RENORMALIZATION GROUP TRAJECTORIES BETWEEN TWO FIXED POINTS - PowerPoint PPT Presentation

RENORMALIZATION GROUP TRAJECTORIES BETWEEN TWO FIXED POINTS Abdelmalek Abdesselam University of Virginia, Department of Mathematics ICMP Prague, Aug 7, 2009

◮ Main reference: A. A. CMP 07’

◮ Main reference: A. A. CMP 07’ ◮ Outline: 1. Global dynamics of Wilson’s RG 2. Rigorous results (selection) 3. The BMS model 4. Good infinite-volume coordinates 5. Idea of the proof 6. Functional analysis, norms 7. Perspectives

1. Global Dynamics of Wilson’s Renormalization Group:

1. Global Dynamics of Wilson’s Renormalization Group: QFT functional integrals: a challenge for mathematicians

1. Global Dynamics of Wilson’s Renormalization Group: QFT functional integrals: a challenge for mathematicians e. g., the φ 4 model � � R d [ 1 2 ( ∇ φ ) 2 ( x )+ µφ ( x ) 2 + g φ ( x ) 4 ] dx D φ · · · e − F

1. Global Dynamics of Wilson’s Renormalization Group: QFT functional integrals: a challenge for mathematicians e. g., the φ 4 model � � R d [ 1 2 ( ∇ φ ) 2 ( x )+ µφ ( x ) 2 + g φ ( x ) 4 ] dx D φ · · · e − F F infinite-dimensional space of functions R d → R D φ Lebesgue measure on F

◮ Construction by scaling limit of lattice theories on ( a Z ) d ⊂ R d ⇐ ⇒ cut-off 1 a on momenta in Fourier space ◮ rescale to unit lattice ◮ approximants to continuum theory = points in the space of all possible unit cut-off theories

◮ Construction by scaling limit of lattice theories on ( a Z ) d ⊂ R d ⇐ ⇒ cut-off 1 a on momenta in Fourier space ◮ rescale to unit lattice ◮ approximants to continuum theory = points in the space of all possible unit cut-off theories ◮ RG = dynamical system on this space

◮ Construction by scaling limit of lattice theories on ( a Z ) d ⊂ R d ⇐ ⇒ cut-off 1 a on momenta in Fourier space ◮ rescale to unit lattice ◮ approximants to continuum theory = points in the space of all possible unit cut-off theories ◮ RG = dynamical system on this space ◮ d ν measure on random φ with ˆ φ ( p ) = 0 if | p | > 1 ◮ introduce magnification ratio L > 1 ◮ split φ = ζ + φ low ⇒ L − 1 < | p | ≤ 1 ζ ⇐ ⇒ | p | ≤ L − 1 φ low ⇐

◮ integrate over ζ − → marginal probability distribution on φ low ◮ rescale ψ ( x ) = L [ φ ] φ low ( Lx ) − → measure d ν ′

◮ integrate over ζ − → marginal probability distribution on φ low ◮ rescale ψ ( x ) = L [ φ ] φ low ( Lx ) − → measure d ν ′ ◮ RG map: d ν − → d ν ′

◮ integrate over ζ − → marginal probability distribution on φ low ◮ rescale ψ ( x ) = L [ φ ] φ low ( Lx ) − → measure d ν ′ ◮ RG map: d ν − → d ν ′ Important features: fixed points, eigenvalues of linearized RG around them, local stable & unstable manifolds. . .

◮ integrate over ζ − → marginal probability distribution on φ low ◮ rescale ψ ( x ) = L [ φ ] φ low ( Lx ) − → measure d ν ′ ◮ RG map: d ν − → d ν ′ Important features: fixed points, eigenvalues of linearized RG around them, local stable & unstable manifolds. . . Local features

◮ integrate over ζ − → marginal probability distribution on φ low ◮ rescale ψ ( x ) = L [ φ ] φ low ( Lx ) − → measure d ν ′ ◮ RG map: d ν − → d ν ′ Important features: fixed points, eigenvalues of linearized RG around them, local stable & unstable manifolds. . . Local features Global features ? e. g. heteroclinic trajectories between fixed points

