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An effective non-commutative loop quantum cosmology To cite this article: Abraham Espinoza-García et al 2015 J. Phys.: Conf. Ser. 654 012003

- Non-commutative Pfaffians Dmitrii V Artamonov and Valentina A Golubeva - Scalar fields in a non-commutative space Wolfgang Bietenholz, Frank Hofheinz, Héctor Mejía-Díaz et al. - Re-entrant phase transitions in noncommutative quantum mechanics Orlando Panella and Pinaki Roy

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

An Eﬀective Non-Commutative Loop Quantum Cosmology Abraham Espinoza-Garc´ıa, M. Sabido, J. Socorro Departamento de F´ısica de la DCeI de la Universidad de Guanajuato-Campus Le´ on C.P. 37150, Guanajuato, M´exico E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Abstract. We construct a Non-Commutative extension of the Loop Quantum Cosmology eﬀective scheme for the open FLRW model. We start from the holonomized Hamiltonian and implement a canonical non-commutativity among the matter degree of freedom and the holonomy variable, in the volume representation. We obtain a noncommutativie extension of the modiﬁed Friedmann equation which arises in Loop Quantum Cosmology for a particular case of the theta deformation.

1. Introduction The idea of Noncommutativity in spacetime is quite old (1947) [1]. It was proposed as an attempt to regularize Quantum Field Theory before the renormalization program was established. Due to its non-local behavior, Noncommutativity was quickly forgotten after renormalization proved to be successful. In the 1970’s M. Flato and co-workers proposed an alternative path to quantization [2], in which a deformation of the Poisson structure of classical phase space is performed and is encoded in the moyal -product [3] and generalizations of it (for a review see [4]). In the early 1980’s mathematicians led by A. Connes succeeded in formulating what they called Noncommutative Geometry [5], motivated by generalizating a classic theorem characterizing C ∗ -algebras. Recently, the noncommutative paradigm has resurrected, mainly due to results in String Theory [6, 7], in which Yang-Mills theories in a noncommutative space arise in diﬀerent circumstances as eﬀective theories when taking certain simple limits (for instance, the low energy limit). This renewed interest has led to a deeper understanding, from the physical and mathematical points of view, of noncommutative ﬁeld theory (for a review see [8]). It is believed that in a full quantum theory of gravity the continuum picture of spacetime would no longer be consistent at distances comparable to the Planck length p ∼ 10−35 cm, a quantization of spacetime itself could be in order. Furtheremore, since quite ago, it has been argued [9] that measurements of position can not be performed to better accuracies than the Planck length, since spacetime itself would be modiﬁed due to the energy required to peform such measurments. A possible way to model this eﬀects could be via an uncertainty relation for the spacetime coordinates of the form [xi , xj ] = iθij (1) Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

