Hopf algebras and renormalization in physics
Alessandra Frabetti
Banff, september 1st, 2004
Contents
Renormalization Hopf algebras
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 . . . . . . . . .
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2
2
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Feed back in mathematics and in physics
Combinatorial Hopf algebras . . . . . . . . .
Relation with operads . . . . . . . . . . . .
Combinatorial groups . . . . . . . . . . . . .
Non-commutative Hopf algebras and groups
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18
19
20
21
22
Physics
classical:
Matter
(particles)
Forces
(fields)
• galaxies
• planets, stars
• gravitational
• cosmic rays
• weak
(acts on mass, int. 10
−40
Interactions
)
macro:
position and velocity
certainty
(acts on “flavour”, int. 10−5 )
• molecule = group of atomes • residual electromagnetic
(chemical link = exchange of
electrons)
• atom = kernel + electrons
• electromagnetic
e− Fem
q JJ
^
T Ftot
t
N+
(acts on electric charge, int. 10−2 )
• X rays
quantum:
• kernel = group of nucleons
uncertainty
• γ rays
• residual strong
micro:
position or velocity
W−
n −→ p + e− + ν̄e
• 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
Particles
• leptons :
e−
νe
Bosons
µ
νµ
τ
ντ
(mass?, charge, 3 flavours)
• photon γ (QED)
Feynman graphs
elementary
particles
Fermions
• W + , W − , Z 0 (EW)
u
d
c
s
t
b
3 colours = red, blu, green)
• gluon g ? (QCD)
• graviton ?
W−
µ− −→ νµ + e− + ν̄e
g
u −→ d + u + d¯
(mass, charge, 3 flavours,
• quarks :
= quantum
fields
γ
e− +e− −→ e− +e−
• Higgs H + , H − , H 0 ?
(spontaneous symmetry break)
hadrons =
• baryons (3 quarks)
• mesons (2 quarks)
++
¯ π − = ūd...
groups of quarks p = uud, n = udd, ∆
= π + = ud,
uuu...
g
∆++ −→ p + π +
Quantum = (canonical + path integrals) quantization of classical
Fields
Observables
classical
functionals F of field ϕ
quantum self-adjoint operators O
on states v ∈ Hilbert
Measures
values F (ϕ) ∈ R
expectation value v t Ov ∈ R
enough: G(xµ − y µ ) = probability from xµ to y µ
where xµ ∈ Minkowski = R4 with metric (−1, 1, 1, 1), µ = 0, 1, 2, 3
From classical to quantum
Lagrangian L(ϕ, ∂µϕ) = Lf ree + Lint
ϕ0(t, x) =
=⇒
Lint = 0
∂L
− ∂µ
Euler equation
∂ϕ
=⇒
Z
∂L
∂(∂µϕ)
d3 p
1 i(px−ωp t)
∗ −i(px−ωp t)
p
ap e
+ ap e
(2π)3
2Ep
=0
=⇒
(wave)
classical: ap , a∗p = numbers ∈ R or C
quantum: ap , a∗p = annihilation and creation operators
µ
µ
ϕ(x ) = ϕ0(x ) +
Lint = j ϕ
j = source field
=⇒
Z
G0 (xµ − y µ )j(y µ)d4 y µ
classical: G0 (xµ − y µ ) = Green function (resolvant)
quantum: G(xµ − y µ ) = G0 (xµ − y µ ) !
