Memorial Conference in Honor of Julius Wess
Paolo AschieriCentro Fermi, Roma,
U.Piemonte Orientale, Alessandria
Munich 06.11.2008
Noncommutative Spacetime Geometries
I present a general program in NC geometry where noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.
• *-differential geometry *-Lie derivative, *-covariant derivative
• *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess]
[P.A., Dimitrijevic, Meyer, Wess]
I present a general program in NC geometry where noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.
• *-differential geometry *-Lie derivative, *-covariant derivative
• *-Riemannian geometry metric structure on NC space
*-gravity
• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess]
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess] [P.A., Dimitrijevic, Meyer, Wess]
I present a general program in NC geometry where
noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.
• *-differential geometry *-Lie derivative, *-covariant derivative
• *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess]
• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess]
• *-Poisson geometry Poisson structure on NC space
*-Hamiltonian mechanics [P.A., Fedele Lizzi, Patrizia Vitale] Application: Canonical quantization on noncommutative space, deformed
algebra of creation and annihilation operators.
II
I
[P.A., Dimitrijevic, Meyer, Wess]
Why Noncommutative Geometry?
• Classical Mechanics
• Quantum mechanics It is not possible to know
at the same time position and velocity. Observables become noncommutative.
= 6.626 x10-34 Joule s Panck constant
• Classical Gravity
• Quantum Gravity Spacetime coordinates become noncommutative
LP = 10-33 cm Planck length
==h/mc
La=h/mc
La=h/mc
La=h/mc
• Below Planck scales it is natural to conceive a more general spacetime structure. A noncommutative one where (as with quantum mechanics phase-space) uncertainty relations and discretization naturally arise.
• Space and time are then described by a Noncommutative Geometry
• In this framework it is relevant to construct a theory of gravity on noncommutative spacetime.
• Below Planck scales it is natural to conceive a more general spacetime structure. A noncommutative one where (as with quantum mechanics phase-space) uncertainty relations and discretization naturally arise.
• Space and time are then described by a Noncommutative Geometry
• In this framework it is relevant to construct a theory of gravity on noncommutative spacetime.
We consider a -product geometry,
The class of -products we consider is not the most general one studied by Kontsevich. It is not obtained by deforming directly spacetime. We first consider a twist of spacetime symmetries, and then deform spacetime itself.
Ingredients for a twist F
• g Lie algebra
• Ug universal enveloping algebra of g (i.e. complex numbers C and sumsof products of elements t ! g, where tt" # t"t = [t, t"]).
Ug is an associative algebra with unit. It is a Hopf algebra:
!(t) = t$ 1 + 1$ t
!(t) = 0
S(t) = #t
The coproduct ! is extended to all Ug by requiring ! and ! to be linearand multiplicative (e.g. !(tt") := !(t)!(t") = tt" $ 1 + t $ t" + t" $t + 1$ tt").
2
Examples of NC spaces
xi ! xj ! xj ! xi = i"ij canonical
y ! xi ! xi ! y = iaxi , xi ! xj ! xj ! xi = 0 Lie algebra type
xi ! xj ! qxj ! xi = 0 quantum (hyper)plane
cf. Leibnitz rulet(ab) = t(a) b + a t(b)
2
The twist F
[Drinfeld ’83, ’85]
F ! Ug " Ug
F is invertible
F " 1(!" id)F = 1" F(id"!)F in Ug " Ug " Ug
Example 1:
F = e#i2!"abXa"Xb
where [Xa, Xb] = 0. Here g is the abelian Lie algebra generated by the Xa’s."ab is a constant (antisymmetric) matrix.
F ! Ug[[!]]" Ug[[!]], formal power series in !. Deformation quantization.
F = 1" 1#1
2!"abXa "Xb + O(!2) .
3
This is a cocycle condition it implies associativity of the --product
Example 2: a nonabelian example, Jordanian deformations, [Ogievetsky, ’93]
F = e12H!log(1+!E) , [H, E] = 2E
4
!-Noncommutative Manifold (M, g,F)
M smooth manifold
g Lie algebra acting on M (we have " : g ! !)
F " Ug # Ug and therefore F " U!# U!
A algebra of smooth functions on M
$A! noncommutative algebra
Usual product of functions:
f # hµ%! fh
!-Product of functions
f # hµ!%! f ! h
µ! = µ & F%1
A and A! are the same as vector spaces, they have different algebra structure
5
.
