Quantum Field Theory I - uni-mainz.de

1 October 12, 2011 Quantum Field Theory IUlrich HaischRudolf Peierls Centre for Theoretical Physics, University of OxfordOX1 3PN Oxford, United KingdomAbstractThis course deals with modern applications of Quantum Field Theory with emphasize onthe quantization of theories involving scalar and spinor Books and ResourcesThere is a vast array of Quantum Field Theory texts, many of them with redeeming I mention a few of them, mostly the ones that I used or looked at when preparing thiscourse. To a large extent, I will follow the first section of M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory This is a very clear and comprehensive book, covering essentially everything in this courseas well as many advanced aspects of Quantum Field Theory that go (far) beyond the scope ofthis lecture.

2 S. Weinberg, The Quantum Theory of Fields: Volume 1, Foundations This is the first in a three volume series by one of the masters of Quantum Field takes a unique route through the subject, focussing initially on particles rather than it has a very particular viewpoint, it is difficult to digest , but certainly worth reading. L. Ryder, Quantum Field Theory This elementary text has a nice discussion of much of the material in this course. It isgood for a first reading. A. Zee, Quantum Field Theory in a Nutshell This is a charming book, where emphasis is placed on physical understanding and theauthor isn t afraid to hide the ugly truth when necessary.

3 It contains many browsing the web, I also found interesting material. Nice introductions to quantumfield Theory (of different length and viewpoint) have been written by C. Anastasiou and The corresponding scripts can be found at: babis/Teaching/ links to useful resources can be found on the web page of D. Tong: completeness, I will also give relevant references at the end of each section of thisscript. The interested reader can consult them for further details on the discussed Why QFT? .. Scales and Units ..52 Elements of Classical Field Dynamics of Fields.

4 Noether s Theorem .. Example: Electrodynamics .. Space-Time Symmetries .. Problems ..193 Klein-Gordon Klein-Gordon Field as Harmonic Oscillators .. Structure of Vacuum .. Particle States .. Two Real Klein-Gordon Fields .. Complex Klein-Gordon Field .. Heisenberg Picture .. Klein-Gordon Correlators .. Non-Relativistic Limit .. Problems ..524 Interacting Classification of Interactions .. Interaction Picture .. First Look at Scattering Processes .. Wick s Theorem .. Second Look at Scattering Processes.

5 Feynman Diagrams .. Third Look at Scattering Processes .. Yukawa Potential .. Connected and Amputated Feynman Diagrams .. From Correlation Functions to Scattering Matrix Elements .. Decay Widths and Cross Sections .. Problems .. 1015 Dirac Spinor Representation .. Discrete Symmetries of Dirac Theory .. Continuous Symmetries of Dirac Theory .. Solutions to Dirac Equation .. Quantization of Dirac Theory .. Problems .. 14321 IntroductionAs the term Quantum Field Theory (QFT) suggests, QFT is the application of Quantum me-chanics (QM) to dynamical systems offields, in the same sense that QM is concerned mainlywith the quantization of dynamical systems ofparticles.

6 QFT is not only a subject that isabsolutely essential to understand the current state of elementary particle physics as well asmodern aspects of cosmology, but also plays a crucial role in many active areas of research,ranging from atomic over nuclear and condensed-matter physics to pure mathematics. Sincethe ultimate goal of this course is to gain a basic understanding of the fundamental laws ofnature, we will in the following focus mainly on the physics of elementary particles and hencedeal mostly Why QFT?The primary reason for introducing the concept of fields in classical physics is to construct lawsof nature that arelocal.

7 The old laws of Newton (Coulomb) involve action at a distance .This means that the force felt by a planet (an electron) changes immediately if a distantstar (proton) moves. The laws of Newton and Coulomb thus Field theories of Einstein (general relativity) and Maxwell (electrodynamics) remedied thesituation, with all interactions mediated in a local fashion by fields. The requirement of localityremains a strong motivation for studying QFTs. However, there are further good reasons totreat the Quantum Field (and not the particle) asfundamental(or as Steven Weinberg puts itin [1]: Quantum fields are the basic ingredients of the universe, and particles are just bundlesof energy and momentum made out of them.)

8 QM and Special RelativityA first reason is that the combination of QM and special relativity implies that particle numberis not conserved. Consider a particle of massmtrapped in a box of sizeL. Heisenberg suncertainty principle tells us that the uncertainty in the momentum of our particle is p ~/L. In the relativistic limit, momentum and energy can be treated on equivalent footing,and one has an uncertainty in the energy of order E ~c/L. Yet, if E= 2mc2, there isenough energy available to create avirtualparticle-antiparticle pair from the vacuum (Diracsea).

9 This little exercise shows that when a particle with massmis localized within a distance Compton=~/(mc), talking about a single particle loses its sense. For distances smaller thanthisCompton wavelengththere is a high probability that we will detect particle-antiparticlepairs swarming around the single particle that we initially put into the box. Notice that Comptonis always smaller than thede Broglie wavelengthgiven by de Broglie=~/|p|.1 Ifyou like, the de Broglie wavelength is the distance at which the wavelike nature of particlesbecomes apparent, while the Compton wavelength is the distance at which the concept ofa single pointlike particle breaks down and one has to start thinking about how to describemultiparticle this course we will use boldface type (ordinary italic type) to denote 3-vectors (4-vectors).

10 3 The presence of a multitude of particles and antiparticles at short distances (or highenergies) tells us that any attempt to write down a relativistic version of the one-particleSchr odinger equation is doomed to fail. There is no mechanism in standard non-relativisticQM to deal with changes in the particle number. Indeed, any attempt to naively write down arelativistic version of the one-particle Schr odinger equation meets serious problems: negativeprobabilities, infinite towers of negative energy states, or a breakdown of causality are thecommon issues that and CausalityLet us have a closer look at the issue ofbreakdown of causality.