classical: perturbative solutions in g
k
Lint = g ϕ
=⇒
g = coupling constant
quantum: perturbative series in g indexed by Feynman graphs
G(xµ−y µ ) =
X
n≥0
Gn (xµ−y µ ) g n =
XX
n |Γ|=n
Uxµ−yµ (Γ) g n =
X
Uxµ−yµ (Γ) g |Γ|
Γ
Uxµ −yµ (Γ) = amplitude (integral) of Feynman graph Γ with n loops
Free theories
Field
Free Lagrangian
Euler equation
Green function
on momentum pµ
φ(xµ ) ∈ C
boson (spin 0, mass m)
LKG = 21 ∂µ φ2 − 12 m2 φ2
Klein-Gordon:
∂2
2
2
( ∂t
2 − ∇ + m )φ = 0
G0 (p) =
i
p2 −m2 +i
ψ(xµ ) ∈ C4
fermion (spin 21 , mass m)
LDir = ψ̄(iγ µ∂µ − m)ψ
Dirac:
(iγ µ∂µ − m)ψ = 0
S0 (p) =
i
γ µ pµ −m+i
∈C
∈ M4(C)
γ µ = 4 × 4 Dirac matrices
Aµ (xν ) ∈ C4
boson (spin 1, mass 0)
LM ax = − 41 (∂ µAν − ∂ ν Aµ )2
Maxwell:
D0 (p) =
− λ2 (∂ν Aµ )2 ∂µ (∂ µAν − ∂ ν Aµ )
+λ∂µ ∂ ν Aµ = 0
λ = mass parameter
−igµν
p2 +i
p p
µ ν
+ i λ−1
λ (p2 +i)2
∈ M4 (C)
Interacting theories
Interacting
theory
4
φ
Lagrangian
LKG (φ) −
1
4!
Feynman graphs
4
gφ
Green function
= series
G(p) =
X
Up (Γ) g |Γ|
Γ∈φ4
S(p) =
QED
µ
µ
Dµν (p) =
X
Γ boson
More: φ3 , 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
Upe (Γ) e2|Γ|
Γ fermion
LDir (ψ) + LM ax (A ) − e ψ̄γ ψAµ
= abelian Gauge
X
Upγ (Γ) e2|Γ|
Divergent Feynman integrals
1PI Feynman graph
Feynman rule:
1PI graphs:
= graph Γ without bridges
The Feynman amplitude is multiplicative with respect to junction of graphs
=⇒ Upe
!
!
connected graphs: =⇒ Upe
Problems:
U (Γ) = ∞ !
In particular:
= S0 (p) (−ie)
=⇒
= Upe
R
4
d k
D0,µν (k)γ ν S0(p − k)γ µ (2π)
4 (−ie) S0 (p)
S0 (p)−1 Upe
find finite R(Γ)
- each cycle in Γ gives a divergent integral,
- each cycle in a cycle gives a subdivergency in a divergency.
g, e, m, ... 6= measured values !
call them bare: g0 , e0, m0 , ...
=⇒
find g0 , e0, m0 from the effective: g, e, m
Dyson renormalization formulas
G(p; g0) =
Bare
Green functions:
X
|Γ|
Up(Γ) g0 =
X
Gn (p) g0n
with
n
Γ
Renormalized
Green functions:
X
Rp (Γ) g |Γ| =
X
Ḡn (p) g n
with
Ḡn (p) =
X
Ḡ(g) = G(g0 ) Z −1/2(g)



Renormalization factors:
α0 = α − αO(α)
Renormalization
group
coupling
constants
renormalization
factors
=
g0 (g) = g Z −1(g)
with
α0 (α) = α Z3−1(α)
Z3 = 1 − O(α),
g0 = g + gO(g 2 ),
n
α=
with
T
T
D̄µν
(α) = Dµν
(α0 ) Z3−1(α)
Coupling constants:
e20
4π
Rp (Γ)
⇒
S̄(α) = S(α0) Z2−1(α)
Z = 1 + O(g 2 ),
α0 =
|Γ|=n
same for S̄(p; e2) and D̄µν (p; e2)
Dyson and Ward:
Up(Γ)
⇒ fine structure constant
n
Γ
X
|Γ|=n
same for S(p; e20) and Dµν (p; e20)
Ḡ(p; g) =
Gn (p) =
e2
4π
Z2 = 1 + O(α)
acts on Green functions:
G(p; g0) 7→ Ḡ(p; g)
'
1
137
Groups of formal diffeomorphisms and invertible series
Group of invertible series
with product:
inv
G (A) =
Group of diffeomorphisms
with composition:
Right action
by composition:
Gdif =
QFT subgroups:
f (x) = 1 +
ren
G
Ggraphs
n
(
∞
X
ϕ(x) = x +
f (x)
×
(ϕ(x), g(x))
dif
∈G
X
−→
inv
nG
)
o
ϕn ∈ C
Gdif n Ginv := Gren
Ginv ,
7→
f ren(x) = f (ϕ(x)) · g(x)
∼
= Gdif
=⇒ f ren(x) = f (ϕ(x)) ·
f (Γ)
for QED also:
⇐⇒
fn =
X
t
f (t)
ϕ(x)
x
Ggraphs ⊂ G
Γ
Gtrees
abelian ⇔ A commutative
A = C for Φ3 , Φ4,
A = M4 (C) for QED
)
Semi-direct
product:
Gren
fn =
fn x , fn ∈ A
ϕn xn+1,
=⇒
×
ϕ(x)
x )
n
n=1
Ginv
= (ϕ(x),
⇐⇒
∞
X
n=1
Ginv × Gdif −→ Ginv
(f, ϕ) 7→ f (ϕ)
Renormalization action:
For bosons (φ, Aµ ):
(
with
f (t) =
X
Γ
f (Γ)
Ggraphs ⊂ Gtrees ⊂ G
BPHZ renormalization formula
To use Dyson formulas, need renormalization Z factors explicitely.