This commutation relation is the starting point of noncommutative ﬁeld theory. Attempts to implement this idea have led to diﬀerent proposals for a noncommutative theory of gravity [10]. Diﬀerent incarnations of noncommutative gravity have a common feature, they are highly non-linear theories, which makes them very diﬃcult to work with. A possible way to study noncommutativity eﬀects in the early universe was proposed by Garcia-Compean et al [11]. They implemented noncommutativity in conﬁguration space (in contrast to noncommutative spacetime), but only after a symmetry reduction of spacetime had been imposed and the Wheeler-DeWitt quantization had been carried, giving rise to a noncommutative quantum cosmology (mathematically similar to noncommutative quantum mechanics [12]). Later, G. D. Barbosa and N. Pinto-Neto [13] introduced this minisuperspace noncommutativity already at the classical level. The idea is that, perhaps, this eﬀective noncommutativity could incorporate novel eﬀects and insights of a full quantum theory of the gravitational ﬁeld, alongside with providing a simple framework for studying the implications of such possible noncommutative eﬀects in the (early) universe through cosmological models. Since the publication of these two seminal investigations, some works along this line have been conducted. For instance, the noncommutativity of the Friedmann-Robertson-Walker cosmology has been studied, as well as some of the Bianchi Class A models [14]. Quantum black holes have also been investigated within this framework [15]. On the other hand, Loop Quantum Gravity (LQG) [16, 17] is an attempt to quantize the gravitational ﬁeld taking seriously the lessons from General Relativity, that is, it aims at a full (non-perturbative) background independent quantization of General Relativity. Loop Quantum Cosmology (LQC) [18, 19] is the quantization of cosmological (symmetry reduced) models following closely the ideas and methods of LQG. In this way, the LQC of the FLRW and some of the Bianchi Class A models in the presence of a massless scalar ﬁeld (employed as internal time) has been constructed [20, 21, 22, 23, 24, 25, 26], in particular, as a result of the underlying quantum geometry, it has been shown that the loop quantization of the FLRW models features a bouncing which enables the resolution of the cosmological singularity [27]. The LQC of the inhomogeneous Gowdy model has also been constructed [28]. As a result of loop quantization, the Wheeler-DeWitt equation is no longer a diﬀerential equation, but a diﬀerence equation, which is diﬃcult to work with even in the simplest models. In order to extract physics, eﬀective equations based on a geometrical formulation of Quantum Mechanics [29] have been employed to study the consecuences of loop quantum corrections in cosmological models. For instance, the eﬀective description of the FLRW models reproduces very well the behavior of the corresponding full loop quantization of such models. The present investigation aims at constructing a noncommutative eﬀective scheme for the open FLRW model, in the pressence of a scalar ﬁeld. The manuscript is organized as follows: In section II we introduce the loop variables for the open FLRW model with a free standard scalar ﬁeld and the corresponding eﬀective scheme. In section III we recall the Weyl-Wigner-Moyal correspondence and the construction of a Noncommutative Quantum Mechanics based on this correspondence. Section IV is devoted to construct a non-commutative model for the eﬀective Loop Quantum Cosmology of the open FLRW model resembling the Noncommutative Quantum Mechanics of section III, along the lines of references [11, 13]. 2. Connection Variables and Eﬀective Dynamics Here we recall the formulation of the open FLRW and Bianchi I models in the AshtekarBarbero variables, with a free massless scalar ﬁeld. The Ashtekar-Barbero variables cast General Relativity in the form of a gauge theory, in which phase space is described by a su(2) gauge connection, the Ashtekar-Barbero connection Aia and its canonical conjugate momentum, the densitized triad Eia (i, j, etc. denote internal su(2) indices while a, b, etc. denote spatial indices).

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

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These quantities are deﬁned as Aia = Γia + γKai ,

Eia =

√ a qei

(2)

where Kai = Kab eaj , with eia and eai the triad and co-triad, respectively (eia eaj = δji , qab = δij eia ejb ); i compatible with the triad (∇ ei + ω i ej = 0, ∇ Γia is deﬁned through the spin connection ωaj a b a aj b i = Γi with being the usual spatial covariant derivative) by the relation ωaj the totally ijk a ijk antisymmetric symbol; γ a real constant called the Barbero-Immirzi parameter. The canonical pair has the following Poisson structure {Aai (x), Ebj (y)} = 8πGγδij δba δ(x, y)

(3)

with δ(x, y) the Dirac delta distribution on the space-like hypersurface Σ. In these variables, the Hamiltonian constraint takes the form i aj bk 1 2 imn m n ijk Fab − (1 + γ ) Ka Kb E E (4) Cgrav = |E| i = ∂ Ai − ∂ Ai + Aj Ak is the curvature of the Ashtekar-Barbero connection and where Fab a b b a ijk a b E the determinant of the densitized triad. While the full gravitational Hamiltonian is Hgrav = d3 xN Cgrav (5)

2.1. Open FLRW model for spatially ﬂat homogeneous models the gravitational Hamiltonian reduces to [19] 1 2 i ijk Fab d3 x E aj E bk Hgrav = −γ N |E| V

