Methods of mathematical physics by Trodden M.

By Trodden M.

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The operator C2 goes by the name of the quadratic Casimir operator and can be defined more generally for any Líe algebra. With the generators of the Líe algebra represented by matrices we can form polynomials out of the generators by multiplying the generators using ordinary matrix multiplication (in contrast to the antisymmetrized multiplication used in the definition of Líe algebras). Then a Casimir operat or is a polynomial formed out of the generators, which commutes with all the generators.

_ 2 . Alternately by thinking about the group space of S0(3) as 18 3 with diametrically opposite points on t he surface identified we can see that there are two types of loops: loops that can be shrunk to a point and loops that can be deformed to a diameter. ·\file express that by saying that S0(3) is doubly connected. The picture below helps to explain why S 1 , in contrast to all the other sn,n>l, is not simply connected. · 50(3) and SU{2) 25 clockwise or going counterclockwise cannot be deformed into each other if we are required to stay in si' the circumference of the circle below.

So we can parameterize every Rotations: 50(3) and SU(2) 23 rotation by a 3-vector whose direction is that of the axis of rotation and whose length is equal to the angle of rotation. 18) which is 18 3 , the three-dimensional ball. However, we now note that while every rotation by an angle smaller than n corresponds to a unique point inside this ball, rotations by n and -n about any axis are the same rotation but correspond to diametrically opposite points on the surface of our ball. Thus, the parameter space of S0(3) is 18 3 with opposite points on its surface identified.

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