Proba

Nekomutativni prostor Kvantizacija

Kiralni fermioni na nekomutativnom prostoru:renormalizabilnost i disperzione relacije

Dusko Latas

Univerzitet u BeograduFizicki fakultet

Zagreb, 22. 12. 2010.

Nekomutativni prostor

[xµ, xν ] 6= 0.

[xµ, xν ] = iθµν

θµν realna, konstantna i antisimetricna matrica.

[xµ, xν ] 6= 0.

Mojal-Weyl-ov ili ?-proizvod

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x∫

f · g ≡∫d4x f ? g

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x∫

f · g ≡∫d4x f ? g

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x∫

f · g ≡∫d4x f ? g

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x∫

f · g ≡∫d4x f ? g

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x

∫f · g ≡

∫d4x f ? g

(Ax , ·)↔ (Ax , ?)

W : Ax → Ax

W (f ) = f = 1(2π)2

∫d4keikµx

W (f ) ·W (g) = f · g = W (f ? g)

(f ? g)(x) = exp(

i2θµν ∂

∂yµ∂∂zν

)f (y)g(z)

∣∣∣y ,z→x∫

f · g ≡∫d4x f ? g

Polja na NC prostoru

Gejdz transformacija: δλψ(x) = iλ(x) ? ψ(x)

Kovarijantni izvod: Dµψ(x) = ∂µψ(x)− iAµ ? ψ(x)

Jacina polja: Fµνψ(x) = i[Dµ?, Dν ]ψ(x)

Fµν = ∂µAν − ∂νAµ − i[Aµ ?, Aν ]

Seiberg-Witten-ovo preslikavanje

→ λ

Aµ(A; θ)→ eiλ(λ,A;θ) ? (A(A; θ) + i∂µ)? e−iλ(λ,A;θ) ≡ Aµ(A′; θ)

Aµ(A; θ) + δλAµ(A; θ) = Aµ(A + δλA; θ)

ψ(ψ,A; θ)→ eiλ(λ,A;θ) ? ψ(ψ,A; θ) ≡ ψ(ψ′,A′; θ)

ψ(ψ,A; θ) + δλψ(ψ,A; θ) = ψ(ψ + δλψ,A + δλA; θ)

δλ1 λ(λ2,A; θ)− δλ2 λ(λ1,A; θ)− i[λ(λ1,A; θ) ?, λ(λ2,A; θ)] =−iλ([λ1, λ2],A; θ)

→ λ

SW jednacine

λ = λ+ λ(1)(λ,A; θ) + λ(2)(λ,A; θ) + . . .

Aµ = Aµ + A(1)µ (A; θ) + A

(2)µ (A; θ) + . . .

ψ = ψ + ψ(1)(ψ,A; θ) + ψ(2)(ψ,A; θ) + . . .

SW jednacina za λ(1):

δλ1λ(1)(λ2)− δλ2λ

(1)(λ1)− i[λ(1)(λ1), λ2]

+i[λ(1)(λ2), λ1] + iλ(1)([λ1, λ2]) = −1

2θµν{∂µλ1, ∂νλ2}

SW jednacine

λ = λ+ λ(1)(λ,A; θ) + λ(2)(λ,A; θ) + . . .

Aµ = Aµ + A(1)µ (A; θ) + A

(2)µ (A; θ) + . . .

ψ = ψ + ψ(1)(ψ,A; θ) + ψ(2)(ψ,A; θ) + . . .

SW jednacina za λ(1):

δλ1λ(1)(λ2)− δλ2λ

(1)(λ1)− i[λ(1)(λ1), λ2]

+i[λ(1)(λ2), λ1] + iλ(1)([λ1, λ2]) = −1

2θµν{∂µλ1, ∂νλ2}

Resenje SW jednacina

λ(1)(λ,A; θ) = −14θµν{Aµ, ∂νλ}

A(1)ρ (A; θ) = −1

4θµν{Aµ,Fνρ + ∂νAρ}

ψ(1)(ψ,A; θ) = −14θµνAµ (Dν + ∂ν)ψ

Nejednoznacnosti SW preslikavanja:

λ′(1) = ic1,λθµν [Aµ, ∂νλ]

A′(1)ρ = ic1,λθ

µν [DρAµ,Aν ]− 2ic1,AθµνDρFµν

ψ′(1) = −c1,λθµνAµAνψ + 1

2 c1,ψθµνFµνψ

λ(1)(λ,A; θ) = −14θµν{Aµ, ∂νλ}

A(1)ρ (A; θ) = −1

ψ(1)(ψ,A; θ) = −14θµνAµ (Dν + ∂ν)ψ

λ(1)(λ,A; θ) = −14θµν{Aµ, ∂νλ}

A(1)ρ (A; θ) = −1

ψ(1)(ψ,A; θ) = −14θµνAµ (Dν + ∂ν)ψ

Nejednoznacnosti SW preslikavanja

A(n)ρ → A

(n)ρ + A(n)

ρ , ψ(n) → ψ(n) + Ψ(n)

∆S (n,A) =∫d4x(DρF ρµ)A(n)