(6)

where the integral is taken in a ﬁducial cell V. When imposing also isotropy, the connection and triad can be described by parameters c and p, respectively, deﬁned by [19] √ (7) Aia = cV0 o eia , Eai = pV0 o q o eai where o qab is a ﬁducial ﬂat metric (with which we can endow Σ due to the symmetries of the model) and o eai , o eia are constant triad and co-triad compatible with o qab ; V0 is the volume of V with respect to o qab . These variables do not depend on the choice of the ﬁducial metric. The relation among these variables and the usual geometridynamical variables is 1/3

c = V0

γ a, ˙

2/3 2

p = V0

a

(8)

We have the canonical relations

8πGγ 3 For convinience we perform the following change of variables {c, p} =

c β=√ , p

3

V = p2

(9)

(10)

with the canonical relations {β, V } = 4πGγ

3

(11)

X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

In these variables the Hamiltonian (with a free massless scalar ﬁeld as the matter content) takes the form (N = 1) p2φ 3 2 H=N − β V + (12) 8πGγ 2 2V The Holonomy correction due to Quantum Gravity eﬀects is coded in the replacement [19] β →

sin λβ λ

(13)

√ where λ2 = 4 3πγ2p is the smallest eigenvalue of the area operator in the full Loop Quantum Gravity [17]. The resulting eﬀective Hamiltonian is thus given by Hef f = −

p2φ 3 2 sin (λβ)V + 8πGγ 2 λ2 2V

(14)

The ﬁeld equations are p2φ ∂Hef f 3 β˙ = 4πGγ =− sin2 (λβ) − 4πGγ 2 ∂V γλ 2V ∂H 3 ef f V˙ = −4πGγ = V sin(λβ) cos(λβ) ∂β γλ ∂Hef f pφ φ˙ = = ∂pφ V ∂Hef f =0 p˙φ = − ∂φ

(15) (16) (17) (18)

˙

V Since V = a3 , 3V = aa˙ = H, where H is the Hubble parameter. Then, taking into account the eﬀective Hamiltonian constraint, Hef f = 0, and the ﬁeld equation for V we have,

2

H = 1 = 2 2 γ λ

2 8πGγ 2 λ2 pφ 3 2V 2

V˙ 3V

2

2 8πGγ 2 λ2 pφ 1− 3 2V 2 8πG ρ = ρ 1− 3 ρmax

(19)

p2

where ρ = 2Vφ2 = 8πGγ3 2 λ2 sin2 (λβ) and ρmax is the maximum value that the ρ can take in view of the eﬀective Hamiltonian constraint, that is, ρmax = 8πGγ3 2 λ2 . The turning points of the volume π , which correspond to a bounce. The last equality is the modiﬁed function occur at β = ± 2λ Friedmann equation, which incorporates holonomy corrections due to Loop Quantum Gravity. In the limit λ → 0 (no area gap) we recover the ordinary Friedmann equation H2 = 8πG 3 ρ. The relational evolution of V in terms of φ is given by √ dV dV dt 3 V = = sin(λβ) cos(λβ) = 12πGV dφ dt dφ γλ pφ

1−

ρ ρmax

1/2 (20)

where we have used the ﬁeld equations for V and φ, and the eﬀective Hamiltonian constraint.

