PHYS 7021, RELATIVISTIC QUANTUM MECHANICS AND INTRODUCTION TO QUANTUM FIELD THEORY, SPRING

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1 PHYS 7021, RELATIVISTIC QUANTUM MECHANICS AND INTRODUCTION TO QUANTUM FIELD THEORY, SPRING 2017 Lecturer: Daniel Phillips Office: Clippinger 242C; Tel: (740) ; Class times: 10 am-11:20 pm, Wednesday and 11 am-12:20 pm Friday, Roger W. Finlay Conference Room, Edwards Accelerator Lab. Office hours: 1 pm-2 pm, Friday, or by appointment. Textbook: An Introduction to Quantum Field Theory, by M. Peskin and D. V. Schroeder (Perseus Books, 1995). Website: Background and goals: In the late 1920s and early 1930s physicists faced a puzzle: how to extend the new, and successful, quantum mechanics of Schrödinger, Heisenberg, and Dirac, to systems that did not consist of a finite number of particles described by a non-relativistic Hamiltonian. Examples of such systems included an electron in an atom of high charge (it could be expected to be moving at speeds approaching the speed of light), and the electromagnetic field (it involved an infinite number of degrees of freedom). First efforts focused on designing a relativistic wave equation, and we ll proceed historically and discuss the successes, and failures, of those efforts in the first half of the course. Quantum field theory was born out of the failures, as it proved to be the best 1 way to marry quantum mechanics and special relativity consistently. Since their invention local quantum field theories have become the basis for essentially all of elementary particle physics, and they also underlie much modern work in theoretical nuclear and condensed matter physics. This course will scratch the surface of this subject: it is intended only as an introduction. Its main goals are for you to: (a) gain an understanding of what it means to construct a relativistic quantum mechanics, and to be able to do some basic computations in such an approach; (b) be able to calculate tree-level graphs in a variety of quantum field theories, and turn the results into a prediction for an experimental cross section or decay rate. This leaves many topics for future investigation. Topics such as loop graphs, regularization and renormalization, and the path-integral formulation of quantum field theory are covered in the sequel to this course. Books: I will probably spend less than 50% of the course working from the official text : I will draw from several sources to write my lectures. Personally, I find a number of different field theory texts to be helpful, depending on the level of sophistication I desire in the presentation, and the topic I want to see treated. Below I've compiled a list of the books I will consult in writing my lectures. These are books you can turn to if you are having trouble understanding the material. Of 1 Some people would disagree with this statement, but I can defend it under a suitable definition of best.

2 course, if you find a book you yourself like that is not on this list then that is just fine. Indeed, I would like to know what that book is! The books on my reading list should be available in our library, or through OhioLink. If they re not, let me know. My reading list: Quantum Mechanics with Applications, I. R. Afnan (Bentham Science e-book, 2010). Quantum Mechanics II: A Second Course in Quantum Theory, R. Landau (Wiley, 2004). Relativistic Quantum Mechanics, J. D. Bjorken and S. D. Drell (McGraw-Hill, 1965). Gauge Theories in Particle Physics, I. J. R. Aitchison and A. J. G. Hey, 2nd edition (Institute of Physics Publishing, 1993). Quantum Field Theory, F. Mandl and G. Shaw, 2nd edition (Wiley, 2009). Quantum Field Theory and Critical Phenomena, J. Zinn-Justin, 2nd edition (Oxford, 1993). Quantum Field Theory, L. Ryder (Cambridge, 1995). Quantum Field Theory, M. Srednicki (Cambridge, 2007). Quantum Field Theory in a Nutshell, A. Zee (Princeton University Press, 2003). Quantum Field Theory, C. Itzykson and J.-B. Zuber (McGraw-Hill, 1985). As you look at the different books on this list you ll discover that there are many ways to present this subject. Likewise, conventions for key quantities, such as scattering amplitudes and Green's functions, often differ by factors of i, or 2π. The key point is that this does not matter, as long as physical quantities, like cross sections, come out the same in everybody's convention. I will attempt to adopt a consistent convention in my lectures, but jumping between books means there is a danger of confusing factors of 2π and i indeed, my conventions are different from those of Peskin and Schroeder. I hope you will check the computations I perform in class, and keep me honest! Richard Feynman reportedly said that factors of -1 were the hardest thing about quantum field theory. This may or may not be true for you, but keeping track of such factors is very important if you want to compute scattering processes from first principles. I hope that, by the end of this course, you will be able to go from a classical theory to tree-level Feynman graphs, and thence to experimental quantities, keeping track of all of the factors along the way! Tentative (i.e. probably unrealistic but I would like to achieve) schedule: Scattering theory: Scattering by a central potential, review: definitions of cross section, scattering amplitude, phase shift, S-matrix, the Born series. Unitarity. Scattering with spin. (1-2 weeks) Relativistic quantum mechanics: Lorentz transformations (review). Symmetries in quantum theory. What is a relativistic quantum mechanics? The Poincaré algebra. The Klein-Gordon equation. Why is it relativistic? Plane-wave solutions. Difficulties with the Klein-Gordon equation. (2 weeks) Relativistic quantum mechanics of spin-1/2 particles: The Dirac equation. Why is it relativistic? Plane-wave solutions. Properties of γ matrices. Spin and the prediction of anti-particles. The hydrogen atom. The ultra-relativistic limit. (2-3 weeks)

