Birkhoff type decompositions and the Baker Campbell Hausdorff recursion Kurusch Ebrahimi-Fard Institut des Hautes Études Scientifiques e-mail: kurusch@ihes.fr Vanderbilt, NCGOA May 14, 2006 D. Kreimer, L. Guo, J. M. Gracia-Bondía, J. Várilly and D. Manchon.
abstarct The algebraic-combinatorial structure of renormalization in perturbative quantum field theory has found a concise formulation in terms of Hopf algebras of Feynman graphs. The process of renormalization is captured by an algebraic Birkhoff decomposition of regularized Feynman characters discovered by Connes and Kreimer. Associative Rota Baxter algebras naturally provide a suitable general underpinning for such factorizations in terms of Atkinson s theorem and Spitzer s identity. This approach gives a different perspective on the original result due to Connes and Kreimer in the context of other renormalization schemes. It enabled us to establish a simple matrix calculus for renormalization. However, a general characterization of the algebraic structure underlying the notion of renormalization schemes needs to be explored in future work. We should underline that our results presented here go beyond the particular application in the context of perturbative renormalization as we presented a general factorization theorem for filtered algebras in terms of a recursion defined using the Baker Campbell Hausdorff formula for which we found a closed formula under suitable circumstances. Moreover, we showed its natural link to a classical result of Magnus known from matrix differential equations. references: KEF, L. Guo and D. Manchon, Birkhoff type decompositions and the Baker Campbell Hausdorff recursion, accepted for publication in Comm. in Math. Phys. [arxiv:math-ph/0602004] KEF and D. Kreimer, Hopf algebra approach to Feynman diagram calculations, J. Phys. A: Math. Gen., 38, R385-R406, 2005. [arxiv:hep-th/0510202]
Motivation Hopf algebra of Feynman graphs: Γ) = Γ 1 + 1 Γ + γ Γ γ Γ/γ is a unital, associative, commutative, coassociative, non-cocommutative, connected, graded Hopf algebra, H F = K n>0 H n).
Renormalization process Factorization problem Bogoliubov s preparation map: φ G A, A comm. unital algebra Γ φγ) := φγ) + φ γ)φγ/γ) γ Γ Counterterm and renormalized character: T : A A, φ Γ) = T φγ) ) φ + Γ) = id A T) φγ) ) φ ± Γ Γ ) = φ ± Γ )φ ± Γ ) Tx)Ty) + Txy) = T xty) + Tx)y ) Birkhoff decomposition in G A : unique for T 2 = T φγ) = φ 1 φ +Γ) φ = expz φ ) = exp T Z φ ) + id T )Z φ ) ) = exp T?) ) exp id T )?) )
Rota Baxter Algebras Recall: A Rota Baxter algebra, A, R), of weight θ K is a K- algebra with a K-linear map R : A A, that fulfills the weight θ Rota Baxter relation Rx)Ry) + θrxy) = R Rx)y + xry) ) for all x, y A. R := θid A R is Rota Baxter map of weight θ. RA) and RA) are subalgebras in A. double Rota Baxter product a R b := Ra)b + arb) θab R a R b ) = Ra)Rb)
Integration-by-parts I : A A, A commutative unital algebra I[f] I[g] = I [ I[f] g ] + I [ f I[g] ] f = 1 A + I[af] f = 1 A + n=0 I [ I[ I[I[a]a]a ]a ] = expi[a]) }{{} n-times I[a]) n =n! I [ I[ I[I[a]a]a a] ] }{{} n-times
If we weight the Integration-by-parts rule with θ K: Rx)Ry) = R Rx)y + xry) ) θr xy ) Y = 1 + RY a) Frank Spitzer: [1956] Spitzer s classical identity exp R log1 A θa) θ ) ) = 1 A + n>0 R R RRa)a)a )a ) }{{} n-times In the limit θ 0 we get back the former case since 1 θ log1 θa) = n>0 θ n 1an n θ 0 a
Multiple-zeta-values f s x) := 1/x s Sf s )c) := i=1 1/i + c) s The map S is a Rota Baxter operator of weight θ = 1. ζs 1,, s n ) := n 1 >n 2 > >n k 1 1 n s 1 1 n s k k = S f s1 S f s2 Sf sn ) )) 0) S log1 + f k t) ) 0) = i=1 1) i 1 t i ζki) i exp i=1 1) i 1 ζik)t i = 1 + i = 1 + n>0 n>0 t i S ) S SSf k )f k )f k )f k 0) }{{} n-times t i ζk,, k) }{{} n times
Non-commutative setting Y = AY, Y 0) = 1, Y = 1 + I[AY ] Y = 1 + n=1 I [ A I[A I[A I[A]] ] }{{} n-times expi[a]) Wilhelm Magnus: Magnus expansion for matrix diff. equation. Y = exp ΩA) ) ΩA) = m 0 B m m! [ Ω,[Ω, [Ω, A]] ], Ω0) = 0. }{{} m-times Generalization to non-commutative Rota Baxter algebra: I?? R Y = 1 + n=1 R a Ra Ra Ra)) ) }{{} n-times = exp R??))