Associativity of this -product follows from the cocycle condition for the twist
!-Noncommutative Manifold (M, g,F)
M smooth manifold
g Lie algebra acting on M (we have " : g ! !)
F " Ug # Ug and therefore F " U!# U!
A algebra of smooth functions on M
$A! noncommutative algebra
Usual product of functions:
f # hµ%! fh
!-Product of functions
f # hµ!%! f ! h
µ! = µ & F%1
A and A! are the same as vector spaces, they have different algebra structure
5
[Connes, Landi] [Connes, Dubois-Violette] isospectral deformations
[Majid, Gurevich], [Varilly,] [Sitarz]
Examples of NC spaces
xi ! xj ! xj ! xi = i"ij canonical
y ! xi ! xi ! y = iaxi , xi ! xj ! xj ! xi = 0 Lie algebra type
xi ! xj ! qxj ! xi = 0 quantum (hyper)plane
2
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
7
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
7
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
= 1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
8
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
= 1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
8
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
= 1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
8
Example: M = R4
F = e!i2!"µ# $
$xµ" $$x#
1" 1!i
2!"µ#$µ " $# !
1
8!2"µ1#1"µ2#2$µ1$µ2 " $#1$#2 + . . .
(f % h)(x) = e! i
2!"µ# $$xµ" $
$y# f(x)h(y)!!!x=y
NotationF = f& " f& F!1 = f̄& " f̄&
f % h = f̄&(f) f̄&(h)
Define F21 = f& " f& and define the universal R-matrix:
R := F21F!1 := R& " R& , R!1 := R̄& " R̄&
It measures the noncommutativity of the %-product:
f % g = R̄&(g) % R̄&(f)
Noncommutativity is controlled by R!1. The operator R!1 gives a represen-tation of the permutation group on A%.
7
F deforms the geometry of M into a noncommutative geometry.
Guiding principle:
given a bilinear map
µ : X ! Y " Z
(x, y) #" µ(x, y) = xy .
deform it into µ! := µ $ F%1 ,
µ! : X ! Y " Z
(x, y) #" µ!(x, y) = µ(f̄ "(x), f̄ "(y)) = f̄ "(x)f̄ "(y) .
The action of F & U!' U! will always be via the Lie derivative.
!-Tensorfields
# '! # ( = f̄"(#)' f̄"(# () .
In particular h !v , h ! df , df ! h etc...7
!-Forms
" !! "" = f̄#(") ! f̄#("") .
Exterior forms are totally !-antisymmetric.
For example the 2-form $ !! $" is the !-antisymmetric combination
$ !! $" = $ #! $" $ R̄#($")#! R̄#($) .
The undeformed exterior derivative
d : A % !
satisfies (also) the Leibniz rule
d(h !g ) = dh !g + h ! dg ,
and is therefore also the !-exterior derivative.
The de Rham cohomology ring is undeformed.
8
!-Action of vectorfields, i.e., !-Lie derivative
Lv(f) = v(f) usual Lie derivative
Lv : !!A " A
(v, f) #" v(f) .
deform it into L! := L $F %1 ,
L!v(f) = f̄"(v)
!f̄"(f)
"!-Lie derivative
Deformed Leibnitz rule
L!v(f ! h) = L!
v(f) ! h + R̄"(f) ! L!R̄"(v)(h) .
Deformed coproduct in U!
"!(v) = v & 1 + R̄"! & R̄"(v)
Deformed product in U!
u !v = f̄"(u)f"(v)
8
_
_
!-Lie algebra of vectorfields !! [P.A., Dimitrijevic, Meyer, Wess, ’06]
Theorem 1. (U!, !,"!, S!, ") is a Hopf algebra. It is isomorphic to Drinfeld’sU!F with coproduct "F .
Theorem 2. The bracket
[u, v]! = [̄f#(u), f̄#(v)]
gives the space of vectorfields a !-Lie algebra structure (quantum Lie algebrain the sense of S. Woronowicz):
[u, v]! = ![R̄#(v), R̄#(u)]! !-antisymmetry
[u, [v, z]!]! = [[u, v]!, z]! + [R̄#(v), [R̄#(u), z]!]! !-Jacoby identity
[u, v]! = L!u(v)
the !-bracket isthe !-adjoint action
Theorem 3. (U!, !,"!, S!, ") is the universal enveloping algebra of the !-Liealgebra of vectorfields !!. In particular [u, v]! = u !v ! R̄#(v) ! R̄#(u).