3 Free-field quantization, scalar field: Re-interpretation of the Klein-Gordon equation. Lagrangian, action, momentum, etc. of the field. Second quantization: canonical commutation relations, creation and annihilation operators. Harmonic Oscillators and the Casimir Effect. Free Green s functions and causality. (2 weeks) Free-field quantization, Dirac field: Second quantization of the Dirac field. Spin and statistics. The Dirac propagator. (1-2 weeks) Perturbation theory: Philosophy. The Interaction picture. A perturbation expansion for the S- matrix: the Dyson expansion. Wick s theorem. How to construct Feynman rules, a first go: φ 4 theory. A second go, for theories with spin-1/2 particles: Yukawa theory. (3 weeks) Quantum Electrodynamics: Gauge invariance and the QED Lagrangian. Quantizing the electromagnetic field: what s the big deal? QED Feynman Rules. The Coulomb potential. Treelevel computations in QED. (2 weeks) The path integral: (if time permits, which apparently is unlikely) Derivation of the path integral. Functional integrals: how to integrate a gaussian. Example: the simple harmonic oscillator. The S-matrix in the path integral. Brief discussion of numerical computation of path integrals. Path integrals in field theory. (1-2 weeks) Assessment: Homework: Since solving problems is an essential part of doing physics I will give you several homework sets over the course of the semester.: approximately one per fortnight. They will be handed out at least one week before they are due. The graded assignments will be returned as soon as possible. I will not discuss the problems in class, but I am happy to schedule extra time, either individually or collectively, in order to discuss problems that are particularly interesting. In grading homework I will be mainly looking to see if you understand how to solve the problems. Therefore partial credit will be given for incomplete solutions, and, conversely, the correct answer without adequate explanation will actually yield very little credit. All steps used in reaching the solution must be properly explained and justified. In particular, the mathematical reasoning should be quite rigorous. Furthermore, the solution should be able to be read as a coherent discussion in English of the problem. I.e., explanatory sentences should be inserted into the mathematical reasoning. Your scores on homework will account for 30% of your overall course grade. Exam: There will be an in-class midterm exam. The midterm will take place on March 3 from 11 am-1 pm in the Edwards Accelerator Lab Conference Room. This exam will be closed book and will last two hours. It contributes 25% of your final grade. Final project: For the final piece of work for this class you will use a particular quantum field theory to compute tree-level cross sections or decay rates for particular processes. I would prefer

