Hopf algebras and renormalization in physics

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1 Hopf algebras and renormalization in physics Alessandra Frabetti Banff, september 1st, 2004 Contents Renormalization Hopf algebras 2 Matter and forces Standard Model and Feynman graphs Quantum = quantization of classical From classical to quantum Free theories Interacting theories Divergent Feynamn integrals Dyson renormalization formulas Groups of formal diffeomorphisms and invertible series BPHZ renormalization formula Hopf algebras of formal diffeomorphisms and invertible series Hopf algebra on Feynman graphs Hopf algebra on rooted trees Alternative algebras for QED Hopf algebras on planar binary rooted trees

2 Feed back in mathematics and in physics 18 Combinatorial Hopf algebras Relation with operads Combinatorial groups Non-commutative Hopf algebras and groups

3 Physics Matter Forces Interactions (particles) (fields) classical: galaxies gravitational macro: planets, stars (acts on mass, int ) position and velocity certainty cosmic rays weak (acts on flavour, int ) molecule = group of atomes residual electromagnetic (chemical link = exchange of atom = kernel + electrons electromagnetic X rays electrons) (acts on electric charge, int ) e F em T F tot N+ quantum: kernel = group of nucleons residual strong micro: position or velocity uncertainty γ rays n W p + e + ν e nucleon = group of 3 quarks strong particles = fields of type u, d (acts on colour, int. 1) (proton p = uud, neutron n = udd) quark (never saw isolated) strong force confinement

4 Particles Fermions Bosons Feynman graphs elementary leptons : particles = quantum fields ( e ν e ) ( µ ν µ ) ( τ ν τ ) (mass?, charge, 3 flavours) photon γ (QED) W +, W, Z 0 (EW) e +e γ e +e µ W ν µ +e + ν e quarks : ( u ) ( c ) ( t d s b) gluon g? (QCD) u g d + u + d (mass, charge, 3 flavours, 3 colours = red, blu, green) graviton? Higgs H +, H, H 0? (spontaneous symmetry break) hadrons = baryons (3 quarks) groups of quarks p = uud, n = udd, ++ = uuu... mesons (2 quarks) π + = u d, π = ūd g p + π +

5 Quantum = (canonical + path integrals) quantization of classical Fields Observables Measures classical functionals F of field ϕ values F(ϕ) R quantum self-adjoint operators O on states v Hilbert expectation value v t Ov R enough: G(x µ y µ ) = probability from x µ to y µ where x µ Minkowski = R 4 with metric ( 1, 1, 1, 1), µ = 0, 1, 2, 3

6 From classical to quantum Lagrangian L(ϕ, µ ϕ) = L free + L int = Euler equation L ( ) L ϕ µ = 0 = ( µ ϕ) ϕ 0 (t, x) = 1 ) (a p e i(px ωpt) + a p e i(px ω pt) d 3 p 2Ep (2π) 3 (wave) L int = 0 = classical: a p, a p = numbers R or C quantum: a p, a p = annihilation and creation operators ϕ(x µ ) = ϕ 0 (x µ ) + G 0 (x µ y µ )j(y µ )d 4 y µ L int = j ϕ j = source field = classical: G 0 (x µ y µ ) = Green function (resolvant) quantum: G(x µ y µ ) = G 0 (x µ y µ )! classical: perturbative solutions in g L int = g ϕ k g = coupling constant = quantum: perturbative series in g indexed by Feynman graphs G(x µ y µ ) = n 0 G n (x µ y µ ) g n = n Γ =nu xµ y µ(γ) gn = Γ U xµ y µ(γ) g Γ U xµ y µ(γ) = amplitude (integral) of Feynman graph Γ with n loops

7 Free theories Field Free Lagrangian Euler equation Green function on momentum p µ φ(x µ ) C L KG = 1 2 µφ m2 φ 2 Klein-Gordon: i G 0 (p) = p 2 m 2 +iɛ C boson (spin 0, mass m) ( m 2 )φ = 0 t 2 ψ(x µ ) C 4 L Dir = ψ(iγ µ µ m)ψ Dirac: i S 0 (p) = γ µ p µ m+iɛ M 4(C) fermion (spin 1, mass m) 2 (iγµ µ m)ψ = 0 γ µ = 4 4 Dirac matrices A µ (x ν ) C 4 L Max = 1 4 ( µ A ν ν A µ ) 2 Maxwell: D 0 (p) = ig µν boson (spin 1, mass 0) λ 2 ( νa µ ) 2 µ ( µ A ν ν A µ ) +λ µ ν A µ = 0 λ = mass parameter p 2 +iɛ + iλ 1 p µ p ν λ (p 2 +iɛ) 2 M 4 (C)