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

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3. Weyl-Wigner-Moyal Correspondence and Noncommutative Quantum Mechanics The Weyl-Wigner-Moyal (WWM) correspondence [30] relates a phase space function f with its operator analog W(f ) = fˆ, it is based on the following map [31] i a a ˆ (21) f (ˆ q , pˆ) = f˜(ξ, η)e (ˆpa ξ +ˆq ηa ) w(ξ, η)dl ξdl η where f˜ is the Fourier transform of f , w a weight function, and dim(T ∗ Q) = 2l; with qˆa , pˆb satisfying the canonical Heisenberg algebra, [ˆ q a , pˆb ] = iδba ;

[ˆ q a , qˆb ] = 0 = [ˆ pa , pˆb ]

(22)

We deﬁne the -product with the help of this correspondence by W(f g) = W(f ) · W(g)

(23)

that is, f g is the Weyl symbol of fˆgˆ. For this case of the Heisenberg algebra (22) and in the symmetric ordering (w = 1) we have f g = exp

i P (f, g) 2

∞ i i r 1 r μν = fg + P (f, g) = f (x) exp ∂μ ω ∂ν g(x) 2 r! 2 r=1

= f g + {f, g} + O(2 ) 2

(24)

where P r (f, g) = ω μ1 ν1 · · · ω μr νr (∂μ1 ...μr f ) (∂ν1 ...νr g) is the rth power of the Poisson bracket bidiﬀerential operator, with ω μν the components of the ﬂat Poisson structure (??); xμ are the phase space coordinates, denoted collectevely as x, the ﬁrst n being the conﬁguration coordinates q a , the second n being the momenta pa . This is the Moyal -product [3], it replaces the ordinary pointwise multiplication in the algebra of functions deﬁned in phase space. The Moyal -bracket f g − g f is thus responsible for realize the canonical Heisenberg algebra (22). Hence, (24) encodes a deformation of the classical phase space which yields the canonical Heisenberg algebra. This product is the cornerstone of deformation quantization [2, 4]. Now, consider the deformation of classical phase space which yields the algebra [ˆ q a , pˆb ] = iδba ;

[ˆ q a , qˆb ] = iθθab ;

[ˆ pa , pˆb ] = iκκab

(25)

with θab , κab antisymmetric constat real matrices. In light of the above prescription, we could encode this deformed Heisenberg algebra in a -product completely analogous to (24). The diﬀerence lies in the ω μν , which in this case would be of the form θΘ I i (26) −I κK This quantization leads us to a Noncommutative Quantum Mechanics, rather than the ordinary Quantum Mechanis obtained above. 4. Eﬀective Non-Commutative Loop Quantum Cosmology In view of the above discussion, we would like to consider eﬀects of the following deformed Poisson algebra {β, φ} = θ, {β, V } = 4πGγ (27)

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The above relations can be implemented by working with the shifted variables β nc = β + aθpφ ,

φnc = φ + bθV

(28)

where a and b satisfy the relation 4πGγb − a = 1. In deed, we have the algebra {β nc , φnc } = θ,

{β nc , V } = 4πGγ

(29)

Since we are not actually performing a deformation of the symplectic structure (the algebra among the basic phase space variables β, φ, V, pφ is the same), the loop quantization of the original variables can be carried as usual, but with a slightly diﬀerent Hamiltonian. Considering the steps above, we can therefore implement the eﬀects of relation (28) in the eﬀective Hamiltonian, which results in nc Hef f =−

p2φ 3 2 nc sin (λβ )V + 8πGγ 2 λ2 2V

(30)

The non-commutative eﬀective ﬁeld equations are 2

p ∂H 3 sin2 (λβ nc ) − 4πGγ 2Vφ2 β˙ = 4πGγ ∂Vef f = − 2γλ ∂H V˙ = −4πGγ ∂βef f =

φ˙ =

∂Hef f ∂pφ

3 γλ V

sin(λβ nc ) cos(λβ nc )

(32)

pφ V

(33)

3aθ nc ) cos(λβ nc ) + = − 4πGγ 2 λ sin(λβ

p˙φ = −

(31)

∂Hef f ∂φ

=0

(34)

In the limit θ → 0 we recover the commutative ﬁeld equations. Due to the ﬁeld equation for φ, p2

˙2

we note that now the matter density ρ = φ2 is not given only by Vφ2 , but by pφ 2 3aθ 1 nc nc nc − sin(λβ ) cos(λβ ) + ρ = 2 4πGγ 2 λ V