2. Rigorous Results (Selection):

2. Rigorous Results (Selection): Local: ◮ RG exponents for Gaussian fixed point, ∃ nontrivial IR fp in “4 − ǫ ” dimensions, local stable/unstable manifolds, for HM: Bleher-Sinai CMP 73’, 75’, Collet-Eckmann CMP 77’, LNP 78’, Gaw¸ edzki-Kupiainen CMP 83’, JSP 84’, Pereira JMP 93’ ◮ HM at ǫ = 1: Koch-Wittwer CMP 86’ 2 ◮ new fps at d = 2 + n − 1 , n = 3 , 4 , . . . in LPA: Felder CMP 95’

2. Rigorous Results (Selection): Local: ◮ RG exponents for Gaussian fixed point, ∃ nontrivial IR fp in “4 − ǫ ” dimensions, local stable/unstable manifolds, for HM: Bleher-Sinai CMP 73’, 75’, Collet-Eckmann CMP 77’, LNP 78’, Gaw¸ edzki-Kupiainen CMP 83’, JSP 84’, Pereira JMP 93’ ◮ HM at ǫ = 1: Koch-Wittwer CMP 86’ 2 ◮ new fps at d = 2 + n − 1 , n = 3 , 4 , . . . in LPA: Felder CMP 95’ ◮ Euclidean model in “4 − ǫ ” dimensions, ∃ nontrivial IR fp and local stable manifold: Brydges-Dimock-Hurd CMP 98’ ◮ Same for nicer model: Brydges-Mitter-Scoppola CMP 03’

2. Rigorous Results (Selection): Local: ◮ RG exponents for Gaussian fixed point, ∃ nontrivial IR fp in “4 − ǫ ” dimensions, local stable/unstable manifolds, for HM: Bleher-Sinai CMP 73’, 75’, Collet-Eckmann CMP 77’, LNP 78’, Gaw¸ edzki-Kupiainen CMP 83’, JSP 84’, Pereira JMP 93’ ◮ HM at ǫ = 1: Koch-Wittwer CMP 86’ 2 ◮ new fps at d = 2 + n − 1 , n = 3 , 4 , . . . in LPA: Felder CMP 95’ ◮ Euclidean model in “4 − ǫ ” dimensions, ∃ nontrivial IR fp and local stable manifold: Brydges-Dimock-Hurd CMP 98’ ◮ Same for nicer model: Brydges-Mitter-Scoppola CMP 03’ Global: ◮ uniqueness of IR fp in LPA for 3 ≤ d < 4: Lima CMP 87’ ◮ Massless GN in “2 + ǫ ” dim: Gaw¸ edski-Kupiainen NPB 85’

2. Rigorous Results (Selection): Local: ◮ RG exponents for Gaussian fixed point, ∃ nontrivial IR fp in “4 − ǫ ” dimensions, local stable/unstable manifolds, for HM: Bleher-Sinai CMP 73’, 75’, Collet-Eckmann CMP 77’, LNP 78’, Gaw¸ edzki-Kupiainen CMP 83’, JSP 84’, Pereira JMP 93’ ◮ HM at ǫ = 1: Koch-Wittwer CMP 86’ 2 ◮ new fps at d = 2 + n − 1 , n = 3 , 4 , . . . in LPA: Felder CMP 95’ ◮ Euclidean model in “4 − ǫ ” dimensions, ∃ nontrivial IR fp and local stable manifold: Brydges-Dimock-Hurd CMP 98’ ◮ Same for nicer model: Brydges-Mitter-Scoppola CMP 03’ Global: ◮ uniqueness of IR fp in LPA for 3 ≤ d < 4: Lima CMP 87’ ◮ Massless GN in “2 + ǫ ” dim: Gaw¸ edski-Kupiainen NPB 85’ ◮ BMS model, construction of discrete heteroclinic trajectories joining Gaussian UV fp to nontrivial IR fp: A. A. CMP 07’

3. The BMS Model:

3. The BMS Model: Scalar field φ : R 3 − → R potential V ( φ ) � � �� � 3+ ǫ D φ e − 1 � 2 � φ, ( − ∆) 4 φ � L 2( R 3) dx ( µ : φ 2 ( x ):+ g : φ 4 ( x ):) − Z = � �� � Gaussian measure

