lecture2 notes - Lecture 2 8.324 Relativistic Quantum Field Theory II Fall 2010 8.324 Relativistic Quantum Field Theory I I MIT OpenCourseWare Lecture

lecture2 notes - Lecture 2 8.324 Relativistic Quantum Field...

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Lecture 2 8.324 Relativistic Quantum Field Theory II Fall 2010 8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 2010 Lecture 2 1 Λ Figure 1: An element Λ of the manifold of the Lie group G , and the Lie algebra g as the tangent space of the identity element. Some facts about Lie groups and Lie algebras: 1. Different Lie groups can have the same Lie algebra. The Lie algebra determines the Lie group up to discrete choices of global structure. For example, SU (2) = S 3 , SO (3) = S 3 / 2 . 2. An invariant subalgebra is a subset of a Lie algebra g g which is closed under the action of g . That is, [ g , g ] g . A simple Lie algebra is a Lie algebra which does not contain invariant subalgebras and which is not Abelian. The complex simple Lie algebras are completely classified: su ( n ) , so (2 n ), so (2 n + 1), sp ( n ) , E 6 , 7 , 8 , F 4 and G 2 are the only possibilities. 3. For a compact Lie group, it is always possible to choose a basis of T a so that f abc = f a bc is truly antisymmetric (there is no distinction between upper and lower indices). All internal symmetry groups are compact. For example, SU ( n ) (the set of n × n unitary matrices): U = exp [ i Λ a T a ] , a = 1 , . . . , n 2 1 , (1) where Tr( T a ) = 0 , ( T a ) = T a , (2) that is, the generators are hermitian and traceless, and hence we can choose [ T a , T b ] = if abc T c , (3) where the f abc are fully antisymmetric.
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