Noncommutative Spacetime Geometries...where (as with quantum mechanics phase-space) uncertainty...

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.



*-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.



*-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").

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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)

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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) .

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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

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!-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

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.

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

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[Connes, Landi] [Connes, Dubois-Violette] isospectral deformations

[Majid, Gurevich], [Varilly,] [Sitarz]





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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%.

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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


f % h = f̄&(f) f̄&(h)


R := F21F!1 := R& " R& , R!1 := R̄& " R̄&


f % g = R̄&(g) % R̄&(f)


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


f % h = f̄&(f) f̄&(h)


R := F21F!1 := R& " R& , R!1 := R̄& " R̄&


f % g = R̄&(g) % R̄&(f)


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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.

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!-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)

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_

_

!-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.

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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 .

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Example, F = e!i2!µ"#µ"#"

#$#µ

(#") = !%µ" $ #%

Tµ"% = !µ"

% ! !"µ%

Rµ"%& = #µ!"%

& ! #"!µ%& + !"%

' $ !µ'& ! !µ%

' $ !"'& .

Ricµ" = R%µ"%.

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"#µ, dx$#! = %$µ

cf. Leibnitz rulet(ab) = t(a) b + a t(b)

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!-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]

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• 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

Noncommutative Spacetime Geometries...where (as with quantum mechanics phase-space) uncertainty...

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