3. The BMS Model: Scalar field φ : R 3 − → R potential V ( φ ) � � �� � 3+ ǫ D φ e − 1 � 2 � φ, ( − ∆) 4 φ � L 2( R 3) dx ( µ : φ 2 ( x ):+ g : φ 4 ( x ):) − Z = � �� � Gaussian measure ◮ propagator ( − ∆) − 3+ ǫ 1 4 ( x , y ) ∼ | x − y | 2[ φ ] ◮ [ φ ] = 3 − ǫ 4 � ∞ � x − y l l − 2[ φ ] u � dl ◮ propagator ∼ 0 l ◮ u finite range, smooth, and nonnegative in x and p � ∞ � x − y l l − 2[ φ ] u � dl ◮ unit cut-off C ( x − y ) = 1 l

◮ split C ( x − y ) = Γ( x − y ) + C L − 1 ( x − y ) with C L − 1 ( x − y ) = L − 2[ φ ] C ( L − 1 ( x − y )) and � L � x − y � dl l l − 2[ φ ] u Γ( x − y ) = l 1 ◮ convolution d µ C = d µ Γ ⋆ d µ C L − 1 � � Z = d µ C ( φ ) Z ( φ ) = d µ C L − 1 ( ψ ) d µ Γ ( ζ ) Z ( ψ + ζ ) � = d µ C ( φ ) ( RZ )( φ ) � d µ Γ ( ζ ) Z ( φ L − 1 + ζ ) and φ L − 1 ( x ) = L − [ φ ] φ ( L − 1 x ) ( RZ )( φ ) =

◮ split C ( x − y ) = Γ( x − y ) + C L − 1 ( x − y ) with C L − 1 ( x − y ) = L − 2[ φ ] C ( L − 1 ( x − y )) and � L � x − y � dl l l − 2[ φ ] u Γ( x − y ) = l 1 ◮ convolution d µ C = d µ Γ ⋆ d µ C L − 1 � � Z = d µ C ( φ ) Z ( φ ) = d µ C L − 1 ( ψ ) d µ Γ ( ζ ) Z ( ψ + ζ ) � = d µ C ( φ ) ( RZ )( φ ) � d µ Γ ( ζ ) Z ( φ L − 1 + ζ ) and φ L − 1 ( x ) = L − [ φ ] φ ( L − 1 x ) ( RZ )( φ ) = RG map: Z − → RZ

4. Good Infinite-Volume Coordinates: Brydges et al. Z 3 ⊂ R 3 = ⇒ cell decomposition X polymer

In finite box Λ Z (Λ , φ ) = � � ∞ � 1 � � dx { g : φ 4 ( x ) : C + µ : φ 2 ( x ) : C } exp − n ! Λ \ ( ∪ X i ) n =0 X 1 ,..., Xn disjoint in Λ × K ( X 1 , φ | X 1 ) · · · K ( X n , φ | X n ) ◮ Z ← → ( g , µ, K ) ◮ K = ( K ( X , · )) X polymer collection of local functionals

In finite box Λ Z (Λ , φ ) = � � ∞ � 1 � � dx { g : φ 4 ( x ) : C + µ : φ 2 ( x ) : C } exp − n ! Λ \ ( ∪ X i ) n =0 X 1 ,..., Xn disjoint in Λ × K ( X 1 , φ | X 1 ) · · · K ( X n , φ | X n ) ◮ Z ← → ( g , µ, K ) ◮ K = ( K ( X , · )) X polymer collection of local functionals ◮ need to extract second order perturbation theory: K ( X , φ ) = g 2 [ explicit complicated formula ] e − V ( X ,φ ) + R ( X , φ ) ◮ R of order g 3

In finite box Λ Z (Λ , φ ) = � � ∞ � 1 � � dx { g : φ 4 ( x ) : C + µ : φ 2 ( x ) : C } exp − n ! Λ \ ( ∪ X i ) n =0 X 1 ,..., Xn disjoint in Λ × K ( X 1 , φ | X 1 ) · · · K ( X n , φ | X n ) ◮ Z ← → ( g , µ, K ) ◮ K = ( K ( X , · )) X polymer collection of local functionals ◮ need to extract second order perturbation theory: K ( X , φ ) = g 2 [ explicit complicated formula ] e − V ( X ,φ ) + R ( X , φ ) ◮ R of order g 3 → ( g ′ , µ ′ , R ′ ) ◮ RG map: ( g , µ, R ) −