Introduction to groups having infinite dimension

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Presentation transcript:

Introduction to groups having infinite dimension PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 11: Introduction to groups having infinite dimension Reading: Eric Carlson’s lecture notes Additional reference: Symmetry Principles.. Melvin Lax, Wiley (1964) 3-dimensional rotation group Generators of the group Algebraic relationships 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Notion of a generator for the three-dimensional rotation group Consider rotation by an angle a about the z-axis y a f x 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Connection to angular momentum in quantum theory 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Generator operator for rotations -- continued 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Commuter relations for rotation generator operators 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

The group of 3-dimensional rotations about a point is called SO(3). Irreducible representations in terms of angular momentum eigenfunctions The group of 3-dimensional rotations about a point is called SO(3). 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Group of all unitary matrices of dimension 2 – SU(2) 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Group of all unitary matrices of dimension 2 – SU(2) --continued 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Generalization of angular momentum to include ½ integers 2/3/2017 PHY 745 Spring 2017 -- Lecture 10

Irreducible representations in terms of angular momentum-spin eigenfunctions 2/3/2017 PHY 745 Spring 2017 -- Lecture 10