To every undeformed infinitesimal transformation there correspond one and only one deformed
infinitesimal transformation: Lv " L!v.
9
!-Lie algebra of vectorfields !! [P.A., Dimitrijevic, Meyer, Wess, ’06]
Theorem 1. (U!, !,"!, S!, ") is a Hopf algebra. It is isomorphic to Drinfeld’sU!F with coproduct "F .
Theorem 2. The bracket
[u, v]! = [̄f#(u), f̄#(v)]
gives the space of vectorfields a !-Lie algebra structure (quantum Lie algebrain the sense of S. Woronowicz):
[u, v]! = ![R̄#(v), R̄#(u)]! !-antisymmetry
[u, [v, z]!]! = [[u, v]!, z]! + [R̄#(v), [R̄#(u), z]!]! !-Jacoby identity
[u, v]! = L!u(v)
the !-bracket isthe !-adjoint action
Theorem 3. (U!, !,"!, S!, ") is the universal enveloping algebra of the !-Liealgebra of vectorfields !!. In particular [u, v]! = u !v ! R̄#(v) ! R̄#(u).
To every undeformed infinitesimal transformation there correspond one and only one deformed
infinitesimal transformation: Lv " L!v.
9
!-Lie algebra of vectorfields !! [P.A., Dimitrijevic, Meyer, Wess, ’06]
Theorem 1. (U!, !,"!, S!, ") is a Hopf algebra. It is isomorphic to Drinfeld’sU!F with coproduct "F .
Theorem 2. The bracket
[u, v]! = [̄f#(u), f̄#(v)]
gives the space of vectorfields a !-Lie algebra structure (quantum Lie algebrain the sense of S. Woronowicz):
[u, v]! = ![R̄#(v), R̄#(u)]! !-antisymmetry
[u, [v, z]!]! = [[u, v]!, z]! + [R̄#(v), [R̄#(u), z]!]! !-Jacoby identity
[u, v]! = L!u(v)
the !-bracket isthe !-adjoint action
Theorem 3. (U!, !,"!, S!, ") is the universal enveloping algebra of the !-Liealgebra of vectorfields !!. In particular [u, v]! = u !v ! R̄#(v) ! R̄#(u).
To every undeformed infinitesimal transformation there correspond one and only one deformed
infinitesimal transformation: Lv " L!v.
9
We can now proceed in our study of differential geometry on noncommutativemanifolds.
Covariant derivative:
!!h!uv = h !!!
uv ,
!!u(h !v ) = L!
u(h) ! v + R̄"(h) !!!R̄"(u)v
our coproduct implies that R̄"(u) is again a vectorfield
On tensorfields:
!!u(v "! z) = !!
u(v)"! z + R̄"(v)"!!!R̄"(u)(z) .
The torsion T and the curvature R associated to a connection!! are the linearmaps T : !! #!! $ !!, and R! : !! #!! #!! $ !! defined by
T(u, v) := !!uv %!!
R̄"(v)R̄"(u)% [u, v]! ,
R(u, v, z) := !!u!!
vz %!!R̄"(v)!
!R̄"(u)z %!
![u,v]!z ,
for all u, v, z & !!.13
!-antisymmetry property
T(u, v) = !T(R̄"(v), R̄"(u)) ,
R(u, v, z) = !R(R̄"(v), R̄"(u), z) .
A!-linearity shows that T and R are well defined tensors:
T(f ! u, v) = f ! T(u, v) , T(u, f ! v) = R̄"(f) ! T(R̄"(u), v)
and similarly for the curvature.
Local coordinates description
We denote by {ei} a local frame of vectorfields (subordinate to an open U "M ) and by {#j} the dual frame of 1-forms:
#ei , #j$! = $ji .
#ei , #j$! = #f̄ "(ei) , f̄ "#j)$ .
Connection coefficients !ijk,
%eiej = !kij ! ek
T =1
2!j !" !i " Tij
l "" el ,
R =1
2!k "" !j !" !i " Rijk
l "" el .
