Introduction to Quantum Field Theory - Stony Brook …

1 Introduction toQuantum Field TheoryMarina von SteinkirchState University of New York at Stony 3, 20112 PrefaceThese are notes made by a graduate student for graduate and undergrad-uate students. The intention is purely educational. They are a review ofone the most beautiful fields on Physics and Mathematics, the QuantumField Theory , and its mathematical extension, Topological Field status ofreviewis necessary to make it clear that one who wants tolearn Quantum Field Theory in a serious way should understand that she/heis not only required to read one book or review. Rather, it is importantto keep studies on many classical books and their different approaches, andrecent publications as well. Quantum Field theories, together with topologi-cal Field theories, are fields in evolution, with uncountable applications anduncountable approaches of learning idea of these notes initially started during my first year at StonyBrook University, when I was very well exposed to the subject, duringthe courses taught by Dr.

2 George Sterman, [STERMAN1993], and by Kharzeev, [KHARZEEV2010] . However, most of the first part ofthese notes was studies from classical books, mainly [PS1995], [SREDNICKI2007],[STERMAN1993], [WEINBERG2005], [ZEE2003]. This is just a tasting ofa huge and intense Field . In the continuation of the journey, I m workingon some derivations on topological Quantum Field theories, from classicalrefereces and books such as [IVANCEVIC2008], [LM2005], [DK2007], andthe pioneering work of Edward Witten, [WITTEN1982], [WITTEN1988],[WITTEN1989], and [WITTEN1998-2].I have divided this book into two parts. The first part is theold-school(and necessary) way of learning Quantum Field Theory , and I shall call thissectionFundamentals of Quantum Field Theory . In this part, in the firstthree chapters I write about scalar fields, fields with spin, and non-abelianfields.

3 The following chapters are dedicated toquantum electrodynamicsandquantum chromodynamics, followed by second part is dedicated toTopological Field Theories. A topologicalquantum Field Theory (TQFT) is a metric independent Quantum Field theory34that introduces topological invariants of the background manifold. The bestknown example of a three-dimensional TQFT is the Chern-Simons-Wittentheory. In these notes I start with an Introduction of the mathematicalformalism and the algebraic structure and axioms. The following chaptersare the Introduction of path integral and non-abelian theories in the newformalism. The last chapters are reserved to the three-dimensional Chern-Simons-Witten Theory and the four-dimensional topological gauge theoryand invariants of four-manifolds (the Donaldson and Seiberg-Witten theo-ries).

4 I do not believe it is possible to ever finish this book, and probablythis is exactly the fun about it. One property of Science is that there isalways more to learn, more to think and more to discovery. That s whatmakes it so delightful! I conclude this preface citing Dr. Mark Srednick on[SREDNICKI2007],You are about to embark on tour of one of humanity s greatestintellectual endeavors, and certainty the one that has producedthe most precise and accurate description of the natural world aswe find it. I hope you enjoy your notes were made during my work as a PhD student at State Uni-versity of New York at Stony Brook , under the support of teaching assistant,and later of research assistent, for the department of Physics of this univer-sity. I d like to thank Prof. Dima Kharzeev, Prof. George Sterman, Meade, Prof.

5 Sasha Kirillov, Prof. Dima Averin, and Prof. BarbaraJacak, for excellent courses and discussions, allowing me to have the basicunderstanding to start this von Steinkirch,January of Fundamentals of Quantum Field Theory71 Spin Quantization of the Point Particle .. Form Invariant Lagrangians .. Noether s Theorem .. The Poincare Group .. Quantization of Scalar Fields .. Transforming States and Fields .. Momentum Expansion of Fields .. States and Fock Space .. Scattering and the S-Matrix .. Path Integral and Feynman Diagrams .. 202 Fields with Dirac Equation and Algebra .. Spinors .. Vectors .. Majorana Spinors .. Weyl Equation and Dirac Equation .. Lorentz Transformations .. Symmetries of the Dirac Lagrangian.

