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18.747: Problem set 5; due Tuesday, March 8 1. (optional; for those who like calculations!) Prove identity (7.18) in KacRaina. 2. B (0) be the Fock space and a(z) be the basic quantum field (a(z) = P Let−n−1 an z ). Let : a(z1 )...a(zn ) : be the usual product in which the annihilation operators an n > 0, have been moved to the right (we agree that this is 1 if n = 0). (a) Express a(z1 )...a(zn ) as a linear combination of : a(zj 1 )...a(zj r ) : with coefficients depending on zj (e.g. a(z1 )a(z2 ) =: a(z1 )a(z2 ) : +1/(z1 − z2 )2 ). (b) Prove the formula for < 1∗ , a(z1 )...a(z2n )1 > from a previous homework using (a). 3. Let d be the degree operator in Fock space B (0) (i.e. it multiplies a homogeneous vector by its degree, where deg(xi ) = i). Let Γ(u, v) be the operator on this space introduced in Kac-Raina. Show that tr(Γ(u, v)q d ) = Y n≥1 (1 − 1 − qn − q n v/u) q n u/v)(1 (as formal series). Hint: Compute the trace of the operator eax eb∂ q x∂ in the space of polynomials in one variable, then obtain the answer for infinitely many variables by tensoring and algebraic manipulations. 1