrangians. Section 4.1 is a very compact outline of essential group theory ideas needed for Unification. The rest of chapter 4 gives a compact discussion of more group theory as needed for MSc students.] S. Weinberg, Gravitation and Cosmology (Wiley, 1972). [Chapter 2, sections 1-9 provides a good background on Special Relativity.] T. Kibble and F. Berkshire, Classical Mechanics" (Longman, Harlow, 1996). [This book covers Lagrangians etc. at a level accessible to Imperial students.]

6 D. Vvedensky, Group Theory" [Notes from a previous lecturer of the undergraduate Group Theory course.] Howard Georgi, Lie Algebras in Particle Physics" (Perseus Books, Reading, MA, Second Edition 1999, ISBN ). [A standard particle physics text which focuses on compact Lie groups and algebras (about 90% of the text), especially SU(2), SU(3), SU(N), SO(N), SU(5), and SO(10). It is quite useful for MSc students. Make sure that you get the second edition as it is much better and includes an excellent forty-page survey of finite groups. Read this introductory survey even if you don't buy the book.] M.Hamermesh, Group Theory and its applications to physical Problems" (Pergamon/Addison-Wesley, 1962), but available now as a cheap Dover (New York, 1989) paperback, ISBN Ch. 1-3,8 [A classic mathematical presentation of Group theory which accessible to physicists. Good for a serious foundation but written in a rather old-fashioned style] Gordon Kane, Modern Elementary Particle Physics" (Addison-Wesley, Redwood City CA, 1987, ISBN ). [Contains little on history or experimental details, but has a very good description of the Standard Model of particle physics and beyond without using quantum field theory.] David Griffths, Introduction to Elementary Particles" (John Wiley, N.Y., 1987). [Has rather more details than Kane, but without much actual quantum field theory.] I.S.Hughes, Elementary Particles" (Cambridge University Press, Cambridge, 3rd edition 1991, ISBN ). D.H.Perkins, Introduction to High Energy Physics" (Adison-Wesley, Redwood City CA, 3rd edition 1986). [Many descriptions of experimental methods as well as the Standard Model of particle physics.] I.J.R.Aitchison and A.J.G. Hey, Gauge Theories in Particle Physics" (Adam-Hilger, Bristol, 1982). [A good introductory text about all aspects of quantum field theory. It stops short of teaching you how to use quantum field theory but it will explain why one uses quantum field theory, what it means and how it relates to the real world. The recent two-volume edition is highly recommended as a comprehensive and readable book.] Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman" (Cambridge University Press, Cambridge, 1985) ISBN [A superb text about certain key topics in field theory but not a complete QFT course. Chapter 1 on SU(2), SU(3) and SU(N) symmetry is a good way to learn about these topics in a particle-physics context, although the applications are by now rather dated. Chapter 5 is the classic introduction to spontaneous symmetry breaking in the context of QFT including gauge fields.]