Lectures on Quantum sine-gordon Models

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Ἁλκυόνη Παπακωνσταντίνου
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1 Lectures on Quantum sine-gordon Models Juan Mateos Guilarte 1,2 1 Departamento de Física Fundamental (Universidad de Salamanca) 2 IUFFyM (Universidad de Salamanca) Universidade Federal de Matto Grosso Cuiabá, Brazil, 2010

2 Outline 1 The model 2 3

3 Lagrangian Massless L(x) = i ψ(x)γ µ µψ(x) 1 2 g ψ(x)γ µ ψ(x) ψ(x)γ µψ(x) ( ) [ψ] = L 1 2, [g] = 1, x (x 0, x 1 ), x 2 x µx µ ψ1 (x), ψ =, ψ(x) = ψ ψ 2 (x) (x)γ 0 γ 0 = σ 1, γ 1 = iσ 2, γ 5 = γ 0 γ 1 = σ 3, γ µ γ ν + γ ν γ µ = 2g µν γ µ γ 5 + γ 5 γ µ = 0, γ µ γ ν = g µν + ε µν γ 5, γ µ = g µνγ ν Invariance and canonical currents ψ(x) U V 1 e iα V ψ (x), j µ (x) = ψ(x)γ µ ψ(x) U A 1 ψ(x) e iα Aγ 5 ψ (x), j µ 5 (x) = ψ(x)γ µ γ 5 ψ(x) { ˆψ(x), ˆψ (y)} = x 0 =y 0 δ(x1 y 1 ), [ĵ 0 (x, t), ˆψ(y, t)] = δ(x y) ˆψ(y, t) Fierz transformation ψγ µ ψ ψγ µψ = ψ 1 ψ 1ψ 2 ψ 2 = ( ) ( ) 2 2 ψψ ψiγ 5 ψ

4 Renormalized chiral eigen-densities Massive ˆσ ± (x) = Z ˆ ψ(x)(1 ± γ 5 ) ˆψ(x), Ĥ Ĥ + m f 2 (ˆσ +(x) + ˆσ (x)) Space-like 2n-point Wightman functions 0 Π n i=1ˆσ +(x i )ˆσ (y i ) 0 = A 2n Π i>j[(x i x j ) 2 (y i y j ) 2 ] δ Π i,j [(x i y j ) 2 ] δ, δ = g π

5 2n-point functions in s-g 2-point Wightman function of a free massless scalar field ˆφ(x) ˆφ(y) = N µ( ˆφ(x) ˆφ(y)) + 0 ˆφ(x) ˆφ(y) 0, + (x, µ) t <<1 = dk 2π + (x, µ) = 0 ˆφ(0) ˆφ(x) 0 e ik(t x) k 2 + µ 2 = 1 2π K 0(µ t 2 + x 2 ) t 2 +x 2 <<1 = 1 4π log( cµ2 x µx µ ) + O( x µx µ ) Wick theorem and normal ordering F( ˆφ) = d 2 x 1 d 2 x n K(x 1,, x n) ˆφ(x 1 ) ˆφ(x n) [ ( )] } 1 = N µ {exp d 2 x d 2 δ δ y + (x y, µ) F( ˆφ) 2 δφ(x) δφ(y) Normal ordered free Hamiltonian and vertex Ĥ 0 = 1 [ 2 Nµ ˆπ ( 1 ˆφ) ] = N µh 0, Ŝ ± = N µe ±i m ( 2 ) t 2 2 x 2 + (x y, µ) = δ (2) (x y) ˆφ

6 Space-like sine-gordon 2n-point vertex functions MTM/sGM equivalence 0 Π n i=1ŝ+(x i )Ŝ (y i ) 0 = A 2n Π i>j[(x i x j ) 2 (y i y j ) 2 ] δ Π i,j [(x i y j ) 2 ] δ, δ = 4πm 2 MTM/sGM identifications m f 2 (ˆσ + + ˆσ ) = m4 Nµ(cos m 2 = 4π2 π + g ˆφ) m, g = 4π 2 m2 π Proof. Infrared cutoff and N ν-ordering of the vertex [ ] [ ] N µ Ĥ 0 N µ Ĥ 0 + ν2 ˆφ 2 = N ν Ĥ 0 + ν2 ˆφ 2 + constant 2 2 N µe i( 1)a m ˆφ = ( ν 2 µ 2 ) 8πm 2 N νe i( 1)a m ˆφ, a = 0, 1

