Klein-Gordon-Wheeler-DeWitt-Schroedinger Equation
aa r X i v : . [ g r- q c ] S e p Klein-Gordon-Wheeler-DeWitt-Schr¨odingerEquation
Matej PavˇsiˇcJoˇzef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia;email: [email protected]
Abstract
We start from the Einstein-Hilbert action for the gravitational field in the presence of a“point particle” source, and cast the action into the corresponding phase space form. Thedynamical variables of such a system satisfy the point particle mass shell constraint, theHamilton and the momentum constraints of the canonical gravity. In the quantized theory,those constraints become operators that annihilate a state. A state can be represented by awave functional Ψ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schr¨odinger equation. The latter equation, besides the term due to gravity, also containsthe Schr¨odinger like term, namely the derivative of Ψ with respect to time, that occursbecause of the presence of the point particle. The particle’s time coordinate, X , serves therole of time. Next, we generalize the system to p -branes, and find out that for a quantizedspacetime filling brane there occurs an effective cosmological constant, proportional to theexpectation value of the brane’s momentum, a degree of freedom that has two discretevalues only, a positive and a negative one. This mechanism could be an explanation for thesmall cosmological constant that drives the accelerated expansion of the universe. The meaning of time in quantum gravity is still a matter of debate (for a recentreview see [1]). A possible resolution of this problem is to consider matter degrees offreedom from which, upon quantization, one can obtain the derivative of the wavefunctional with respect to a time variable [2] in the Wheeler-DeWitt equation [3]. Theidea is to introduce a reference fluid [4], which enables the identification of spacetimepoints and the occurrence of a time variable. Instead of a fluid, one can considera model with one point particle only [2]. In this letter we will further explore andadapt that model. We start with the Einstein-Hilbert action for gravity in the pres-ence of a “point particle” source that is in fact an extended object, like a ball, whosecenter of mass worldline satisfies the equations of motion for a point particle. Thenwe cast the action into the phase space form that involves the set of Lagrange mul-tipliers: α , the einbein on the particle’s worldline, N , the lapse and N i , the shiftfunctions that occur in the ADM decomposition [5] of the spacetime metric tensor1 µν . Variation of the action with respect to α , N and N i gives the mass shell con-straint, the Hamilton and the momentum constraint. In the quantized theory, suchsystem is described by a wave functional Ψ[ X µ , q ij ] that satisfies the Klein-Gordonand the Wheeler-DeWitt-Schr¨odinger equation. The latter equation contains, besidesthe usual Wheeler-DeWitt terms due to gravity, also the term δ ( x − X ) i∂ Ψ /∂X due to the point particle. In addition, the wave functional also satisfies the quantummomentum constraint that contains the term δ ( x − X ) i∂ Ψ /∂X i , i = 1 , , x = X ( τ ) of the worldline parametric equation x µ = X µ ( τ ), and fix theparameter τ by requiring X ( τ ) = τ . It turns out that the particle coordinate X serves as evolution parameter, just like in field theories. That X , which is not theparticle dynamical degree of freedom, serves as time is in agreement with the wellknown fact that time in quantum mechanics is not a dynamical degree of freedom, butmerely a parameter. According to this line of reasoning, we do not need to worry howto find a dynamical variable with the role of time. It comes out that t ≡ x = X ,i.e., the quantity that in special and general relativity