Download Hopf algebraic structures of quantum field and many

Hopf algebraic structures of quantum
field and many-body theories (II)
Chris&an
Brouder
Ins&tut
de
Minéralogie
et
de
Physique
des
Milieux
Condensés,
Paris
Cargese
2009
Outline
•  Lecture
1:
From
Hopf
to
Feynman
diagrams
• 
• 
• 
• 
Coproduct
Hopf
algebra
Twisted
product
Feynman
diagrams
(with
proofs)
•  Lecture
2:
• 
• 
• 
• 
Cumulant
expansion
A
new
coproduct
for
connected
diagrams
Structure
of
Green
func&ons
Renormaliza&on
Cargese
2009
Hopf algebra definition
A
1
•  A
Hopf
algebra
is
an
algebra
with
unit
and
with
!
•  A
coproduct
,
,
is
∆ : A → A ⊗ A ∆a = a(1) ⊗ a(2)
an
algebra
morphism
∆(ab) = (∆a)(∆b)
•  A
counit
such
that
ε:A→C
!
!
ε(a(1) )a(2) = a(1) ε(a(2) ) = a
•  The
counit
is
an
algebra
morphism
ε(ab) = ε(a)ε(b)
•  An
an&pode
S : A → A
!
!
S(a(1) )a(2) = a(1) S(a(2) ) = ε(a)1
Cargèse
2009
Hopf algebra of fields
•  A
is
the
polynomial
algebra
generated
by
the
fields
at
dis&nct
points.
•  The
coproduct
is
!
"
n1
nk
∆ ϕ (x1 ) . . . ϕ (xk )
=
$ %
nk $ %
#
n1
nk
···
···
j1
jk
=0
j =0
n1
#
j1
k
ϕj1 (x1 ) · · · ϕjk (xk ) ⊗ ϕn1 −j1 (x1 ) · · · ϕnk −jk (xk )
•  The
counit
sa&sfies
! n
"
n
ε ϕ 1 (x1 ) . . . ϕ k (xk ) = δn1 ,0 . . . δnk ,0
•  The
an&pode
!
"
S ϕ (x1 ) . . . ϕ (xk ) = (−1)n1 +···+nk ϕn1 (x1 ) . . . ϕnk (xk )
n1
nk
Cargèse
2009
Laplace pairing
•  A
Laplace
pairing
is
a
map
(.|.) : A ⊗ A → C
!
•  (a|bc) =
(a(1) |b)(a(2) |c)
!
(ab|c) =
(a|c(1) )(b|c(2) )
!
"
ϕn (x)|1 = δn,0
•  In
par&cular
(a|1) = (1|a) = ε(a)
!
•  Wick's
theorem
a ◦ b = (a(1) |b(1) ) a(2) b(2)
•  Examples
ϕ(x) ◦ ϕ(y) = ϕ(x)ϕ(y) + g(x, y)
ϕ2 (x) ◦ ϕ(y) = ϕ2 (x)ϕ(y) + 2g(x, y) ϕ(x)
ϕ2 (x) ◦ ϕ2 (y) = ϕ2 (x)ϕ2 (y) + 4g(x, y) ϕ(x)ϕ(y) + 4g 2 (x, y)
Cargèse
2009
Twisted products
!
"
g(x, y) = !0|T ϕ(x)ϕ(y) |0"
•  If
(properly
regularized),
the
twisted
product
is
the
(commuta&ve)
chronological
(or
&me‐ordered)
product
! n "
•  Previous
iterated
&me‐ordered
product
T ϕ (x) = ϕn (x)
!
"
T ϕ (x1 ), . . . , ϕ (xk ) = ϕn1 (x1 ) ◦ · · · ◦ ϕnk (xk )
! 2
"
•  Example:
T ϕ (x), ϕ(y) = ϕ2 (x) ◦ ϕ(y)
n1
nk
T :A→A
•  New
defini&on
such
that
!
"
T (ab) = T (a) ◦ T (b)
T ϕ(x) = ϕ(x)
and
! 2
"
•  Example:
T ϕ (x)ϕ(y) = ϕ(x) ◦ ϕ(x) ◦ ϕ(y)
•  Feynman
integral
or
func&onal
deriva&ve
approaches
Cargèse
2009
Examples
! " # 3
$" %
3
2
"
•  For
0 T φ (x)φ (y)φ (z) "0
•  Previous
defini&on
g(x, x)
•  New
defini&on:
add
loops
,
etc.
+
+
+
+
+
Cargèse
2009
Exponential form
u∈A
•  A
primi%ve
element
is
an
element
such
that
∆u = u ⊗ 1 + 1 ⊗ u
•  The
fields
are
the
primi&ve
elements
ϕ(x)
g(x, y)
T :A→A
•  For
symmetric
,
define
by
T (ab) = T (a) ◦ T (b), T (u) = u
•  Does
not
work
for
the
operator
product
!
T (a) = t(a(1) )a(2)
T !
•  The
map
is
a
coregular
ac&on:
"
with
t(a) = ε T (a)
•  Exponen&al
form:
t(a) = e!τ (a)
Cargèse
2009
Convolution
σ
τ
A→C
•  If
and
are
linear
maps
,
their
convolu%on
is
σ " τ (a) =
!
σ(a(1) )τ (a(2) )
!
!
•  Our
coproduct
is
cocommuta&ve
a(2) ⊗ a(1) = a(1) ⊗ a(2)
•  The
convolu&on
product
is
associa&ve
and
commuta&ve
τ
•  The
convolu&on
powers
of
are
τ !0 (a) = ε(a)
τ !1 (a) = τ (a)
!
!(n+1)
τ
(a) =
τ (a(1) )τ !n (a(2) )
τ
•  The
convolu&on
exponen&al
of
is
∞
! 1
e (a) =
τ !n (a)
n!
n=0
!τ
Cargèse
2009
Convolution exponential
!
"
"
τ ϕ(x1 ) . . . ϕ(xk ) = 0 k != 2 τ ϕ(x1 )ϕ(x2 ) = g(x1 , x2 )
•  Let
if
,
!
•  Then
t(a) = !0|T (a)|0" = e!τ (a)
!
"
!
"
∗τ
•  Examples
!0|T ϕ(x)ϕ(y) |0" = g(x, y) = e ϕ(x)ϕ(y)
! 2
"
2
!0|T ϕ (x)ϕ (y) |0" = g(x, x)g(y, y) + 2g(x, y)2
! 2
"
∗τ
2
= e ϕ (x)ϕ (y)
!
•  Equa&on
is
equivalent
to
T (a) = e!τ (a(1) )a(2)
R
T (a) = e
!
δ
dxdy 12 g(x,y) δϕ(x)δϕ(y)
2
"
a
Anderson
(1954),
Brune^,
Dütsch,
Fredenhagen
(2009)
Cargèse
2009
Proof (I)
τ :A→C
a∈A
u
•  For
any
,
any
and
any
primi&ve
e (au) =
!τ
!
τ
•  First,
prove
e!τ (a(1) )τ (a(2) u)
 
