On some possible mathematical connections concerning Noncommutative Minisuperspace Cosmology, Noncommutative Quantum Cosmology in low-energy String Action, Noncommutative Kantowsky-Sachs Quantum Model, Spectral Action Principle associated with a Noncommut

On some possible mathematical connections concerning Noncommutative Minisuperspace Cosmology, Noncommutative Quantum Cosmology in low-energy String Action, Noncommutative Kantowsky-Sachs Quantum Model, Spectral Action Principle associated with a Noncommutative Space and some aspects concerning the Loop Quantum Gravity

Michele Nardelli1,2

1Dipartimento di Scienze della Terra

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy

2_{Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”}

Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

This paper is a review of some interesting results that has been obtained in various sectors of noncommutative cosmology, string theory and loop quantum gravity.

In the Section 1, we have described some results concerning the noncommutative model of the closed Universe with the scalar field. In the Section 2, we have described some results concerning the low-energy string effective quantum cosmology. In the Section 3, we have showed some results regarding the noncommutative Kantowsky-Sachs quantum model. In Section 4, we have showed some results regarding the spectral action principle associated with a noncommutative space and applied to the Einstein-Yang-Mills system. Section 5 is a review of some results regarding some aspects of loop quantum gravity. In Section 6, we’ve described some results concerning the dynamics of vector mode perturbations including quantum corrections based on loop quantum gravity. In Section 7, we’ve described some equations concerning matrix models as a non-local hidden variables theories. In Section 8, we have showed some results concerning the quantum supergravity and the role of a “free” vacuum in loop quantum gravity. In Section 9, we’ve described various results concerning the unifying role of equivariant cohomology in the Topological Field Theories.

In conclusion, in Section 10 we have showed the possible mathematical connections between the arguments above mentioned and some relationship with some equations concerning some sectors of Number Theory.

1. On some equations concerning the Noncommutative model of the closed Universe with the scalar field. [1]

We remember that for any nD minisuperspace model the ordering parameter ξ in the conformally invariant Wheeler-De Witt (WDW) equation (Planck’s constant h=1):

( )

( )

( )

0 2 1 = Ψ     + + ∆ − = Ψ A A q q U R q H ξ , (1.1)

(where the Laplace-Beltrami operator is

( )

( )

( )

     ∂ ∂ − ∂ ∂ − ≡ ∆ ab _B A q q G q G q q G 1 , R is the scalar Riemannian curvature in minisuperspace M , U

( )

q is the minisuperspace potential and Ψ

( )

qA is the wave function of the universe), is equal to ξ =

(

2−n

) (

/

[

81−n

)

]

for n≥2.

The Non-commutative Wheeler-De Witt equation (NWDW) is given by the following expression:

(

)

(

)

1

(

,

)

0 2 4 2 2 _ =   − + + ∂ − ∂ + ∂ − ∂x y y x xC yC xC yC i C C C C α β α β ψ θ . (1.2)

It is possible to find the particular solutions of the NWDW equation by applying the Hartle-Hawking condition and using the θ-modified method of path integrals.

Under the Hartle-Hawking (H-H) condition and the gauge condition N& =0, the non-commutative quantum mechanical propagator _G

(

_q~''A,_N0,0

)

C HH

E

θ can be exactly found from the Pauli formula

(

)

( )

'' '

( )

~ 0 2 2 / '' ' ~ exp ~ ~ ~ det 2 1 00 , ~ = −         ∂ ∂ ∂ − = A C q A C A C n A C HH E I q q I N q G_θ θ _θ π , (1.3)

where n=2 is the dimension of minisuperspace. The propagator is:

(

)

(

)

      − − + + − − = 2 2 3 2 '' 2 '' '' '' 6 2 4 1 exp 8 1 0 , 0 , 0 , , α β π θ N N y x N N y x G C C C C HH E

(

)

(

)

    + ×              − + − − − 2 2 '' '' 2 '' '' 16 exp 32 2 x_C y_C i x_C y_C N α β θ α β θ β α . (1.4)

When α=β the propagator (1.4) does not contain the θ2 dependence so that integration over the complex lapse parameter by applying the method of fastest descent yields the particular solutions of the NWDW equation:

(

x_C y_C

)

= dN

∫

NB '' '' , θ ψ

(

)

      − − + + − − 2 2 3 2 '' 2 '' 6 2 4 1 exp 8 1 β α π N N y x N C C

(

2 2

)

(

'' ''

)

(

'' ''

)

(

'' ''

)

2 '' '' , 16 exp 16 exp 32 2 x_C y_C i x_C y_C i x_C y_{C NB} x_C y_C N α β θ α β θ β α = _ θα + _ψ     + ×              − + − − − . (1.5)

where ψ_NB

(

x_C'',y_C''

)

are all particular solutions of the WDW equation.

Now, we consider the noncommutative geometry of the minisuperspace of the quantum model of the closed universe. It is important to note that the WDW equation (1.1) of the standard (commutative) nD minisuperspace model may be obtained from the action:

[

( )

]

∫

[

( )

]

=       Ψ + Ψ + Ψ ∇ Ψ ∇ − − = Ψ Ψ M B A AB n AB nG q d q G G R U q S ξ 2 1 , ,

∫

( )

Ψ =−

∫

− Ψ Ψ     + + ∆ − Ψ − − = M M n n H G q d q U R G q d ξ 2 1 , (1.6)

by the Lagrange variation of Ψ

( )

qA . Physically, action (1.6) is related to the expectation value of the energy of the Universe, which is invariant to the conformal transformation for the fixed value of the ordering parameter ξ. In the case when nD minisuperspace M possesses Weyl geometry, we •

have the following action:

[

( )

]

' ' ' ' '

( )

' , ' ' ' ' . 2 1 ' , ' , ' , ' ' ' ' ' ' _            Γ =       Γ       Ψ       +       Γ Ψ + Ψ ∇ − − = Ψ Ψ

∫

• • • • w B AB M n A AB w n G q w d q G G R U q R R S ξ (1.7) The action (1.7) is invariant on Weyl rescaling:

( )

q

( )

q G

( )

q

( )

q

( )

q

( ) ( )

q q

( )

q

( )

q U

( )

q

( ) ( )

qU q G A n A A n A AB AB , ' , ' , ' 2 2 1 2 1 ' 2 − − − Ω = Ψ Ω = Ψ Ψ Ω = Ψ Ω = , (1.8)

where n is the dimension of the minisuperspace and the rescaling = + Ω−∂ Ω

A A

A w

w ' 2 1 , (1.9)

for any value of the ordering parameter ξ due to the validity of the following relation:

( )

      Γ Ω =       Γ =       Γ =       Γ =       Γ =       Γ =       Γ• ' •' •' ' •' −2 ' •' ' R q R G R G R R R R AB _AB AB _AB w w . (1.9b)

When minisuperspace is (conformally) flat, then action (1.7) acquires the following form:

[

( )

]

( )

    Ψ Ψ + Ψ ∂ Ψ ∂ − − = Ψ Ψ

∫

• • • ' ' ' ' ' 2 1 ' , ' , ' , ' ' ' ' ' q U G G q d w q G S AB _{A B} M n A AB w n , (1.10)

and by applying the Weyl rescaling (1.8) and the rescaling wA =w'A+2Ω−1∂AΩ=0, (1.11) one obtains the action:

[

( )

]

( )

    Ψ Ψ + Ψ ∂ Ψ ∂ − − = = Ψ Ψ w

∫

d q G G U q q G S AB _{A B} M n A AB n 2 1 0 , , , , (1.12)

which is same as action (1.6) for which the Riemann scalar curvature vanishes. If we apply the Weyl rescaling to the following noncommutative Hamiltonian

(

)

    − − + + + − = ≡ 2 2 1 2 2 4 2 2 2 C C C C C a sh ch a a p a p N NH H C C φ β φ α φ θ θ

[

(

2 2

)

(

2 2

)

]

0 4 2 ≈  + + + − − _{a C C C C C} C sh ch p sh ch a p a C β φ α φ C α φ β φ θ φ , (1.12b)

(

2 2

)

0 4 2 2 ' ' ' ₃ 3 2 2 ' ≈       + − Θ − − + + − = ≡ Θ Θ _{α φ} α _{φ φ} φ φ ch p ash p a ch a a p a p N H N H a _{a . (1.13)}

Furthermore, with regard the eq. (1.12b), we remember that:

(

)

(

)

( )

3 7 2 3 4 4 32 8 4 1 4 1 ln 4 1 θ θ θ φ θ θ φ φ φ φ φ O a p p a p p a p a p a p a C a C a C a C C a C C C C C C C C C C + + + = + − − − + = . (1.13b)

Applying the Legendre transformations to (1.13) we obtain the noncommutative Lagrangian:

        +       _Θ − −       − =     Θ a a dt da N a N N dt da a L 3 2 2 2 2 64 ' 2 ' ' , , ' α α . (1.14)

Solving the equations of motion obtained from (1.14), taking care of the gauge condition N'=1 and the initial conditions t=₀=0,

dt da

( )

0 0 2 2 a dt a d

t= =α we obtain the Lorentz 4-metric determined by the space-time interval:

( )

2 3 2 2 2 2 64 1 Ω       + + − = dt ch td ds θ α α α ,

(

σ =1

)

. (1.15)

The scalar curvature of the θ -deformed de Sitter space-time is:

( )

(

)

           Λ       _Λ + Λ − Λ = t ch t R 3 / 576 1 1152 1 4 2 2 2 2 2 4 θ θ , (1.16)

while the scalar curvature of the 3D subspace is:

( )

(

t

)

ch t R 3 / 576 1 2 2 2 2 3 Λ       _Λ + Λ = θ , (1.17) where Λ=3α.

After applying the Wick rotation to eq. (1.14), and obtaining the corresponding equations of motion from this non-commutative Lagrangian, we obtain the Euclidean 4-metric determined by the space-time interval: 2

(

)

2₃ 2 2 2 sin 64 1 Ω       + + =d d ds_E θ α ατ α τ ,

(

σ =1

)

. (1.18)

From eq. (1.18), the semiclassical non-commutative Hartle-Hawking (H-H) wave functions that corresponds to this 4-geometry are of the form:

(

( )

)

                      + −       + ± ≈ = − ± 2 / 3 1 2 2 2 2 / 3 2 2 64 1 '' 1 64 1 3 1 exp 1 '' ' θ α α θ α α ψθ a a a . (1.19)

From eq. (1.19) we see that the non-commutativity parameter θ increases (for the “+” sign) or decreases (for the “–“ sign) the two corresponding standard semiclassical H-H tunnelling amplitudes. So, from this consideration we may conclude that the canonical non-commutativity prefers the creation of the theta deformed de Sitter universe rather by ψ ' than by _θ+ −

θ

ψ ' .

Hence, at the Hartle-Hawking condition and for different choices of the gauge condition N& =0 (N is the lapse function) the _{θ term either decreases or increases the semiclassical probability} 2

amplitude for tunnelling from nothing to the closed universe with the stable matter potential.

Furthermore, under the Hartle-Hawking condition and when α>0 and β =0 the canonical noncommutativity of the minisuperspace prefers as the most probable the creation of the closed universe with φ =0 by the semiclassical wave function which for θ =0 corresponds to the geometry of filling in the three-sphere with more than half of a four-sphere of radius

(

1/α

)

.

2. On some equations concerning the low-energy string effective quantum cosmology.[2] At low energy, the tree-level, (3+1)-dimensional string effective action can be written as

_S =−

∫

_{d x} −_ge−

(

_R+∂ ∂ +_V

)

s φ φ λ µ µ φ 4 2 1 . (2.1)

Here φ is the dilaton field, a

( )

t =exp

[

β

( )

t / 3

]

, λ is the fundamental string length parameter _s governing the high-derivative expansion of the action and V is a possible dilaton potential. When we consider this theory in the metric of isotropic and homogeneous spacetime, after integrating by

parts, and using the convenient time parametrization φ τ

d e dt = − , reduces to =−

∫

(

− + − φ

)

β φ τ λ 2 2 2 ' ' 2 d Ve S s _{, (2.2) where φ =φ−ln}

∫

(

3 _/λ3

)

_{− 3β} s x d . (2.3)

The Hamiltonian of the system is

(

_{β φ} λ φ

)

λ 2 2 2 2 2 1 Ve H _s s + Π − Π = , (2.4)

where the canonical conjugate momenta are,

Πβ =λsβ', Πφ =−λsφ'. (2.5)

(

,

)

(

,

)

0 2 1 2 2 2 2 2 2 =       + ∂ ∂ − ∂ ∂ _{λ φ β} − _{ψ φ β} β φ λ φ e V s s . (2.6)

We shall assume V =V

( )

φ in order to separate variables. We consider two simple cases of the potential as toy models that allow us to obtain exact solutions

1) Case: V =−V₀e4φ

Therefore the solution of the WDW equation (2.6) is

ψ_ν

(

φ,β

)

=CeiνβY_iν

(

λ_s V₀eφ

)

, where Y is the second class Bessel function. _iν

2) Case: V =−V₀ Now the wave function is

ψ_ν

(

φ β

)

νβ _ν

(

λ 2φ

)

0

, =Cei K_{i s} V e− , (2.7)

where K_iν is the modified Bessel function. We can construct wormhole type solutions by means integrating over the separation constant ν ,

(

)

∫

+∞ ( )

(

)

[ ] ∞ − + − + ₌ = φ λ β µ ν µ β ν φ ν λ β φ ψ cosh2 0 0 , i _{i s} V e WH s e d e V K e , (2.8)

where µ const. For the noncommutative quantum cosmology model, we will assume the =

“cartesian coordinates” φ and β of the Robertson-Walker minisuperspace obey a kind of commutation relation,

[

φ,β

]

=iθ . (2.9)

This is a particular ansatz in these configuration coordinates. The deformation of minisuperspace can be studied in terms of Moyal product,

(

) (

)

(

)

(

φ β

)

φ β β φ θ β φ β φ β φ , 2 exp , , , g f i g f                 ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ = r s r s
. (2.10)

Then the noncommutative WDW equation is

(

,

)

(

,

)

0 2 1 2 2 2 2 2 2 =       + ∂ ∂ − ∂ ∂ − β φ ψ β φ λ β φ λ φ

_sV e s . (2.11)

Now, we take the eq. (2.6) and obtain:

(

)



(

)

= ⇒      + ∂ ∂ − ∂ ∂ − 0 , , 2 1 2 2 2 2 2 2 β φ ψ β φ λ β φ λ φ e V s s

(

)

(

φ

)

β φ λ φ β λ 2 2 2 2 2 2 , 1 2 1 −       ∂ ∂ − ∂ ∂ = ⇒ e V s s . (2.12)

Hence, from the eq. (2.1), we have that:

_S =−

∫

_{d x} −_ge−

(

_R+∂ ∂ +_V

)

⇒ s φ φ λ µ µ φ 4 2 1

(

)

(

φ

)

β φ λ φ β 2 2 2 2 2 2 , 1 −       ∂ ∂ − ∂ ∂ − ⇒ e V s

(

)

∫

_{d x} −_ge− _R+∂ _φ∂µ_φ+_V µ φ 4 _{. (2.13)}

Furthermore, always from the eq. (2.6), we obtain:

(

(

)

)

2 1 1 , 2 2 2 2 2 2 ⋅       ∂ ∂ − ∂ ∂ = − φ β β φ λ λ φ e V s s . (2.14)

Hence, from the eqs. (2.2) and (2.3), we have that

=−λ

∫

_τ

(

_φ'2−_β'2+ −2φ

)

2 d Ve S s

(

(

)

)

      ∂ ∂ − ∂ ∂ − = − 2 2 2 2 2 2 1 , 4 1 φ β β φ λ φ e V s

∫

(

)

− + −_β φ φ τ _'2 _'2 2 Ve d , (2.15) where φ =φ−ln

∫

(

d3x/λ3_s

)

− 3β.

3. On some equations concerning the noncommutative Kantowski-Sachs quantum model.[3]

The Hamiltonian of General Relativity without matter is =

∫

(

Η+ Ηj

)

j N N x d H 3 , (3.1) where Η=G_ijklΠijΠkl −h1/2R( )3 , Ηj =2D_iΠij. (3.2) Hence, we can write the (3.1) as

=

∫

(

Π Π − ( )+ Πij

)

i j kl ij ijkl h R N D NG x d H 3 1/2 3 2 . (3.2b)

Units are chosen such that h=c=16πG=1. The quantity ( )3

R is the intrinsic curvature of the spacelike hypersurfaces, D_i is the covariant derivative with respect to h_ij, and h is the determinant of h_ij. The momentum Π_ij canonically conjugated to hij, and the DeWitt metric G_ijkl are

Π_ij =−h1/2

(

K_ij −h_ijK

)

, (3.2c) G_ijkl = h−1/2

(

h_ikh_jl +h_ilh_jk−h_ijh_kl

)

2 1

where K_ij =−

(

∂_th_ij −D_iN_j −D_jN_i

)

/

(

2N

)

is the second fundamental form. The super-Hamiltonian constraint Η≈0 yields the Wheeler-DeWitt equation

1/2 ( )3 Ψ

[ ]

=0      + ij kl ij ijkl h R h h h G δ δ δ δ . (3.2e)

The Kantowski-Sachs line element is

2 2 2 2

( )

2 2

( )

(

2 2 2

)

sin θ ϕ θ d d t Y dr t X dt N ds =− + + + . (3.3)

In the Misner parametrization, (3.3) is written as

2 2 2 2 3β 2 2 3β 2 3

(

_θ2 _sin2_{θ ϕ}2

)

d d e e dr e dt N ds =− + + − − Ω + _{. (3.4)}

From (3.1) and (3.2), the Hamiltonian of General Relativity for this metric is found to be

(

)

(

)

      Ω − − + − Ω + = Η = Ω _{2exp 2 3} 24 24 3 2 3 exp 2 2 β β P P N N H . (3.5)

The Poisson brackets for the classical phase space variables are

{

Ω,P_Ω

}

=1,

{

β,Pβ

}

=1,

{

PΩ,Pβ

}

=0,

{

Ω,β

}

=0. (3.6)

For the metric (3.4), the super-Hamiltonian constraint Η≈0 is reduced to Η=ξh≈0, (3.7) where = exp

(

3 +2 3Ω

)

24 1 β ξ , h=−P_Ω2+Pβ2−48exp

(

−2 3Ω

)

≈0. (3.8) Hence, the eq. (3.7) can be rewrite as

exp

(

3 2 3

)

[

48exp

(

2 3

)

]

0 24

1 _{+ Ω −} 2₊ 2_{− − Ω ≈}

=

Η β P_Ω P_β . (3.8b)

Now, let us introduce a noncommutative classical geometry in the model by considering a Hamiltonian that has the same functional form as (3.5) but is valued on variables that satisfy the deformed Poisson brackets

{

Ω,PΩ

}

=1,

{

β,Pβ

}

=1,

{

PΩ,Pβ

}

=0,

{

Ω,β

}

=θ . (3.9)

The equations of motion in this case are written as

Ω& =−2P_Ω, _P&Ω=−_{96 3e}−2 3Ω_{, β =2P −96 3θe}−2 3Ω

β

& _{, =0}

β

The solution for Ω

( )

t and β

( )

t are

( )

[

(

)

]

        − = Ω 2 ₀ 2 0 0 3 2 cosh 48 ln 6 3 t t P P t _β β ,

( )

(

₀

)

₀

[

(

₀

)

]

0 0 0 tanh2 3 2P t t P P t t t = β − +β −θ β β − β . (3.11)

Now, we may achieve the solutions above by making use of the auxiliary canonical variables Ω _c and β_c, defined as _c θ P_β 2 + Ω = Ω , _c = − P_Ω 2 θ β β , P_Ω = P_Ω c , Pβc =Pβ. (3.12)

The Poisson brackets for these variables are

{

Ωc,PΩc

}

=1,

{

βc,Pβc

}

=1,

{

PΩc,Pβc

}

=0,

{

Ωc,βc

}

=0. (3.13)

As the equations of motion in the canonical formalism in the gauge N =24exp

(

− 3β−2 3Ω

)

we have c P c =− Ω Ω& 2 , _Ω =− −2 3Ω 3 96 e P c & _, − Ω − = 2 3 3 48 2P e c c θ β& β , Pβc =0 & _{. (3.14)} whose solutions are

( )

[

(

)

]

0 0 0 2 3 2 cosh 48 ln 6 3 0 2 2 β β β θ P t t P P t c +         − = Ω ,

( )

(

₀

)

₀

[

(

₀

)

]

0 0 0 tanh2 3 2 2P t t P P t t t c = β − + − β β − θ β β ,

( )

[

(

₀

)

]

0 0tanh2 3P t t P t P c =− − Ω β β ,

( )

0 β β t P P c = . (3.15)

Finally, from (3.12) and (3.15) we can recover the solution (3.11). The Wheeler-DeWitt equation, for the Kantowski-Sachs universe, is

[

₋ ˆ2₊ ˆ2₋48exp

(

₋2 3_Ω

)

]

_Ψ

(

_Ω,

)

₌0

Ω Pβ β

P , (3.16)

where Pˆ_Ω=−i∂/∂Ω and Pˆβ =−i∂/∂β. Hence, the eq. (3.16) can be rewritten also

48exp

(

2 3

)

(

,

)

0 2 2 = Ω Ψ         Ω − −       ∂ ∂ − +       Ω ∂ ∂ − − β β i i . (3.16b) A solution to equation (3.16) is Ψ

(

Ω

)

= 3

(

− 3Ω

)

4 , eiν βK_iν e ν β , (3.16c)

where K_iν is a modified Bessel function and

ν

is a real constant.

Now, we fix the gauge N =24exp

(

− 3

β

−2 3Ω

)

in (3.5). The Wheeler-DeWitt equation for the noncommutative Kantowski-Sachs model is

[

−PΩ2c +Pβ2c −48exp

(

−2 3Ωc

)

]

Ψ

(

Ωc,

β

c

)

=0, (3.17)

which is the Moyal deformed version of (3.16). By using the properties of the Moyal product, it is possible to write the potential term (which we denote by V to include the general case) as

V

(

_{c c}

)

(

_{c c}

)

V _c i _c i

(

_{c c}

)

V

( )

(

_{c c}

)

c c β β β θ β θ β β β , ˆ, ˆ , 2 , 2 , , Ψ Ω = Ω Ψ Ω      ∂ − ∂ + Ω = Ω Ψ Ω
_Ω , (3.18) where c P c β θ _ˆ 2 ˆ ˆ _{=Ω −} Ω , c P c+ Ω = ˆ 2 ˆ ˆ _β θ β . (3.19)

Equation (3.19) is nothing but the operatorial version of equation (3.12). The Wheeler-DeWitt equation then reads

[

₋ ˆ2 ₊ ˆ2 ₋48exp

(

₋2 3_Ωˆ ₊ 3 ˆ

)

]

_Ψ

(

_Ω ,

)

₌0

Ωc Pc c Pc c c

P β θ β β . (3.20)

In our time gauge N =24exp

(

− 3

β

−2 3Ω

)

, the Hamiltonian H =Nξh, with ξ and h defined in Eq. (3.8), reduces simply to h . We can therefore use h to generate time displacements and obtain the equations of motion for Ω_c

( )

t and

β

_c

( )

t as

( )

[

]

_{( )} ( ) ( )( )t t c t t c c c c c c c c c c S h i B t β β β β = Ω = Ω = Ω = Ω Ω ∂ ∂ − =       Ω = Ω& 1 ˆ ,ˆ 2 , (3.21)

( )

[ ]

_{( )} ( )

(

)

(

)

( ) ( )t t iS iS c c t t c c c c c c c c c c c e R e R i S h i B t β β β β β

θ

θ

β

β

β

==Ω Ω ==Ω Ω                 ⋅ ⋅ ∂ − Ω − − ∂ ∂ =       = 1 ˆ ,ˆ 2 48 3 Re exp 2 3 3 & _{. (3.22)}

As long as Ω_c

( )

t and

β

_c

( )

t are known, the minisuperspace trajectories are given by

( )

t _c

( )

t S

[

_c

( ) ( )

t _c t

]

c β θ β , 2∂ Ω − Ω = Ω , (3.23)

( )

t _c

( )

t S

[

_c

( ) ( )

t _c t

]

c β θ β β , 2∂ Ω + = _Ω . (3.24) A solution to (3.17) is

(

)

. 2 3 3 exp 4 , 3                         − Ω − = Ω Ψ

β

ν

νθ

β ν ν i c i c c e K c _(3.25)

Once a quantum state of the universe is given as a super-position of states

(

)

∑

= ⋅                         − Ω − = Ω Ψ ν ν β ν ν

νθ

β

iS c i i c c C e K R e c 2 3 3 exp 4 , 3 , (3.26)

the universe evolution can be determined by solving the system of equations constituted by (3.21) and (3.22) and substituting the solution in (3.23) and (3.24).

4. On some equations concerning the spectral action principle associated with a noncommutative space, applied to the Einstein-Yang-Mills system. [4]

The basic data of Riemannian geometry consists in a manifold M whose points x∈M are locally labelled by finitely many coordinates xµ∈R, and in the infinitesimal line element ds ,

ds2= g_µνdxµdxν

. (4.1)

The laws of physics at reasonably low energies are well encoded by the action functional, I= I_E +I_SM (4.2) where =

∫

R gd x G I_E 4 16 1

π is the Einstein action, which depends only upon the 4-geometry and

where I_SM is the standard model action, I_SM =

∫

L_SM , L_SM =L_G+L_GH +L_H +L_Gf +L_Hf. The action functional I_SM involves, besides the 4-geometry, several additional fields: bosons G of spin 1 such as γ, W and ± Z , and the eight gluons, bosons of spin 0 such as the Higgs field H and fermions f

of spin 1/2 , the quarks and leptons.

To test the following spectral action functional

Traceχ D+ ψ,Dψ      Λ , (4.3)

we shall first consider the simplest noncommutative modification of a manifold M . Thus we replace the algebra C∞

( )

M of smooth functions on M by the tensor product Α=C∞

( )

M ⊗M_N

( )

C

where M_N

( )

C is the algebra of N×N matrices. We shall compare the spectral action functional (4.3) with the following

I =

∫

R gd x+IYM 4 2 2 1

κ

(4.4)

where I_YM =

∫

(

L_G+L_Gf

)

gd4x _{is the action for an SU}

( )

_{N Yang-Mills theory coupled to fermions}

in the adjoint representation. Hence, the eq. (4.4) can be rewritten also

I=

∫

R gd4x+

∫

(

L_G +L_Gf

)

gd4x

2 2

1

κ . (4.4b)

The coupling of the Yang-Mills field A with the fermions is equal to

ψ

,D

ψ

ψ∈Η. (4.5) The operator D=D₀+A+JAJ∗ is given by

(

)

_            − ⊗ + ⊗ + ∂ = i i N a u T A g i e D _αγ _µ ω_µ 0 _µ 2 1 1 (4.6)

where ω_µ is the spin-connection on M : ωµ ωµabγ_ab 4 1 = and i

T are matrices in the adjoint representation of SU

( )

N satisfying

(

i j

)

ij

T T

Tr =2δ .

With regard to compute the square of the Dirac operator given by (4.6), this can be cast into the elliptic operator form:

P=D =−

(

g ∂ ∂ ⋅ +A ∂µ+B

)

µ ν µ µν ₁ 2 (4.7)

where 1 , Aµ and B are matrices of the same dimensions as D and are given by: Aµ =

(

2ωµ−Γµ

)

⊗1_N −ig014⊗AµiTi

B=

(

∂µω_µ+ωµω_µ−Γνω_ν +R

)

⊗1_N−ig₀ω_µ⊗AµiTi. (4.8) We shall now compute the spectral action for this theory given by

χ

(

ψ Dψ

)

m D Tr ₂ , 0 2 +       (4.9)

where the trace Tr is the usual trace of operators in the Hilbert space Η , and m₀ is a (mass) scale to be specified. The function χ is chosen to be positive and this has important consequences for the positivity of the gravity action.

Using identities:

( )

( )

∫

∞ − − − Γ = 0 1 1 dt Tre t s P Tr s s tP Re

( )

s ≥0 (4.10)

and the heat kernel expansion for

∑

∫

(

) ( )

≥ − − ≅ 0 , n M n d m n tP x dv P x a t Tre (4.11)

where m is the dimension of the manifold in C∞

( )

M , d is the order of P and dv

( )

x = gdmx where gµν is the metric on M appearing in equation (4.7). If s=0,−1,... is a non-positive integer then _Tr

( )

_P−s _{is regular at this value of s and is given by}

( )

− s = P Tr Res

( )

_n d n m s a s ₋ = Γ .

From this we deduce that

( )

∑

( )

≥ ≅ 0 n n na P f P Trχ (4.12)

where the coefficients f_n are given by

=

∫

∞

( )

0 0 u udu f χ , =

∫

∞

( )

0 2 u du f χ , _{2( 2)}

( )

1n ( )n

( )

0 n f ₊ = − χ , n≥0 (4.13)

and a_n

( )

P =

∫

a_n

(

x,P

) ( )

dv x . Hence, the eq. (4.12) can be rewritten also

( )

∑

≥ ≅ 0 n n f P Trχ

∫

a_n

(

x,P

) ( )

dv x . (4.13b)

The Seeley-de Witt coefficients a_n

( )

P vanish for odd values of n . The first three a_n’s for n even are: _a

(

_{x P}

) ( )

m/2_Tr

( )

1 0 , 4 − = π

(

) ( )

      + − = − Tr R E P x a m 1 6 4 , /2 2 π

(

) ( )

= −

[

(

− + − + µνρσ

)

1+ µνρσ µν µν µ µ π Tr R R R R R R P x a m 12 ; 5 2 2 360 1 4 , /2 2 4 −60RE+180E2+60E;_µµ+30Ω_µνΩµν

]

, (4.14) where E and Ω are defined by _µν

E =B−gµν

(

∂_µω'_ν+ω'_µω'_ν−Γ_µνρω'_β

)

, Ω_µν =∂_µω'_ν−∂_νω'_µ+

[

ω'_µω'_ν

]

, ω_µ= _gµν

(

_Aν −Γν ⋅1

)

2 1

' . (4.14a)

The Ricci and scalar curvature are defined by

_{Rµρ =Rµν}ab_eν_beaρ, µ ν µν a b ab_{e e}

R

R= . (4.14b)

Now, it is possible evaluate explicitly the spectral action (4.9). Using equations (4.8) and (4.14) we find: E= R⊗ ⊗ _N + iγµν ⊗gF_µνi Ti 4 4 1 4 1 1 _, i i N ab ab T gF i R_{µν µν} µν = γ ⊗ − ⊗ Ω 4 2 1 4 1 1 _{. (4.15)}

From the knowledge that the invariants of the heat equation are polynomial functions of R , R_µν,

µνρσ

R , E and Ω and their covariant derivatives, it is then evident from equation (4.15) that the _µν spectral action would not only be diffeomorphism invariant but also gauge invariant. The first three invariants are then

( )

=

∫

M gd x N P a_{0 2} 4 4π ,

( )

=

∫

M gRd x N P a_{2 2} 4 48π ,

( )

∫

(

)

    + − − + ⋅ = M i i F F g N R R R R R R g x d N P a µν µν µνρσ µνρσ µν µν µ µ π 2 2 4 2 4 120 7 8 5 ; 12 360 16 1 . (4.16)

For the special case where the dimension of the manifold M is four, we have a relation between the Gauss-Bonnet topological invariant and the three possible curvature square terms:

_R∗_R∗_{=R R −4R R}µν _+R2 µν µνρσ µνρσ (4.17) where R R εµνρσε_αβγδR_µναβRγδ_ρσ 4 1 ≡ ∗

∗ _{. Moreover, we can change the expression for}

_{( )}

P

µνρσ C instead of R_µνρσ where C_µνρσ =R_µνρσ −

(

g_µ[ρR_νσ]−g_ν[ρR_µσ]

)

+

(

g_µρg_νσ −g_µσg_νρ

)

R 6 1 (4.18)

is the Weyl tensor. Using the identity:

2 3 1 2R R R C C R R = + µν µν − µνρσ µνρσ µνρσ µνρσ (4.19)

we can recast a₄(P)into the alternative form:

( )

∫

(

)

      + + + − = ∗ ∗ i i F F N g R R R C C g x d N p a _µνρσ µνρσ _µµ _µν µν π 2 4 2 4 11 12 ; 120 1 20 3 48 (4.20)

and this is explicitly conformal invariant. The Euler characteristic χ_E is related to R∗R∗ by the relation _E = ₂

∫

d4x gR∗R∗ 32 1 π χ . (4.21)

It is also possible to introduce a mass scale m₀ and consider χ to be a function of the dimensionless variable _      2 0 m P

χ . In this case terms coming from a_n

( )

P ,n>4 will be suppressed by

powers of ₂ 0 1 m :      + + − + + =

∫

∫

∫

µµ µνρσ µνρσ π 10 ; 1 20 3 12 48 4 4 4 2 2 0 4 0 4 0 2 m f d x g m f d x gR f d x g C C R N I_b          +    + + ∗ ∗ ₂ 0 2 ₁ 0 20 11 m F F N g R R _µνi µνi . (4.22)

Normalizing the Einstein and Yang-Mills terms in the bare action we then have:

0 2 0 2 2 2 0 8 1 1 24 G f Nm π κ π = ≡ , 12 2 1 2 0 4 ₌ π g f , (4.23) and (4.22) becomes:

∫

_      + + + + + = ∗ ∗ i i b d x g R e a C C c R R d R F F I µν µν µ µ µνρσ µνρσ κ 4 1 ; 2 1 0 0 0 0 2 0 4 _{, (4.24)} where ₂ 0 0 1 80 3 g N a = − , _{0 0} 3 2 a c =− , _{0 0} 3 11 a d =− , _{2 0} 4 0 0 4 f Nm e π = . (4.25)

∫

   +       − − − + + = _R∗_R∗ g N C C g N f Nm R g x d I_{b 2} 0 2 0 0 2 4 0 2 0 4 1 40 1 80 3 4 2 1 µνρσ µνρσ π κ   +       − − i i F F R g N µν µν µ µ 4 1 ; 1 80 11 ₂ 0 . (4.25b)

The action for the fermionic quark sector is given by

(

Q,D_qQ

)

(4.26), while the leptonic action have the simple form

(

L,DlL

)

(4.27). According to universal formula (4.3) the spectral action for

the Standard Model is given by:

Tr

[

χ

(

D2/m₀2

)

]

+

(

ψ,Dψ

)

, (4.28)

where

(

ψ,Dψ

)

will include the quark sector (4.26) and the leptonic sector (4.27). Calculating the bosonic part of the above action, we have the following result:

(

  + +     − + =

∫

∫

∗

∫

µ µ π π π 4 12 ; 5 40 1 4 2 4 5 4 3 4 5 9 4 2 4 2 4 2 2 2 0 4 0 2 4 0 _{f d x g} m _{f d x g R y H H} f _{d x g R} m I

)

+ + +      − + − + ∗ ∗ ∗ ∗ α µνα µν µν µν µ µ µνρσ µνρσC y D H D H RH H g G G g F F C R R 2 ₀₃2 i i ₀₂2 6 1 3 18 11

(

)

(

)

]

_      + − + + ∗ ∗ 2 0 2 2 2 2 01 1 0 ; 3 3 5 m H H y H H z B B g _µν µν _µµ , (4.29)

where we have denoted

      + + = 02 2 0 2 0 2 3 1 e u d k k k Tr y ,       +       ₊ = 04 2 2 0 2 0 2 3 1 e u d k k k Tr z , D_µH _µH i g02A_µασαH i g01B_µH 2 2 − − ∂ = . (4.30)

Normalizing the Einstein and Yang-Mills terms gives:

₂ 0 2 2 2 0 1 4 15 κ π = f m , ₂4 1 2 03 ₌ π f g , ₀₃2 ₀₂2 ₀₁2 3 5 g g g = = . (4.31)

Relations (4.31) among the gauge coupling constants coincide with those coming from SU

( )

5 unification. To normalize the Higgs field kinetic energy we have to rescale H by:

H y g H 03 3 2 → . (4.32)

∫

_

(

)

  + + + + + + + − = ∗ ∗ ∗ i i b d x g R H H aC C b R c R R d R e G G I µ _µνρσ µνρσ _µµ _µν µν κ 4 1 ; 2 1 0 0 0 2 0 0 2 0 2 0 4

(

)

  + − + + + ∗ 2 0 2 0 2 4 1 4 1 H H H R H D B B F F µ ξ λ µν µν µνα α µν , (4.33) where ₂ 0 2 0 3 4 κ µ = , ₂ 03 0 8 9 g a =− , b₀=0, _{0 0} 18 11 a c =− , _{0 0} 3 2 a d =− , 4 0 0 2 0 4 45 m f e π = , ₄ 2 2 03 0 3 4 y z g = λ , 6 1 0= ξ . (4.34)

This action has to be taken as the bare action at some cutoff scale Λ . The renormalized action will have the same form as (4.33) but with the bare quantities κ₀,µ₀,λ₀,a₀ to e₀ and g₀₁,g₀₂,g₀₃

replaced with physical quantities.