we anyway call “time” , isindeed time, since it can serve as an evolution parameter. This happens, if we do notconsider gravity in empty space, but gravity in the presence of a point particle forwhich it is no problem to identify X as time.Next, we consider the gravity in the presence of many particles, and finally inthe presence of a p -brane. Then, instead of one time, we have the many fingeredtime X ( σ a ), a = 1 , , ..., p . The wave functional for the brane satisfies, besides theWheeler-DeWitt-Schr¨odinger equation, also the quantum p -brane constraints thatreplace the Klein-Gordon equation. We explore a special case of a spacetime fillingbrane, and obtain the positive or negative cosmological constant that depends onthe sign of the brane momentum p . The latter momentum, because of the p -braneconstraint, has two discrete values only, p = + µ B √ q and p = − µ B √ q , where µ B is the brane tension and q the determinant of the 3-space metric. The quantizedtheory then gives an expectation value h ˆ p i for the state that is a superposition ofthe eigenstates with positive and negative p . The effective cosmological constant,proportional to h ˆ p i , has thus a continuous range of possible values, including the onethat fits the observed accelerated expansion of the universe. At the end we discussthe possibility that a 3-brane in a higher dimensional bulk space is our world—a“brane world.” 2 The Einstein-Hilbert action with a “point par-ticle” matter term and its quantization
Let us consider the Einstein-Hilbert action for the gravitational field g µν ( x ), µ, ν =0 , , ,
3, in the presence of a “point particle” source, described by variables X µ ( τ ): I [ X µ , g µν ] = m Z d τ ( ˙ X µ ˙ X ν g µν ) / + κ Z d x √− g R (1)where κ ≡ / (16 πG ). It is well known that the Einstein equations with a point likesource have no solution, because a solution in in the vacuum around a source is theblack hole with a horizon, the black hole singularity being spacelike and cannot hencebe interpreted as a point particle worldline. However, for the sake of completeness,let me mention that alternative views can be found in the literature [6]. Leaving suchintricacies aside, we can nevertheless use the action (1) as an approximation to arealistic physical situation in which instead of a point particle we have an extendedsource, described by X µ ( τ, σ a ), with X µ ( τ ) being the center of mass coordinates. Inparticular, if the particle is a ball, then the parameters are σ a = ( R, θ, φ ), 0 < R < R ,0 < θ < π , 0 < φ < π , where R is greater than the Schwarzschild radius.Let us now consider the ADM split of spacetime, M , = R × R , . Then the 4Dmetric can be decomposed as g µν = (cid:18) N − N i N i , − N i − N j , − q ij (cid:19) (2)where N = p /g and N i = − g i , i = 1 , ,
3, are the laps and shift functions. Theinverse metric is g µν = (cid:18) /N , − N i /N − N j /N , N i N j /N − q ij (cid:19) (3)Here q ij is the inverse of q ij and N i = q ij N j .The gravitational part of the action (1) can be cast, by using Ref. [3], into thephase space form [7]: I G [ q ij , π ij , N, N i ] = Z dt d x (cid:2) π ij ˙ q ij − N H ( q ij , π ij ) − N i H i ( q ij , π ij ) (cid:3) , (4)where H = − κ G ij kℓ π ij π kℓ + κ √ qR (3) (5) H i = − j π ij (6)and G ij kℓ = 12 √ q ( q ik q jℓ + q iℓ q jk − q ij q kℓ ) (7)3s the Wheeler-DeWitt metric. If we vary the gravitational action with respect to N , N i , we obtain the constraints H = 0 (8) H i = 0 (9)Variation with respect to π ij gives the relation π ij = κ √ q ( K ij − Kq ij ) (10)where K ij = 12 N (D i N j + D j N i − ˙ q ij ) (11)The matter part of the action can also be cast into the phase space form: I m [ X µ , p µ , α ] = Z d τ h p µ ˙ X µ − α g µν p µ p ν − m ) i (12)To cast