Let A be a complete filtered associative algebra. Then the functions exp : A 1 1 A + A 1, expa) := n=0 a n log : 1 A + A 1 A 1, are well-defined and inverse of each other. n!, log1 A + a) := n=1 a) n n Example: Let A be a commutative algebra. M l n A) n n lower triangular matrices carries a natural decreasing filtration in terms of the number of zero lower subdiagonals. M l n A) 1: ideal of lower triangular matrices with zero main diagonal. M l n A) k>1: ideal of strictly lower triangular matrices with zero main diagonal and zero on the first k 1 subdiagonals. M l na) M l na) 1 M l na) k 1 M l na) k, k < n, M l n A) k M l n A) m M l n A) k+m. M n A) = 1 + M l n A) 1: group of unipotent triangular matrices
expx)expy) = exp x+y+bchx, y) ), Cx, y) := x+y+bchx, y) BCHx, y) = 1 1 [x, y]+ 2 12 [x,[x, y]] 1 12 [y,[x, y]] 1 [x,[y,[x, y]]]+ 24 Now let P : A A be any linear map preserving the filtration of A, PA n ) A n. We define P to be id A P. For a A 1, define χ : A 1 A 1 χa) = a BCH Pχa)), Pχa)) ) Theorem: Let A be a complete filtered associative or Lie) algebra with a linear, filtration preserving map P : A A. For any a A 1, we have expa) = exp Pχa)) ) exp Pχa)) ) Let P : A A be an idempotent linear filtration preserving map. Let A := PA) and A + := PA). Define A 1, := PA 1 ) and A 1,+ := PA 1 ). Then for any η 1 A + A 1 there are unique η exp ) A 1, η + exp ) A 1,+ such that η = η η +
χat) = t k 0 χ k) a)t k. For k = 0,1,2 we find χ 0) a) = a χ 1) a) = 1 2 [Pa), Pa)] = 1 [Pa), a] 2 χ 2) a) = 1 2 [Pχ1) a)), Pa)] 1 2 [Pa), Pχ 1) a))] 1 [Pa),[Pa), ] a] [ Pa),[Pa), a] ]) 12 = + 1 [ P[Pa), a]), Pa) ] + 1 Pa), P[Pa), a]) 4 4[ ] 1 [Pa),[Pa), ] a] [ Pa),[Pa), a] ]) 12 = 1 [ ] 1 [Pa),[Pa), ] [ ] ) P[Pa), a]), a + a] [Pa), a], a. 4 12 The factorization gives rise to a simpler recursion for the map χ, without the appearance of P. χu) := u + BCH Pχu)), u ).
