aa r X i v : . [ phy s i c s . g e n - ph ] O c t Position-Dependent Mass Quantum systems andADM formalism
Davood Momeni * Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khodh123, Muscat, Sultanate of OmanE-mail: [email protected]
The classical Einstein-Hilbert (EH) action for general relativity (GR) is shown to be formallyanalogous to the classical system with position-dependent mass (PDM) models. The analogyis developed and used to build the covariant classical Hamiltonian as well as defining an alter-native phase portrait for GR. The set of associated Hamilton’s equations in the phase space ispresented as a first-order system dual to the Einstein field equations. Following the principlesof quantum mechanics, I build a canonical theory for the classical general. A fully consistentquantum Hamiltonian for GR is constructed based on adopting a high dimensional phase space.It is observed that the functional wave equation is timeless. As a direct application, I present analternative wave equation for quantum cosmology. In comparison to the standard Arnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, I extended my analysis beyond thecovariant regime when the metric is decomposed into the 3 + International Conference on Holography, String Theory and Discrete Approaches in Hanoi PhenikaaUniversityAugust 3-8, 2020Hanoi, Vietnam * Speaker. © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/
DM Quantum systems & ADM formalism
Davood Momeni
1. Introduction
Classical mechanics is Galilean invariant, i.e, time parameter t and position coordinate q ( t ) are explicitly functions of each other. Since quantum mechanics is Galilean invariant there isno simple way to build a locally Lorentz invariant theory with single particle interpretation (thepossible version known as Klein-Gordon has field theoretic realization). In GR, we have difficultyto interpret time as we did in classical mechanics. Furthermore GR is locally Lorentz invariant. Thesimple reason is that in the context of GR (or any other classical gauge theory for gravity ),time t is just a coordinate and is no longer considered as a parameter (in non-relativistic mechanics andQM the time t is an evolution parameter). As a result ,the analogue to coordinate (in the variationalprocess), the Riemannian metric g ab ( x c ) , , a , b = ... x c =( t , x A ) , A = , ..
3. That makes quantum gravity difficult to construct. A way to address quantumgravity is string theory [1] and the other well studied candidate is loop quantum gravity [2]. There isalso semi quantum gravity, when we keep the background classical and let fields simply propagateon it. With both of these nice ideas , still we have the problem of “disappearance of time” [3]. Ifwe adopt any of these proposals, it seems that time problem remains unsolved both in the quantumgravity and cosmology [4]. Time problem has a rather long history [5] . There is a simple way toaddress quantum gravity without time considered in canonical quantization in metric variables:[6].With all of the above historical backgrounds and many others, we are still looking for a fullycovariant canonical quantum theory for gravity which make sense same as we know for usual qmechanics. It is necessary to find an appropriate representation for Lagrangian of the gravity (hereGR as the best tested one ).With a suitable covariant definition of the conjugate momentum we define a Hamiltonian. Fur-thermore we need to adopt a well defined phase space. In that phase space one can build Poissonbrackets easily, and then by replacing the classical bracket with the Dirac bracket, we can find asuitable fully consistent Hamiltonian for quantum GR. Later, one can build an associated (func-tional) Hilbert space and develop all the concepts of ordinary quantum mechanics systematically.This is a plan to find a successful quantum theory for gravity or as it is known, quantum gravity.During studying non standard classical dynamical systems I found a class of Lagrangian modelswith second order time derivative of the position ¨ q ( t ) (configuration coordinate q ( t ) ). It is easy toshow that a wide class of such models reduce to the position dependence mass (PDM) models as itwas investigated in literature [7]-[9]. It is obviously interesting to show that whether GR reduces tosuch models. This is what I investigate in this letter. I show that the classical Einstein-Hilbert (EH)Lagrangian reduces to the position-dependent -mass (PDM) model up to a boundary term. ThenI adopted the standard quantization scheme for a PDM system and I suggested a fully covariantquantum Hamiltonian for GR. The functional wave equation for the metric proposed naturally andthen it was developed for quantum cosmology.My observations initiated when I studied GR as a classical gauge theory [10]. As everyoneknew, GR has a wider class of symmetries provided by the equivalence principle. It respects gaugetransformations (any type of arbitrary change in the coordinates, from one frame to the other x a → ˜ x a ) [10]. Consequently GR is considered just a classical gauge theory for gravity. There is alsoa trivial hidden analogue between GR and classical mechanics (see table I). In GR as a classicaldynamical system (but with second order derivative of the position ), if we make an analogous1 DM Quantum systems & ADM formalism
Davood Momeni
Riemannian metric g ab with the coordinate q ( t ) and if we use the spacetime derivatives of themetric ∂ c g ab (which is proportional to the Christoffel symbols Γ dab ), instead of the velocity , i.e,the time variation of the coordinate ˙ q , and by adopting a symmetric connection , we can rewrite EHLagrangian in terms of the metric, first and second derivatives of it. It looks like a classical systemin the form of L ( q , ˙ q , ¨ q ) (see TABLE I). In this formal analogy, the classical acceleration term inthe classical models under study now ¨ q is now replaced with the second derivative for metric i.e, ∂ e Γ dab . Integration part by part from this suitable representation of the EH Lagrangian reduces it toa PDM system where we will need to define a super mass tensor as a function of the metric insteadof the common variable mass function m ( q ( t )) in classical mechanic. Table 1:
Analogy between classical mechanics and GR
Model Position first derivative second derivative massClassical PDM q ( t ) ˙ q ( t ) ¨ q ( t ) scalar m ( q ) GR g ab ∂ d g ab ∂ e ∂ d g ab super mass tensor M abldeh given in eq.(2.6)In this letter, I focus on the classical EH action for GR as an analogy to the model investigatedin the former above. The notation for Einstein-Hilbert action is S EH = Z d x L GR (1.1) We don’t consider matter fields action S matt , although the matter fields are playing crucialroles as source for non vacuum classical solution in GR. There is a reason on why we focusedon the vacuum GR: If one quantizes the matter sector, the same quantization technique can’tbe applied to the geometrical part. The resulting theory will be considered as a semi classicalqunatum gravity, i.e, it defines a theory where the quantized matter fields propagate on afully classical background. There is no quantization in any form imposed on the geometryof the spacetime. For example we don’t consider non commutative of the spacetime struc-tures or etc. The resulting theory is fully consistent and works as a semi classical regime asit proposed and developed for example in Ref. [11]. Furthermore, the matter Lagrangian isconsidered non-minimally coupled to the kinetic sector of the theory,we believe the canonicalapproach used in this paper can’t easily explored for theories with matter Lagrangian withnon-minimal coupling between geometry and matter fields because they will violate equiv-alent principle as well as they can’t easily be interpreted as unforced position dependencequnatum mechanical system. By the above reasons we only construct a canonical quantiza-tion for GR as an empty space theory with non trivial( at least one) classical geometry. Thetechnique can be applied if one consider locally distributions of the matter field with a lo-cal and singular matter energy momentum tensor T µν ∝ δ ( ~ x − ~ x ′ ) at singularity point x = x ′ .For example when the geometry is static, time independent and spherically symmetric withSchwarzschild vacuum solution. The key idea is to realize GR as a classical dynamical systemwith second order derivative. The Einstein field equations are derived as a standard variationalproblem subjected to a set of appropriate boundary conditions. Since the idea of GR is to find thebest geometry for a given source of matter fields, it is formally equivalent to the classical mechan-ics. By combining all these similarities I end up to an equivalent representation of EH action as a2
DM Quantum systems & ADM formalism
Davood Momeni classical Lagrangian in the form : L GR = L ( | g | , ∂ a | g | , g bc , g bc , Γ abc , ∂ d Γ abc ) . (1.2)here | g | ≡ det ( g µν ) is determinant of the metric tensor, and ∂ β is coordinate derivative. We canwrite the above Lagrangian formally in a more compact form as L GR = L GR ( | g | , ∂ | g | , g , g , ∂ g , ∂ g ) . (1.3)here we abbreviated by g ≡ g ab , g ≡ g ab , | g | ≡ det ( g ab ) = ( g ) , g . ∂ g ≡ Γ abc , ∂ g = Γ bla ≡ g bl , a + g la , b − g ba , l . We adopt metricity condition ∇ a g bc = | g | − / ∂ a ( | g | / g bc ) ≡ | g | − / ∂ ( | g | / g ) = ∂ . g = − g . ∂ g . g . The plan of this letter is as following: In Sec. (2) I showed that GR Lagrangianreduces to a PDM fully classical system with a super mass tensor of rank six. In Sec. (3) I constructa consistent super phase space as well as a set of Poisson brackets. As an attempt to break thecomplexity of the field equations, I show that gravitational field equations reduced top a set offirst order Hamilton’s equations. In Sec. (4) I define quantum Hamiltonian simply by replacingthe classical brackets with Dirac brackets. The functional wave equation will be proposed and bysolving it, we can obtain generic wave function for a fully canonical quantized Riemannian metric.As a concrete example,in Sec. (5) I solve functional wave equation for a cosmological background.Some asymptotic solutions are presented. The last section is devoted to summarize results.
2. Super Mass Tensor for GR as PDM classical system
We adopt the conversion of indices as Ref. [16]. The EH action for GR In units 16 π G ≡ S EH = Z M d x √ gR ≡ Z M d x L GR . (2.1)The Ricci scalar R is composed of the metric and its first and second derivatives. The first aimis to express the integrand (Lagrange density L GR ) is as the form from which PDM kinetic termis obvious. We note that the Lagrangian density eq. (2.2) is a purely kinetic form, with a PDMeffective mass. This adequate representation can be obtained from the definition of Γ bla , this will beclear if we rewrite the Lagrangian in following equivalent form(note that the Lagrangian enjoys anexchange of indices symmetry a → d in the first two terms),in action presented in eq.(1.3) one caneliminate the second derivative term ∂ de g ab simply by integrating by part and using the metricitycondition ∇ a g bc =
0, by taking into the account all the above requirements a possible equivalentform for Lagrangian of the GR is given by: L GR = p | g | (cid:16) g al g be Γ bla ∂ h g eh + g be g dh Γ bld ∂ l g eh + g al g bd g te Γ tld Γ bea − g al g bd g te Γ tla Γ bed (cid:17) . (2.2)and S EH = R d x L GR + B . T here by B . T we mean boundary term defined as B . T = Z ∂ M p | h AB | h BD h AL Γ BLA | x D = constant + Z ∂ M p | h AB | h BD h AL Γ BLD | x A = constant . (2.3)We can re express the above GR Lagrangian in our convenient notations as L GR = p | g | (cid:16) g . ∂ g . g . ∂ g + g . g . ∂ g . ∂ g + g . g . g . ∂ g . ∂ g − g . ∂ g . g . g . ∂ g (cid:17) . (2.4)3 DM Quantum systems & ADM formalism
Davood Momeni
Note that by ” . ” we mean tensor product(we adopt Einstein summation rule). From the aboverepresentation we can realize { g , g } as two fields , in analogy to the Dirac Lagrangian where thefermionic pairs ψ , ¯ ψ appeared . The difference here is due to the fact that the pair of objects g , g depend on each other as we know g . g = δ , the Kronecker delta, however in the Dirac Lagrangianthe norm ψ ˙ ψ = I . In our program we wont use this duality and we will focus on the coordinatesrepresentation of the GR Lagrangian, i.e, eq.(2.2) . If we substitute the definition of Gamma termsand combine the theory, we obtain the final form for the Lagrangian as a PDM system for coordinate g ab (or as a tensor version for k-essence [12]): L GR = p | g | M abldeh ∂ a g bl ∂ d g eh . (2.5)here M abldeh = | g | − / ∂ L GR ∂ ( ∂ a g bl ) ∂ ( ∂ d g eh ) is defined as super mass tensor. This super mass tensor pre-viously introduced in [13],[14]. In both the Refs. [13],[14], the authors the reduction of thegravitational action in the favor of the horizon thermodynamics and mainly to explain the fea-tures of the emergent gravity paradigm.
An alternative form for (2.5) is L GR = √ | g | M ∂ g ∂ g . Itis equivalent to the classical Lagrangian of PDM systems L = M ( q ) ˙ q for one dimensional, posi-tion dependent mechanical system. As we expected in GR, the mass term transformed to a higherorder (here rank six) tensor. The explicit form for the super mass tensor is expressed as following: | g | / M = | g | / M a b l d e h = g al g bd g te × (2.6) (cid:16) δ a a δ b b δ l e + δ b a δ a b δ l e − δ l a δ b b δ a e (cid:17) (cid:16) δ d d δ h l δ e t − δ h d δ d l δ e t + δ e d δ h l δ d t (cid:17) − g al g bd g te (cid:16) δ a b δ b d δ l e + δ b b δ a d δ l e − δ b b δ l d δ a e (cid:17) (cid:16) δ d a δ h l δ e t − δ h a δ d l δ e t + δ e a δ h l δ d t (cid:17) + g al g bd g te (cid:16) δ d a δ e b δ h e + δ e a δ d b δ h e − δ h a δ e b δ d e (cid:17) (cid:16) δ a d δ l l δ b t − δ l d δ a l δ b t + δ b d δ l l δ a t (cid:17) ) − g al g bd g te (cid:16) δ d b δ e d δ h e + δ e b δ d d δ h e − δ e b δ h d δ d e (cid:17) (cid:16) δ a a δ l l δ b t − δ l a δ a l δ b t + δ b a δ l l δ a t (cid:17) )+ g al δ d a g be g dh δ e e δ h h (cid:16) δ b b δ a d δ l l + δ a b δ l d δ b l − δ b b δ l d δ a l (cid:17) + g al g be g dh δ d d δ e e δ h h (cid:16) δ a a δ b b δ l l + δ l a δ a b δ b l − δ l a δ b b δ a l (cid:17) + g al δ a a g be g dh δ b e δ l h (cid:16) δ e b δ d d δ h l + δ d b δ h d δ e l − δ e b δ h d δ d l (cid:17) + g al g be g dh δ a d δ b e δ l h (cid:16) δ d a δ e b δ h l + δ h a δ d b δ e l − δ h a δ e b δ d l (cid:17) . Having the Lagrangian of GR given in eq. (2.5), one can define a canonical pair of position conju-gate momentum ( g , p ) and construct a phase space. This is what we are going to do in next section. Before to have more process, we would like to motivate of our study. If one can reduce theGr to a Hamiltonian form, the reduced form can be used for example in the numerical rela-tivity and canonical quantization. The Hamiltonian formulation is a preferred approach tostudy dynamics of the classical systems as well as an attempt to quantization of the models.As we know the Hamiltonian formulation of GR requires a + decomposition of the spaceand time coordinates. We refer the readers to the mini review on the subject presented by DM Quantum systems & ADM formalism
Davood Momeni [15] for more discussions and seeing of how the Hamiltonian formalism works in classicalbackgrounds.
3. Super phase space
The phase space description of the classical model presented in eq.(2.5) is very straightfor-wardly done, by defining the super conjugate momentum tensor is p rst = ∂ L GR ∂ ( ∂ r g st ) = √ g (cid:16) M rstdeh ∂ d g eh + M ablrst ∂ a g bl (cid:17) . (3.1)Note that the mass tensor M rstdeh ∂ d g eh = M ablrst ∂ a g bl . A possible classical Hamiltonian willbe H GR = p | g | M abldeh M rstabl p rst M uvwdeh p uvw . (3.2)A possible Poisson’s bracket { F , G } P . B adopted to this system is: { F ( g mn , p stu ) , G ( g mn , p stu ) } P . B = ∑ (cid:16) ∂ F ∂ g ab ∂ G ∂ p rst − ∂ F ∂ p rst ∂ G ∂ g ab (cid:17) .. (3.3)or in our notation it simplifies to the following expression { F ( g , p ) , G ( g , p ) } P . B = ∑ (cid:16) ∂ F ∂ g ∂ G ∂ p − ∂ F ∂ p ∂ G ∂ g (cid:17) . (3.4)and specifically for our super phase coordinates ( g ab , p rst ) , I I postulate that { g ab , p rst } P . B = c r δ rsab . (3.5)Here δ rstab is the generalized Kronecker defined as [17] δ rstab = δ s [ a δ tb ] (3.6)In the above Poisson’s bracket, with structure constants c r provide a classical minimal volume forsuper phase space (zero for Poisson’s bracket same objects ). We have now full algebraic structuresin the super phase space and canonical Hamiltonian. As a standard procedure, we can write downHamilton’s equations as first order reductions of the Euler-Lagrange equations derived from theLagrangian given in eq. (2.5)(Einstein field equations). This is one of the main results of this letterand I will address it in the next short section. The set of Hamilton’s equations derived from the Hamiltonian (6.3), are defined automaticallyusing the Poisson’s bracket are given as following: ∂ a g bl = { g bl , H GR } P . B = ∂ H GR ∂ p abl , (3.7) ∂ a p abl = { p abl , H GR } P . B = − ∂ H GR ∂ g bl . (3.8)5 DM Quantum systems & ADM formalism
Davood Momeni
We explicitly can write this pair of Hamilton’s equations given as follows:2 √ g ∂ a g bl = M a ′ b ′ l ′ deh M rsta ′ b ′ l ′ M uvwdeh × (cid:16) δ ua δ vb δ wl p rst + δ ra δ sb δ tl p uvw (cid:17) . (3.9) − √ g ∂ a p abl = M a ′ b ′ l ′ deh p rst M rsta ′ b ′ l ′ p uvw ∂ M uvwdeh ∂ g bl + M a ′ b ′ l ′ deh p rst M uvwdeh ∂ M rsta ′ b ′ l ′ ∂ g bl (3.10) + ∂ M a ′ b ′ l ′ deh ∂ g bl p rst M rsta ′ b ′ l ′ p uvw M uvwdeh + M a ′ b ′ l ′ deh ∂ p rst ∂ g bl M rsta ′ b ′ l ′ p uvw M uvwdeh + M a ′ b ′ l ′ deh p rst M rsta ′ b ′ l ′ ∂ p uvw ∂ g bl M uvwdeh . This set of first order partial differential equations are considered the first phase space alterna-tive to the original gravitational field equations. When we succeed to write a covariance Hamilto-nian, the Hamilton’s equations are first order version of the Einstein field equations.
Reduction ofthe Einstein field equation to a system of the first order Hamiltonian equations were investi-gated in several former works. For example in the classical Ref. [18], a + decompositiontechnique introduced to explore a possible Hamiltonian formulation of the GR. The generalideas about how to write Hamilton constraints discussed by Dirac in [19].Following Dirac’sidea, later the authors in Ref.[20] proved that for GR, a set of the canonical variables existedonly for a given restricted submanifold. This submanifold is specified by putting all the canon-ical variables equal to zero. In the important work by DeWitt [21] , the author presented aconventional canonical formulation of GR. The classical field theory perspective introducedin Ref.[22] along the standard other gauge theories. In comparison to the CMB anisotropiesin the standard cosmology, the Hamiltonian formalism used to compute the anisotropies [23]. They showed that how such Hamiltonian formalism helps us to calculate the model predic-tions in linear theory for any standard classical cosmological background. The Hamiltonianformalism extensively introduced and applied to calculate cosmological perturbations in cos-mological backgrounds in Ref [24]. The set of equations given in (3.7,3.8) are defined whena first first order Hamiltonian version of the field equations for a generic Lorentzian metric. I be-lieve that one can integrate this system as a general non autonomous dynamical system for a givenset of the appropriate initial values of the metric and super momentum given as a specific initialposition x a (not specific time as is commonly considered as the initial condition in QG literature).A remarkable observation that the system may possess chaotic behavior and doesn’t suffer fromCauchy’s problem. We have now the classical Hamiltonian and the set of Poisson brackets. Nowwe can develop a qunatum version and obtain qunatum Hamiltonian for GR. This will be done inthe next section.
4. Quantization of GR
In this section, I’m going to define appropriate forms for Dirac brackets simply by defining,ˆ π rst ≡ − i ¯ h r ∂∂ ˆ g st , (4.1) h ˆ g ab , ˆ π rst i = i ¯ h r δ stab (4.2)6 DM Quantum systems & ADM formalism
Davood Momeni
Instead of the usual fundamental reduced Planck’s constant ¯ h we required to define a vector formof the reduced Planck’s constant ¯ h r . The reason is that even the classical phase space spannedby the (cid:16) g ab , p rst (cid:17) has more degrees of freedom (dof), basically the total dof is 10 (= × ) dimensional for a Riemannian manifold. The Dirac constant ¯ h is proportional to the minimumvolume of the phase space V defined as ω = Z D ( g ab , p rst ) . (4.3)where the D ( g ab , p rst ) is a measure for the super phase space and D ( g ab , p rst ) is a covariant volumeelement. We obviously see that the ω is related to the dof of the system, for example if thesystem has f numbers of dof, then the minimal volume of the phase space is given as ¯ h f and here¯ h ∝ f − log ( ω ) , note that in our new formalism f = ≫
1, as a result the effective || ¯ h r || ≪ ¯ h .A remarkable observation is that the super mass tensor M = M abldeh is a homogeneous (or-der 6) of the metric tensor. Using the formalism of quantization for PDM systems the canonicalquantized Hamiltonian for GR is: ∧ H GR ( ˆ g ab , ∂∂ ˆ g st ) = − f / rstuvw ¯ h r ∂∂ ˆ g st h f / rstuvw ¯ h u ∂∂ ˆ g vw i . (4.4)here the auxiliary, scaled super mass tensor f rstuvw is defined as bellows, f rstuvw ≡ | g | − / M abldeh M rstabl M uvwdeh . (4.5)It is adequate to write the quantum Hamiltonian in the following closed form: ∧ H GR ( ˆ g ab , ˆ π mnp ) = h | g | − / MMM i / ˆ π h | g | − / MMM i / ˆ π . (4.6)where p is contravariant component of the super momentum p , etc. The above quantization ofHamiltonian is covariant since we didn’t specify time t from the other spatial coordinates x A . Themodel is considered as a timeless model, i.e, there is no first order time derivative in the final waveequation like ∂∂ t , and the associated functional second order wave equation which is fully locallyLorentz invariant as well as general covariant is expressed as: − f / rstuvw ¯ h r ∂∂ ˆ g st h f / rstuvw ¯ h u ∂∂ ˆ g vw Ψ ( ˆ g ab ) i = E Ψ ( ˆ g ab ) Note that in our suggested functional wave equation for Ψ ( ˆ g ab ) , we end up by the covariant (nofirst order derivative) of the functional Hilbert space, furthermore all the physical states are static(i.e., no specific time dependency) and consequently we have a covariant full evolution for ourfunctional. We should emphasize here that the theory which we studied in this paper is con-sidered as an attempt to construct quantum mechanics on a classical GR background (see forexample independent works in Ref. [25]). There is no simple field-theoretic interpretationfor the Hamiltonian which we obtained in this work as well as any other brackets are simplynon-quantum field theoretical ones. In his approach, we can’t reach the renormalizability asit has been investigated in many other alternative quantum gravity scenarios.
We believe thatmy model is a subclass of the timeless models of QG. Building quantum gravity via timeless phasespace investigated in the past by some authors mainly recent work [26]. Our approach is com-pletely different and independent from the others. We will study qunatum cosmology as a directapplication of our wave equation in the next section.7
DM Quantum systems & ADM formalism
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5. Quantum cosmology
In flat, Friedmann-Lemaître-Robertson-Walker (FLRW) model with Lorentzian metric g ab = diag ( , − a ( t ) Σ ) where Σ is the unit metric tensor for flat space, in coordinates x a = ( t , x , y , z ) ,the non vanishing elements of the super mass tensor defined in eq.(2.6), M abldeh = − a − δ a δ d δ BL δ EH (5.1)here B , L , E , H = , , f rstuvw f rstuvw = − a / δ u δ r δ VW δ ST . (5.2)The functional wave equation reduces to the hypersurfaces σ coordinates X A = ( x , y , z ) :3¯ h a / ∂ Ψ ( ˆ g AB ) ∂ ˆ g SS ∂ g VV = E Ψ ( ˆ g AB ) (5.3)and in the coordinates for FLRW metric it reduces simply to the following ordinary differentialequation a Ψ ′′ ( a ) − Ψ ′ ( a ) − Ea / h Ψ ( a ) = a . If we know boundary conditions, one canconstruct an orthonormal set of eigenfunctions using the Gram Schmidt process. Furthermore theabove single value wave equation can be reduce to a standard second order differential equation forwave function Ψ ( a ) = √ a φ ( a ) , φ ′′ ( a ) − (cid:0) Ea / h + a (cid:1) φ ( a ) = φ ( a ) ∝ ( a if a → [ √ E √ h a ] if a → ∞ (5.6)and one can build an exact solution via Poincare’s asymptotic technique, i.e, by suggesting φ ( a ) = ζ ( a ) a e √ E √ h a (5.7)and ζ ( a ) will come as a transcendental (hypergeometric) function ζ ( a ) = / ¯ h / e − (cid:16) √ ea / h + (cid:17) √ a C I − q e a / h + C I q e a / h (5.8)8 DM Quantum systems & ADM formalism
Davood Momeni here I ν ( y ) is the modified Bessel function of the first kind and the eigenvalue E (positive, negative orzero ) can be discrete as well as continuous (bound states for E < E =
0, the generic wave function is given by Ψ ( a ) = N + Na , a ∈ [ , ∞ ) (5.9) The above-approximated wave functions obtained for an empty spacetime. In classical cos-mology, there are no dynamics for a cosmological background without including matter fields.The resulting FLRW equations simply lead to a static Universe with zero Hubble’s parameter H = . It shows that there is no stable equilibrium around this static Universe. The solutionis fully rejected by considering cosmological facts. In our quantum mechanical model, we aredealing with wave function instead of the unique classical scale factor or Hubble’s parameter.In comparison to the simple Harmonic oscillator in standard QM, the wave function obtainedin the above cases give us a non zero scale factor or Hubble’s parameter. That is consider-able because in ordinary QM, the wave function provides possible probabilistic access to theforbidden regions of the system with inaccessible classical energy regimes. In our case, thefollowing wave functions simply can be understood as we do in QM. In quantum FLRW for-malism, the wave function even for ground state as it has been obtained in expression (5.9)gives non-classical expectation values for different quantities. If for example one computethe < ∧ H GR ( ˆ g ab , ˆ π mnp ) > , it shows extended amplitudes for the quantum wave function for theUniverse. Is also possible to define (formally) uncertainty expressions for ∆ a , ∆ p a and basedon the above discussions those exact solution for wave functions are somehow still useful inthe absence of a full quantum gravity scenario based on a preferred time foliation.
6. Note about ADM decomposition formalism and reduced phase space
Working with an extended phase space with a conjugate momentum with one more indexdoesn’t look very friendly at all, although that is the unique way to define a fully covariant form forthe phase space as well as a purely kinetic Lagrangian for GR. If one adopt the ADM decompositionof the metric g ab as follows[27, 28], ds = g ab dx a dx b = h AB dx A dx B + N A dx A dx + ( − N + h AB N A N B )( dx ) (6.1)here x is time, A , B = , , h AB is spatial metric. It is easy toshow that the set of the first order Hamilton’s equations presented in the previous section reducesto the ADM equations, only if one consider t as dynamical time evolution. Basically if we recallthe super conjugate momentum π rs = ∂ L GR ∂ ˙ g st = √ g (cid:16) M stdeh ∂ d g eh + M abl st ∂ a g bl (cid:17) . (6.2)Builiding the Hamiltonian in a standard format as H ADMGR = p | g | M abldeh M stabl π st M vwdeh π vw . (6.3)9 DM Quantum systems & ADM formalism
Davood Momeni
Briefly I wanna to mention here that although my formalism is worked with covariant derivativewithout specifying any coordinate as time (so technically is a timeless technique) if one turnsback to the standard metric decomposition in ADM and use the 0 component of super conjugatemomentum, again we can recover ADM Hamiltonian. I emphasis here that my construction wasbased on purely geometrical quantization of the GR action rather opting a standard time coordinate.
7. Final remarks
The canonical covariant quantization which I proposed here is a consistent theory. I startedit by basic principles, just by rewriting the GR action in a suitable form the Lagrangian reducedto a purely kinetic theory with position dependence mass term. In this equivalent form of the La-grangian, gradient of the metric tensor appears as a hypothetical scalar field. With such a simplequadratic Lagrangian, I defined a conjugate momentum corresponding to the metric tensor. Themass term for graviton derived as tensor of rank six . I developed a classical Hamiltonian using themetric and its conjugate momentum. It is remarkable that one can write classical Hamilton’s equa-tions for metric and momentum (super phase space coordinates) are analogous to the second ordernonlinear Einstein field equations. Later I replaced Poisson’s brackets with Moyal(Dirac) and Idefined a quantum Hamiltonian for GR. There is no time problem in this formalism because theoryis fully covariant from the beginning. As a direct application I investigated qunatum cosmology,i.e and wave function for a homogeneous and isotropic Universe. I showed that wave equationsimplifies to a linear second order ordinary differential equation with appropriate asymptotic solu-tions for very early and late epochs. In my letter I used an integration part by part to reduce GRLagrangian to a form with first derivatives of the metric. The price is to define two boundary termson the spatial boundary regions. Those terms vanish in any asymptotic flat(regular) metric. I noticehere that even if we didn’t remove second derivative terms using integration by part, it was possibleto define a second conjugate momentum r abcd = ∂ L GR ∂ ( ∂ a ∂ b g cd ) corresponding to the second derivativeof the metric ∂ a ∂ b g cd . If I impose a Bianchi identity between g ab , p rst , r abcd , it is possible to fix thisnew momentum in terms of the other one and the metric. A suitable Legendre transformation fromthe GR Lagrangian H GR = p rst ∂ r g st + r abcd ( ∂ a ∂ b g cd ) − L GR (7.1)with the Bianchi identity, { g ab , { p cde , r f ghi } P . B } P . B + { p cde , { r f ghi , g ab } P . B } P . B + { r f ghi , { g ab , p cde } P . B } P . B = . (7.2)I obtain a standard Hamiltonian without this new higher order momentum. Consequently themethod of finding a Hamiltonian based on the super phase spaces which I defined by g ab , p rst in Sec. II and the one with one more conjugate momentum i.e, g ab , p rst , r abcd leads to the sameresult.
8. Acknowledgments
This work supported by the Internal Grant (IG/SCI/PHYS/20/07) provided by Sultan QaboosUniversity . I thank Profs. Lawrence P. Horwitz, John. R. Morris for carefully reading my first10
DM Quantum systems & ADM formalism
Davood Momeni draft, very useful comments, corrections and discussions. The tensor manipulations of this doneusing WX M AXIMA platform. I thank Prof. T. Padmanabhan for remembering his previous relevantworks.
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