UNIVERSIDAD DE GUANAJUATO DEPARTAMENTO DE FISICAhidalgo/segundo_taller_es_files/sabido.pdf ·...

Seminario del Departamento de Física de la DCeI

August 6, 2014

Cuernavaca.

UNIVERSIDAD DE GUANAJUATO

M. Sabido

Efectos del Minisuperespacio Deformado en Cosmología

DEPARTAMENTO DE FISICA


August 6, 2014

Cuernavaca.

PLAN OF THE TALK

INTRODUCTION, MOTIVATION !NC-GRAVITY !NC-COSMOLOGY !NC AND LAMBDA !FINAL REMARKS


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INTRODUCTION


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INTRODUCTION

Noncommutativity of space time is an old idea (Snyder Phys. Rev. 71 (1947)


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INTRODUCTION


There are physical systems that can easily be described by assuming that coordinates do not commute.


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INTRODUCTION



For the last decade it has attracted a lot of attention in connection to string theory. N. Seiberg and E. Witten, JHEP 9909:032 (1999).


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INTRODUCTION




An immense amount of work has been done in noncommutative gauge theory.


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INTRODUCTION




An immense amount of work has been done in noncommutative gauge theory.

Unfortunately on the gravity side of things, the story has been more complex.


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NC- GRAVITY


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It was shown from string theory that noncommutative space-time arises in the low energy limit, of open strings in the presence of NS B field (N.

Seiberg and E. Witten JHEP 2000).

Using different regularization methods either a commutative or a non commutative gauge theory arises.

NC- GRAVITY


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Consider a theory invariant under the action of a Lie group G, with gauge group Aμ and a mater field Φ, that transforms in the adjoint representation

NC- GRAVITY

Aµ = µ + i [, Aµ] , = i [, ] ,


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NC- GRAVITY

Aµ = µ + i [, Aµ] , = i [, ] ,


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For the noncomutative theory, the gauge transformations are:

NC- GRAVITY

Aµ = µ + i [, Aµ] , = i [, ] ,

bAµ = µ

+ i , Aµ

, b

= i ,

.


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For the noncomutative theory, the gauge transformations are:

NC- GRAVITY


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NC- GRAVITY


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Because the result should be independent of the regularization scheme both theories should be equivalent the a map between them must exist. This is the Seiberg-Witten map, which lets us write the noncommutative gauge fields from the commutative ones.

NC- GRAVITY

(, A) = +14µ µ, A+O

2

,


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NC- GRAVITY

(, A) = +14µ µ, A+O

2

, Fµ = Fµ +

14

2 Fµ, FA, DFµ + Fµ

+O

2

.


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Cuernavaca.


NC- GRAVITY

(, A) = +14µ µ, A+O

2

, Fµ = Fµ +

14


+O

2

.

(, A) = 14µ Aµ, (D + )+O

2


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NC- GRAVITY

(, A) = +14µ µ, A+O

2

,

Aµ (A) = Aµ 14 A , Aµ + Fµ+O

2

,

Fµ = Fµ +14


+O

2

.

(, A) = 14µ Aµ, (D + )+O

2


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NC- GRAVITY

(, A) = +14µ µ, A+O

2

,

Aµ (A) = Aµ 14 A , Aµ + Fµ+O

2

,

Fµ = Fµ +14


+O

2

.

(, A) = 14µ Aµ, (D + )+O

2


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From these equations we can write the noncommutative lagrangian and the noncommutative theory. This results where reproduced by J. Wess independent of string theory (Wess , EPJC 2005).

NC- GRAVITY


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GR

(O. Obregon, H.Compean, C. Ramirez and M.S., PRD 68 (2003) .

NC- GRAVITY


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MOTIVATION


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MOTIVATION

We need an alternative


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MOTIVATION


Minisuperspace Models

+


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MOTIVATION



+Canonical Quantization

+


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MOTIVATION




+Noncommutative Quantum Mechanics


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MOTIVATION





_________________________


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MOTIVATION





_________________________Noncommutative Quantum Cosmology

H. Compeán, O.Obregon and C. Ramirez PRL 88 (2002)


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In quantum mechanics we have

MOTIVATION


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Being gravity the theory of space time, quantization gravity could be interpreted as a quantization of space time

MOTIVATION


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Being gravity the theory of space time, quantization gravity could be interpreted as a quantization of space time

What happens if space time does not commute?

