15 Elitzur’s Theorem

Problems: Quantum Fields on the Lattice

Prof. Dr. Andreas Wipf WiSe 2019/20

MSc. Julian Lenz

Sheet 6

15 Elitzur’s Theorem

Elitzur’s theorem states that it is impossible to break spontaneously a local symmetry. Verify this in the following simple setup: Consider aZ2-gauge theory coupled to a scalar field with the action

S=−β∑

p

U_p−κ∑

⟨x,y⟩

ϕ_xU_⟨x,y⟩ϕ_y +h∑

x

ϕ_x+V(ϕ_x) (1)

where we have included a generalZ2-gauge invariant potentialV as well as a source term parametrized byh. Prove that

hlim↘0⟨ϕ_x⟩= 0 (2)

uniformly in the volume and the couplings.

16 Some group integrals

1. Show that

∫

SU(N)

dU U = 0. (3)

2. LetF be anN×N matrix. Prove that ifΛFΛ⁻¹=F holds for allΛ∈SU(N)thenF =c1. Hint: Start withN = 2. Find two specialSU(2)matrices which allow to showF =c1. Embed SU(2)intoSU(N)and use theN = 2property to show it for allN ∈N.

3. Use the previous result to calculate f_ijkl=

∫

SU(N)

dU Uij

( U^†

)

kl (4)

and determine the constantcfor this case. (Hint: You can crosscheck parts of your result by the use of the gluing property from Problem 18.)

17 Conjugacy slasses of SU(3)

Characterize the conjugacy classes ofSU(3).

18 Applications of the Peter-Weyl theorem

Use the Peter-Weyl theorem to prove the following properties:

1. orthogonality:

(R^a_b, χ_R′) = δ_RR′

dR

δ^a_b, (5)

2. gluing:

∫

dΩχR(UΩ⁻¹)χ_R′(ΩV) =δRR^′

d_R χR(U V), (6) 3. separation:

∫

dΩχ_R(ΩUΩ⁻¹V) = 1 dR

χ_R(U)χ_R(V), (7) 4. decomposition of1:

∑

R

dRχR(U) =δ(1, U). (8)