5. Review of some equations concerning various aspects of Loop Quantum Gravity. [5] We introduce the volume associated with a region Ω⊂Σ, where Σ is a spatial manifold of fixed topology (and no boundary). We have that:

( )

∫

∫

∫

Ω = Ω ≡ Ω = Ω p c n b m a abc mnp E E E x d E x d xe d V ~ ~ ~ ! 3 1 ~ 3 3 3 _{ε ε . (5.1)}

Writing V ≡ V

( )

Σ , we first use the substitution

e_ma

( )

x _mnp abcE E_bnE_cp

( )

x

{

A_ma

( )

x,V

}

4 1 ~ ~ ~ 1/2 γ ε ε = = − _(5.2)

to recover the spatial dreibein. The second trick is to eliminate the extrinsic curvature using a doubly nested bracket. The first bracket is introduced by rewriting

K_ma

( )

x 1

{

A_ma

( )

x,K

}

γ = where

( )

∫

Σ = Σ ≡ am a m E xK d K K : 3 ~ . (5.3)

The second bracket comes in through identity

( )

( )

      = F x V E E E x K mnc abc n b m a _, ~ ~ ~ 1 2 / 3 ε γ . (5.4)

Canonical quantisation in the “position space representation” now proceeds by representing the dreibein as a multiplication operator, and the canonical momentum by the functional differential operator

( )

( )

x e i x _a m m a δ δ h = Π . (5.5)

With these replacements, the classical constraints are converted to quantum constraint operators which act on suitable wave functionals. The diffeomorphism and Lorentz constraints become H_a

( )

xΨ e

[ ]

=0, L_ab

( )

xΨ e

[ ]

=0. (5.6)

They will be referred to as “kinematical constraints” throughout. Dynamics is generated via the Hamiltonian constraint, the Wheeler-De Witt (WDW) equation

H₀

( )

xΨ e

[ ]

=0. (5.7)

It is straightforward to include matter degrees of freedom, in which case the constraint operators and the wave functional Ψ

[ ]

e,... depend on further variables (indicated by dots). The functional

[ ]

e,...

Ψ is sometimes referred to as the “wave function of the Universe”, and is supposed to contain the complete information about the Universe “from beginning to end”.

Now we represent the connection A_ma by a multiplication operator, and sets

( )

( )

x A i x E _a m m a δ δ h = ~ . (5.8)

The WDW functional depending on the spatial metric (or dreibein) is replaced by a functional

[ ]

A

Ψ living on the space of connections (modulo gauge transformations). The spatial metric must be determined from the operator for the inverse densitised metric

( )

( )

x A

( )

x A x gg _a n a m mn δ δ δ δ 2 h − = . (5.9)

Furthermore, the spatial volume density is obtained from

( )

( )

( )

x A

( )

x A

( )

x A i x E x g _c p b n a m mnp abc δ δ δ δ δ δ ε ε 6 ~ _h3 = = . (5.10)

For the quantum constraints the replacement of the metric by connection variables leads to a Hamiltonian which is simpler than the original WDW Hamiltonian. Allowing for an extra factor of

e (and assuming e≠ ,0 ∞) the WDW equation becomes

(

( )

)

( )

x A

( )

x Ψ A

[ ]

=0 A x A F _c n b m mna abc δ δ δ δ ε . (5.11)

There is at least one interesting solution if one allows for a non-vanishing cosmological constant Λ . Using an ordering opposite to the one above, and including a term Λ with the volume density g

(5.10), the WDW equation reads

( )

( )

(

( )

)

6

( )

__Ψ

[ ]

=0      _Λ − _Λ A x A i x A F x A x A_ma _nb mnc mnp _pc abc δ δ ε δ δ δ δ ε h . (5.12) This is solved by

[ ]

( )

      Λ = Ψ

∫

Σ Λ d xL A i A exp 3 _CS h , (5.13)

with the Chern-Simons Lagrangian L_CS = A∧dA+iA∧A∧A. Thence, the eq. (5.12) can be also rewritten

( )

      ∧ ∧ + ∧ Λ

∫

Σd xA dA iA A A A i 3 exp h

( )

( )

(

( )

)

6

( )

_=0      _Λ − x A i x A F x A x A_ma _nb mnc mnp _pc abc δ δ ε δ δ δ δ ε h . (5.13b)

Loop Quantum Gravity makes use of wave functions which have singular support in the sense that they only probe the gauge connection on one-dimensional networks embedded in the three-dimensional spatial hypersurface Σ . By definition, each network is a graph Γ embedded in Σ and consisting of finitely many edges e_i∈Γ and vertices v∈Γ. The edges are connected at the vertices. Each edge e carries a holonomy h_e

[ ]

A of the gauge connection A . The wave function on the spin network over the graph Γ can be written as

[ ]

(

[ ] [ ]

, ,...

)

,

2 1

,ψ A =ψ he A he A

ΨΓ (5.14)

where the ψ is some function of the basic holonomies associated to the edges e∈Γ.

The wave functionals (5.14) are called cylindrical, because they probe the connection A only “on a set of measure zero”. With regard the definition of the space of spin network states, we introduce a suitable scalar product. In Loop Quantum Gravity this is the scalar product of two cylindrical functions Ψ_{Γ,{ } { }j,C}

[ ]

A and Ψ_{Γ,'{ } { }j',C'}

[ ]

A and it is defined as

ΨΓ_{,{ } { }}j_,C Ψ'Γ_{',{ } { }}j_',C_' =0 if Γ≠Γ' _{{ } { } { } { }}

∫∏

_{{ } { }}

(

)

_{{ } { }}

(

)

Γ ∈ Γ Γ Γ Γ Ψ = Ψ i i e e C j e C j e C j C j, ' ,' ', ' dh , , h1,... ' ', ', ' h1,... , ψ ψ if Γ=Γ', (5.15)

where the integrals

∫

dh_e are to be performed with the SU

( )

2 Haar measure.

With regard the form of the quantum Hamiltonian one starts with the classical expression written in loop variables. Despite the simplifications brought about by the following equation

( )

(

2

) (

2 2

)