MOTIVATION


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MOTIVATION


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In analogy with quantum mechanics

MOTIVATION


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We have a minimal scale we can probe

MOTIVATION


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This deformation can be encoded in Moyal product of functions

MOTIVATION


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This deformation can be encoded in Moyal product of functions

which yields the noncommutative Schrodinger equation (L. Mezincescu ArXiv:hep-th\0007046)

MOTIVATION


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NONCOMMUTATIVE QM


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We may recover the original commutation relations with the variables

NONCOMMUTATIVE QM


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so the effects are encoded only on the potential

NONCOMMUTATIVE QM


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And the NC equation takes a very simple form

NONCOMMUTATIVE QM

r2 + V

x+

2P

y

, y

2P

x

(x, y) = E (x, y).


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And the NC equation takes a very simple form

NONCOMMUTATIVE QM


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The original proposal for NCQC was the Kantowski-Sachs model

WDW equation for KS

NC COSMOLOGY

2

posal in order to explore the influence of noncommuta-tivity at early times, by the introduction of an effectivenoncommutativity in quantum cosmology. Instead of adeformation of space-time, we consider a “deformationof minisuperspace”.

We will assume that the minisuperspace variables donot commute, as it has been proposed for the space-time coordinates. Our proposal is inspired in variousrelated results in the literature, some of them alreadyreferred here, as is the case of the Seiberg-Witten map[9], where by demanding gauge invariance, a redefinitionof the gauge fields as an expansion on the noncommu-tativity parameters is obtained. As a consequence ofspace-time noncommutativity, the fields do not commuteamong them in a specific manner dictated by the Moyalproduct. In our ansatz, we propose a simple and di-rect noncommutativity among certain components of thegravitational field.

It is well known that already in the usual spacetime,noncommutativity is usually defined in the “preferred”frame of cartesian coordinates, where the noncommuta-tive parameters are taken to be constant. For any othercoordinate systems, the corresponding noncommutativ-ity will be, in general, in terms of parameters relatedin a complicated manner to those of cartesian coordi-nates. For the even-dimensional case, the spacetimes canbe interpreted as symplectic manifolds with the noncom-mutativity parameters playing the role of a symplecticform. In this case, Darboux’s theorem ensures that al-ways, locally, there exists a coordinate system in whichthe components of the symplectic form are constant. Instring theory this is assumed when a ‘constant’ B-field isconsidered.

On the other hand, the Seiberg-Witten map for gravi-tation has been proposed in [13], where noncommutativetetrads and connections are computed. In the case ofquantum cosmology, the minisuperspace variables playthe role of the “coordinates” of the configuration space.Thus, it seems reasonable to propose a kind of noncom-mutativity among these specific gravitational variables,as it is the case in standard spacetime, when cartesian co-ordinates are selected. From the canonical quantizationof a cosmological model a quantum mechanical version,with a finite number of degrees of freedom, of the generalWheeler-de Witt equation for general relativity arises. Inthis way, quantum cosmology is usually understood as aquantum mechanical model of the universe. We will thenfurther assume, that it can be proceeded as in standardnoncommutative quantum mechanics.

The noncommutativity we propose, can be reformu-lated in terms of a Moyal deformation of the Wheeler-DeWitt equation, similar to the case of the noncommu-tative Schrodinger equation [23]. Actually, cosmology de-pends only on time and a Moyal product of functions ofonly one variable is trivially realized. Other authors haveworked out noncommutativity in the early universe, how-

ever without considering the gravitational field. In Ref.[24] the noncommutativity of space-time is interpretedas a magnetic field on a horizon scale. In [25, 26], it isargued that noncommutativity affecting matter or gaugefields could have played an important role to produceinflation.

As an example of our proposal, we will consider thecosmological model of the Kantowski-Sachs metric. Inthe parametrization due to Misner, this metric looks like[27]:

ds2 = −N2dt2 + e2√

3βdr2 + e−2√

3βe−2√

3Ω(dθ2 + sin2θdϕ2).(1)

The corresponding Wheeler-DeWitt equation, in a par-ticular factor ordering, can be written as

exp(√

3β + 2√

3Ω)[−P 2Ω + P 2

β − 48exp(−2√

3Ω)]ψ(Ω,β) = 0,

(2)

where PΩ = −i ∂∂Ω and Pβ = −i ∂

∂β. Thus, in this

parametrization the Wheeler-DeWitt equation has a sim-ple form, which can be formally identified with usualquantum mechanics in cartesian coordinates.