6 Gauge Invariance .. Canonical Quantization .. Grassmanian Variables .. Discrete Symmetries .. Chirality .. The Parity Operators .. Propagators in The Field .. 5456 CONTENTS3 Non-Abelian Field Gauge Transformations .. Lie Algebras .. Spontaneous Symmetry Breaking .. 644 Quantum Functional Quantization of Spinors Fields .. Path Integral for QED .. Feynman Rules for QED .. Reduction .. Compton Scattering .. The Bhabha Scattering .. Cross Section .. Dependence on the Spin .. Diracology and Evaluation of the Trace .. 785 Electroweak The Standard Model .. 816 Quantum Corrections given by QCD .. The Gross-Neveu Model .. The Parton Model.

7 The DGLAP Evolution Equations .. Jets .. Flavor Tagging .. Quark-Gluon Plasma .. 1007 Gamma and Beta Functions .. Dimension Regularization .. Terminology for Renormalization .. Classification of Diagrams for Scalar Theories .. Renormalization for 34.. Guideline for Renormalization .. 1138 Sigma Model115II Topological Field Theories117 Part IFundamentals of QuantumField Theory7 Chapter 1 Spin ZeroIn this first chapter we will begin studyingscalar fields, particles with nospin. They are the simplest way of getting the first techniques of quantumfield Theory , even though there is no known scalar particles in we start our journey in Quantum Field Theory , I would like to saytwo important things. One of them is that it is always useful to performdimensional analysis on our Lagrangians, operators, etc.

8 In the naturalunits, where[l] = [t] = [m] 1= [E] 1,we have [L] = we are working on phenomenological problems, it is also useful toremember that 200 MeV 1 fm Quantization of the Point ParticleSupposing a particle with only a defined momentum, with no charges and nospin, such as the Dirac s original photon. The recipe of quantization from aclassical picture is:1. Start with a coordinateq(t), and the classical LagrangianL(q(t), q(t)).2. Write the HamiltonianHcl(p,q) =p q Lcl, wherepis the PostulateH( p, q) = i~ q .1 There are theoretical candidates for scalar fields in nature, such as some models in-cluding theHiggs 1. SPIN ZERO4. For the nonrelativistic case, the Hamiltonian of the free particle isH=p22m, for the relativistic case it isH=c p2+ (mc) the same logic one can restrict the global Field to a local Field theoryonx, writing the Lagrangian asL(t) = d3xL( a(x,t),d a(x,t)).

9 The action is thenSv= t2t1dtL(t) = t2t1dt V(t )d3xL[ a].ThePrinciple of Minimum Actionsays that any variation on the actionshould be zero S= 0. (x,t1) = (x,t2) = 0,and from performing this variation on the action,0 = dtdx3( L a a+ L ( a)L( a)),= dtdx3( L i x L ( i)) a,one gets theLagrange s equationsof motion for this Field , L i x L( i)= 0.( )Example: The real Klein-Gordon EquationThe Klein-Gordon Lagrangian3is2L=12[( )( ) m2 2],( )and by varying the action, one can find its equation of motion(2+m2) = is the density of the Lagrangian, but since it is a common practice to call it justLagrangian, we will follow this convention FORM INVARIANT LAGRANGIANS11 Example: Complex Scalar FieldTreating the Field and its complex conjugate as independent, = ( 1+i 2),one can writes a Lagrangian asL= m2 .( ) Form Invariant LagrangiansA transformation on the Lagrangian is relevant for Quantum Field theorywhen this transformation keeps the Lagrangian form invariant, L( , i y ) =L( i(x), i x)d4xd4y.

10 ( ) Ld4y= Ld4x.( )For these cases of transformations, solutions in the equation of motiononximplies solutions iny. The most general invariant transformation isgiving by modifying equation ( ) by any term that lives on the surface,for instancedF dy the case of our local Field , we can generalize the invariance studyingthe infinitesimal transformations of coordinates and fields. Introducing a setof parameters{ }N =1, in terms of a small : x=x + x ( ) =x +N =1 ( x ) ( ) ,( ) i= i(x) + i( ) = i(x) +N =1 ( i) ( ) .( )One example it is the translation transformation where x = a andone has ( x ) ( a )=g = . Noether s TheoremEvery time we have a transformation on the Lagrangian that keeps it in-variant, we can say that we have asymmetry. From theNoether s Theorem,12 CHAPTER 1. SPIN ZEROL( i, i) formsNconserved currents and charges.