µ , ∆S (n,ψ) = i∫d4xψγµ(DµΨ(n))

Nejednoznacnosti SW preslikavanja

A(n)ρ → A

(n)ρ + A(n)

ρ , ψ(n) → ψ(n) + Ψ(n)

∆S (n,A) =∫d4x(DρF ρµ)A(n)

µ , ∆S (n,ψ) = i∫d4xψγµ(DµΨ(n))

Dejstvo na NC prostoru

LNCf = i(ϕγµ∂µϕ− iϕγµAµ ? ϕ)

LNCYM = −1

4 Fµν ? Fµν

S(1)f = 1

4θµν∫d4x (−iϕγρFµν(Dρϕ) + 2iϕγρFµρ(Dνϕ))

S(1)YM = 1

8θµνTr

∫d4x (−4FρµFσνF ρσ + FµνFρσF ρσ)

Dejstvo na NC prostoru

LNCf = i(ϕγµ∂µϕ− iϕγµAµ ? ϕ)

LNCYM = −1

4 Fµν ? Fµν

S(1)f = 1

4θµν∫d4x (−iϕγρFµν(Dρϕ) + 2iϕγρFµρ(Dνϕ))

S(1)YM = 1

8θµνTr

∫d4x (−4FρµFσνF ρσ + FµνFρσF ρσ)

Metod pozadinskog polja

Γ[φi ] = Scl[φi ]− 12i Tr log S [2][φi ]

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L0 = iϕσµ(Dµϕ)− 14 FµνFµν

L1,A = −12 θ

µν(FµρFνσF ρσ − 1

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

µν∆αβγµνρ Fαβ ϕ σ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

2 ψγµ(∂µ − iγ5Aµ)ψ − 1

4 FµνFµν

L1,ψ = − i16 θ

µν∆αβγµνρ Fαβψγ

ρ(∂γ − iγ5Aγ)ψ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

L0 = i2 ψγ

µ(∂µ − iγ5Aµ)ψ − 14 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

φi → φi + Φi

S [φi + Φi ]

S[2]ij = δ2S

δφiδφj

L1,A = −12 θ

4 FµνFρσF ρσ)

L1,ϕ = − i16 θ

ρ(Dγϕ) + h.c.

(ϕαϕα

)L0 = i

4 FµνFµν

L1,ψ = − i16 θ

Γ[Aµ, ψ] = Scl[Aµ, ψ]− 12i STr logB[Aµ, ψ]

S (2) =∫d4x

(Aκ Ψ

)B(AλΨ

)B = B0 + B1

B0 = 12

(gκλ� ψγκγ5

γλγ5ψ i/∂ + /Aγ5

S (2) =∫d4x

(Aκ Ψ

)B(AλΨ

B = B0 + B1

B0 = 12

(gκλ� ψγκγ5

S (2) =∫d4x

(Aκ Ψ

)B(AλΨ

)B = B0 + B1

B0 = 12

(gκλ� ψγκγ5

S (2) =∫d4x

(Aκ Ψ

)B(AλΨ

)B = B0 + B1

B0 = 12

(gκλ� ψγκγ5

Bkin = 12

(gκλ� 0

0 i/∂

(gκλ 0

)BkinC = IC = 2

(1 00 −i/∂

)STr (logB) = STr

(log�−1BC

)− STr

(log C�−1

Bkin = 12

(gκλ� 0

0 i/∂

(gκλ 0

BkinC = IC = 2

(1 00 −i/∂

)STr (logB) = STr

(log�−1BC

)− STr

(log C�−1

Bkin = 12

(gκλ� 0

0 i/∂

(gκλ 0

)BkinC = I

(1 00 −i/∂

)STr (logB) = STr

(log�−1BC

)− STr

(log C�−1

Bkin = 12

(gκλ� 0

0 i/∂

(gκλ 0

)BkinC = IC = 2

(1 00 −i/∂

STr (logB) = STr(log�−1BC

)− STr

(log C�−1

Bkin = 12

(gκλ� 0

0 i/∂

(gκλ 0

)BkinC = IC = 2

(1 00 −i/∂

)STr (logB) = STr

(log�−1BC

)− STr

(log C�−1

BC = �I + N1 + T1 + T2

Kvantna korekcija:

Γ(1) =i

2STr log

(I + �−1N1 + �−1T1 + �−1T2

∑ (−1)n+1

(�−1N1 + �−1T1 + �−1T2

)nN1 =

(0 −iψγ5γ

λ/∂−γ5γ

κψ iγ5 /A/∂

BC = �I + N1 + T1 + T2

Kvantna korekcija:

Γ(1) =i

2STr log

(I + �−1N1 + �−1T1 + �−1T2

∑ (−1)n+1

(�−1N1 + �−1T1 + �−1T2

(0 −iψγ5γ

λ/∂−γ5γ

κψ iγ5 /A/∂

BC = �I + N1 + T1 + T2

Kvantna korekcija:

Γ(1) =i

2STr log

(I + �−1N1 + �−1T1 + �−1T2

∑ (−1)n+1

(�−1N1 + �−1T1 + �−1T2

)nN1 =

(0 −iψγ5γ

λ/∂−γ5γ

κψ iγ5 /A/∂

(V κλ −1

4θµν∆αβγ

µνρ δκα(∂βψ)γρ∂γ/∂

− i4θµν∆αβγ

µνρ δλαγρ(∂βψ)∂γ −1

8θµν∆αβγ

µνρFαβγρ∂γ/∂

V κλ = −∂σV σκ,τλ∂τ

V σκ,τλ = 12 (gστgκλ − gσλg τκ)θαβFαβ − gκλ(θξσFξ

θξτFξσ)− gστ (θξκFξ

λ + θξλFξκ) + gκτ (θξλFξ

σ + θξσFξλ) +

gσλ(θξκFξτ + θξτFξ

κ)− θκλF στ + θκτF σλ + θσλFκτ +θσκF τλ + θτλF σκ − θστFκλ

(V κλ −1

4θµν∆αβγ

µνρ δκα(∂βψ)γρ∂γ/∂

− i4θµν∆αβγ

µνρ δλαγρ(∂βψ)∂γ −1

8θµν∆αβγ

µνρFαβγρ∂γ/∂

V κλ = −∂σV σκ,τλ∂τ

V σκ,τλ = 12 (gστgκλ − gσλg τκ)θαβFαβ − gκλ(θξσFξ

θξτFξσ)− gστ (θξκFξ

λ + θξλFξκ) + gκτ (θξλFξ

σ + θξσFξλ) +

gσλ(θξκFξτ + θξτFξ

κ)− θκλF στ + θκτF σλ + θσλFκτ +θσκF τλ + θτλF σκ − θστFκλ

(δκαδ

λβ(∂γψγ5γ

ρψ + ψγ5γρψ∂γ) iδκα(2∂βAγ + Fβγ)ψγ5γ

ρ/∂

δλαγ5γρ(2Aγ∂β − Fβγ) iFαβAγγ5γ

ρ/∂

Divergentni deo efektivnog dejstva

Γ(1)|div = Γ 2 + Γ3 + Γ4 = Γ2 + Γ3

Γ2 = − i

2STr(�−1N1�

−1T1)|div

(4π)2εθµν

12εµν

ρσ(∂ρψ)γσ(�ψ) +1

12εµρστFρσ(�Fντ )

Divergentni deo efektivnog dejstva: 3-point

Γ3 =i

(− (�−1N2�

−1T1)∣∣div

+ STr(�−1N1�−1N1�

−1T1)∣∣div− STr(�−1N1�

−1T2)∣∣div

STr(�−1N2�−1T1)∣∣div

STr(�−1N1�−1T2)

∣∣div

(4π)2εθµν

(− i

3Aρ(∂µψ)γρ(∂νψ)

3Aµ(∂νψ)γρ(∂ρψ)− i

3Aρ(∂µψ)γν(∂ρψ)

3εµρστAρ(∂νψ)γ5γσ(∂τψ)− 4i

3Fµρψγ

ρ(∂νψ)

− 4i

3Fµρψγν(∂ρψ)− 4

3εµρστFνρψγ5γσ(∂τψ)

− 2i

3Aµψγν(�ψ) +

6εµν

ρσAρψγ5γσ(�ψ)

3AρFµν(∂σFρσ) −4

3AµFνρ(∂σF ρσ)

STr(�−1N1�−1N1�

−1T1)∣∣div

(4π)2εθµν

(− i

3Aρ(∂µψ)γρ(∂νψ)

3Aµ(∂νψ)γρ(∂ρψ)− 5i

3Aρ(∂µψ)γν(∂ρψ)

3εµρστAρ(∂νψ)γ5γσ(∂τψ) +

3Fµρψγ

ρ(∂νψ)

3Fµρψγν(∂ρψ) +

6Fµνψγ

ρ(∂ρψ)

− 2i

3(∂ρAρ)ψγµ(∂νψ) +

3εµν

ρσAτ (∂ρψ)γ5γσ(∂τψ)

6εµν

ρσ(∂τAτ )ψγ5γρ(∂σψ) +1

12εµρστFρσψγ5γτ (∂νψ)

6εµρστFνρψγ5γσ(∂τψ)− 1

4εµν

ρσFρσψγ5γτ (∂τψ)

Γ3 =1

(4π)2εθµν

6FµνF ρσFρσ −

3FµρF νσFρσ +

6Fµρψγ

ρ(∂νψ)

6Fµρψγν(∂ρψ)− 2i

3Fµνψγ

ρ(∂ρψ) +4

3εµρστFρσψγ5γτ (∂νψ)

2εµν

ρσFρτ ψγ5γσ(∂τψ) +1

8εµν

ρσFρσψγ5γτ (∂τψ)

12εµν

ρσAρψγ5γσ(�ψ)− 1

6εµν

ρσAτ (∂ρψ)γ5γσ(∂τψ)

12εµν

ρσ(∂τAτ )ψγ5γρ(∂σψ)

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