The solutions to this Wheeler-DeWitt equation aregiven by [27]

ψ±ν (β, Ω) = e±iν

√3βKiν(4e−

√3Ω), (3)

where Kiν is the modified Bessel function. Packet wavesof these solutions have been constructed as superpo-sitions of these solutions. Summing over eiν

√3β and

e−iν√

3β to make real trigonometric functions, the “Gaus-sian” state

Ψ(β, Ω) = 2iN! ∞

0ν"

ψ+ν (β, Ω) − ψ−ν (β, Ω)

#

dν

= N e−√

3Ωsinh(√

3β)exp[−2√

3e−√

3Ωcosh(√

3β)], (4)

has been obtained [27, 28]. A possible connection withquantum black holes [28, 29] and quantum wormholes[30, 31] has been suggested.

For our noncommutative proposal of quantum cosmol-ogy, we will follow the procedure outlined above. We willassume that the “cartesian coordinates” Ω and β of theKantowski-Sachs minisuperspace obey a kind of commu-tation relation, like the ones in noncommutative quantummechanics [23]

[Ω,β] = iθ. (5)

As stated above, this is a particular ansatz in these con-figuration coordinates. The relation with other minisu-perspace coordinates would follow in a similar way as instandard spacetime.


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NC COSMOLOGY


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NC COSMOLOGY


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NC COSMOLOGY


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from this we can find the NCWDW equation for the KS (H. Compeán, O.Obregon and C. Ramirez PRL 88 (2002), J. Lopez,O. Obregon M. S., Phys Rev D 74 084024, 2006)

NC COSMOLOGY


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4

02

46

810

Ω-10

-5

0

5

10

β0

0.1

0.2#Ψ#2

02

46

8Ω

FIG. 1: Variation of |Ψ|2 with respect to Ω and β, at the valueθ = 0. It shows only one possible universe around Ω = 4.812and β = 0.

02

46

810

Ω-10

-5

0

5

10

β0

0.10.20.30.4

#Ψ#2

02

46

8Ω

FIG. 2: Variation of |Ψ|2 with respect to Ω and β, at thevalue θ = 4. It shows many peaks corresponding to differentpossible universes.

4 6 8 10 Ω

0.1

0.2

0.3

0.4

#Ψ#2

FIG. 3: Variation of |Ψ|2 for θ = 4, along the line of the mainpeaks in figure 2. It shows the tunneling possibility amongthese states.

map was used to construct a deformed Einstein gravity.By means of this result, one could try to find the corre-sponding noncommutative Wheeler-DeWitt equation forspecific cosmological models. In that case the θ termscorresponding to spacetime noncommutativity could beconsidered in order to search for another way to definea noncommutative cosmological model, even though cos-mology depends only on time. The computation of thisWheeler-DeWitt equation is a very complicated task, be-cause at each θ order higher derivative terms will appear.If such a noncommutative Hamiltonian model could bedefined, its quantum cosmological solutions and the cor-responding states could be compared with those obtainedby means of our noncommutative minisuperspace pro-posal. In this context the θ parameter we have proposed(5), could be a kind of effective noncommutative param-eter. Such an approach is currently under exploration.

Further work in more realistic cosmological models, in-cluding matter, will be needed to search for constraintson the range of values of the θ parameter in the earlystages of evolution of the universe. In our proposal thisis in principle possible, because it means to enlarge theminisuperspace configuration space and then consider anappropriate commutativity among its coordinates. Fol-lowing these lines, we will consider also the supersym-metric extension [32] to these models.

In previous works, the Wheeler-DeWitt equation ofthe Kantowski-Sachs model has been related to quan-tum black holes [28, 29] and wormholes [30, 31]. It couldbe interesting to search for an extension of our resultsto possible noncommutative versions of these quantumgravitational systems.

Our simple proposal provides a picture of the dramaticinfluence that noncommutativity could have played atearly stages of the universe. We have been able to ob-tain and work with an exact quantum solution for theKantowski-Sachs metric, without the need to expandon the θ parameter. The corresponding wave packetexhibits new stable states of the universe with similarprobabilities, showing peaks around β values differentfrom zero. A tunneling process could happen betweenthese states. By these means there are different possi-ble universes (states) from which our present universecould have evolved and also could have tunneled in thepast, from one universe (state) to other one. Furtherwork is needed to analyze other physical consequences ofthis and other more realistic quantum cosmological mod-els, taking into account the influence of matter. It willbe also necessary to extend these ideas to other, moregeneral gravitational models. In particular, it would bevery interesting to reinterpret the results of reference [33],concerning the influence of a primordial magnetic field inclassical cosmology. The corresponding quantum modelshould be constructed and compared with our proposalin terms of the noncommutativity of the minisuperspace.Results in these directions and those mentioned above

4

02

46

810

Ω-10

-5

0

5

10

β0

0.1

0.2#Ψ#2

02

46

8Ω

FIG. 1: Variation of |Ψ|2 with respect to Ω and β, at the valueθ = 0. It shows only one possible universe around Ω = 4.812and β = 0.

