Download A Course in Mathematical Physics, Vol. 1: Classical by Walter E Thirring PDF

By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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Iz> J The second expression for ponential to the bra vector I ZI=Z + w~l ZW=Z is obtained by moving the ex-

Z,)2(al~z ,) + 2j~,,] f(z,) = = [ (2J)! ] I/2 (z')j-m (j-m) ! (j +m) ! and the reproducing kernel : (1 + z'~) 2j Proof: The commutation relations are easily verified from the explicit form of the generators. The hermitian properties may be shown by use of the reproducing kernel. One finds 1 2jz' = f(z) g(z') du(z) d~(z') 1 + z ~' and = : ((-z2~ + 2jz)f(z)) f(z')((-z 2 ~ g(z) du(z) + 2jz)) f(z') g(z) 2j~ g(z) du(z') du(z) d~(z') d~(z) I + z'~ and therefore - = O.

We parametrize the coset space SU(2)/U(1) by two Euler angles and write u3(a) = exp (eA3) = [ exp(ia/2)O exp(-ia/2)O ] COSB/2 sin 8/2 -sin 8/2 cos 6/2 U2(8) = exp (6A 2) = ] An arbitrary element of SU(2) may now be decomposed in the form u = u(m6y) = u3(m) u2(6) u3(Y) For the representation we write U = U((~SY) = U3(c~) U2(8) U3(Y) and obtain for the matrix elements = : <6'Y'16Y> . 5 Proposition: The symplectic structure for the representation D j, j~O of SU(2) is given by {F , G } : i C j s i n 6 ) -1 [(BIB6) F (~/ay) ~ - (a/a~) T (~/~6) g) , the symplectic generators are 41(6y) = <6yIAll6Y> = ijsin6 cosy A2(6y) = (6YIA216y> = ijsin6 siny A3(6Y) = <6yIA316y> = i j c o s 6 This symplectic structure coincides with the one derived by geometric means if the radius of the spheres on is taken as r = j = I/2, 2/2,3/2 ....

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