 
(au) = (n + 1)
!
τ !n (a(1) )τ (a(2) u)
!
τ (au) = ε(a(1) )τ (a(2) u)
n=0
This
is
true
for
because
!(n+1)
n
If
this
is
true
up
to
,
then
!
τ !(n+1) (au)
=
τ !n ((au)(1) )τ ((au)(2) )
!
!
!n
=
τ (a(1) u)τ (a(2) ) +
τ !n (a(1) )τ (a(2) u)
!
!
!(n−1)
= n
τ
(a(1) )τ (a(2) u)τ (a(3) ) +
τ !n (a(1) )τ (a(2) u)
•  Conclude
by
∞
!
1 !n
!τ
e (au) =
τ (au)
n!
n=0
Cargèse
2009
Proof (II)
!
"
τ τ ϕ(x1 )ϕ(x2 ) = g(x1 , x2 )
•  If
the
only
non‐zero
value
of
is
then
t(a) = e!τ (a)
•  Induc&on
on
the
degree
of
a
 
a = ϕ(x1 )ϕ(x2 )
a = 1 a = ϕ(x)
True
for
,
and
 
a
If
this
is
true
up
to
degree
,
take
of
degree
n
n
 
(a(2) |u) = τ (a(2) u)
u
a(2)
is
of
degree
1,
so
that
is
of
degree
1
and
 
!
"
!
"
t(au) = ε T (au) = ε T (a) ◦ u
#
#
=
t(a(1) )ε(a(2) ◦ u) =
t(a(1) )(a(2) |u)
The
recursion
hypothesis
gives
us
!
!
!τ
t(au) =
e (a(1) )τ (a(2) u) =
e!τ (au)
Cargèse
2009
Convolution inverse
τ
t:A→C
•  If
sa&sfies
,
there
is
a
such
that
t(1) = 1
t(a) = e!τ (a)
t
t !t −1 = ε
•  The
convolu&on
inverse
of
sa&sfies
•  We
have
t−1 (a) = e!(−τ ) (a)
!
−1
T (a) = t(a(1) )a(2)
T, T : A → A
•  Let
with
and
!
"
! −1
−1
−1
T (a) = t (a(1) )a(2)
,
then
T
T (a) = a
•  Recover
the
product
of
normal
products
by
! −1
"
−1
a ◦ b = T T (a)T (b)
Brune^,
Dütsch,
Fredenhagen
(2009)
Cargèse
2009
Many-body physics
•  Hamiltonian
H = H0 + V
|Φ0 ! H0
•  Ini&al
state:
eigenstate
of
•  When
the
ini&al
state
is
non‐degenerate,
it
can
be
considered
as
a
new
vacuum
(occupied
and
unoccupied
orbitals)
•  In
the
presence
of
symmetry,
the
ini&al
state
is
o`en
degenerate
•  QFT
(Wightman
and
Challifour)
•  Degeneracy
has
very
important
technological
consequences
(giant
magnetoresistance)
Cargèse
2009
Gemstones
•  Chromium
in
an
oxygen
octahedron
(ruby
and
emerald)
•  Fe
in
beryl
(heliodor
and
aquamarine)
Cargese
2009
Many-body physics
•  Calculate
and
the
Green
func&ons
for
a
!Φ0 |T (a)|Φ0 "
|Φ0 !
general
ini&al
state
•  Many
QFT
tools
break
down
 
Wick's
theorem
fails
(pairings
are
not
enough)
 