02

46

810

Ω-10

-5

0

5

10

β0

0.10.20.30.4

#Ψ#2

02

46

8Ω

FIG. 2: Variation of |Ψ|2 with respect to Ω and β, at thevalue θ = 4. It shows many peaks corresponding to differentpossible universes.

4 6 8 10 Ω

0.1

0.2

0.3

0.4

#Ψ#2

FIG. 3: Variation of |Ψ|2 for θ = 4, along the line of the mainpeaks in figure 2. It shows the tunneling possibility amongthese states.

map was used to construct a deformed Einstein gravity.By means of this result, one could try to find the corre-sponding noncommutative Wheeler-DeWitt equation forspecific cosmological models. In that case the θ termscorresponding to spacetime noncommutativity could beconsidered in order to search for another way to definea noncommutative cosmological model, even though cos-mology depends only on time. The computation of thisWheeler-DeWitt equation is a very complicated task, be-cause at each θ order higher derivative terms will appear.If such a noncommutative Hamiltonian model could bedefined, its quantum cosmological solutions and the cor-responding states could be compared with those obtainedby means of our noncommutative minisuperspace pro-posal. In this context the θ parameter we have proposed(5), could be a kind of effective noncommutative param-eter. Such an approach is currently under exploration.

Further work in more realistic cosmological models, in-cluding matter, will be needed to search for constraintson the range of values of the θ parameter in the earlystages of evolution of the universe. In our proposal thisis in principle possible, because it means to enlarge theminisuperspace configuration space and then consider anappropriate commutativity among its coordinates. Fol-lowing these lines, we will consider also the supersym-metric extension [32] to these models.

In previous works, the Wheeler-DeWitt equation ofthe Kantowski-Sachs model has been related to quan-tum black holes [28, 29] and wormholes [30, 31]. It couldbe interesting to search for an extension of our resultsto possible noncommutative versions of these quantumgravitational systems.

Our simple proposal provides a picture of the dramaticinfluence that noncommutativity could have played atearly stages of the universe. We have been able to ob-tain and work with an exact quantum solution for theKantowski-Sachs metric, without the need to expandon the θ parameter. The corresponding wave packetexhibits new stable states of the universe with similarprobabilities, showing peaks around β values differentfrom zero. A tunneling process could happen betweenthese states. By these means there are different possi-ble universes (states) from which our present universecould have evolved and also could have tunneled in thepast, from one universe (state) to other one. Furtherwork is needed to analyze other physical consequences ofthis and other more realistic quantum cosmological mod-els, taking into account the influence of matter. It willbe also necessary to extend these ideas to other, moregeneral gravitational models. In particular, it would bevery interesting to reinterpret the results of reference [33],concerning the influence of a primordial magnetic field inclassical cosmology. The corresponding quantum modelshould be constructed and compared with our proposalin terms of the noncommutativity of the minisuperspace.Results in these directions and those mentioned above

NC COSMOLOGY


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NONCOMMUTATIVE COSMOLOGY


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If we consider that the noncommtuative field can be written as a series expansion on the noncommutative parameter, furthermore this expansion can be calculated from the full noncommutative theory.



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based on this facts a new commutation relation is proposed


[,] = i,


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[,] = i,


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This is an effective noncommutativity between the minisuperspace fields, and is similar to the modification used in noncommutative quantum mechanics. H. Compean, O. Obregón and C. Ramírez PRL 88 (2002), W. Guzmán, M.S, J. Socorro Phys. Rev D (2007), )



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One of a noncommutative mini-superspace point to the possibility of late time acceleration of the universe. W. Guzmán, M.S. and J.Socorro PRD (2007).

0.2 0.4 0.6 0.8 1t

0.2

0.4

0.6

0.8

1

Ε


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=2

2

"1 µ

1

p6

tanh [3µ(2 6)t]

!#,

0.2 0.4 0.6 0.8 1t

0.2

0.4

0.6

0.8

1

Ε


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=2

2

"1 µ

1

p6

tanh [3µ(2 6)t]

!#,

0.2 0.4 0.6 0.8 1t

0.2

0.4

0.6

0.8

1

Ε


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One can ask if the noncommutative deformation can be related to the late time acceleration or to the cosmological constant. !There was also evidence that noncommutativity reduced the number of states for a Schwarzchild black Hole (J. Lopez,O. Obregon M. S., Phys Rev D 74 084024, 2006)


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By the same procedure as in the black hole, we use the appropriate metric for the universe!