Example: Let H be a connected graded Hopf algebra. Let A = HomH, K),, ǫ) denote its complete filtered associative algebra of linear maps containing the group G of characters respectively its corresponding Lie algebra g of derivations. H = H H +, π : H H. We have for all φ = expz) G, Z g the unique characters φ 1 := exp πχz)) ) and φ + := exp ) id H π) χz)) such that φ = expz) = exp π Z) + π + Z) ) }{{} =:π + = exp π χz)) ) exp π + χz)) ) = φ 1 φ +. From the factorization we derive a closed form for the BCH-recursion χz) = Z + BCH π Z) 1 2 BCH Z, Z 2π Z) ), Z )
Atkinson s factorization theorem Theorem: [Atkinson 1963] Let A be an associative, unital Rota Baxter algebra. Assume a A fix and X and Y to be solutions of the recursive equations X = 1 A RX a) Y = 1 A Ra Y ) } X1 A + θa)y = 1 A XY = 1 A RX a) Ra Y ) + RX a) Ra Y ) = 1 A RX a) Ra Y ) + R Xa Ra Y ) ) + R RX a) ay ) = 1 A R Xa 1 A Ra Y )) ) R 1 A RX a)) ay ) ) = 1 A RXaY ) RXaY ) = 1 A θxay Canonical Rota Baxter Factorization: 1 A + θa) = X 1 Y 1
Let A be a complete filtered algebra, P a linear filtration preserving map, s.t. P + P = θid A. Let u A 1 χ θ u) = u 1 θ BCH P χ θ u) ), P χ θ u) )). χ θ u) = u + 1 θ BCH P χ θ u) ) ), θu. Such that for all expθu) 1 A + θa 1 =, u A 1 expθu) = exp Pχ θ u)) ) exp Pχ θ u)) ) We call χ θ the BCH-recursion of weight θ K, or θ-bch-recursion. A, R) complete filtered Rota Baxter algebra. exp R χ θ log1a + θa) θ ))) = exp Rχ θ u)) ) = 1 A + R exp R χθ u) ) 1 A ) = 1 A R exp Rχ θ u))a ) = 1 A + n=1 1) n R R Ra)a)... a ) } {{ } n times χ θ id A for commutative algebras: Spitzer s classical identity.
Generalized Spitzer identity: A, R) Rota Baxter algebra. ))) log1a + θa) X = exp R χ θ X = 1 A RXa) θ Y = exp log1a + θa) R χ θ θ ))) Y = 1 A RaY ) X 1 Y 1 = 1 A + θa Ca, b) = a + b + BCHa, b) = n 0 H n a, b), H n a, b): homogenous of degree n with respect to b, H 0 a, b) = a. For n = 1 we have H 1 a, b) = ad a 1 A e ad ab). In the limit θ 0: χ θ a) = a + 1 θ BCH R χ θ a) ), θa ) χ 0 a) = adr χ 0 a) ) 1 A e adrχ 0a)) a) = 1 A + B n [adr χ 0 a) )] n ) a) n>0
Magnus: Y t) = 1 + I[aY ]t) Y t) = exp Ω[a]t) ) d dt Ω[a]t) = adω[a] e adω[a] 1 a)t). exp R )) θ 0 non com. exp R χ 0 a) Magnus com. ))) χ log1a +θa) θ θ θ 0, non com. com. θ 0 θ 0 com. θ 0 exp Ra) ) θ=0, com. exp R )) log1a +θa) θ cl. Spitzer
A matrix representation of the combinatorics of renormalization in pqft Let A, R) be a commutative Rota Baxter algebra with idempotent Rota Baxter map. H is the Hopf algebra of Feynman graphs. Theorem: HomH, A),, R) is a complete filtered Rota Baxter algebra with Rota Baxter operator Rφ) := R φ and filtration from H. We denote its unit by e A := η A ǫ. For an A-valued character φ = exp Z φ ) G A we have: 1. φ G A and φ 1 + G A are solutions to φ = e A R φ φ e A ) ) such that φ = φ 1 φ + φ 1 + = e A R φ e A ) φ 1 ) + 2. φ and φ + are algebra morphisms, i.e., A-valued characters in G A := e A + RA 1 ) and G + A := e A + RA 1 ), respectively, and φ = exp R χlog φ)) )) resp. φ + = exp R χlog φ)) )).
Coproduct matrix Let H be the Hopf algebra of Feynman graphs. Recall Γ) = Γ 1 + 1 Γ + γ Γ γ Γ/γ F H F Fix a list of graphs J F such that J is a right coideal. In fact, the elements in J are indexed by I = N or I = {1,, m} and ordered according to the grading in H The coproduct matrix with respect to J is the I I matrix MJ) with entries in H defined by : γ i ) = j I MJ) ij γ j. MJ) ij H is a lower triangular matrix with unit diagonal. For the cograph we have degγ j ) < γ i.