4 you do this in a field theory that includes least one particle of spin > 0. You should start thinking about what you would like to calculate soon after the semester begins. No later than Friday, March 17th you must send me a proposal ( is fine) that states -What field theory you intend to study; -What process(es) you will calculate. If we agree that this is a reasonable and doable project you will have a bit more than one month to complete the calculation. You will write a report that explains the problem (including why you think it s interesting), the field theory you used, your computations, and the results. That report is due Wednesday, April 26th at 10 am. The quality of your idea for the project, the calculations you did, and the write-up itself will all be factored into the grade for the project, which is then 30% of your overall grade. Presentation: We will also arrange a time during exam week where you can give a minute talk in which you present the findings from your project to the rest of the class. This oral presentation is worth 5% of your overall grade. Participation: The final 10% of your grade will be based on participation in class: questions, discussion, etc. Since we are a small group I hope there will be lots of questions, discussion, etc. As the semester progresses I will run some sessions in seminar format, i.e. ask you to read material before the meeting, and then use the class time to discuss any questions you have and push further into the topic under consideration. Grading summary: Homework: 30% Midterm (Week 7): 25% Project (Due Week 15): 30% Presentation (Week 15): 5% Participation: 10% Attendance and make-up work: Late or make-up work is accepted only in case of illness, death in the immediate family, religious observance, jury duty, and involvement in University-sponsored activities (departmental trip, music or debate activity, ROTC function, or athletic competition) and service or training for military reserves, including reasonable travel time to the training location. These reasons also constitute legitimate reasons for missing a discussion session. You may document reasons for your absence as explained in the Academic Policies and Procedures portion of the Undergraduate Catalog ( Academic misconduct: Academic misconduct is an A1 violation of the Ohio University Student Code of Conduct and is defined as dishonesty or deception in fulfilling academic requirements. It includes, but is not limited to: cheating, plagiarism, un-permitted collaboration, forged attendance (when attendance is required), fabrication (e.g., use of invented information or falsification of research or other

5 findings), using advantages not approved by the instructor (e.g., unauthorized review of a copy of an exam ahead of time), knowingly permitting another student to plagiarize or cheat from one's work, or submitting the same assignment in different courses without consent of the instructor. Cheating is defined as any attempt by a student to answer questions on a test, quiz, or assignment by means other than his or her own knowledge. Examples of cheating include: using the textbook or other materials, such as a notebook, not authorized for use during an examination; using technology (i.e. cell phones, laptop computers, social media, text messages, etc.) to aid in the completion of work when not permitted to do so; observing the work of another student or allowing another student to plagiarize, copy, or observe your work; using unauthorized material during a test, notes, formula lists, notes written on clothing, etc.; taking a quiz, exam, or similar evaluation in the place of another person; providing or requesting assistance from another person in a manner prohibited by the instructor; using a laboratory, computer, or calculator improperly or without authorization; changing material on a graded exam and then requesting a regarding of the exam; acquiring unauthorized knowledge of an examination or any part of an examination. Plagiarism is defined as the presentation of the ideas or the writing of someone else as one's own. Examples of plagiarism include: reproducing another person's work, whether published or unpublished (this also includes using materials from companies that sell research papers); submitting as your own any academic exercise (written work, computer printout, sculpture) prepared totally or in part by another; allowing another person to substantially alter or revise your work and submitting it as your own; using another's written ideas or words without properly acknowledging the source. If a student uses the words of someone else, he or she must put quotation marks around the passage and add indication of its origin, such as a footnote; simply changing a word or two while leaving the organization and content substantially intact and failing to cite the source is plagiarism. Students should also take note that failure to acknowledge study aids such as Cliff's Notes or common reference sources, such as Wikipedia constitutes plagiarism. (From Ohio U. website In practice, this means you cannot copy solutions to problems from those with whom you are taking the class, or from others who have taken the class before you. Plagiarism can also occur if you use a solution or approach that you read in a book or on a website and fail to properly cite the source. If you base your answer to a problem substantially on something you have read you must cite that source. Note that I do think it s a good idea for you to discuss together how to solve the problems. Operationally, this means that you are welcome to talk together about the homework assignments, but then you should each go off and generate the solutions you will hand in on your own. In general, if you are unsure about a question of plagiarism or cheating, you are obliged to consult me on the matter before submitting the material. Plagiarism and cheating will not be tolerated; penalties will be applied. You may appeal any academic sanctions I impose through the

6 grade appeal process. I may also report you to the University Judiciaries, which could choose to impose additional sanctions. Accommodations for students with disabilities: If you suspect that you may need an accommodation based on the impact of a disability please contact us privately to discuss your specific needs. We will need written documentation from the Office of Student Accessibility Services in order to determine the appropriate accommodation. If you have not yet registered as a student with a disability then you should contact the Office of Student Accessibility Services.

Physics 610: Quantum Field Theory I

Physics 610: Quantum Field Theory I Fall 2011 Instructor: George Sterman (MT-6108, george.sterman@stonybrook.edu) Room: Physics P-112 TuTh 8:20-9:40 This course will introduce quantum field theory, emphasizing