Green
func&ons
cannot
be
wriben
in
terms
of
Feynman
diagrams
 
The
Gell‐Mann
and
Low
formula
does
not
converge
 
The
Dyson
equa&on
and
the
Bethe‐Salpeter
equa&on
do
not
hold
•  Slow
reconstruc&on
of
these
tools
•  Combinatorics
is
tough
Cargèse
2009
Example
•  Perturba&ve
expansion
for
G(x, y)
=
x
y
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ ...
•  Cumulants
of
the
ini&al
state
Dn (x1 , . . . , xn )
•  Example
Cargèse
2009
Hopf algebra
•  In
terms
of
Hopf
algebra,
we
have
simply
!Φ0 |T (a)|Φ0 " = e!τ (a)
•  The
only
change
is
!
"
τ ϕ(x1 ) . . . ϕ(xn ) = Dn (x1 , . . . , xn )
•  Decomposi&on
of
Green
func&on
in
terms
of
connected
Green
func&ons
•  One‐par&cle
irreducible
structure
of
Green
func&ons
•  Dyson
equa&on
•  Bethe‐Salpeter
equa&on
•  Effec&ve
Hamiltonians
Cargèse
2009
Connected graphs
•  How
to
describe
a
connected
graph
?
!
x1
γ
x2
γ
+
+
=
x4
+
+
+
x3
•  Define
a
second
coproduct
!
δϕn (x) = ϕn (x) ⊗ 1 + 1 ⊗ ϕn (x)
"
δ ϕ (x)ϕ (y)
n
m
=
!
"! m "
δϕ (x) δϕ (y)
n
= ϕn (x)ϕm (y) ⊗ 1 + ϕn (x) ⊗ ϕm (y)
+ϕm (y) ⊗ ϕn (x)1 ⊗ ϕn (x)ϕm (y)
•  The
convolu&on
product
for
this
coproduct
is
Ruelle's
product
Cargèse
2009
Connecting coproduct
•  Dis&nguish
the
two
coproducts
" #
n
!
n k
n
∆ϕ (x) =
ϕ (x) ⊗ ϕn−k (x)
k
k=0
n
δϕn (x) = ϕ (x) ⊗ 1 + 1 ⊗ ϕn (x)
•  Graphical
representa&on
∆
=
⊗
+3
δ
=
⊗1 + 1⊗
⊗
•  Sweedler's
nota&on
+3
∆a =
δa =
Cargèse
2009
+
⊗
!
!
⊗
a(1) ⊗ a(2)
a[1] ⊗ a[2]
A bit of Hopf theory
δ)
•  (A,
is
a
coalgebra
with
the
counit
defined
by
! n "
n
,
for
all
εδ (1) = 1 εδ ϕ (x) = 0
(A, δ)
(A, ∆)
•  The
map
is
a
right
coac&on
of
on
β=∆
(β ⊗ Id)β
(Id ⊗ ε)β
= (Id ⊗ ∆)β
= Id
δ)
(A, ∆)
•  (A,
is
a
‐comodule
coalgebra
(εδ ⊗ Id)β
(δ ⊗ Id)β
=
1εδ
= (Id ⊗ Id ⊗ µ)(Id ⊗ τ ⊗ Id)(β ⊗ β)δ
•  Sweedler's
nota&on
!
!
εδ (a(1) )a(2)
a(1) [1] ⊗ a(1) [2] ⊗ a(2)
= εδ (a)1
!
=
a[1] (1) ⊗ a[2](1) ⊗ a[1] (2) a[2] (2)
Cargèse
2009
Connected graphs
a = ϕn1 (x1 ) . . . ϕnk (xk ) t(a) = !0|T (a)|0"
•  For
,
is
k
the
sum
over
all
Feynman
diagrams
with
ni
xi
ver&ces,
with
edges
abached
to
vertex
•  The
sum
over
connected
diagrams
is
tc (a)
∞
•  The
rela&on
is
!
(−1)n !
tc (a) = −
!
where
n=1
n
t(a[1] ). . .t(a[n] )
a[1] ⊗ a[2] = ∆a = ∆a − a ⊗ 1 − 1 ⊗ a
•  Examples
tc (u)
tc (uv)
tc (uvw)
= t(u)