7 N µ-ordered 2n-point vertex functions ( ν 2 µ 2 ( ν 2 = µ 2 ν<<1 = Strings of vertex vev ) 8πm 2 ( i ( 1)a i ) 2 0, ν Π i N νe i( 1)a i m ˆφ(xi ) 0, ν = ) 8πm 2 ( i ( 1)a i ) 2 Π i>j e ( 1)a i ( 1) aj + (x i x j,ν) ν<<1 = ( ν 2 µ 2 ) 8πm 2 ( i ( 1)a i ) 2 Π i>j [cν 2 (x i x j ) 2 ] ( 1)a i ( 1) a j Normal-ordering no contractions at the same point Survivors at the ν 0 limit ( ν 2 µ 2 ) 4πm 2 8πm 2 ( i ( 1)a i ) 2 Π i>j [cν 2 (x i x j ) 2 ] ( 1)a i ( 1) a j 4πm 2 f (µ) iif (a i = 0) = (a i = 1) (Ŝ + ) = (Ŝ )

8 Position operator eigen-states Field configuration states ˆx x = x x, 0 x = δ(x), ˆp p = p p, 0 p = δ(p) p x = 1 2π eipx, dp p p = 1 e iˆpa x = e iˆpa dp p p x = 1 dp e i(x a)p p = x a 2π Field operator eigen-states ˆφ(x) φ(y) = δ(x y)φ(y) φ(y), 0 φ(x) = δ[φ(x)] ˆπ(x) π(y) = δ(x y)π(y) π(y), 0 π(x) = δ[π(x)] π(x) φ(y) = 1 2π eiπ(x)φ(y), dx Π xdπ(x) π(x) π(x) = 1 : e i dyˆπ(y)f (x) : φ(x) = =: e i dyˆπ(y)f (x) : dx Π xdπ(x) π(x) π(x) φ(x) = φ(x) f (x)

9 Mandelstam Creation/annihilation soliton : e i dy ˆπ(y)φ K (x) : 0 = φk (x), ˆψ(x) m =: Â(x)exp{ 2πi dξ ˆπ(ξ)θ(x ξ)} : [ ˆφ(y), ˆψ(x)] = 2π m ˆψ(y), y < x ; [ ˆφ(y), ˆψ(x)] = 0, y > x Soliton annihilation ˆψ(x) φ K (y x) φ K (y x) 2π m θ(y x) = 0 Different points commutators ˆψ α(x) : eŝα(x) :, α = 1, 2 Ŝ α(x) = 2πi m x dξ ˆπ(ξ) + ( 1) α i ˆφ(x) 2m [Ŝ 1 (x), Ŝ 1 (y)] = iπ[θ(x y) θ(y x)] = { +iπ [Ŝ α(x), Ŝ β (y)] = iπ, α, β { +iπ, x > y iπ, x < y

10 Different points anti-commutator Baker-Haussdorf-Campbell formula eâeˆb = e [Â,ˆB] eˆb eâ if [Â, ˆB] = c number { ˆψα(x) ˆψ β (y) = ˆψ β (y) ˆψ α(x) ˆψ α(x) ˆψ β (y) = ˆψ β (y) ˆψ, x y, α, β α(x) Preparing the dangerous lim y x coalescence ˆψ 1 (x) = N( cm 2π ) 2 1 exp[ 2πi m x dξ ˆπ(ξ) i ˆφ(x)] 2m ˆψ 2 (x) = in( cm 2π ) 2 1 exp[ 2πi m x dξ ˆπ(ξ) + i ˆφ(x)] 2m N: infinite renormalization constant. Cutoff in the ξ integration Formal current algebra { ˆψ α(x), ˆψ β (y)} = Zδ αβδ(x y) ĵ µ 1 (x) = lim ˆ ψ(x)γ µ ˆψ(y) y x Z Does [ĵ µ (x), ˆψ(y)] = (g µ0 + ε µ0 γ 5 ) ˆψ(x)δ(x y) really hold?

11 Renormalized currents Short distance operator behavior { ĵ µ (x) = lim cm(x y) σ ˆ ψ(x)γ } µ ˆψ(y) + F(x y) y x Choose σ and F(x y) in such a way that ĵ µ (x) exists Normal ordered Baker-Haussdorf-Campbell formula : eâ :: eˆb :=: eâ+ˆb : e [Â+,ˆB ], if [Â +, ˆB ] = c number ˆψ α (x) ˆψ α(y) = ( 1) α {2π(x y) 1 } cm(x y) 4 1 8πm 2 : exp[ 2πi m y ( ) dξ ˆπ(ξ) + ( 1) α i ˆφ(y) ˆφ(x) + O((x y) 2 )] :, no sum in α(1) x REMARKS 1 The factor cm(x y) 1 4 8πm 2 comes from the ˆφ ˆφ terms. Note the different critical exponent with respect to the bosonic case. The 1 is due to the dimension of the 4 Fermi fields. 2 The factor 2π(x y) 1 is due to the ˆπ ˆφ terms: derivation of the short-time Wightman function with respect to the time and integration in ξ. The ( cµ 2π ) 1 2 factor cancels because the normalization. The result can be hinted directly at the lim y x. See Appendix 3 The ˆπˆπ terms give contributions of O((x y)) 2 order.