Connection forms #ji , the torsion forms !l and the curvature forms "k
l by
#ij := !k " !ki
j ,
!l := #1
2!j !" !i " Tij
l ,
"kl := #
1
2!j !" !i " Rkij
l ,
Cartan structural equations hold
!l = d!l # !k !" #kl ,
"kl = d#k
l # #km !" #m
l .
Bianchi identities
d!i + !j !" #ji = !j !" "j
i ,
d" lk + "k
m !" #ml # #k
m !" "ml = 0 .
14
T =1
2!j !" !i " Tij
l "" el ,
R =1
2!k "" !j !" !i " Rijk
l "" el .
Connection forms #ji , the torsion forms !l and the curvature forms "k
l by
#ij := !k " !ki
j ,
!l := #1
2!j !" !i " Tij
l ,
"kl := #
1
2!j !" !i " Rkij
l ,
Cartan structural equations hold
!l = d!l # !k !" #kl ,
"kl = d#k
l # #km !" #m
l .
Bianchi identities
d!i + !j !" #ji = !j !" "j
i ,
d" lk + "k
m !" #ml # #k
m !" "ml = 0 .
14
T =1
2!j !" !i " Tij
l "" el ,
R =1
2!k "" !j !" !i " Rijk
l "" el .
Connection forms #ji , the torsion forms !l and the curvature forms "k
l by
#ij := !k " !ki
j ,
!l := #1
2!j !" !i " Tij
l ,
"kl := #
1
2!j !" !i " Rkij
l ,
Cartan structural equations hold
!l = d!l # !k !" #kl ,
"kl = d#k
l # #km !" #m
l .
Bianchi identities
d!i + !j !" #ji = !j !" "j
i ,
d" lk + "k
m !" #ml # #k
m !" "ml = 0 .
14
T =1
2!j !" !i " Tij
l "" el ,
R =1
2!k "" !j !" !i " Rijk
l "" el .
Connection forms #ji , the torsion forms !l and the curvature forms "k
l by
#ij := !k " !ki
j ,
!l := #1
2!j !" !i " Tij
l ,
"kl := #
1
2!j !" !i " Rkij
l ,
Cartan structural equations hold
!l = d!l # !k !" #kl ,
"kl = d#k
l # #km !" #m
l .
Bianchi identities
d!i + !j !" #ji = !j !" "j
i ,
d" lk + "k
m !" #ml # #k
m !" "ml = 0 .
14
Example, F = e!i2!µ"#µ"#"
#$#µ
(#") = !%µ" $ #%
Tµ"% = !µ"
% ! !"µ%
Rµ"%& = #µ!"%
& ! #"!µ%& + !"%
' $ !µ'& ! !µ%
' $ !"'& .
Ricµ" = R%µ"%.
14
Examples of NC spaces
xi ! xj ! xj ! xi = i"ij canonical
y ! xi ! xi ! y = iaxi , xi ! xj ! xj ! xi = 0 Lie algebra type
xi ! xj ! qxj ! xi = 0 quantum (hyper)plane
"#µ, dx$#! = %$µ
cf. Leibnitz rulet(ab) = t(a) b + a t(b)
2
!-Riemaniann geometry
!-symmetric elements:
" !! "" + R̄#("")!! R̄#(") .
Any symmetric tensor in ! !! is also a !-symmetric tensor in !! !! !!, proof: expansionof above formula gives f̄ #(")! f̄ #("") + f̄ #("")! f̄ #(").
g = gµ$ ! dxµ !! dx$
"µ$% =
1
2(&µg$' + &$g'µ # &'gµ$) ! g!'%
g$' ! g!'% = (%$ .
Noncommutative Gravity
Ric = 0
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess][P.A., Dimitrijevic, Meyer, Wess]
15
• This theory of gravity on noncomutative spacetime is completely geometric. It uses deformed Lie derivatives and covariant derivatives. Can be formulated without use of local coordinates.
• It holds for a quite wide class of star products. Therefore the formalism is well suited in order to consider a dynamical noncommutativity. The idea being that both the metric and the noncommutative structure of spacetime could be dependent on the matter/energy content.
• Exact solutions of this theory are under investigations
• First order formalism and coupling to spinors, supergravity.
Conclusions and outlook
Dynamical nc in flat spacetime [P.A., Castellani, Dimitrijevic]Lett. Math.Phys 2008