lecturenotes.pdf - Lecture Notes for Quantum Field Theory III Spring 2011 Lecturer Professor Erick Weinberg Transcriber Alexander Chen Contents 1

lecturenotes.pdf - Lecture Notes for Quantum Field Theory...

This preview shows page 1 - 4 out of 125 pages.

Lecture Notes for Quantum Field Theory III Spring 2011 Lecturer: Professor Erick Weinberg Transcriber: Alexander Chen July 17, 2011 Contents 1 Lecture 1 3 1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Lecture 2 7 2.1 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Lecture 3 11 3.1 Continued SU (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The Algebra of SO ( N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Classification of Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Lecture 4 18 4.1 Roots Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Exceptional Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Classification According to Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Lecture 5 24 5.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 Lecture 6 29 6.1 Proof of Goldstone Theorem in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7 Lecture 7 33 8 Lecture 8 38 9 Lecture 9 42 1
Image of page 1
10 Lecture 10 46 11 Lecture 11 51 11.1 Anomalies Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 12 Lecture 12 55 13 Lecture 13 58 13.1 Grand Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 14 Leture 14 62 14.1 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 15 Lecture 15 67 15.1 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 16 Lecture 16 73 16.1 Kink Solution Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 16.2 Multikink Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 17 Lecture 17 78 18 Lecture 18 83 19 Lecture 19 87 20 Lecture 20 91 21 Lecture 21 95 21.1 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 22 Lecture 22 99 23 Lecture 23 103 23.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 24 Lecture 24 107 24.1 Wess-Zumino Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 24.2 Notation Transmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 25 Lecture 25 111 26 Lecture 26 115 27 Lecture 27 120 28 Lecture 28 123 2
Image of page 2
Quantum Field Theory III Lecture 1 1 Lecture 1 1.1 Structure We will start with a bit of group theory, and we will talk about spontaneous symmetry broken. Then we will talk about anomalies and grand unification. Then we will cover solitions & duality instantons. Finally we wil talk about supersymmetry. 1.2 Group theory Suppose we have a group G with elements g 1 , g 2 , . . . with operation g 1 g 2 = g 3 . We require the operation to be associative, there is an identity and every element has an inverse gI = Ig = g, gg - 1 = g - 1 g = I (1.1) A group can have finite number of elements or infinite. For the infinite case we can have discrete elements (group of integers) or continuous (real numbers). The most important kind of groups in high energy physics is Lie groups, which is a continuous group. A group element will be labeled by a number of labels g ( x 1 , x 2 , . . . ) = g ( x ). We have the group multiplication law g ( x ) g ( y ) = g ( z ) (1.2) where z is a continuous differentiable function of x and y . We can also think of the Lie group as a manifold. The dimension of the manifold, which is also the dimension of the Lie group, is equal to the number of parameters necessary to specify each group element. The manifold can be compact or non-compact. For example, a rotation group is compact, and if you keep rotation you will come back to the original position. However the Lorentz group is not compact, and you can boost forever. The Lorentz group will likely be the only non-compact group that we will make reference to. References can be found in the book by Georgi. Another one written by Gilmore. Let’s look at some examples. Consider the group SO ( N ). This can be defined as the group of N × N orthogonal matrices with determinant 1, or can be defined as the rotation group in N dimensions. From N × N we have N 2 parameters, but from orthogonality we have the constraint R ij R jk = δ ik (1.3) This gives N conditions when i = k and N ( N - 1) / 2 conditions when i 6 = k . So in the end we have N (
Image of page 3
Image of page 4

You've reached the end of your free preview.

Want to read all 125 pages?

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes