Effects of GUP in Quantum Cosmological Perfect Fluid Models
aa r X i v : . [ g r- q c ] A p r IISER(Kolkata)/GR-QCJune 11, 2018
Effects of GUP in Quantum Cosmological PerfectFluid Models
Barun Majumder Department of Physical Sciences,Indian Institute of Science Education and Research (Kolkata),Mohanpur, Nadia, West Bengal, Pin 741252,India
Abstract
Very recently authors in [5] proposed a new Generalized Uncertainty Principle(or GUP) with a linear term in Plank length. In this Letter the effect of thisGUP is studied in quantum cosmological models with dust and cosmic string asthe perfect fluid. For the quantum mechanical description it is possible to find thewave packet which resulted from the superposition of the stationary wave functionsof the Wheeler-deWitt equation. However the norm of the wave packets turnedout to be time dependent and hence the model became non-unitary. The loss ofunitarity is due to the fact that the presence of the linear term in Plank length inthe Generalized Uncertainty Principle made the Hamiltonian non-Hermitian.Keywords: quantum cosmology, GUP, minimal length
The idea that the uncertainty principle could be affected by gravity was first given by Mead[1]. Later modified commutation relations between position and momenta commonlyknown as Generalized Uncertainty Principle ( or GUP ) were given by candidate theoriesof quantum gravity ( String Theory, Doubly Special Relativity ( or DSR ) Theory andBlack Hole Physics ) with the prediction of a minimum measurable length [2, 3]. Similarkind of commutation relation can also be found in the context of Polymer Quantizationin terms of Polymer Mass Scale [4].The authors in [5] proposed a GUP which is consistent with DSR theory, String theoryand Black Hole Physics and which says[ x i , x j ] = [ p i , p j ] = 0 , (1) [email protected] x i , p j ] = i ~ (cid:20) δ ij − l (cid:18) pδ ij + p i p j p (cid:19) + l (cid:0) p δ ij + 3 p i p j (cid:1)(cid:21) , (2)∆ x ∆ p ≥ ~ (cid:2) − l < p > +4 l < p > (cid:3) ≥ ~ " l p h p i + 4 l ! ∆ p + 4 l h p i − l p h p i , (3)where l = l l pl ~ . Here l pl is the Plank length ( ≈ − m ). It is normally assumed that thedimensionless parameter l is of the order unity. If this is the case then the l dependentterms are only important at or near the Plank regime. But here we expect the existenceof a new intermediate physical length scale of the order of l ~ = l l pl . We also note thatthis unobserved length scale cannot exceed the electroweak length scale [5] which implies l ≤ . These equations are approximately covariant under DSR transformations butnot Lorentz covariant [3]. These equations also imply∆ x ≥ (∆ x ) min ≈ l l pl (4)and ∆ p ≤ (∆ p ) max ≈ M pl cl (5)where M pl is the Plank mass and c is the velocity of light in vacuum. It can be shown thatequation (2) is satisfied by the following definitions x i = x oi and p i = p oi (1 − l p o + 2 l p o ),where x oi , p oj satisfies [ x oi , p oj ] = i ~ δ ij . Here we can interpret p oi as the momentum atlow energies having the standard representation in position space ( p oi = − i ~ ∂∂x oi ) with p o = P i =1 p oi p oi and p i as the momentum at high energies. We can also show that the p term in the kinetic part of any Hamiltonian can be written as [5] p = ⇒ p o − l p o + O ( l ) + . . . . (6)Here we neglect terms O ( l ) and higher in comparison to terms O ( l ) to study the effectof the linear term in l in the first approximation as l = l l pl . The effect of this proposedGUP is studied for some well known quantum mechanical Hamiltonians in [5, 6].In this Letter we are going to study the effect of this GUP [5] ( only upto a linearterm in l ) in some selected quantum cosmological perfect fluid models with dust andcosmic string. For brief discussion on quantum cosmological perfect fluid models we cansee [7, 8, 9, 10, 11, 12, 13]. The expression for action in these quantum cosmological models with perfect fluid can bewritten as A = Z M d x √− g R + 2 Z ∂M d x √ h h ab K ab + Z M d x √− g P, (7)where h ab is the induced metric over three dimensional spatial hypersurface which is theboundary ∂M of the four dimensional manifold M and K ab is the extrinsic curvature.2ere units are so chosen that c = 16 πG = ~ = 1. The second term was first obtained in[14]. P is the pressure of the fluid and satisfies the equation of state P = αρ where ρ isthe energy density and − ≤ α <
1. In Schutz’s formalism [15, 16] the fluid’s four velocitycan be expressed in terms of three potentials ǫ , θ and S (here we are studying spatiallyflat FRW model so other potentials are absent in this model because of its symmetry), u ν = 1 h ( ǫ ,ν + θS ,ν ) . (8)Here h is the specific enthalpy, S is the specific entropy, ǫ and θ have no direct physicalmeaning. The four velocity also satisfy the normalization condition u ν u ν = 1 . (9)The metric for the spatially flat FRW model is ds = N ( t ) dt − a ( t ) (cid:2) dr + r ( dϑ + sin ϑdϕ ) (cid:3) , (10)where N ( t ) is the lapse function and a ( t ) the scale factor. Using Schutz’s formalism [15, 16]along with some thermodynamic considerations [10] it is possible to simplify the action.The final form of the super-Hamiltonian after using some canonical transformations [10, 7]can be written as H = N (cid:20) − p a a + p T a α (cid:21) . (11)The lapse function N plays the role of a Lagrange multiplier leading to the constraint H = 0. Here the only canonical variable associated with matter is p T and it appearslinearly in the super-Hamiltonian. The equation of motion ˙ T = ∂ H ∂p T = N a − α revealsthat in the gauge N = a α , T may play the role of cosmic time. Using usual quantizationprocedure we can get the Wheeler-deWitt equation for our super-Hamiltonian believingthat the super-Hamiltonian operator annihilates the wave function. So with p a → − i∂ a , p T → i∂ t and ˆ H Ψ( a, t ) = 0 we get ∂ Ψ ∂a + i a (1 − α ) ∂ Ψ ∂t = 0 . (12)Here we have considered a particular choice of factor ordering and our final results willbe independent of the different choices of factor ordering. Any two wave functions Φ andΨ must take the form [11, 9, 7] h Φ | Ψ i = Z ∞ a (1 − α ) Φ ∗ Ψ da (13)to make the Hamiltonian operator self-adjoint and the restrictive boundary conditionsbeing Ψ(0 , t ) = 0 or ∂ Ψ( a, t ) ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 = 0 . (14)To solve equation (12) we can use the method of separation of variables. WritingΨ( a, t ) = e − iEt φ ( a ) (15)3nd using (12) we get ∂ φ∂a + 24 Ea (1 − α ) φ = 0 . (16)The solutions of this equation can be written in terms of Bessel functions and we can nowwrite the stationary wave functions asΨ E = e − iEt √ a (cid:20) c J − α ) (cid:18) √ E − α ) a − α )2 (cid:19) + c Y − α ) (cid:18) √ E − α ) a − α )2 (cid:19)(cid:21) (17)where c , are the integration constants. To satisfy the first boundary condition of (14)we consider c = 0 and c = 0 (to avoid the divergence of the wave function in thelimit a → c to be a gaussian function of the parameter E . Setting s = √ E − α ) the expression for the wave packet can be written asΨ( a, t ) = √ a Z ∞ s ν +1 e − γs − i s (1 − α ) t J ν ( sa − α )2 ) ds = a (2 η ) − α − α ) e − a − α )4 η (18)where η = γ + i (1 − α ) t , ν = − α ) and γ is an arbitrary positive constant in thegaussian factor. To find the norm of the wave function for α = 0 (dust) we use equation(13) and we finally get h Ψ | Ψ i = Z ∞ a Ψ ∗ Ψ da = Γ( )3(2 γ ) . (19)So the norm is finite and independent of time. Similarly for α = − (cosmic string) wesee that h Ψ | Ψ i = Z ∞ a Ψ ∗ Ψ da = Γ( )4(2 γ ) (20)which is also finite and time independent. Now we are going to study the effect of the Generalized Uncertainty Principle (or GUP)in the context of the quantum cosmological models described above. Here we will study4wo cases, model with dust as the Schutz’s perfect fluid and the model with an equationof state P = − ρ (cosmic string). Throughout this whole process we will keep in mindthat equation (2) and (3) have a linear term in Plank length as l = l l pl . So we willneglect terms O ( l ) and higher in the first approximation whenever they appear in thecalculation. Due to GUP the p a term of the super-Hamiltonian (11) should be corrected.Following the arguments in [5] and using (6) we rewrite (11) as H = N (cid:20) − a ( p o − lp o ) + p T a α (cid:21) . (21)Here we have neglected terms O ( l ). Using usual quantization procedures we find ∂ Ψ ∂a + i l ∂ Ψ ∂a + i a (1 − α ) ∂ Ψ ∂t = 0 . (22)Using Ψ( a, t ) = e − iEt φ ( a ) we separate the variables and we get ∂ φ∂a + i l ∂ φ∂a + 24 Ea (1 − α ) φ = 0 . (23)As mentioned before we will study two cases. One with α = 0 and another with α = − . α = 0 (Dust) With α = 0 equation (23) reduces to ∂ φ∂a + i l ∂ φ∂a + 24 Eaφ = 0 . (24)This third order equation is very difficult to solve analytically. So we will try to solvethis equation approximately [17] in the region a ≈ l term can be written as φ = d √ a J (cid:18)r E a (cid:19) , (25)where d is one integration constant while there is a second one which is assigned to theBessel function of second kind, i.e. Y , and is set to zero to avoid the divergence in small a limit. As we are studying early universe cosmology so in the region a ≈ φ ≈ d √ a " (cid:0) (cid:1) (cid:18)r E (cid:19) a − (cid:0) (cid:1) (cid:18)r E (cid:19) a + . . . ≈ D a − D a , (26)where D = d (cid:0) (cid:1) (cid:18)q E (cid:19) and D = d (cid:0) (cid:1) (cid:18)q E (cid:19) . So clearly ∂ φ∂a = − D a .From (26) we see that for small a we can also consider the approximation φ ≈ D a andthe result we get is ∂ φ∂a = − Eφ . (27)5f we incorporate this result in equation (24) we get ∂ φ∂a + 24 Eaφ − i lEφ = 0 . (28)The solution of this equation is known in terms of Bessel functions and we can write thefinal form of the stationary wave functions asΨ E = c e − iEt √ a − i l J (cid:18) √ E ( a − i l ) (cid:19) (29)where c is one integration constant while there is a second one which is assigned to theBessel function of second kind, i.e. Y , and is set to zero to avoid the divergence in small a limit. In this case also we should construct the wave packet superposing these solutions.So for the wave packet we can writeΨ( a, t ) = Z ∞ A ( E )Ψ E ( a, t ) dE. (30)Defining s = √ E and considering A ( E ) to be a gaussian function (here we have chosen A = s e − γs ), the expression for the wave packet can be written asΨ( a, t ) = √ a − i l Z ∞ e − s (cid:0) γ + i t (cid:1) s J (cid:0) s ( a − i l ) (cid:1) ds . (31)This is a known integral [18] and finally we can writeΨ( a, t ) = ( a − i l )2 (cid:0) γ + i t (cid:1) e − ( a − i l )34 (cid:0) γ + i t (cid:1) . (32)A straightforward calculation givesΨ ∗ Ψ = (2 A ) − a e − γ A a e lt A a (33)where A = (cid:0) γ + t (cid:1) . As we are interested in the norm of the wave packet we haveto follow equation (13) and in this case we have to evaluate h Ψ | Ψ i = Z ∞ a Ψ ∗ Ψ da. (34)Using equation (33) we evaluate the square of the norm as h Ψ | Ψ i = (2 A ) − Z ∞ a e − γ A a e lt A a da = Γ( )3(2 γ ) + 332 2 ltγ (cid:0) γ + t (cid:1) . (35)Throughout this whole process we have neglected all the terms O ( l ) and higher. Clearlywe can see from equation (35) that the norm is time dependent and hence we can concludethat this quantum model is non-unitary. If we set