the matter part into a form comparable to the gravitational part of the action,we insert the integration over δ ( x − X ( τ )) d x , which gives identity. In both partsof the action, I m and I G , now stands the integration over d x . We identify x ≡ t .Splitting the metric according to (2), we have I m [ X µ , p µ , α, N, N i , q ij ]= Z d τ (cid:16) p µ ˙ X µ − α (cid:2) N ( p ) − q ij ( p i + N i p ) ( p j + N j p ) − m (cid:3)(cid:17) (13)Varying the total action I = I G + I m (14)with respect to α , N and N i we obtain the following constraints : δα : N ( p ) − q ij ( p i + N i p ) ( p j + N j p ) − m = 0 (15) δN : H = Z d τ αN δ ( x − X ( τ ))( p ) = − δ ( x − X ) N p (16) δN i : H i = Z d τ αN δ ( x − X ( τ )) q ij ( p j + N j p ) p = δ ( x − X ) q ij ( p j + N j p ) (17) Since a realistic source is extended, e.g. like a “ball”, R d τ δ ( x − X ( τ )) should be considered asan approximation to R d τ d σδ ( x − X ( τ, σ a )), so that, e.g., the constraint H = − δ ( x − X ) N p isan approximation to H = − R d σδ ( x − X ( σ a )) N p . H and H i = q ij H j are given in Eqs. (5),(6). Eq. (15), of course, is nothing butthe ADM splitting of the mass shell constrain g µν p µ p ν − m = 0 (18)In Eqs. (16),(17) we have performed the integration over τ , and used the equation p µ = ˙ X µ /α , that results from varying the action (12) with respect to p µ .Let us now use the relations p µ = g µν p ν and p µ = g µν p ν with the metrics (2),(3),and rewrite (16),(17) into the form with covariant components of momenta p , p i : H = − δ ( x − X ) 1 N ( p − N i p i ) (19) H i = − δ ( x − X ) p i (20)In a quantized theory, the constraints (15)–(17) become operator equations actingon a state vector. In the Schr¨odinger representation, in which X µ and q ij ( x ) arediagonal, the momentum operators are ˆ p µ = − i∂/∂X µ and ˆ π ij = − iδ/δq ij . Moreprecisely, momentum operators have to satisfy the condition of hermiticity, thereforethe latter definitions are not quite correct in curved spaces, and have to be suitablymodified. For instance, a possible definition [8] that renders ˆ p µ hermitian, and alsohelps to resolve the factor ordering ambiguity, is ˆ p µ = − i (cid:2) ∂ µ + ( − g ) − / ∂ µ ( − g ) / (cid:3) .An alternative procedure was proposed in Ref. [9]. Analogous holds for ˆ π ij .Choosing a gauge in which N = 1 , N i = 0, we have (cid:0) g µν ( X )ˆ p µ ˆ p ν − m (cid:1) Ψ = 0 (21)ˆ H Ψ = δ ( x − X ) i ∂ Ψ ∂T (22)ˆ H i Ψ = δ ( x − X ) i ∂ Ψ ∂X i (23)A state vector is represented by Ψ[ T, X i , q ij ( x )] that depends on the time parameter T ≡ X , the particle center of mass coordinates X i , and the 3-metric q ij ( x ). In otherwords, Ψ is a function of T, X i , and a functional of q ij ( x ). It satisfies simultaneouslythe Klein-Gordon equation (21), the Wheeler-DeWitt-Schr¨odinger like equation (22),and the quantum momentum constraint (23) in the presence of a point particle source.However, Eq. (22) is not the Schr¨odinger evolution equation; it is a constraint that We assume that the coupled system actually describes an extended particle whose center ofmass coordinates are X µ . This system can be envisaged to describe, e.g., the neutron that certainlyis extended, and yet only its center of mass coordinates can be considered in the wave function. Ifwe wish to use the above coupled system for description of electron and other fundamental particles,one has to assume that they are as well extended beyond their Schwarzschild radia. Otherwise those“particles” would be black holes. Since the underlying physical system whose description we havein mind, is in fact extended, it has classical solutions. x . Since x runs over the 3-manifold, we have in factan infinite set of constraints.Usually, for a quantum description of