Hopf algebras and renormalization in physics Alessandra Frabetti

Alessandra Frabetti

Description

Hopf algebras and renormalization in physics Alessandra Frabetti Banff, september 1st, 2004 Contents Renormalization Hopf algebras 2 Matter and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Standard Model and Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Quantum = quantization of classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 From classical to quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Free theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Interacting theories .

• path integrals


• quantum self-adjoint operators


• e? particles


• residual strong


• hopf algebras


• e?


• dyson renormalization

Sujets

• Path integral

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 29
• Langue: English

Extrait

Hopf algebras and renormalization in physics
Alessandra Frabetti
Banff, september 1st, 2004
Contents
Renormalization Hopf algebras 2
Matter and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Standard Model and Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Quantum = quantization of classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
From classical to quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Free theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Interacting theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Divergent Feynamn integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Dyson renormalization formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Groups of formal diffeomorphisms and invertible series . . . . . . . . . . . . . . . . . . . . . . . . . . 11
BPHZ renormalization formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Hopf algebras of formal diffeomorphisms and invertible series . . . . . . . . . . . . . . . . . . . . . . . 13
Hopf algebra on Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Hopf algebra on rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Alternative algebras for QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Hopf algebras on planar binary rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1Feed back in mathematics and in physics 18
Combinatorial Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Relation with operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Combinatorial groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Non-commutative Hopf algebras and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Physics Matter Forces Interactions
(particles) (fields)
classical: • galaxies • gravitational macro:
−40
• planets, stars (acts on mass, int. 10 ) position and velocity
certainty
• cosmic rays • weak
−5
(acts on “flavour”, int. 10 )
• molecule = group of atomes • residual electromagnetic
−
+
e F
em
N
(chemical link = exchange of
-
q t
J
electrons)
J^
T F
tot
• atom = kernel + electrons • electromagnetic
−2
(acts on electric charge, int. 10 )
• X rays
quantum: • kernel = group of nucleons • residual strong micro:
position or velocity
uncertainty • γ rays
−
W
−
n−→p+e +ν¯
e
•nucleon=groupof3quarks • strong
particles= fields of typeu,d (acts on “colour”, int. 1)
(protonp =uud, neutron n =udd)
• quark (never saw isolated) • strong force confinementParticles Fermions Bosons Feynman graphs

−
γ
e μ τ
− − − −
elementary • leptons : • photonγ (QED) e +e −→e +e
ν ν ν
e μ τ
particles
(mass?, charge, 3 flavours)
= quantum
fields
−
W
+ − 0 − −
• W , W , Z (EW) μ −→ν +e +ν¯
μ e

g
u c t
¯
• quarks : • gluong ? (QCD) u−→d+u+d
d s b
(mass, charge, 3 flavours,
3 colours = red, blu, green) • graviton ?
+ − 0
• HiggsH , H , H ?
(spontaneous symmetry break)
g
++ +
hadrons = • baryons (3 quarks) • mesons (2 quarks) Δ −→p+π
++ + −
¯
groups of quarks p = uud, n = udd, Δ = π =ud, π =u¯d...
uuu...






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

∂L ∂L
LagrangianL(ϕ,∂ ϕ) =L +L
=⇒ Euler equation −∂ = 0 =⇒
μ free int μ
∂ϕ ∂(∂ ϕ)
μ
Z

3
1 d p
i(px−ω t) ∗ −i(px−ω t)
p p
ϕ (t,x) = p a e +a e (wave)
0 p
p
3
(2π)
2E
p
L = 0 =⇒
int ∗
classical: a ,a = numbers∈R orC
p
p
∗
quantum: a ,a = annihilation and creation operators
p
p
Z
μ μ μ μ μ 4 μ
ϕ(x ) =ϕ (x )+ G (x −y )j(y )d y
0 0
L =j ϕ
int
=⇒ μ μ
classical: G (x −y ) = Green function (resolvant)
0
j = source field
μ μ μ μ
quantum: G(x −y ) =G (x −y ) !
0
classical: perturbative solutions ing
quantum: perturbative series in g indexed by Feynman graphs
k
L =g ϕ
int
=⇒ X XX X
μ μ μ μ n n |Γ|
g = coupling constant
μ μ μ μ
G(x−y ) = G (x−y )g = U (Γ)g = U (Γ)g
n x−y x−y
n≥0 n
|Γ|=n Γ
μ μ
U (Γ) = amplitude (integral) of Feynman graph Γ withn loops
x −yFree theories
Field Free Lagrangian Euler equation Green function
μ
on momentum p
μ 1 2 1 2 2 i
φ(x )∈C L = ∂ φ − m φ Klein-Gordon: G (p) = ∈C
KG μ 0 2 2
2 2 p−m +i
2
∂
2 2
boson (spin 0, mass m) ( −∇ +m )φ = 0
2
∂t
i
μ 4 μ
¯
ψ(x )∈C L =ψ(iγ ∂ −m)ψ Dirac: S (p) = ∈M (C)
Dir μ 0 μ 4
γ p −m+i
μ
μ
1
fermion (spin , mass m) (iγ ∂ −m)ψ = 0
μ
2
μ
γ = 4×4 Dirac matrices
−ig p p
1 μν λ−1 μ ν
μ ν 4 μ ν ν μ 2
A (x )∈C L =− (∂ A −∂ A ) Maxwell: D (p) = +i
Max 0 2 2 2
4 p +i λ (p +i)
λ
μ 2 μ ν ν μ
boson (spin 1, mass 0) − (∂ A ) ∂ (∂ A −∂ A )
ν μ
2
ν μ
+λ∂ ∂ A = 0 ∈M (C)
μ 4
λ = mass parameterInteracting theories
Interacting Lagrangian Feynman graphs Green function
theory = series
X
4 1 4 |Γ|
φ L (φ)− gφ G(p) = U (Γ) g
KG p
4!
4
Γ∈φ
X
e 2|Γ|
S(p) = U (Γ) e
p
Γ fermion
μ μ
¯
QED L (ψ)+L (A )−eψγ ψA
Dir Max μ
X
γ 2|Γ|
= abelianGauge D (p) = U (Γ) e
μν
p
Γ boson
3
More: φ , scalar QED, QCD = non-abelian Gauge, Yukawa, ...
Feynman graph = graph Γwith - arrows depending on the fields
- valence of vertices depending on the interaction term
Feynman amplitude = integral U(Γ) computed from the graphDivergent Feynman integrals
1PI Feynman graph = graph Γ without bridges
TheFeynmanamplitudeismultiplicativewithrespecttojunctionofgraphs
Feynman rule:
!
R
4
e ν μ d k
1PI graphs: =⇒ U =S (p) (−ie) D (k)γ S (p−k)γ (−ie) S (p)
0 0,μν 0 4 0
p
(2π)
!

e e −1 e
connected graphs: =⇒ U =U S (p) U
0
p p p
U(Γ) =∞ ! =⇒ find finiteR(Γ)
Problems:
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 !
=⇒ find g ,e ,m from the effective: g,e,m
0 0 0
call them bare: g ,e ,m ,...
0 0 0
6Dyson renormalization formulas
X X X
|Γ|
n
G(p;g ) = U (Γ) g = G (p) g with G (p) = U (Γ)
0 p n n p
0
0
Bare
n
Γ |Γ|=n
Green functions:
2
e
2 2
0
same for S(p;e ) and D (p;e ) ⇒ fine structure constant α =
μν 0
0 0
4π
X X X
|Γ| n
¯ ¯ ¯
G(p;g)= R (Γ) g = G (p) g with G (p) = R (Γ)
p n n p
Renorma