0 2 3 2 2 2 1 4 1 2 1 ~ ~ K K K e e R e F E E_am _bn _{mnc ab ab ab ab} abc _− =− Η − + −     Π − Π Π − = γ γ γ ε , (5.16)

the Hamiltonian constraint is:

[ ]

∫

(

)

_{[ ]} Σ     + − = b n a m mnc abc n b m aE F K K E E xN d N H 3 1 2 2 1 ~ ~ ~ det 1 γ ε . (5.16b)

In order to write the constraint in terms of only holonomies and fluxes, one has to eliminate the inverse square root as well as the extrinsic curvature factors. This can be done using the relations (5.2) – (5.4). Inserting these into the Hamiltonian constraint one obtains the expression

[ ]

∫

{

}

(

)

{

}{

}

{

}

Σ     + − = d xN Tr F A V A K A K A V N H mnp _{mn p} 1 _m, _n, _p, 2 1 , 2 3 _{ε γ . (5.17)}

6. On some equations concerning the dynamics of vector mode perturbations including quantum corrections based on Loop Quantum Gravity. [6]

The perturbed densitized triad and Ashtekar connection around a spatially flat Friedmann-Robertson-Walker (FRW) background are given by

a i a i a i p E E = δ +δ ,

(

i

)

a i a i a i a i a i a K k K A =Γ +γ =γ δ + δΓ +γδ , (6.1)

where p and kγ are the background densitized triad and Ashtekar connection. In a canonical formulation, the Einstein-Hilbert action can be written equivalently using the Ashtekar connection and densitized triad as

∫ ∫

(

)

Σ _     + + Λ − = E LA G N C NC G x d dt S_{EH i}a _{t a}i i _i a _a γ π 8 1 3 _{, (6.2)} where _{Λ ,} i a

N and N are Lagrange multipliers of the Gauss, diffeomorphism and Hamiltonian

constraints. In triad variables, a Gauss constraint appears which generates internal gauge rotations of phase space functions because triads whose legs are rotated at a fixed point correspond to the same spatial metric. This constraint is given by

( )

∫

∫

(

)

Σ Λ = Σ Λ ∂ + = Λ a k j a k ij a i a i i i _{d x E A E} G G x d G ε γ π 3 3 8 1 : . (6.3)

Using the perturbed form of basic variables (6.1), it can be reduced to

( )

∫

(

)

Σ Λ + = Λ a k k ia j a a ij i E k K p x d G G ε δ ε δ π 3 8 1 . (6.4)

The diffeomorphism constraint generates gauge transformations corresponding to spatial coordinate transformations of phase space functions. Its general contribution from gravitational variables is given by

[ ]

∫

∫

[

]

Σ = Σ − = i i a b i i ab a a a a G d xN F E AG G C xN d N D 3 3 8 1 : γ π , (6.5)

where the subscript “G” stands for “gravity” to separate the term from the matter contribution. Using the expression of the perturbed basic variables (6.1), one can reduce the diffeomorphism constraint to

[ ]

∫

[

(

)

(

)

]

Σ − ∂ − ∂ = d k d k c k c k c a G d x N p K k E G N D δ δ δ δ π 3 8 1 . (6.6)

In a canonical formulation, the Hamiltonian constraint generates “time evolution” of the spatial manifold for phase space functions satisfying the equations of motion. Its gravitational contribution in Ashtekar variables is (see also (5.16b))

[ ]

∫

(

(

)

_{[ ]}

)

Σ − + = k d j c i cd jk i d k c j G E E F K K E xN d G N H 3 21 2 det 1 16 1 γ ε π . (6.7)

Using the general perturbed forms of basic variables and the expression of curvature k b j a jk i i a b i b a i ab A A A A

F =∂ −∂ +ε , one can simplify (6.7). Up to quadratic terms is given by

[ ]

_∫

(

)

(

)

(

)

Σ           − + − − = j c c j d j c k k d j c j d k c d k c j G E K p k K K p E E p k p k N x d G N H δ δ δ δ δ δ δ δ δ δ π 2 2 6 16 1 2 3 2 2 3 , (6.8)

with δN =0 for vector modes.

With regard the quantum corrected Hamiltonian constraint, this is given by

[ ]

_∫

(

)

(

)

(

)

(

)

Σ           − + − − = j c c j d j c k k d j c j d k c d k c j a i Q G E K p k K K p E E p k p k E p N x d G N H α δ δ δ δ δ δ δ δ δ δ δ π 2 2 6 , 16 1 2 3 2 2 3 (6.9), where α

(

p,δE_ia

)

is the correct function, now also depends on triad perturbations.

To study quantum gravity effects, we have introduced a quantum correction function

(

a

)

i

E p δ

α ,

which depends on phase space variables. Having a new expression for the Hamiltonian constraint, there could be an anomaly term of quantum origin in the constraint algebra. A non-trivial anomaly in the algebra could occur in the Poisson bracket between HQ

[ ]

N

G and

[ ]

a G N

D . This bracket turns out to be

{

[ ]

[ ]

}

(

)

( )

(

)

_      ∂ ∂ + ∂ ∂ ∂ =

∫

Σ j d k c d k c j c j a G Q G E p p E p k N N p x d G N D N H α δ δ δ δ α δ π 3 1 3 8 1 , 3 2 . (6.10)

With regard the quantum dynamics, there are also the holonomy corrections. Hence, it is possible write the following expression for the corrected Hamiltonian constraint

[ ]

∫

(

)

Σ      +       −       − = j d k c d k c j Q G E E k p k p N x d G N H δ δ δ δ γ µ γ µ γ µ γ µ π 2 2 3 2 3 sin 2 1 sin 6 16 1

(

)

(

)

          − + j c c j d j c k k d j c E K k p K K p δ δ γ µ γ µ δ δ δ δ 2 2 sin 2 . (6.11)

Further, a non-trivial anomaly in the algebra can occur between the Poisson bracket between

[ ]

N HQ G and

[ ]

a G N D

{

[ ]

[ ]

}

[ ]

∫

(

)

Σ ∂ ⋅ +       − = j c a G a G Q G d xp N G N D k k p N N D N H δ π γ µ γ µ 3 8 1 2 2 sin , _               −       +       ∂ ∂ ⋅ p E k k k p p p N δ cj γ µ γ µ γ µ γ µ 2 2 2 sin sin . (6.12)