gravity in the presence of matter, one doesnot take the matter action in the form (12). Instead, one takes [10] for I m an actionfor, e.g., a scalar or spinor field, and then attempts to quantize the total action follow-ing the established procedure of quantum field theory. Here I have pointed out thatwe can nevertheless start from the point particle action (12) together with the corre-sponding gravitational action (4). After quantization, we arrive at the Klein-Gordonequation (21) and the equations (22),(23) that are the Wheeler-DeWitt equation,and the momentum constraint, with the terms due to the presence of point particlesource.The presence of the δ -distribution can be avoided, if we perform the Fouriertransform. The classical constraints (19),(20), with N = 1 , N i = 0, then become H ( k ) = − e i kX p (24) H i ( k ) = − e i kX p i (25)where H ( k ) = Z d x e i kx H , H i ( k ) = Z d x e i kx H i (26)and X ≡ X i , i = 1 , ,
3, is the particle’s position at fixed time T . Notice that k ≡ k i are the Fourier partners of the spacetime coordinates x , not of the particle position X . The quantum constraint are Z d x e i k ( x − X ) (cid:18) κ G ij kℓ δ δq ij δq kℓ + κ √ qR (3) (cid:19) Ψ = i ∂ Ψ ∂T (27) − Z d x e i k ( x − X ) q iℓ D j (cid:18) − i δδq jℓ (cid:19) Ψ = i ∂ Ψ ∂X i (28)The above k -dependent set of constraints (27),(28) replaces the set of constraints(22),(23). For a fixed k , Eq. (27) has the form of the Schr¨odinger equation, with theHamilton operator that contains the functional derivatives − iδ/δq ij . In the Hamil-tonian we have the integration over x , just as in the Hamiltonians of the usual fieldtheories.The zero mode Schr¨odinger equation, for k = , is Z d x (cid:18) κ G ij kℓ δ δq ij δq kℓ + κ √ qR (3) (cid:19) Ψ = i ∂ Ψ ∂T (29) We are not interested here in the issues of hermiticity and factor ordering, therefore the expres-sions with − iδ/δq ij have symbolical meaning only. In actual calculation one has to take suitablehermitian operators, and choose a factor ordering. of Eq. (29) is an approximate solution of our dynamical system.Correction terms to Ψ come from the contributions from the higher modes, k = ,in Eq. (27). Bear in mind that the momentum constraints are a consequence [11] ofthe conservation of the Hamiltonian constraint with respect to x . If instead of one, there are many particle sources, then the matter part of the action, I m , consists of the sum over single particle sources: I m [ X µn , p nµ , α n ] = X n Z d τ n h p nµ ˙ X µn − α n g µν p nµ p nν − m n ) i (30)As a consequence, in the constraints (19),(20), instead of a single δ -distribution, wehave a sum. The quantum equations (21)–(23) become (cid:0) g µν ( X n )ˆ p nµ ˆ p nν − m n (cid:1) Ψ = 0 (31)ˆ H Ψ = X n δ ( x − X n ) i ∂ Ψ ∂T n (32)ˆ H i Ψ = X n δ ( x − X n ) i ∂ Ψ ∂X in (33)The Wheeler-DeWitt equation thus becomes a multi fingered time equation. ItsFourier transform is Z d x e i kx H Ψ = i X n e i kX n ∂ Ψ ∂T n (34)We can single out one particle, denote its time and spatial coordinates as T and X ,respectively, and rewrite Eq. (34) according to Z d x e i k ( x − X ) H Ψ = i N − X n =1 e i k ( X n − X ) ∂ Ψ ∂T n + i ∂ Ψ ∂T (35)One particle, in the above case the N th one, was singled out and chosen as a clock thatmeasures a time T . The name ‘particle’ in the quantum equation (35) should be takenwith caution. In fact we have a system of many gravitationally interacting particles,described by Ψ[ T n , X n , q ij ( x )], n = 1 , , ..., N , and if particles are indistinguishable,one cannot say which particle is at which position. What we have singled out was infact one of the parameters T n , namely T N ≡ T .7nstead of a system of many particles, we can consider an extended source, forinstance, a p -brane. Then we have [12]–[16]: I m [ X µ , p µ , α, α a ] = Z d τ d p σ " p µ ˙ X µ − α µ B p | ¯ f | ( g µν p µ p ν + µ B ¯ f ) − α a ∂ a X µ p µ (36)Here µ B is the brane tension, τ, σ a , a = 1 , , ..., p , the brane time like and space likeparameters, α, α a , Lagrange multipliers, i, j = 1 , , ..., D −
