# 2D Quantum Gravity and SC at high Tc by Polyakov A.

By Polyakov A.

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**Example text**

8. 18. Instead of the classical ﬁeld, use the corresponding operator. Prove the following relations: (a) [Mµν , φ(x)] = −i(xµ ∂ν − xν ∂µ )φ(x) , (b) [Mµν , Pλ ] = i(gλν Pµ − gλµ Pν ) , (c) [Mµν , Mρσ ] = i(gµσ Mνρ + gνρ Mµσ − gµρ Mνσ − gνσ Mµρ ) . 9. Prove that φk (x) = k|φ(x)|0 satisﬁes the Klein–Gordon equation. 10. 15. (a) Prove that in both cases the charges satisfy the commutation relations of the SU(2) algebra. 15. 11. 19, it is shown that the action of a free massless scalar ﬁeld is invariant under dilatations.

Calculate 0| ψ(x in terms of vacuum expectation value of two ﬁelds. ¯ µ ψ := 1 [ψ, ¯ γ µ ψ]. 15. 16. Prove that 0| T (ψ(x)Γ ψ(y)) |0 is equal to zero for Γ = {γ5 , γ5 γµ }, while for Γ = γµ γν one gets the result −4imgµν ∆F (y − x). 17. The Dirac spinor in terms of two Weyl spinors ϕ and χ is of the form ψ= ϕ −iσ2 χ∗ . (a) Show that the Majorana spinor equals ψM = χ −iσ2 χ∗ . (b) Prove the identities: ψ¯M φM = φ¯M ψM , ψ¯M γ µ φM = −φ¯M γ µ ψM , ψ¯M γ5 φM = φ¯M γ5 ψM , ψ¯M γ µ γ5 φM = φ¯M γ µ γ5 ψM , ψ¯M σµν φM = −φ¯M σµν ψM .

B) is the so–called, Bianchi identity and is a kinematical condition. • Electrodynamics is invariant under the gauge transformation Aµ → Aµ + ∂ µ Λ(x) , where Λ(x) is an arbitrary function. The gauge symmetry can be ﬁxed by imposing a ”gauge condition”. The following choices are often convenient: Lorentz Coulomb Time Axial gauge gauge gauge gauge ∂µ Aµ = 0 , ∇·A=0 , A0 = 0 , A3 = 0 . C) where ωk = |k|, µλ (k) are polarization vectors. The transverse polarization vectors which satisfy (k) · k = 0 we denote by µ1 (k) and µ2 (k).