= t(uv) − t(u)t(v)
= t(uvw) − t(u)t(vw) − t(v)t(uw) − t(w)t(uv) + 2t(u)t(v)t(w)
Cargèse
2009
Connected graphs
a = ϕ2 (x1 )ϕ2 (x2 )ϕ2 (x3 )ϕ2 (x4 )
•  For
,
the
graphs
are
x
x
!
1
γ
2
γ
+
+
=
x4
+
+
+
x3
•  The
sum
over
connected
graphs
is
indeed
tc (a)
!
"! 2
"
2
= t(a) − t ϕ (x1 )ϕ (x2 ) t ϕ (x3 )ϕ (x4 )
! 2
"! 2
"
2
2
−t ϕ (x1 )ϕ (x3 ) t ϕ (x2 )ϕ (x4 )
! 2
"! 2
"
2
2
−t ϕ (x1 )ϕ (x4 ) t ϕ (x2 )ϕ (x3 )
2
2
•  Well
known:
Haag
(1958),
Epstein
and
Glaser
(1973),Fredenhagen
et
al.
(1997,2001),
Mestre
and
Oeckl
(2006)
Cargèse
2009
Connected time-ordering
a = ϕ(x1 ) . . . ϕ(xk ) T (a) = ϕ(x1 ) ◦ · · · ◦ ϕ(xk )
•  For
,
"
T (a) = t(a(1) )a(2)
•  We
have
,
with
t(a) = ε T (a)
!
•  We
defined
the
connected
&me‐ordering
∞
!
(−1)n !
Tc (a) = −
T (a[1] ). . .T (a[n] )
n
n=1
!
•  Using
the
comodule
coalgebraic
structure,
Tc (a) =
with
!
tc (a(1) )a(2)
∞
!
(−1)n !
tc (a) = −
t(a[1] ). . .t(a[n] )
n
n=1
Cargèse
2009
Exponential form
•  The
Ruelle
product
is
f ! g(a) =
!
f (a[1] )g(a[2] )
t(a) = e!tc (a)
tc (1) = 0
•  If
we
put
,
then
•  A
local
Lagrangian
is
primi&ve:
if
L(x) = λϕ4 (x)
δL(x) = L(x) ⊗ 1 + 1 ⊗ L(x)
!
u = dxL(x)
•  If
,
then
δeu = eu ⊗ eu
u
tc (eu )
t(e ) = e
•  Thus,
.
We
recover
Z = eW
•  How
about
Green
func&ons?
Cargèse
2009
Green functions
a = ϕ(x1 ) . . . ϕ(xn )
•  For
,
the
Green
func&on
is
t(aeu )
G(a) = G(x1 , . . . , xn ) =
t(eu )
t(a) = e!tc (a) δeu = eu ⊗ eu
•  The
rela&ons
,
!
!tc
e (va) = tc (va[1] )e!tc (a[2] )
and
imply
t(ϕ(x)eu ) = tc (ϕ(x)eu )t(eu )
•  In
other
words
where
G(x) = Gc (x)
Gc (a) = tc (aeu )
•  More
generally,
assuming
,
the
rela&on
Gc (1) = 0
between
standard
and
connected
Green
func&ons
is
G(a) = e!Gc (a)
Cargèse
2009
Renormalization
•  Johann
Melchior
Ernst
Karl
Gerlach
Stückelberg
von
Breidenbach
zu
Breidenstein
und
Melsbach
(1905‐1984)
•  Nikolay
Nikolaevich
Bogoliubov
(1909‐1992)
•  Henri
Epstein
•  Vladimir
Jurko
Glaser
(1924‐1984)
•  Gudrun
Pinter
Cargèse
2009
Transformation of T-products
Λ:A→A
•  For
a
linear
map
,
we
define
∞
!
#
1 "
TΛ (a) =
T Λ(a[1] ) . . . Λ(a[n] )
n!
n=1
δa = δa − a ⊗ 1 − 1 ⊗ a
•  The
reduced
coproduct
is
•  Its
itera&on
is
δ 0 = Id
δ1
δ n+1
= δ
=
!
(δ ⊗ Id⊗n )δ n
a[1] ⊗ · · · ⊗ a[n]
•  Nota&on
δ n−1 a =
! !Λ "
Λ(1) = 0
•  If
,
then