1, the spatial indices ofthe D -dimensional spacetime in which the brane is embedded, and ¯ f ≡ det ¯ f ab thedeterminant of the induced metric ¯ f ab ≡ ∂ a X µ ∂ b X ν g µν .If we vary the action (36) with respect to α, α a , we obtain the p -brane con-straints [13, 15]: g µν p µ p ν + µ B ¯ f = 0 , ∂ a X µ p µ = 0 (37)and if we vary the total action I g + I m with respect to N, N i , we obtain the constraints H = − Z d p σ p δ D − ( x − X ( σ )) (38) H i = − Z d p σ p i δ D − ( x − X ( σ )) (39)Varying the action (36) with respect to p µ , we obtain the relation between mo-menta and velocities: p µ = µ B ˙ X µ p − ¯ fα (40)Squaring the latter equation and combining it with Eq. (37), we obtain α =˙ X µ ˙ X ν g µν .As in the case of a point particle, there are difficulties with classical equations ofmotion of the branes coupled to the gravitational field in all cases except with theappropriate codimension [14]. But again, instead of infinitely thin branes, we can con-sider thick branes, and inteprete the distribution δ D − ( x − X ( σ )) as an approximationof the corresponding distribution for the thick brane.In the quantized theory, we replace p µ ( σ ) → − iδ/δX µ ( σ ). Instead of Eqs. (31)–(33), we have (cid:18) − g µν δ δX µ ( σ ) δX ν ( σ ) + µ B ¯ f (cid:19) Ψ = 0 , ∂ a X µ δ Ψ δX µ ( σ ) = 0 (41)ˆ H Ψ = i Z d p σ δ D − ( x − X ( σ )) δ Ψ δT ( σ ) (42)ˆ H i Ψ = i Z d p σ δ D − ( x − X ( σ )) δ Ψ δX i ( σ ) (43) See footnote 1 and text after Eq. (20) T ( σ ) ≡ X ( σ ), X ( σ ) ≡ X i ( σ ), and σ ≡ σ a , a = 1 , , ..., p . In general, Ψ =Ψ[ X µ ( σ ) , q ij ( x )] ≡ Ψ[ T ( σ ) , X ( σ ) , q ij ( x )] Since now we have a spacetime of arbitrarydimension D , the definitions (5),(6) of H and H i have to be modified accordingly: R (3) should be replaced by R ( D − , and the Wheeler-DeWitt metric is now G ij kℓ =(1 / (2 √ q ))[ q ik q jℓ + q iℓ q jk − (2 / ( D − q ij q kℓ ].Of particular interest are the following special cases:( i) The spacetime filling brane . Then p = D −
1, and i = a = 1 , , ..., D − σ a such that X i ( σ ) = δ ia σ a . Then we have ∂ a X i ( σ ) = δ ia . The second constraint (41) then reads δ Ψ /δX i = 0. This means thatΨ = Ψ[ T ( σ ) , q ij ( x )], i.e., it does not depend on spatial functions X i ( σ ). Therefore,the first constraint (41) retains the T -derivatives only: (cid:18) − g δ δT ( σ ) δT ( σ ) + µ B ¯ f (cid:19) Ψ = 0 (44)If we now assume ∂ a T ( σ ) = 0, the functional derivative can be replaced by thepartial derivative according to the relation δ/δT ( σ ) → p − ¯ f ∂/∂T . Then, instead of(44), we have − N ∂∂T (cid:18) N ∂ Ψ ∂T (cid:19) − µ B Ψ = 0 (45)The factor ordering has been chosen in order to achieve covariance in the one dimen-sional space comprised of T . The constraints (42),(43) becomeˆ H Ψ = √ q i ∂ Ψ ∂T , ˆ H i Ψ = 0 (46)where we have taken into account the relation ¯ f ab ≡ ∂ a X µ ∂ b X ν g µν = g ab = − q ab = − q ij , and ¯ f = − q ≡ − det q ij .Eq. (45), in which we take N = 1, implies that a general solution Ψ[ T, q ij ( x )] is asuperposition of particular solutions Ψ + = e + iµ b T ψ [ q ij ( x )] and Ψ − = e − iµ b T ψ [ q ij ( x )]that are eigenfunctions of the operator ˆ p / √ q = − i∂/∂T with eigenvalues ± µ B . Forsuch particular solutions, the quantum Hamilton constraint equation becomesˆ H Ψ ± = ∓√ q µ b Ψ ± (47)The expectation value of the operator ˆ p / √ q in a superposition state Ψ = α Ψ + + β Ψ − is h ˆ p / √ q i = ( | α | − | β | ) µ B , where | α | + | β | = 1.That there must be plus or minus sign in Eq. (47), can be seen already at theclassical level. For a spacetime filling brane, the Hamilton constraint (38) becomes H = − p . From Eq. (40) we have p = µ B ˙ X µ √ q/α , where α = q ˙ X ˙ X g = | ˙ X |√ g = | ˙ X | is taken to be a positive quantity. In the last step we have used g = N − N i N j and set N = 1, N i = 0. Thus we obtain that p = µ B √ q ˙ X / | ˙ X | =9 µ B √ q , depending on whether ˙ X is positive or negative. In other words, the signof p depends on whether the spacetime filling brane moves forward or backwardsin time. Despite that the momentum p of a spacetime filling brane, because of theconstraint (37), which now reads p = µ b q , is not a continuous dynamical degree offreedom, there still remains a freedom for p to be either positive or negative, moreprecisely, to be p = µ b √ q or p = − µ B √ q .It is illustrative to look at the situation from another angle. The p -brane phasespace action (36) is equivalent to the minimal surface action I m [ X µ ( ξ )] = µ B Z d p +1 ξ p − det ∂ A X µ ∂ B X ν g µν (48)where ξ A = ( τ, σ a ). Performing the ADM split on the brane’s world manifold, thiscan be written as [15] I m [ X , X i ] = µ B Z d τ d p σ q ˙ X µ ˙ X ν g µν q − ¯ f (49)For a spacetime filling brane we have X i = δ ia σ a , D = p + 1, and the latter actionbecomes I m [ X ] = µ B Z d τ d p σ q ˙ X ˙ X g √ q = µ B Z d X d p X | ˙ X | ˙ X N √ q = ± µ B Z d x d p x N √ q (50)where we have used g = N , N i = 0 , ¯ f = − q , and identified X µ with x µ . Thevariation of the latter action with respect to N gives δI m δN = ±√ q µ B (51)The function X ( τ ), where τ is a monotonically increasing parameter, has no physicalmeaning; it depends on choice of coordinates. Therefore, the derivative ˙ X has nophysical meaning as well. However, there exist two possibilities. One possibilityis that X ( τ ) increases with τ . Another possibility is that X ( τ ) decreases with τ We assume that these two different possibilities correspond to physically differentsituations, because they lead, respectively, to the positive and negative cosmologicalconstant. They provide an explanation for the positive or negative sign in Eq. (47).We can look at the situation even more directly. Since in the case of a spacetimefilling brane its world volume fils the embedding spacetime, we can choose coordinatesin the action (48) so that X µ ( ξ A ) = δ µA ξ A = ξ µ . Bear in mind that now µ =0 , , , ..., D − A = 0 , , , ..., p = D −
1. By such choice of coordinates, weobtain I m = µ B R d x √− g . But we may as well choose X ( ξ A ) = − τ , where τ ≡ , which means that X increases in the opposite direction than τ does, and so,figuratively speaking, our brane “moves backward in time”. Then we obtain I m = − µ B R d x √− g . This corresponds to the term with the cosmological constant Λ = ± πG µ B = 16 πG p / √ q . The spacetime filling brane is thus responsible for thecosmological constant, which can be positive or negative. In the quantized theory, ageneric state is a superposition of those two possibilities, Ψ = α Ψ + + β Ψ − , wherethe eigenstates Ψ ± simultaneously satisfy Eq. (45) and (46). We have thus verifiedthat, in the case of the spacetime filling brane the system of equations (41)–(43) hasa consistent solution. For the expectation value of the cosmological constant in thesuperposition state state Ψ we obtain h ˆΛ i = 16 πG h ˆ p √ q i = ( | α | − | β | )16 πGµ B (52)It can be any value between 16 πGµ B and − πGµ B , including zero or a small valuethat fits the accelerated expansion of the universe. That a spacetime filling brane givesthe cosmological constant was considered by Bandos [17], but he took into accountone sign only.(ii) The brane is a brane world.