X
Rp (Γ) = Up (Γ) + Cp (Γ) +
Bogoliubov,
Parasiuk,
Hepp,
Zimmermann:
Cp (Γ) =

Up(Γ/γ1...γl ) Cp1 (γ1) · · · Cpl (γl ),
1PI γ1 ,...,γl ⊂Γ
γi ∩γj =∅
deg(Γ) 
−Tfixed p Up(Γ)
X
+
1PI γ1 ,...,γl ⊂Γ
γi ∩γj =∅


Up (Γ/γ1...γl ) Cp1 (γ1) · · · Cpl (γl ).
Here, the 1PI subgraphs γ’s contain all the cycles which give subdivergencies.
Then:
Z(g) =
X
1PI Γ
C(Γ) g
|Γ|
and
2
Z2 (e ) =
X
1PI Γ
e
2|Γ|
C (Γ) e
,
2
Z3 (e ) =
X
1PI Γ
=⇒ Dyson global formulas not enough, need computations on Feynman graphs!
C γ (Γ) e2|Γ| .
Hopf algebras of formal diffeomorphisms and invertible series
Toy model A = C :
group G
G∼
= HomAlg (C(G), C)
⇐⇒
coordinate ring C(G) := Fun(G, C)
= commutative Hopf algebra
C(Ginv ) ∼
= C[b1, b2, ...]
Hopf algebra
of invertible series:
inv
∆ bn =
n
X
1 dn f (0)
bn(f ) = fn =
n! dxn
bn−m ⊗ bm
h∆inv (bn), f × gi = bn (f · g)
m=0
Hopf algebra
of diffeomorphisms
(Faà di Bruno):
C(Gdif ) ∼
= C[a1 , a2 , ...]
dif
∆ an =
n
X
an (ϕ) = ϕn =
an−m ⊗ polynomial
1
dn+1ϕ(0)
(n + 1)! dxn+1
h∆dif (an ), ϕ × ψi = an (ϕ ◦ ψ)
m=0
Right
coaction:
δ : C(Ginv ) −→ C(Ginv ) ⊗ C(Gdif )
=⇒
hδ(bn ), f × ϕi = bn(f ◦ ϕ)
Renormalization coaction:
For bosons:
Hren ∼
= C(Gdif )
Semi-direct
coproduct:
δ ren : C(Ginv ) −→ C(Ginv ) ⊗ Hren,
and
δ ren bn = ∆dif bn
!!!
Hren := C(Gdif ) n C(Ginv )
∆ren(am ⊗ bn ) = ∆dif (am ) (δ ⊗ Id)∆inv (bn )
δ ren bn = (δ ⊗ Id)∆inv (bn)
Hopf algebra on Feynman graphs
Question:
Since
ren
Gren
graphs ,→ G
=⇒
ren
Hren −→ Hgraphs
:= C[1PI Γ]
via
an , bn
7→
X
Γ,
|Γ|=n
can we define the renormalization coproduct on each Γ?
Theorem. [Connes-Kreimer] For the scalar theory φ3 :
1) HCK = C[1PI Γ] is a commutative and connected graded Hopf algebra, with coproduct
X
∆CK Γ = Γ ⊗ 1 +
Γ/(γ1...γl ) ⊗ (γ1 · · · γl ) + 1 ⊗ Γ.
