Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Next, we associate with the two operations two operators ^O Ep and ^O Ip ,
which act on the real or complex function f …x 1 ; y 1 ; z 1 ; x 2 ; y 2 ; z 2 † with the
following e€ects:
^O Ep f ˆ f ; ^O Ip f …x 1 ; y 1 ; z 1 ; x 2 ; y 2 ; z 2 †ˆf …x 2 ; y 2 ; z 2 ; x 1 ; y 1 ; z 1 †:
Show that the two operators form a group that is isomorphic to G 2 .
12.11. Verify that the multiplication table of S 3 has the form:
P 1 P 2 P 3 P 4 P 5 P 6
P 1 P 1 P 2 P 3 P 4 P 5 P 6
P 2 P 2 P 1 P 6 P 5 P 6 P 4
P 3 P 3 P 4 P 5 P 6 P 2 P 1
P 4 P 4 P 5 P 3 P 1 P 6 P 2
P 5 P 5 P 3 P 4 P 2 P 1 P 6
P 6 P 6 P 2 P 1 P 3 P 4 P 5
12.12. Show that an n n orthogonal matrix has n…n 1†=2 independent
elements.
12.13. Show that the 2 2 matrix 2 can be obtained from the rotation matrix
R…† by di€erentiation at the identity of SO…2†, that is, ˆ 0.
12.14. Show that an n n unitary matrix has n 2 1 independent parameters.
12.15. Show that det e ~A ˆ e Tr ~A where ~A is any square matrix.
12.16. Show that the Lorentz transformation
x 0 1 ˆ …x 1 ‡ ix 4 †;
x 0 2 ˆ x 2 ;
x 0 3 ˆ x 3 ;
x 0 4 ˆ …x 4 ix 1 †
corresponds to an imaginary rotation in the x 4 x 1 plane. (A detailed discussion
of this can be found in the book Classical Mechanics, by Tai L.
Chow, John Wiley, 1995.)
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