Another possibility is that a 3-brane, embedded in a higher dimensional spacetime(bulk), is our observable world (“brane world”) [18]. Then everything that directlycounts for us as observers are points on the brane. It does not matter that pointsoutside the brane cannot be identified, and that Eqs. (42),(43) read H Ψ = 0, H i Ψ = 0,which implies that the wave functional is ”timeless”, with no evolution. What mattersis that the wave functional on the brane, i.e., at x = X ( σ ), has evolution due to theterm with δ Ψ /δT ( σ ) on the right hand side of Eq. (42). However, strictly speaking,Eq. (42) is not a true evolution (Schr¨odinger) equation; it is a set of constraints, validat any point x , that can be on the brane or in the bulk. If we perform the Fouriertransform, then we obtain the brane analogue of Eq. (27), and the zero mode equationhas the form of the Schr¨odinger equation, with the Hamilton operator H | k =0 = R d x H . Because of the integration over x , the quantity H | k =0 has the correct formof a field theoretic Hamiltonian. However, such zero mode Schr¨odinger equation doesnot provide a complete description of the system, for which also all higher modeswith k = 0 are necessary. We have considered a “point particle” coupled to the gravitational field. The classi-cal constraints become after quantizations a system of equations that comprises theKlein-Gordon, Wheeler-DeWitt and Schr¨odinger equation. Then we generalize the11heory to p -branes, in which case the Klein-Gordon equation is replaced by the p -brane quantum constraints. In our approach we start from a classical theory in whichthe point particle or the brane coordinates X µ and the spatial metric q ij are on thesame footing in the sense that they are the quantities that describe the system. Inthe quantized theory, the system is described by a wave functional Ψ[ X µ , q ij ] thatsatisfies the system of equations (41–(43). In the case of a point particle, the lattersystem becomes (21)–(23). A benefit of such approach is that there is no problemof time. The matter coordinate X ≡ T is time. Moreover, the Wheeler-DeWittequation has the part i∂ Ψ /∂T , just as the Schr¨odinger equation.The wave function(al) Ψ[ X µ , q ij ], satisfying the Klein-Gordon equation, is a gen-eralization of the Klein-Gordon field that depends on X µ only. In quantum fieldtheory, the Klein-Gordon field, after second quantization, becomes an operator fieldthat, roughly speaking, creates and annihilates particles at spacetime points X µ .Analogously, we can envisage, that the function(al) Ψ[ X µ , q ij ] should also be consid-ered as a field that can be (secondly) quantized and promoted to an operator thatcreates or annihilates a particle (in general, a p -brane) at X µ , together with thegravitational field q ij . An action functional for Ψ[ X µ , q ij ] that leads to Eqs. (21)–(23)should be found and its quantization carried out, together with the calculation of thecorresponding vacuum energy density due to the quantum field ˆ ψ [ X µ , q ij ]. It has tobe investigated anew, how within such generalized theory vacuum energy influencesthe gravitational field and what is its effect on the cosmological constant.Instead of the point particle action (12) that leads to the Klein-Gordon equa-tion, we could take the spinning particle action [19] that leads to the Dirac equation.Then, in the system (21)–(23) we would have the Dirac instead of the Klein-Gordonequation, and Ψ[ X µ , q ij ] would be a generalization of the Dirac field.We have thus a vision that the quantum field theory of a scalar or spinor fieldin the presence of a gravitational field could be formulated differently from whatwe have been accustomed so far. Usually, we have an x µ -dependent field, e.g., ascalar field ϕ ( x ) or a spinor field ψ ( x ), that is a “source” of the gravitational field g µν ( x ), decomposed, according to ADM, into N ( x ), N i ( x ) and q ij ( x ). The actionis a functional of those fields, e.g., I [ ϕ ( x ) , N, N i , q ij ( x )] or I [ ψ ( x ) , N, N i , q ij ( x )], andin the quantized theory we have a wave functional Ψ[ ϕ ( x ) , q ij ( x )] or Ψ[ ψ ( x ) , q ij ( x )].In this paper, we investigated an alternative approach, in which the classical actionwas I [ X µ ( τ ) , N, N i , q ij ( x )], and, after quantizing it, we arrived at the wave functionalΨ[ X µ , q ij ( x )], i.e., a generalized field that did not depend on the particle’s position X µ in spacetime only, but also on the dynamical variables of gravity, q ij ( x ). Quan-tum field theory of the generalized field Ψ[ X µ , q ij ( x )] is an alternative to the usualquantum field theoretic approaches to gravity coupled to matter. Since the usualapproaches have not yet led us to a consistent theory of quantum gravity, it is worthto investigate what will bring the new approach, conceived in this Letter.12e also considered the case in which, instead of a point particle, X µ ( τ ), we havea brane X µ ( τ, σ a ). In a particular case of a spacetime filling brane, we obtainedpositive or negative cosmological constant, as a consequence of the fact that thebrane’s momentum p has two discrete values only, namely ± µ B √ q . In the quantizedtheory, a state is a superposition of those two possibilities, and the expectation valueof the operator ˆ p is proportional to the effective cosmological constant that can besmall or can vanish. The spacetime filling brane could thus be an explanation for asmall cosmological constant driving the accelerated expansion of the universe. Finally,there is a possibility that our world is a 3-brane moving in a higher dimensional bulkspace. The wave functional, describing such brane, satisfies the Wheeler-DeWittequation with a Schr¨odinger like term iδ Ψ /δT ( σ ) that governs the evolution on thebrane, whereas there is no evolution in the bulk. Acknowledgment
This work has been supported by the Slovenian Research Agency.