And the NCWDW equation is!


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By comparing these equations we arrive to the relationships

From which the vacuum density energy is

And the ratio of the observed to calculated densities is

For kmax=Mp we only need ΘPφ of order 240!! (E. Mena, O. Obregon, MS, MPLA 2009)


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Is there a classical interpretation and formulation of NC Quantum Cosmology?

We want

This can be constructed from Hamiltonian Manifolds (W. Guzman,M.S., J. Socorro PLB 2011)

PRELIMINARIES: NC COSMOLOGY

In order to find the NC metric, we needed that the noncommutative model had a classical limit.


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PRELIMINARIES: NC CLASSICAL MECHANICS


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Lets start with the manifold (M2n,ω) and ω in invertible.



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In local coordinates xμ we have X= Xμ∂μ. We say is locally Hamiltonian if



LX d(iX) = 0.


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LX d(iX) = 0.


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If iXω is exact then M is a Hamiltonian manifold


iX = dH = µ H

x

LX d(iX) = 0.


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If iXω is exact then M is a Hamiltonian manifold



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The Poisson brackets are defined as


f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x


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and for the coordinates

f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x


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f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ


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f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ


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we have in the T*M for the coordinates



f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ

xµ = (q1, q2, .., qn, p1, p2, ..., pn)


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f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ

xµ = (q1, q2, .., qn, p1, p2, ..., pn)


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The canonical 2-form



f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ

xµ = (q1, q2, .., qn, p1, p2, ..., pn)

c =12c

µdxµdx , cµ =

0 ij

ij 0

.


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f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ

xµ = (q1, q2, .., qn, p1, p2, ..., pn)

c =12c

µdxµdx , cµ =

0 ij

ij 0

.


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From which we calculate Hamilton’s equations



f, q = p (Xf , Xg) f, q (p) =f

xµµ g

x

xµ, x = µ

xµ = (q1, q2, .., qn, p1, p2, ..., pn)

c =12c

µdxµdx , cµ =

0 ij

ij 0

.

iXH c = dH dqi

dt = Hpi

,dpj

dt = Hpj .


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From which we calculate Hamilton’s equations



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A noncommutative classical mechanics, is a Hamiltonian manifold, for which the exact 2-form is given by a non degenerate ωnc.



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The equations of motion are


dqi

dt= ij H

pj+ ij H

qj,

dpi

ddt= ij H

qj+ ij H

pj.


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dqi

dt= ij H

pj+ ij H

qj,

dpi

ddt= ij H

qj+ ij H

pj.


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and the Poisson algebra


dqi

dt= ij H

pj+ ij H

qj,

dpi

ddt= ij H

qj+ ij H

pj.

qi, qj = ij , qi, pj = i

j , pi, p

j = ij


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and the Poisson algebra



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COSMOLOGIA NO CONMUTATIVA




________________________

NC Quantum Cosmology


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________________________



+


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________________________


Simplectic Manifods


+


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________________________


Simplectic Manifods

_________________________


+


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________________________


Simplectic Manifods

_________________________

NC Classical Cosmology. Deformed phase space

cosmology.


+


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Deformed Phase Space Cosmology


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We can apply this formalism to scalar field cosmology, the approach is straight forward.



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Again we start with Einstein-Hilbert action and find the Hamiltonian



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H = e3

112

P 2

12P 2

e6V ()

,





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H = e3

112

P 2

12P 2

e6V ()

,



for the noncommmutative model we use exact 2-form



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H = e3

112

P 2

12P 2

e6V ()

,

µ =

0 1 0 0 0 11 0 0 00 1 0 0






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H = e3

112

P 2

12P 2

e6V ()

,

µ =

0 1 0 0 0 11 0 0 00 1 0 0

, = , p, p = 0,

, p = 1, , p = 1.






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With the new commutation relations we can find the modified Friemann equation and the KG equation.



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3

a

a

2

=122 + V () 6a3

a

a

dV

d+ V

32a6

dV

d

2

6V 2

,




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3

a

a

2

=122 + V () 6a3

a

a

dV

d+ V

32a6

dV

d

2

6V 2

,

+ 3

a

a

= dV

d+ 6a3

dV

d+ 6

a

a

V

.




August 6, 2014

Cuernavaca.

3

a

a

2

=122 + V () 6a3

a

a

dV

d+ V

32a6

dV

d

2

6V 2

,

+ 3

a

a

= dV

d+ 6a3

dV

d+ 6

a

a

V

.


We also have the Hamiltonian constraint. (W. Guzmán, M.S,, J. Socorro PLB (2011)



August 6, 2014

Cuernavaca.

NONCOMMUTATIVITY AND LAMBDA


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Cuernavaca.


The results of the dynamical systems analysis have revealed is that, independently of the kind of self- interaction potential considered: i) exponential potential, or ii) cosh-like potential, the noncommutative effects of the kind considered here9 affect not only the early-times dynamics but also modify the late-time behaviour. I. Quiros and O. Obregón, PRD (2011).


August 6, 2014

Cuernavaca.


In the a more general case the noncommutative Friedmann equation for scalar field cosmology where derived, and it is found that noncommutativity could be relevant in the late time dynamics of the universe. W. Guzmán, M.S. and J.Socorro PLB (2011).

The results of the dynamical systems analysis have revealed is that, independently of the kind of self- interaction potential considered: i) exponential potential, or ii) cosh-like potential, the noncommutative effects of the kind considered here9 affect not only the early-times dynamics but also modify the late-time behaviour. I. Quiros and O. Obregón, PRD (2011).


August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.

Let us consider the action for a flat FRW universe a scalar field and cosmological constant as the matter content of the the model.


S =

Zdt

(3aa2

2N+ a3

2

2N+N

!),


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Cuernavaca.



S =

Zdt

(3aa2

2N+ a3

2

2N+N

!),


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Cuernavaca.



making the change of variables

S =

Zdt

(3aa2

2N+ a3

2

2N+N

!),

x = m

1a

3/2sinh(m),

y = m

1a

3/2cosh(m),


August 6, 2014

Cuernavaca.




S =

Zdt

(3aa2

2N+ a3

2

2N+N

!),

x = m

1a

3/2sinh(m),

y = m

1a

3/2cosh(m),


August 6, 2014

Cuernavaca.




The action takes the form

S =

Zdt

(3aa2

2N+ a3

2

2N+N

!),

x = m

1a

3/2sinh(m),

y = m

1a

3/2cosh(m),

S =

Z 1

2N

x

2 y

2N

!

2

2

x

2 y

2

dt.


August 6, 2014

Cuernavaca.




The action takes the form


August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


The Hamiltonian for this model is

Hc

= N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2,


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Cuernavaca.



Hc

= N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2,


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Cuernavaca.



Where ω2 is proportional to Λ. For the noncommutative case, we impose noncommutativity among the minisuperspace coordinates and canonical momentum,

Hc

= N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2,

y, x = , u, px

= y, py

= 1 + , py

, p

x

= ,


August 6, 2014

Cuernavaca.




Hc

= N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2,

y, x = , u, px

= y, py

= 1 + , py

, p

x

= ,


August 6, 2014

Cuernavaca.



For the noncommutative case we use the shift


Hc

= N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2,

y, x = , u, px

= y, py

= 1 + , py

, p

x

= ,

x = x+

2p

y

, y = y

2p

x

,

p

x

= p

x

2y, p

y

= p

y

+

2x,


August 6, 2014

Cuernavaca.



For the noncommutative case we use the shift



August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


The resulting noncommutative Hamiltonian for this model is

bH =1

2

p2x

p2y

+ !21(xpy + yp

x

) + !22(x

2 y2) .


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Cuernavaca.



bH =1

2

p2x

p2y

+ !21(xpy + yp

x

) + !22(x

2 y2) .


August 6, 2014

Cuernavaca.



Where we defined

bH =1

2

p2x

p2y

+ !21(xpy + yp

x

) + !22(x

2 y2) .

!21 =

4( !2)

4 !22, !2

2 =4(!2 2/4)

4 !22,


August 6, 2014

Cuernavaca.



Where we defined

bH =1

2

p2x

p2y

+ !21(xpy + yp

x

) + !22(x

2 y2) .

!21 =

4( !2)

4 !22, !2

2 =4(!2 2/4)

4 !22,

˙x = x, bH =1

2[2p

x

+ !21y], ˙y = y, bH =

1

2[2p

y

+ !21x],

˙px

= px

, bH =1

2[!2

1 py 2!22x], ˙p

y

= py

, bH =1

2[!2

1 px + 2!22y],


August 6, 2014

Cuernavaca.



Where we defined



August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


To solve the equations of motion we propose the change of variables


August 6, 2014

Cuernavaca.


To solve the equations of motion we propose the change of variables = x+ y, = y x


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Cuernavaca.



An arrive to a simplified set of equations

!21 +

1

4(4!2

2 + !41) = 0,

+ !21 +

1

4(4!2

2 + !41) = 0,


August 6, 2014

Cuernavaca.