8 Interacting theories Interacting Lagrangian Feynman graphs Green function theory = series φ 4 L KG (φ) 1 4! g φ4 G(p) = Γ φ 4 U p (Γ) g Γ S(p) = Γ fermion QED L Dir (ψ) + L Max (A µ ) e ψγ µ ψa µ = abelian Gauge D µν (p) = Γ boson Up e (Γ) e2 Γ Up γ (Γ) e2 Γ More: φ 3, scalar QED, QCD = non-abelian Gauge, Yukawa,... Feynman graph = graph Γ with - arrows depending on the fields - valence of vertices depending on the interaction term Feynman amplitude = integral U(Γ) computed from the graph

9 Divergent Feynman integrals 1PI Feynman graph = graph Γ without bridges Feynman rule: The Feynman amplitude is multiplicative with respect to junction of graphs 1PI graphs: = U e p ( ) = S 0 (p) ( ie) D 0,µν (k)γ ν S 0 (p k)γ µ d4 k (2π) 4 ( ie) S 0 (p) ( ) ( ) ( ) connected graphs: = U e p = U e p S 0 (p) 1 U e p Problems: U(Γ) =! = find finite R(Γ) In particular: - each cycle in Γ gives a divergent integral, - each cycle in a cycle gives a subdivergency in a divergency. g, e, m,... measured values! call them bare: g 0, e 0, m 0,... = find g 0, e 0, m 0 from the effective: g, e, m

10 Dyson renormalization formulas Bare Green functions: Renormalized Green functions: G(p; g 0 ) = Γ U p (Γ) g Γ 0 = n G n (p) g n 0 with G n (p) = Γ =n U p (Γ) same for S(p; e 2 0) and D µν (p; e 2 0) fine structure constant α 0 = e2 0 4π Ḡ(p; g) = Γ R p (Γ) g Γ = n Ḡ n (p) g n with Ḡ n (p) = Γ =n R p (Γ) same for S(p; e 2 ) and D µν (p; e 2 ) α = e2 4π Dyson and Ward: Ḡ(g) = G(g 0 ) Z 1/2 (g) with g 0 (g) = g Z 1 (g) S(α) = S(α 0 ) Z 1 2 (α) with α 0 (α) = α Z 1 D µν T (α) = DT µν (α 0) Z3 1 (α) 3 (α) Renormalization factors: Z = 1 + O(g 2 ), Z 3 = 1 O(α), Z 2 = 1 + O(α) Coupling constants: g 0 = g + go(g 2 ), α 0 = α αo(α) Renormalization group = coupling constants renormalization factors acts on Green functions: G(p; g 0 ) Ḡ(p; g)

11 Groups of formal diffeomorphisms and invertible series Group of invertible series with product: G inv (A) = { f(x) = 1 + } f n x n, f n A n=1 abelian A commutative A = C for Φ 3, Φ 4, A = M 4 (C) for QED Group of diffeomorphisms with composition: G dif = { ϕ(x) = x + ϕ n x n+1, n=1 ϕ n C } Right action by composition: G inv G dif G inv (f, ϕ) f(ϕ) = Semi-direct product: G dif G inv := G ren Renormalization action: G inv G ren G inv, f(x) (ϕ(x), g(x)) f ren (x) = f(ϕ(x)) g(x) For bosons (φ, A µ ): G ren = { } (ϕ(x), ϕ(x) x ) Gdif G inv = G dif = f ren (x) = f(ϕ(x)) ϕ(x) x QFT subgroups: G graphs f n = Γ f(γ) G graphs G for QED also: G trees f n = t f(t) with f(t) = Γ f(γ) G graphs G trees G

12 BPHZ renormalization formula To use Dyson formulas, need renormalization Z factors explicitely. Bogoliubov, Parasiuk, Hepp, Zimmermann: R p (Γ) = U p (Γ) + C p (Γ) + C p (Γ) = T deg(γ) fixed p U p (Γ) + 1PI γ 1,...,γ l Γ γ i γ j = 1PI γ 1,...,γ l Γ γ i γ j = U p (Γ/γ 1...γ l ) C p1 (γ 1 ) C pl (γ l ), U p (Γ/γ 1...γ l ) C p1 (γ 1 ) C pl (γ l ). Here, the 1PI subgraphs γ s contain all the cycles which give subdivergencies. Then: Z(g) = C(Γ) g Γ and Z 2 (e 2 ) = C e (Γ) e 2 Γ, 1PI Γ 1PI Γ Z 3 (e 2 ) = 1PI Γ C γ (Γ) e 2 Γ. = Dyson global formulas not enough, need computations on Feynman graphs!