(35)

In order to construct the Friedmann equation we would need to obtain a relation for pφ in terms ˙ but since the ﬁeld equation for φ˙ is now more involved, such relation can not be obtained; of φ, and so the Friedmann equation can not be constructed. The relational evolution of V in terms of φ is now given by

pφ −2 dV 2 3V 2 2 3θ nc 2 nc 2 nc 2 nc = sin (λβ ) cos (λβ ) (36) sin (λβ ) cos (λβ ) + dφ γλ 8πGγ 2 λ2 V When taking θ → 0 this relation reduces to (20). 4.1. Addition of potential term In the case of a scalar ﬁeld with a potential term V (φ) we have nc Hef f =−

p2φ 3 2 nc + V (φnc )V sin (λβ )V + 8πGγ 2 λ2 2V

(37)

The non-commutative eﬀective ﬁeld equations are 2

p ∂H 3 sin2 (λβ nc ) − 4πGγ 2Vφ2 + 4πGγV (φnc ) + 4πGγbθV β˙ = 4πGγ ∂Vef f = − 2γλ ∂H V˙ = −4πGγ ∂βef f =

φ˙ =

∂Hef f ∂pφ

3 γλ V

sin(λβ nc ) cos(λβ nc )

3aθ nc ) cos(λβ nc ) + = − 4πGγ 2 λ V sin(λβ

p˙ φ = −

∂Hef f ∂φ

= −V

6

∂V (φnc ) ∂φ

pφ V

∂V (φnc ) ∂V

(38) (39) (40) (41)

X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

These ﬁeld equations reduce to the commutative ones when taking θ → 0. The matter density φ˙ 2 nc 2 + V (φ ) is given by ρ

nc

1 = 2

−

pφ 3aθ sin(λβ nc ) cos(λβ nc ) + 4πGγ 2 λ V

2

+ V (φnc )

(42)

In the particular case in which a = 0 the whole noncommutativity is encoded in the shifted variable φnc = φ + bθV , in this case we have the Hamiltonian nc Hef f =−

p2φ 3 2 + V (φnc )V sin (λβ)V + 8πGγ 2 λ2 2V

(43)

The non-commutative eﬀective ﬁeld equations are 2

p ∂H 3 sin2 (λβ) − 4πGγ 2Vφ2 + 4πGγV (φnc ) + 4πGγbθV β˙ = 4πGγ ∂Vef f = − 2γλ

V˙ =

∂H 3 V sin(λβ) cos(λβ) −4πGγ ∂βef f = γλ ∂H p φ˙ = ∂pefφ f = Vφ

p˙ φ = −

∂Hef f ∂φ

= −V

∂V

(φnc )

∂φ

∂V (φnc ) ∂V

(44) (45) (46) (47)

These ﬁeld equations reduce to the commutative ones when taking θ → 0. The matter density φ˙ 2 nc 2 + V (φ ) is given by p2φ nc ρ = + V (φnc ) (48) 2V 2 In this particular case, it is possible to construct the Friedmann equation, which takes the form 2

H =

V˙ 3V

2

8πG 2 nc ρnc = ρ 1− 3 ρmax

(49)

This equation reduces to the standard one by taking θ → 0. The Klein-Gordon equation for the scalar ﬁeld takes the form 1 ∂V (φnc ) φ¨ = −3H φ˙ − V ∂V

(50)

which reduces to the standard one by taking θ → 0. The relational evolution of V in terms of φ is in this case the same as the commutative one since the ﬁeld equations for V and φ remain unchanged. 4.2. First order non-commutative quantum corrections In the following we will neglect terms of O(θ2 ). Expanding the Hamiltonian (30) in λ, keeping only up to the leading term (λ2 ), we have nc Hef f =−

p2φ 3V V λ2 2 4 3 (β + 2aθβp ) + (β + 4aθβ p ) + φ φ 8πGγ 2 8πGγ 2 2V

(51)