!21 +

1

4(4!2

2 + !41) = 0,

+ !21 +

1

4(4!2

2 + !41) = 0,


August 6, 2014

Cuernavaca.




This equations are very easy to solve for any value of Λ, θ and β.

a3(t) = V0 cosh2

1

4

t

,

-2 -1 1 2 3 4t

2

4

6

8

10

12

a3@tD

2.


August 6, 2014

Cuernavaca.


Let analyse the case when the cosmological constant is zero, as well as the parameter θ. In this case the solution is very simple, in particular the scale factor is (S. Perez, M.S, C. Yee,Phys. Rev D 2013)

Figure 1: Dynamics of the phase space deformed model for the values X0 = Y0 = 1,δ2 = δ1 = 0,ω = 0andβ = 1. Thesolidlinecorrespondstothe volume of the universe, calculated with the noncommutative model. The dotted line corresponds to the volume of the de Sitter spacetime. For large values of t the behavior is the same.

Comparing the to models we arrive to the relationship


August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


With this result lets find a model from which to derive Λ. We start with a (4+1) gravity.

I =

Z pg (R ) dtd3xd,


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Cuernavaca.



I =

Z pg (R ) dtd3xd,


August 6, 2014

Cuernavaca.



After dimensional reduction, the effective 4 dimensional theory is

I =

Z pg (R ) dtd3xd,

L =1

2

aa2 + a2a a+

1

3a3


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Cuernavaca.




I =

Z pg (R ) dtd3xd,

L =1

2

aa2 + a2a a+

1

3a3


August 6, 2014

Cuernavaca.



The hamiltonian for the model is, after an appropriate change of variables


I =

Z pg (R ) dtd3xd,

H =1

2

x2 + !2x2

y2 + !2y2

L =1

2

aa2 + a2a a+

1

3a3


August 6, 2014

Cuernavaca.



The hamiltonian for the model is, after an appropriate change of variables



August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


Following the same steps as before we calculate the noncommutative hamiltonian

H =

(pu 2( !2)

4 !22v

2

pv +

2( !2)

4 !22u

2

+

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

u2

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

v2


August 6, 2014

Cuernavaca.



H =

(pu 2( !2)

4 !22v

2

pv +

2( !2)

4 !22u

2

+

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

u2

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

v2


August 6, 2014

Cuernavaca.



This can be written in simpler form if we define a 2 dimensional vector potential

H =

(pu 2( !2)

4 !22v

2

pv +

2( !2)

4 !22u

2

+

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

u2

4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

v2

H =nh

(pu Au)2 + !02u2

i

h

(pv Av)2 + !02v2

io


August 6, 2014

Cuernavaca.



This can be written in simpler form if we define a 2 dimensional vector potential


August 6, 2014

Cuernavaca.


!02 4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

Au 2( !2)

4 !22v, Av 2( !2)

4 !22u


August 6, 2014

Cuernavaca.


!02 4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

Au 2( !2)

4 !22v, Av 2( !2)

4 !22u


August 6, 2014

Cuernavaca.


We can calculate the effective magnetic field

!02 4( !2)2

(4 !22)2+

4(!2 2/4)

4 !22

Au 2( !2)

4 !22v, Av 2( !2)

4 !22u

B =4( !2)

4 !22


August 6, 2014

Cuernavaca.


We can calculate the effective magnetic field


August 6, 2014

Cuernavaca.



August 6, 2014

Cuernavaca.


We can now find the relationship between the commutative and noncommutative cosmological constants

eff = eff +32

8


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Cuernavaca.



eff = eff +32

8


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Cuernavaca.



So even if in the original theory Λ=0, we find that the noncommutative universe has an effective Λ

eff = eff +32

8

eff =32

8


August 6, 2014

Cuernavaca.



So even if in the original theory Λ=0, we find that the noncommutative universe has an effective Λ


August 6, 2014

Cuernavaca.


3

analogous to Eq.(4) but constructed with the variablesthat obey the modified algebra Eq.(10)

H = N

1

2P 2x

+!2

2x2

N

1

2P 2y

+!2

2y2. (11)

After using the change of variables Eq.(9) and makingthe following definitions

!21 =

!2

1 !22/4, !2

2 =!2 2/4

1 !22/4, (12)

we get

H =1

2

P 2x

P 2y

!2

1(xPy

+ yPx

) + !22(x

2 y2).