1PI γ1 ,...,γl ⊂Γ
γi ∩γj =∅
2) The BPHZ formula is equivalent to the coproduct ∆CK:
R(Γ) = hU ⊗ C, ∆CKΓi.
3) The group of characters GCK := HomAlg (HCK , C) is the renormalization group.
Example:
∆
CK
=
⊗1+
⊗
+2
Conclusions: 1) Feynman graphs = natural local coordinates in QFT
2) By Feynman rules:
⊗
+1⊗
(= basis for the algebra of functions)
connected graph = junction of 1PI graphs
junction = disjoint union = free product
3) Green functions = characters of the algebra of connected Feynman graphs.
Hopf algebra on rooted trees
Hierarchy
of divergences:
1PI Feynman graphs
−→ Rooted trees decorated with simple divergencies
x
Problem for overlapping divergences
7→
x
@
@
@
@
@x
x
: need the difference of trees
⇒ forests
Theorem. [Kreimer] For the scalar theory φ3 :
1) HR = C[T rooted trees] is a commutative and connected graded Hopf algebra, with coproduct
X
∆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 ) = hU ⊗ C, ∆KT i.
Alternative algebras for QED
For QED, fn and f (Γ) ∈ A = M4 (C) non-commutative:
Fun(Ginv (A), C) 6= C[bn] or C[1PI Γ]!
1) Matrix
elements:
Basis Γij for the matrix elements f (Γij ) := (f (Γ))ij .
Then Fun(Ginv (A), C) = C[1PI Γij ], but junction 6= free product!
2) Non-commutative
characters:
Green functions = “characters with values in A”: Gp ∼
= HomAlg (Ch1PI Γi, A).
Then Ch1PI Γi is an algebra, but not necessarily Hopf.
Hinv := Fun(Ginv (A), A) ∼
= Chbn, n ∈ Ni is a Hopf algebra with
Simple
Lemma.
inv
∆ bn = bn ⊗ 1 + 1 ⊗ bn +
Ginv (A) ∼ HomAlg (Hinv , A).
inv
Moreover Hab
= C(Ginv ).
Hren := C(Gdif ) n Hinv = non-comm. Hopf algebra
Question:
bn−m ⊗ bm ,
m=1
and
Then
n−1
X
Since
ren
Gren
trees ,→ G
=⇒
⇒ electron renormalization with
ren
Hren −→ Htrees
:= C[t ∈ Y ]
via
an , bn
7→
∆ren, δ ren
X
|t|=n
can we define the renormalization coproduct on each QED tree?
t,
Hopf algebras on planar binary rooted trees
Theorem. [Brouder-F.]
1) Charge renormalization: commutative Hopf
∆α
t
=
X
t
Hα = C[
t
“what remains of
, t ∈ Y ] with coproduct
” ⊗ “\-branches of t”.
∆e := (δ e ⊗ Id)∆inv
e
2) Electron renormalization: coaction
of Hqed on He , where
• He = ChY i/(1 − ) is non-commutative Hopf with ∆inv
e dual to product under
• δ e : He −→ He ⊗ Hα
• renormalization group:
coaction extended from
Hqed := Hα n He
3) Photon renormalization: coaction
δ
with
• H = ChY i/(1 − ) is non-commutative Hopf with
• Hα n Hγ
H α
= ∆α
t
−
⊗
t
;
∆qed = ∆α × ∆e .
∆γ := m323 (δ γ ⊗ σ)∆inv
γ
γ
• δ γ : Hγ −→ Hγ ⊗ Hα
t
s
/
t\s = t ;
∆inv
γ
of Hα on Hγ , where
dual to product over
t/s =
coaction extended from δ;
induced by the 1-cocycle
Remark: ∆γ ≡ ∆α on single trees t ∈ Y !
σ : Hγ −→ Hα
σ(t1 . . . tn ) = t1 / · · · /tn .
t\
s;
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,...
P
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...
1) Combinatorial Hopf algebras
[Foissy,
Holtkamp]
HPRD = ChT planar rooted decorated treesi
X
“what remains of T ” ⊗ “branches of T ”
∆T =
Hopf algebra
non-commutative version of HR
admissible
cuts
[Brouder-F.]