This Hamiltonian incorporates the ﬁrst quantum correction as well as the ﬁrst non-commutative correction. The ﬁrst addend is composed of the classical term β 2 plus its ﬁrst non-commutative

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

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correction; whereas the second addend is composed of the ﬁrst quantum correction (β 4 term) plus its ﬁrst non-commutative correction. The ﬁeld equations are ∂H 3 (β 2 + 2aθβpφ ) + β˙ = 4πGγ ∂Vef f = − 2γ ∂H V˙ = −4πGγ ∂βef f =

φ˙ =

∂Hef f ∂pφ

3V γ

λ2 4 2γ (β

2V λ2 3 γ (β + p aλ2 θβ 3 V + Vφ 2πGγ 2

(β + aθpφ ) −

3aθβV = − 4πGγ 2 +

p˙φ = −

∂Hef f ∂φ

=0

p2

+ 4aθβ 3 pφ ) − 4πGγ 2Vφ2 3aθβ 2 pφ )

(52) (53) (54) (55)

When taking θ → 0 these ﬁeld equations reduce to the ones incorporating ﬁrst order quantum corrections. ˙2 The energy density φ2 is in this case given by 2 p2φ βpφ β 3 ρ= − (56) + aθ 2V 2 πGγ 2 2 4 Employing the Hamiltonian constraint we have 4λ2 4 8πG 1 2 ρ + ρθ 2 3 θ (ρ + ρ ) 1 − (57) H = 2 β + 2aθβpφ − 2 β + 4aθβ pφ = γ 3γ 3 ρmax 3βp λ2 β 3 p where ρθ = aθ 4πGγφ2 − 2πGγ 2φ . Equation (57) is the Friedmann equation with ﬁrst order non-commutative and loop quantum corrections. We note that when θ → 0 we obtain the leading term of the modiﬁed Friedmann equation (19). 5. Discusion A simple noncommutative extension of the open FLRW loop quantum cosmology has been constructed, by introducing a theta deformation at the eﬀective scheme of Loop Quantum Cosmology. This model thus incorporates eﬀective corrections from both Loop Quantum Cosmology and Noncommutative Geometry. In the general case it was not possible to construct the corresponding Friedmann equation; however it was possible to construct it for the ﬁrst order non-commutative corrections. Also, a noncommutative modiﬁed Friedmann equation could be constructed for the case in which the noncommutativity is encoded just in the matter degree of freedom. The physical implications of such noncommutative corrections are currently being investigated and will be reported elsewhere. In the general case (eq. (30)) it is seen from the equation for V˙ that a bounce ocurs when π β = 2π − aθpφ , morover, at this value of β, the noncommutative density is equal to the value given by Eﬀective Loop Quantum Comology, ρmax . This means that the noncommutative extension constructed in this work does not modify the way in which a bounce is reached; however, the time at which a bounce ocurs can be shifted by tunning the noncommutative parameter θ. It is also seen from the equation for ρnc , that this new energy density incorporates a geometric (gravitational) term, which could kill the usual matter term for particular values in the noncommutative parameter. Naturally, the same conclusions can be drawn from the ﬁrst order approximation (eq. (51)). For the particular case in which the whole noncommutativity is encoded in the matter degree of fredom (eq. (43)) we observe that noncommutativity amounts to only modifying the potential term, and therefore there are no modiﬁcations to the Eﬀective Loop Quantum Cosmology scheme; we can conclude that in order to obtain a real modiﬁcation of Eﬀective Loop Quantum Comology, noncommutativity should be implemented in the gravitational degrees of freedom. However, it is interesting to investigate if a cosmological constant term could arise when implementing noncommutativity only in the matter sector.

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

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X-DGFM Journal of Physics: Conference Series 654 (2015) 012003

IOP Publishing doi:10.1088/1742-6596/654/1/012003

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