(13)Written in terms of the original variables the Hamilto-nian explicitly has the e↵ects of the phase space defor-mation. These e↵ects are encoded by the parameters and . In [29], the late time behaviour of this modelwas studied. From this formulation two di↵erent physi-cal theories arise, one that considers the variables x andy and a di↵erent theory based on x and y. The firsttheory, is interpreted as a “commutative” theory with amodified interaction, this theory is referred as being re-alised in the “C-frame” [27]. The second theory whichprivileges the variables x and y, is a theory with “non-commutative” variables but with the standard interac-tion and is referred to as realised in the “NC-frame”.One usually privileges one of the frames or assumes thatthe di↵erences between them are negligible, but in somecases there can be dramatic di↵erences in the physicson each frame. Although the physical implications be-tween the frames can be startlingly di↵erent [27], thereare cases when the predictions are very similar. In thoseinstances, one can take advantage of the formulation inthe “C-frame” and have very clear interpretation of thedeformation.

In the “C-frame” our deformed model has a very niceinterpretation, that of a ghost oscillator in the presence ofconstant magnetic field and allows us to write the e↵ectsof the noncommutative deformation as minimal couplingon the Hamiltonian and write the Hamiltonian in termsof the magnetic B-field.

To obtain the dynamics for our model, we derive theequations of motion from the Hamiltonian Eq.(13)

x = = Px

1

2!21y, y = P

y

1

2!21x,

Px

=1

2!21Py

!22x, P

y

=1

2!21Px

+ !22y, (14)

and solve them to get

x+ !21 y +

!22 +

1

4!41

x = 0, (15)

y + !21 x+

!22 +

1

4!41

y = 0. (16)

C-Framede-SitterNC-FrameNC-Frame

-2 2 4 6t

5

10

15

20

a3HtL

FIG. 1. Dynamics of the deformed phase space model. In allthe plots of the deformed model, 2 = 1 = 0. The thin lineand dashed lines are for 0 = 0 = 1, = 1,! = 0 and = 5;the dotted line is for 0 = 5/8, 0 = 4, = 4

p6/5,! = 1, =

1/p5. Finally the thick line corresponds to the volume of the

de Sitter space time. We can see that for large values of t thebehaviour is exponential.

Defining new variables and as = x+y, = yx, wecan decouple and solve equations Eqs.(16). The solutionsfor the variables x(t) and y(t) in the “C-frame” are

x(t) = 0 e!2

12 t cosh (!0t+ 1) 0 e

!212 t cosh (!0t+ 2) ,

y(t) = 0 e!2

12 t cosh (!0t+ 1) + 0 e

!212 t cosh (!0t+ 2) ,

(17)

where !0 =q

24!2

4!

2

2 . To compute the volume of the

universe in the “C-frame” we use Eq.(17) and Eq.(3)

a3(t) = V0 cosh2 (!0t) , (18)

where we have taken 1 = 2 = 0.There is a di↵erent solution for = 2!, for this case

the solutions in the “C-frame” are

x(t) = (a+ bt)e!2

12 t + (c+ dt)e

!212 t, (19)

y(t) = (a+ bt)e!2

12 t (c+ dt)e

!212 t.

and the volume of the universe is given by

a3(t) = V0 +At+Bt2, (20)

where V0, A and B are constructed from the integrationconstants. There is another solution for !

02 < 0, thehyperbolic functions are replaced by harmonic functions.To find the solutions in the “NC-frame” we start fromthe “C-frame” solutions and use Eq.(9),we get for the

0

2

4

b

02

4

q

-10

-5

0

L

III

III

IV

.0


August 6, 2014

Cuernavaca.

FINAL REMARKS


August 6, 2014

Cuernavaca.

FINAL REMARKS

- Using the ideas of noncommutative quantum mechanics and effective noncommutative in the minisuperspace is used for noncommutative quantum cosmology.


August 6, 2014

Cuernavaca.

FINAL REMARKS

- A consistent formulation of non commutative classical mechanics has been constructed.



August 6, 2014

Cuernavaca.

FINAL REMARKS

- A consistent formulation of non commutative classical mechanics has been constructed.


- The late time influence of phase space deformations in cosmological evolution have been stablished.*


August 6, 2014

Cuernavaca.

FINAL REMARKS


August 6, 2014

Cuernavaca.

- There is a posibility that Λ is related to the noncommutative parameter. This was found in a simple cosmological model.

FINAL REMARKS


August 6, 2014

Cuernavaca.

- A simple 5 dimensional model was proposed, where the origin of Λ is related to the noncommutativity between the compact dimension and the radius of the non compact dimension.

- There is a posibility that Λ is related to the noncommutative parameter. This was found in a simple cosmological model.

FINAL REMARKS

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