TΛ (a) = T e (a)
Cargèse
2009
Transformation of T-products
•  Example
a = ϕ3 (x)ϕ3 (y)
•  Reduced
coproduct
δ 0 a = ϕ3 (x)ϕ3 (y)
δ 1 a = ϕ3 (x) ⊗ ϕ3 (y) + ϕ3 (y) ⊗ ϕ3 (x)
δ n a = 0,
n>1
•  Transformed
T‐product
TΛ (a)
$
! "
$
! "
#
#
"
#
= T Λ ϕ3 (x)ϕ3 (y) + T Λ ϕ3 (x) Λ ϕ3 (y)
•  is
called
a
generalized
vertex
by
Bogoliubov
Λ
Cargèse
2009
Resummation
∞
!
#
1 "
•  Bogoliubov‐Pinter
TΛ (a) =
T Λ(a[1] ) . . . Λ(a[n] )
n!
n=1
! n
•  For
a
local
Lagrangian
u = ϕ (x)dx
δ(eu − 1) = (eu − 1) ⊗ (eu − 1)
•  Thus,
TΛ (eu )
= TΛ (1) + TΛ (eu − 1)
! Λ(eu −1)
"
= TΛ (1) + T e
−1
TΛ (1) = T (1) = 1
•  If
we
assume
,
then
"
TΛ (e ) = T e (e ) = T (euΛ )
•  The
renormalized
Lagrangian
is
uΛ = Λ(eu − 1)
u
!
!Λ
u
Cargèse
2009
Renormalized Lagrangian
•  The
renormalized
Lagrangian
is
uΛ = Λ(eu − 1)
Λ
•  The
image
of
is
local
(i.e.
primi&ve)
•  For
example
!
"
Λ ϕ4 (x1 ) . . . ϕ4 (xn )
=
#
!
dxδ(x1 − x) . . . δ(xn − x)
"
Cn1 ϕ2 (x) + Cn2 ϕ4 (x) + Cn3 ϕ(x)!ϕ(x)
•  In
general
is
the
bare
Lagrangian
and
Λ(u) = u
are
the
counterterms
uΛ − u
•  Compa&ble
with
connec&vity
Cargèse
2009
Renormalization group (I)
uΛ
•  The
renormalize
Lagrangian
is
related
to
the
bare
u
Lagrangian
by
e!Λ (eu ) = euΛ
!Λ!
•  We
compose
two
renormaliza&ons
e
•  For
example
!
!Λ
e
"
(a)
e!Λ (uv) = Λ(uv) + Λ(u)Λ(v)
!Λ!
e
!
!Λ
e
"
!
"
!
"
!
" "!
"
"
"
"
(uv) = Λ Λ(uv) + Λ Λ(u)Λ(v) + Λ Λ(u) Λ Λ(v)
!
•  We
define
the
product
Λ • Λ(a) = Λ e
!
•  We
have
e
!Λ!
!
!Λ
e
"
!(Λ! •Λ)
(a) = e
(a)
Cargèse
2009
!
"Λ
"
(a)
Renormalization group (II)
A
L
•  Let
be
the
set
of
linear
maps
from
to
the
A
primi&ve
elements
of
L
•
•  endowed
with
the
product
is
a
unital
associa&ve
algebra
Λ0
•  The
unit
of
this
algebra
is
the
map
that
is
the
iden&ty
for
primi&ve
elements
and
zero
otherwise
•  The
inver&ble
elements
of
this
algebra
are
the
maps
that
are
inver&ble
over
the
primi&ve
elements
Λ(u) = u
•  In
the
standard
case
and
we
have
a
group
Cargèse
2009
Conclusion
•  Gauge
theory:
the
Slavnov‐Taylor
iden&&es
generate
a
Hopf
ideal
of
the
Hopf
algebra
of
renormaliza&on
(van
Suijlekom)
•  Noncommuta&ve
analogues
of
connec&vity,
one‐
par&cle
irreducibility,
renormaliza&on
•  Extension
to
non‐rela&vis&c
case
(same
&me
instead
of
same
space&me
point)
•  Open
problems
 
