By Martin Schottenloher (auth.)
Read Online or Download A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg PDF
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Additional resources for A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg
Of biholomorphic maps ¢ " I? ---, I? of the Riemann sphere I?. ) using the compactification C ~ I? 6. 8 says even more: Mb is also isomorphic to the proper Lorentz group SO(3, 1). An interpretation of the 2. The Conformal Group 30 isomorphism Aut(IP) =~ S0(3, 1) from a physical viewpoint was given by Penrose, cf. g. [Sch95, p. 210]. g. [BPZ84, p. 335] "The situation is somewhat better in two dimensions. " [FQS84, p. 4 2 0 ] "Two dimensions is an especially promising place to apply notions of conformal field invariance, because there the group of conformal transformations is infinite dimensional.
1 Con[ormal Compactification o [ R p'q 23 is a conformal transformation and a diffeomorphism. The inverse transformation ¢-1 = CA-~ is also conformal. The map A ~-, CA is not injective. However, CA = CA' implies A = A ~ or A = - A ' . e. ~/(h~) e N p'q. e. e. A ~ ~ A~. Altogether, ¢ : N p'q --, N p'q is well-defined. Because of the fact that the metric on R p÷l'q÷l is invariant with respect to A, CA turns out to be conformal. if P is . represented by E Sp x S q. ) Obviously, CA = ¢-A and ¢~1 = Ch-~.
Of the Hilbert space ]E one defines the set U(]E) of unitary operators on ]HI. U(]HI) is the set of all C-linear bijective maps U" H ---, ]HI with the following property: f, g E ]I"]I ~ (U f , Ug) = (f ,g) . It is easy to see that the inverse U -1 : ]I-lI---, IE of a unitary operator U :]E ---, IE is unitary as well and that the composition U o V of two unitary operators U, V is always unitary. Hence, the composition of operators defines the structure of a group on U(IE). U(]HI) with this structure is called the unitary group of IE.