l = 0 we will get back equation (19)and there the norm is time independent. So, keeping in mind this interesting result let usstudy the quantum model with cosmic string as the perfect fluid.6 .2 α = − (Cosmic String) If we consider a cosmic string fluid then equation (23) reduces to ∂ φ∂a + i l ∂ φ∂a + 24 Ea φ = 0 . (36)Approaching in the same way as we did in the dust case we can write φ = d √ a J (cid:0) √ E a (cid:1) (37)for l = 0. In the limit a → φ ≈ d (cid:0) (cid:1) (cid:18) E (cid:19) a − d (cid:0) (cid:1) (cid:18) E (cid:19) a + . . . ≈ D a − D a , (38)where D and D are the coefficients of a and a respectively. So clearly ∂ φ∂a = − D a .For small enough a the approximation φ ≈ D a yields ∂ φ∂a = − Eaφ . (39)Putting this in equation (36) we get ∂ φ∂a + (24 Ea − i lEa ) φ = 0 . (40)If we take x = a − i l the equation (40) reduces to ∂ φ∂x + (24 Ex + 216 l E ) φ = 0 . (41)Here also we will neglect the term O ( l ) and find the solution of equation (41). Thesolution is known and we now write the final form of the stationary wave functions:Ψ E = c e − iEt p ( a − i l ) J (cid:0) √ E ( a − i l ) (cid:1) . (42)To construct the wave packet superposing these solutions we have to evaluate equation(30) again in this case. Here we define s = √ E and choose A ( E ) in such a manner sothat we can easily do the integration as before. After a straightforward calculation wenow write the final form of the wave packet:Ψ( a, t ) = ( a − i l )2 (cid:0) γ + i t (cid:1) e − ( a − i l )44( γ + i t ) . (43)This equation implies Ψ ∗ Ψ = (2 A ) − a e − γ A a e l tA a (44)7here A = (cid:0) γ + t (cid:1) . Now using equation (13) in this case we evaluate the square ofthe norm of the wave packet and it turns out to be h Ψ | Ψ i = (2 A ) − Z ∞ a e − γ A a e l tA a da = Γ( )4(2 γ ) + lt γ (cid:0) γ + t (cid:1) . (45)In the whole process of the calculation we have neglected terms O ( l ) and higher. If l = 0we get back equation (20). So this model like the dust model is also non-unitary as thesquare of the norm is time dependent.Anisotropic quantum cosmological models are not unitary as the Hamiltonian operatorin those anisotropic models is Hermitian but not self-adjoint [11, 19, 12]. But in ourcase if we carefully study equations (21) and (22) we can understand that the effective-Hamiltonian operator which is defined by H eff = N (cid:0) ∂ ∂a + i l ∂ ∂a (cid:1) is not Hermitian orvery weakly Hermitian in the limit l →
0. So the loss of unitarity is due to the fact thatthe presence of a linear term in Plank length in the Generalized Uncertainty Principle ismaking the effective-Hamiltonian operator non-Hermitian.
With the very recently proposed Generalized Uncertainty Principle (or GUP) [5] we havestudied the flat minisuperspace FRW quantum cosmological model with dust and cosmicstring as the perfect fluid. This GUP has a linear term in Plank length and here we havestudied the effect of this term in the context of very early universe. In both the cases(dust and cosmic string) Schutz’s mechanism has allowed us to obtain the Wheeler-deWittequation for this minisuperspace in our early universe. Well behaved wave packet can beconstructed from the linear superposition of the stationary wave functions of the Wheeler-deWitt equation. While solving the Wheeler-deWitt equation we considered a particularchoice of factor ordering of the position and momentum operators present in the equationand it is seen that the behaviour of the constructed wave packet remains same for otherfactor orderings. The presence of the linear term in Plank length in the GUP made thenorm of the wave packet time dependent. So the model became non-unitary. But in thelimit l pl → Acknowledgements
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