13 Hopf algebras of formal diffeomorphisms and invertible series Toy model A = C : group G G = Hom Alg (C(G), C) coordinate ring C(G) := Fun(G, C) = commutative Hopf algebra Hopf algebra of invertible series: C(G inv ) = C[b 1, b 2,...] n inv b n = b n m b m m=0 b n (f) = f n = 1 d n f(0) n! dx n inv (b n ), f g = b n (f g) Hopf algebra of diffeomorphisms (Faà di Bruno): C(G dif ) = C[a 1, a 2,...] n dif a n = a n m polynomial m=0 a n (ϕ) = ϕ n = 1 d n+1 ϕ(0) (n + 1)! dx n+1 dif (a n ), ϕ ψ = a n (ϕ ψ) Right coaction: δ : C(G inv ) C(G inv ) C(G dif ) δ(b n ), f ϕ = b n (f ϕ) = Semi-direct coproduct: H ren := C(G dif ) C(G inv ) ren (a m b n ) = dif (a m ) [ (δ Id) inv (b n ) ] Renormalization coaction: δ ren : C(G inv ) C(G inv ) H ren, δ ren b n = (δ Id) inv (b n ) For bosons: H ren = C(G dif ) and δ ren b n = dif b n!!!

14 Hopf algebra on Feynman graphs Question: Since G ren graphs Gren = H ren H ren graphs := C[1PI Γ] via a n, b n can we define the renormalization coproduct on each Γ? Γ =n Γ, Theorem. [Connes-Kreimer] For the scalar theory φ 3 : 1) H CK = C[1PI Γ] is a commutative and connected graded Hopf algebra, with coproduct CK Γ = Γ 1 + Γ/(γ 1...γ l ) (γ 1 γ l ) + 1 Γ. 1PI γ 1,...,γ l Γ γ i γ j = 2) The BPHZ formula is equivalent to the coproduct CK : R(Γ) = U C, CK Γ. 3) The group of characters G CK := Hom Alg (H CK, C) is the renormalization group. Example: CK ( ) = Conclusions: 1) Feynman graphs = natural local coordinates in QFT (= basis for the algebra of functions) 2) By Feynman rules: connected graph = junction of 1PI graphs junction = disjoint union = free product 3) Green functions = characters of the algebra of connected Feynman graphs.

15 Hopf algebra on rooted trees Hierarchy of divergences: 1PI Feynman graphs Rooted trees decorated with simple divergencies Problem for overlapping divergences : need the difference of trees forests Theorem. [Kreimer] For the scalar theory φ 3 : 1) H R = C[T rooted trees] is a commutative and connected graded Hopf algebra, with coproduct T = T 1 + what remains of T branches of T + 1 T. admissible cuts 2) The BPHZ formula is equivalent to the coproduct K : R(T) = U C, K T.

16 Alternative algebras for QED For QED, f n and f(γ) A = M 4 (C) non-commutative: Fun(G inv (A), C) C[b n ] or C[1PI Γ]! 1) Matrix elements: Basis Γ ij for the matrix elements f(γ ij ) := (f(γ)) ij. Then Fun(G inv (A), C) = C[1PI Γ ij ], but junction free product! 2) Non-commutative characters: Green functions = characters with values in A : G p = Hom Alg (C 1PI Γ, A). Then C 1PI Γ is an algebra, but not necessarily Hopf. H inv := Fun(G inv (A), A) = C b n, n N is a Hopf algebra with Simple Lemma. inv b n = b n b n + n 1 m=1 b n m b m, and G inv (A) Hom Alg (H inv, A). Moreover H inv ab = C(G inv ). Then H ren := C(G dif ) H inv = non-comm. Hopf algebra electron renormalization with ren, δ ren Question: Since G ren trees G ren = H ren H ren trees := C[t Y ] via a n, b n t =n t, can we define the renormalization coproduct on each QED tree?

17 Hopf algebras on planar binary rooted trees Theorem. [Brouder-F.] 1) Charge renormalization: commutative Hopf H α = C[ t, t Y ] with coproduct α t = what remains of t \-branches of t. 2) Electron renormalization: coaction e := (δ e Id) inv e H e = C Y /(1 of H qed on H e, where ) is non-commutative Hopf with inv e dual to product under t\s = t δ e : H e H e H α coaction extended from δ t = α t t ; renormalization group: H qed := H α H e with qed = α e. s / ; 3) Photon renormalization: coaction γ := m 3 23 (δγ σ) inv γ of H α on H γ, where H γ = C Y /(1 t ) is non-commutative Hopf with inv \ γ dual to product over t/s = s ; δ γ : H γ H γ H α coaction extended from δ; H α H γ H α induced by the 1-cocycle σ : H γ H α σ(t 1...t n ) = t 1 / /t n. Remark: γ α on single trees t Y!