Reduced list of 1PI graphs: take the following simple set of electron self-energy graphs borrowed from QED J := { Γ 1 := 1, Γ 2 :=, Γ 3 :=, Γ 4 :=, Γ 5 :=, } 1 Γ 2 Γ 3 Γ 4 Γ 5 = MJ) ij ) Γ 1. Γ 5 = 1 0 0 0 0 Γ 2 1 0 0 0 Γ 3 Γ 2 1 0 0 Γ 4 Γ 3 Γ 2 1 0 Γ 5 Γ 2 Γ 2 2Γ 2 0 1 1 Γ 2 Γ 3 Γ 4 Γ 5, MJ) ij : Γ j Γ i /MJ) ij ) = 1 + + 2 + 1 Let f HomH, A) a linear map f := f MJ) ) = fmj) ij ) ) 1 j i J Ml J A) with f 1i = fγ i ) for all graphs Γ i, i = 1,..., J possibly infinite). The matrix f is in the algebra of lower triangular matrices of size J with entries in the commutative Rota Baxter algebra A.
Feynman rules character φ G A : Feynman rules matrix φ = 1 A 0 0 0 0 φγ 2 ) 1 A 0 0 0 φγ 3 ) φγ 2 ) 1 A 0 0 φγ 4 ) φγ 3 ) φγ 2 ) 1 A 0 φγ 5 ) φγ 2 )φγ 2 ) 2φΓ 2 ) 0 1 A Lie group ĜA := 1 + M l J A) 1 M l J A). For the Lie algebra L A of infinitesimal characters we see that Z L A applied to MJ) maps the unit diagonal and non-linear entries to zero Ẑ = 0 0 0 0 0 ZΓ 2 ) 0 0 0 0 ZΓ 3 ) ZΓ 2 ) 0 0 0 ZΓ 4 ) ZΓ 3 ) ZΓ 2 ) 0 0 ZΓ 5 ) 0 2ZΓ 2 ) 0 0
Rota Baxter structure on M l n A): Rα) := Rα ij ) ) 1 j i J. Representation: Rota Baxter homomorphism Ψ A,J : HomH, A), R ) M l J A),R), Ψ A,J [f] = f Ψ[f] : A J id A A H J id A f id J A A J m A id J A J Ψ[f]Γ i ) = f id J Γ) = Ψ[f g] = Ψ[f] Ψ[g] = f ĝ, i j=1 fγ ij ) Γ j A J. Ψ[Rf)] = RΨ[f]) = R f) M ij ) = J k=0 M ik M kj. Ψ J [f g]γ j ) = i = i f g)m ij ) γ i k fm ik )gm kj ) γ i = Ψ J [f]ψ J [g]γ j ),
Factorization of ĜA M l J A) into the subgroups: and Ĝ A 1 + R M l J A)1) M l J A) Ĝ + A 1 + R M l J A)1) M l J A), that is, for each φ := Ψ[φ] ĜA, φ G A there exist unique φ Ĝ A and φ + Ĝ+ A, such that: φ = φ 1 φ +. The factors are unique solutions of the equations φ = 1 R φ + = 1 R φ φ 1) ) φ + φ 1 1) The matrix entries can be calculated from: α := φ φ ) ij = Rα ij ) j i k=2 j i φ+ ) ij = Rα 1 ) ij ) i>l 1 >l 2 > >l k 1 >j k=2 i>l 1 >l 2 > >l k 1 >j, 1) k+1 R ) R Rα il1 )α l1 l 2 ) α lk 1 j ) 1) k+1 R R ) Rα 1 ) il1 )α 1 ) l1 l 2 ) α 1 ) lk 1 j
Bogoliubov s preparation map φ = exp RχẐφ) ) φ + = exp RχẐφ) ) Now observe φ + = 1 + R φ φ ) 1) φ = 1 R φ φ ) 1) We may therefore define the matrix φ := φ φ 1) such that we get Bogoliubov s matrix formulae for the counter term and renormalized Feynman rules matrix ) φ = 1 R φ and φ ) + = 1 + R φ
Recalling the double Rota Baxter product in this context a R b := arb) + Ra)b ab and the fact that Ra R b) = Ra)Rb) gives exp ))) R χ Ẑ φ = 1 + R exp R χẑφ) )) Remember that R is idempotent i.e. R1) = 0). This leads to the following matrix representation of Bogoliubov s preparation map φ := Ψ J [ φ] = Ψ J [φ φ e A )] = Ψ J [exp R χzφ ) ) ] = exp ) R χẑφ) THANK YOU!!