By American Mathematical Society
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Additional info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco
I have seen unpublished 1960s course notes of J. G. Belinfante [ 6, 19], very much like the birdtracks drawn here in chapters 6–9, and there are surely many other such doodles lost in the mists of time. But, citing Frege [ 128], “the comfort of the typesetter is certainly not the summum bonum,” and now that the typesetter is gone, it is perhaps time to move on. The methods used here come down to us along two distinct lineages, one that can be traced to Wigner, and the other to Feynman. Wigner’s 1930s theory, elegantly presented in his group theory monograph [ 345], is still the best book on what physics is to be extracted from symmetries, be it atomic, nuclear, statistical, many-body, or particle physics: all physical predictions (“spectroscopic levels”) are expressed in terms of Wigner’s 3n-j coefficients, which can be evaluated by means of recursive or combinatorial algorithms.
0 0 .. . . 6). In the matrix C(M − λ2 1)C † the eigenvalues corresponding to λ 2 are replaced by zeroes: ⎞ ⎛ λ1 − λ2 ⎟ λ1 − λ2 ⎜ ⎟ ⎜ λ1 − λ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟, ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎠ ⎝ .. and so on, so the product over all factors (M − λ 2 1)(M − λ3 1) . . , with exception of the (M − λ1 1) factor, has nonzero entries only in the subspace associated with λ1 : ⎞ ⎛ 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 1 ⎟ ⎜ ⎟ ⎜ † 0 C (M − λj 1)C = (λ1 − λj ) ⎜ ⎟. ⎟ ⎜ 0 j=1 j=1 ⎟ ⎜ ⎟ ⎜ 0 0 ⎠ ⎝ ..
57) j (i) Now the characteristic equation for M 2 (if nontrivial) can be used to decompose Vi subspace. 46) has more than one distinct eigenvalue; otherwise it is proportional to the unit matrix and commutes trivially with all group elements. A rep is said to be irreducible if all invariant matrices that can be constructed are proportional to the unit matrix. 40) is a statement that the defining rep is assumed irreducible. 31). 58) (remember that P i are also invariant [d×d] matrices). Hence, a [d×d] matrix rep can be written as a direct sum of [d i ×di ] matrix reps: G = 1G1 = Pi GPj = i,j Pi GPi = i Gi .