18 Feed back in mathematics 1) Combinatorial Hopf algebras: Foissy, Holtkamp, Brouder, F., Krattenthaler, Loday, Ronco, Grossmann, Larson, Hoffman, Painate... 2) Relation with operads: Chapoton, Livernet, Foissy, Holtkamp, van der Laan, Loday, Ronco,... 3) Combinatorial groups: invent group law on tree-expanded series of the form f(x) = t Y f(t) xt. Interesting composition! 4) Non-commutative Hopf algebras and groups: look for new duality between groups and non-commutative Hopf algebras. Need a new coproduct with values in the free product of algebras. Bergman, Hausknecht, Fresse, F., Holtkamp,... Feed back in physics 1) More computations and developpements: Kreimer, Broadhurst, Delbourgo, Ebrahimi-Fard, Bierenbaum,... 2) Hopf algebras everywhere: Connes, Kreimer s school, Pinter et al., Brouder, Fauser, F., Oeckl, Schmitt in QFT; Patras and Cassam-Chenai in quantum chemistry...

19 1) Combinatorial Hopf algebras [Foissy, Holtkamp] H PRD = C T planar rooted decorated trees T = what remains of T branches of T admissible cuts Hopf algebra non-commutative version of H R [Brouder-F.] H dif = C a n, n N n dif a n = a n m m=0 m ( n k) k=1 m 1 + +m k =m m 1 >0,...,m k >0 a m1 a mk Hopf algebra non-commutative version of C(G dif ) n 1 [F.-Krattenthaler] S dif a n = ( 1) k+1 k=0 n 1 + +n k+1 =n n 1,...,n k+1 >0 m 1 + +m k =k m 1 + +m h h h=1,...,k 1 ( n1 +1 m 1 ) ( nk +1 m k ) an1 a nk a nk+1 explicit non-commutative antipode [Brouder-F.] H α extends naturally to Hα := C t, t Y Hopf algebra analogue to H dif non-commutative version of H α

20 2) Relation with operads [Chapoton-Livernet] L := Prim((H R ) ) = Lie algebra from the free pre-lie algebra on one generator [Foissy, Holtkamp, Patricia?] H α = ( H α ) = H LR related to the Loday-Ronco Hopf algebra = free dendriform Hopf algebra [van der Laan] P operad = S( P(n) Sn ) commutative Hopf algebra P non-σ operad = T( P(n)) Hopf algebra operadic version [van der Laan] F operad of Feynman graphs S( F(n) Sn ) = H CK

21 3) Combinatorial groups Tree-expanded series: formal symbols x t, for any tree t Y. { Invertible series G e := f(x) = } f(t) x t, f( ) = 1 for the electron: t Y { Invertible series G γ = f(x) = } f(t) x t, f( ) = 1 for the photon: t Y { Diffemorphisms G α = ϕ(x) = } ϕ(t)x \t, ϕ( ) = 1 for the charge: t Y group with the product under f(x)\g(x) := f(t) g(s) x t\s t,s Y group with the product over f(x)/g(x) := f(t) g(s) x t/s t,s Y group with the composition law (ϕ ψ)(x) := ϕ(ψ(x)) Composition: ψ(x) t := µ t (ψ(x)) ψ(x) in each vertex of t where µ t is the monomial which describes t as a sequence of over and under products of For instance: = ( \ )/ hence µ (s) = (s\s)/s Theorem. [F.] 1) The sets G e, G γ and G α form non-abelian groups. 2) QED renormalization at tree-level: G α = HomAlg (H α, C) and G α G e = HomAlg (H qed, A) 3) The order map : Y N induces group projections G α G dif trees Gdif, G γ G inv trees Ginv, G e G inv trees Ginv.

22 4) Non-commutative Hopf algebras and groups Fact: G inv (A) still group if A non-commutative, and H inv = Fun(G inv (A), A) = C b n, n N non-commutative Hopf Question: which duality group G non-commutative Hopf H? Answer: replace coproduct : H H H with : H H H, where = free product, such that T(U V ) = T(U) T(V ). Call H = (H, ) and look for duality G = Hom Alg (H, A). Group of invertible series: Co-groups in associative algebras: [Bergman-Hausknecht,Fresse] inv : H inv H inv H inv, inv (b n ) = inv (b n ) well defined and co-associative! Moreover G inv (A) = Hom Alg (H inv, A). Group of diffeomorphisms: G dif (A) not a group, because not associative. dif : H dif H dif H dif, dif (a n) = dif (a n ) well defined but not co-associative! [Holtkamp] on trees.

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