12 Canonical anti-commutation relations Expand the exponential in (1) and select σ = 1 4 8πm 2 (killing the divergent term) ĵ 0 (x) = ˆψ 1 (x) ˆψ 1 (x) + ˆψ 2 (x) ˆψ 2 (x) = 1 ˆφ 2π m x ĵ 1 (x) = ˆψ 1 (x) ˆψ 1 (x) ˆψ 2 (x) ˆψ 2 (x) = 2 ˆφ m t fulfilling the right anti-commutation relations even in coinciding points. ĵ 0 (x) and ĵ 1 (x) are not the two components of a vector. Lorentz covariant current densities ĵ µ (x) = lim {[δ µ y x π m 2 δµ 1 ] cm(x y) σ ˆ ψ(x)γ µ ˆψ(y) + F(x y)} Anti-commutation relations [ĵµ (x), ˆψ(y)] { = g µ0 + 1 } 4π m 2 εµ0 γ 5 ˆψ(x)δ(x y) ĵ µ (x) = 1 2π εµν ν ˆφ(x)

13 Renormalized MTM field equations The MTM field equations I ( iγ µ µ m f0 ) ˆψ(x) g = lim δx 0 2 γµ {ĵ µ(x + δx) + ĵ µ(x δx)} ˆψ(x), m 0 ˆψ = [ ˆM, ˆψ] ˆM = Zm f0 dx ˆ ψ(x) ˆψ(x) = dx lim cm(x y) δ m f ˆ ψ(x) ˆψ(y) From the sine-gordon to the model {[ ˆψ 1 (x) = 2πi m x ] } dξ ˆφ(ξ) i ˆφ(x), ˆψ 1 (x) 2m {[ = N 2πi m ˆφ x (x) i ˆφ(x) + 2πi m4 x ] } dξ sin ˆφ(ξ) i ˆφ(x), ˆψ 1 (x) 2m 2m N : normal ordering, always with respect to sine-gordon, but not with respect to the ˆψ s Use of [ ˆφ(y), ˆψ(x)] = 2π m θ(x y) ˆψ(y) to show that, see Appendix i m4 [ dξ : cos ˆφ(ξ) :, ˆψ 1 (x)] = 2πi m4 x m N { dξ sin ˆφ(ξ), ˆψ 1 (x)} m

14 Fermi mass term from soliton The MTM field equations II ˆψ 2 (x) ˆψ 1 (y) cm = ( 2π ) cm(x y) δ : e i m ˆφ(x) : ˆψ 1 (x) ˆψ 2 (y) = ( cm 2π ) cm(x y) δ : e i m ˆφ(x) :, δ = 1 4 8πm 2 Bosonization of the MTM: sine-gordon potential m f = π m 3, ˆM = c c π mm f dξ : cos ˆφ(ξ) : m { N 2πi m4 x } dξ sin ˆφ(ξ), ˆψ 1 (y) = i[ ˆM, γ 0 ˆψ2 (y)] m PDE equations of the normal ordered Mandelstam ˆψ { 1 x =: ( 2πi ˆφ i ˆφ } m t 2m x ), ˆψ 1 (x) ˆψ 1 (x) + ˆψ { 1 (x) + i[ ˆM, γ 0 ˆψ2 (x)] = i 2πm g : ( ˆφ (x) ˆφ(x)), } ˆψ1 (x) : (ĵ0 ( lim (x + δx) + ĵ 0 (x δx) + ĵ 1 (x + δx) + ĵ 1 (x δx)) = i ˆφ (x) ˆφ(x) ) δx 0 πm

15 Appendix and References ( x x ) exp{ π [ ˆφ(y), dξ ˆπ(ξ)] [ dξ ˆπ(ξ), ˆφ(x)] } = x y = exp[ πi( dξ δ(y ξ) dξ δ(x ξ))] = = e y x dξ d ln(2πi(z ξ))) dξ = e ln(2πi(y x)) {Â 3, ˆB} = Â[Â 2, ˆB] + [ˆB, Â]Â 2 ( 1) n (2n)! ( m )2n [ ˆφ 2n (ξ), ˆψ 1 (x)] = n=0 ( 1) k 2π (2k + 1)! ( m )2k+1 { ˆφ 2k+1 (ξ), ˆψ 1 (x)}θ(x ξ) k=0 Sidney F. Coleman, Classical lumps and their quantum descendants, in Aspects of Symmetry, Cambridge University Press, 1985 Stanley Mandelstam, Soliton for the quantized sine-gordon equation, in Extended Systems in Field Theory, Physics Reports 23C, (1976)

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