Direct
connec&on
with
BPHZ
 
Rigorous
formula&on
of
the
quantum
field
Hopf
algebra
over
space&me
Cargèse
2009
Analytical questions
•  Product
of
fields
at
a
point
ϕn (x)ϕm (x) = ϕn+m (x)
!
•  Coproduct
of
smoothed
fields
ϕ(g) = dxϕ(x)g(x)
n " #$
!
n
n
∆ϕ (g) =
dx ϕk (x) ⊗ ϕn−k (x)g(x)
k
k=0
•  Product
of
the
polynomial
algebra
in
the
variable
by
ϕ
the
manifold
R4
•  For
a
general
manifold,
define
a
Hopf
algebra
bundle
Cargese
2009
Analytical questions (II)
•  Dütsch
and
Fredenhagen:
Non
linear
evalua&on
n
n
ϕ
(x)[h]
=
h
(x)
func&onals:
•  Smoothed
fields:
ϕn (g)[h] =
•  Coproduct
∆ϕn (x)[h1 , h2 ]
=
!
dx hn (x) g(x)
n " #
!
n
k=0
n
k
ϕk (x)[h1 ] ⊗ ϕn−k (x)[h2 ]
= ϕ (x)[h1 + h2 ]
•  Smoothed
coproduct
∆ϕ (g)[h1 , h2 ]
n
=
n " #$
!
n
k=0
n
k
% k
dx h1 (x) ⊗ h2n−k (x))g(x)
= ϕ (g)[h1 + h2 ]
Cargese
2009