References [1] E. Anderson, The Problem of Time in Quantum Gravity, 1009.2157 [gr-qc].[2] C. Rovelli, Class. Quant. Grav. 8 (1991) 297;C. Rovelli, Class. Quant. Grav. 8 (1991) 317.[3] B.S. DeWitt, Phys. Rev. D 160 (1967) 1113.[4] B.S. DeWitt, Gravitation: an Introduction to Current Research, ed. L. Witten,Wiley, New York, 1962;K.V. Kuchaˇr and C. Torre, Phys. Rev. D 43 (1991) 419;S. Mercuri and G. Montani, Mod. Phys. Lett. A 19 (2004) 1519.[5] R. Arnowitt, S. Deser and C. Misner Phys. Rev. 116 (1959) 1322;R. Arnowitt, S. Deser and C. Misner, 1961 The Dynamics of General Relativity,in
Gravitation , Wiley New York,1961 , pp. 227–265.[6] C. Castro, Adv. Sud. Theor. Phys. 1 (2007) 119;C. Castro, J. Math. Phys. 49 (2008) 042501;P. Fiziev, gr-qc/041213;H. Crater, 11106.2040 [gr-qc].[7] See e.g., J. Greensite, Class. Quant. Grav. 13 (1996) 1339;[8] B.S. DeWitt, Rev. Mod. Phys. 29 (1952) 377;B.S. DeWitt, Phys. Rev. 85 (1952) 653.139] M. Pavˇsiˇc, Class. Quant. Grav. 20 (2003) 2697, gr-qc/0111092.[10] V.G. Lapschinsky and V.A. Rubakov, Acta Physica Polonica B 10 (1979) 1041;See also G. Montani, Nucl. Phys. B 634 (2002) 370.[11] V. Moncrief and C. Teitelboim, Phys. Rev. D 6 (1972) 966.[12] M. Pavˇsiˇc, Phys. Lett. B 197 (1987) 327.[13] R. Capovilla, J. Guven and E. Rojas, hep-th/0404178;R. Capovilla, J. Guven and E. Rojas, Nucl. Phys. Proc. Suppl. 88 (2000) 337;N. Turok, M. Perry and P.J. Steinhard, Phys. Rev. D 70 (2004) 106004,hep-th/0408083.[14] R. Geroch and J. Traschen, Phys. Rev. D 36 (1987) 1017.[15] M. Pavˇsiˇc, Class. Quant. Grav. 9 (1992) L13;M. Pavˇsiˇc, The Landscape of Theoretical Physics: A Global View; From PointParticles to the Brane World and Beyond, in Search of a Unifying Principle,Kluwer, Dordrecht, 2001.[16] A.O. Barut and M. Pavˇsiˇc, Mod. Phys. Lett. A 7 (1992) 1381.[17] I.A. Bandos, “Spacetime Brout-Englert-Higgs effect in General Relativity inter-acting with p-brane matter,” J. Phys. Conf. Ser.229