Hdif = Chan , n ∈ Ni
n
m
X
X
dif
∆ an =
an−m ⊗
m=0
[F.-Krattenthaler]
dif
S an =
n−1
X
k=0
[Brouder-F.]
α
k=1
(−1)
k+1
n
k
X
am1 · · · amk
m1 +···+mk =m
m1 >0,...,mk >0
X
X
fα := Ch
H
t
n1 +1
m1
n1 +···+nk+1 =n m1 +···+mk =k
n1 ,...,nk+1 >0 m1 +···+mh ≥h
h=1,...,k−1
H extends naturally to
Hopf algebra
non-commutative version of C(Gdif )
,t ∈ Yi
···
nk +1
mk
an1 · · · ank ank+1
explicit
non-commutative
antipode
Hopf algebra analogue to Hdif
non-commutative version of Hα
2) Relation with operads
[Chapoton-Livernet]
[Foissy, Holtkamp,
Patricia?]
L := Prim((HR)∗ )
fα ∼
fα )∗ ∼
H
= (H
= HLR
P operad
[van der Laan]
[van der Laan]
=⇒ Lie algebra from the free pre-Lie algebra on one generator
related to the Loday-Ronco Hopf algebra
=⇒ free dendriform Hopf algebra
=⇒ S(⊕P(n)Sn ) commutative Hopf algebra
P non-Σ operad
=⇒ T (⊕P(n)) Hopf algebra
F operad of Feynman graphs ⇒ S(⊕F (n)Sn ) = HCK
operadic version
3) Combinatorial groups
Tree-expanded series:
Invertible series
for the electron:
formal symbols xt , for any tree t ∈ Y .
)
(
group with the
X
Xproduct under
t
e
f (t) x , f ( ) = 1
G := f (x) =
f (x)\g(x) :=
f (t) g(s) xt\s
t∈Y
Invertible series
for the photon:
γ
G =
(
(
t,s∈Y
X
f (x) =
f (t) xt , f ( ) = 1
t∈Y
X
Gα = ϕ(x) =
ϕ(t)x
Diffemorphisms
for the charge:
\t
, ϕ( ) = 1
t∈Y
ψ(x)t := µt (ψ(x))
Composition:
= ( \ )/
hence
µ
group with the
Xproduct over
f (x)/g(x) :=
f (t) g(s) xt/s
t,s∈Y
)
group with the composition law
(ϕ ◦ ψ)(x) := ϕ(ψ(x))
where µt is the monomial which describes t as
a sequence of over and under products of
⇔ ψ(x) in each vertex of t
For instance:
)
(s) = (s\s)/s
Theorem. [F.]
1) The sets Ge , Gγ and Gα form non-abelian groups.
2) QED renormalization at tree-level:
3) The “order” map
Gα
:Y
N
Gdiftrees ⊂ Gdif ,
Gα ∼
= HomAlg (Hα , C)
and
Gα n Ge ∼
= HomAlg (Hqed, A)
induces group projections
Gγ
inv
Ginv
trees ⊂ G ,
Ge
inv
Ginv
trees ⊂ G .
4) Non-commutative Hopf algebras and groups
Fact:
Ginv (A) still group if A non-commutative, and
Hinv = Fun(Ginv (A), A) = Chbn , n ∈ Ni non-commutative Hopf
Question:
which duality
“ group G
Answer:
replace coproduct
←→ non-commutative Hopf H ”
∆ : H −→ H ⊗ H with
?
∆∗ : H −→ H ? H ,
? = free product, such that T (U ⊕ V ) ∼
= T (U ) ? T (V ).
Call H∗ = (H, ∆∗) and look for duality G ∼
= HomAlg (H∗, A).
where
⇒ Co-groups in associative algebras: [Bergman-Hausknecht,Fresse]
Group of
invertible series:
inv
∆inv
−→ Hinv ? Hinv ,
∗ : H
Moreover
inv
∆inv
∗ (bn ) = ∆ (bn ) well defined and co-associative!
Ginv (A) ∼
= HomAlg (H∗inv , A).
Gdif (A) not a group, because ◦ not associative.
Group of
diffeomorphisms:
dif
∆dif
−→ Hdif ? Hdif ,
∗ : H
⇒ [Holtkamp] on trees.
dif
∆dif
∗ (an ) = ∆ (an ) well defined but not co-associative!