A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time
aa r X i v : . [ phy s i c s . g e n - ph ] O c t A Novel Approach to Quantum Gravity in thePresence of Matter without the Problem of Time
Matej PavˇsiˇcJoˇzef Stefan Institute, Jamova 39, 1000 Ljubljana, Sloveniae-mail: [email protected]
Abstract
An approach to the quantization of gravity in the presence matter is examinedwhich starts from the classical Einstein-Hilbert action and matter approximated by“point” particles minimally coupled to the metric. Upon quantization, the Hamil-ton constraint assumes the form of the Schr¨odinger equation: it contains the usualWheeler-DeWitt term and the term with the time derivative of the wave function. Inaddition, the wave function also satisfies the Klein-Gordon equation, which arises asthe quantum counterpart of the constraint among particles’ momenta. Comparison ofthe novel approach with the usual one in which matter is represented by scalar fieldsis performed, and shown that those approaches do not exclude, but complement eachother. In final discussion it is pointed out that the classical matter could consist ofsuperparticles or spinning particles, described by the commuting and anticommutingGrassmann coordinates, in which case spinor fields would occur after quantization. . Keywords: Canonical quantum gravity; problem of time; wave function of theuniverse; quantum field theory; configuration space
Quantum gravity is an unfinished project. In one actively investigated approach, thestate of the universe is represented by a wave functional that satisfies the Wheeler-DeWitt equation and comprises matter degrees of freedom as well as the gravitationalfield. Matter degrees of freedom are typically given in terms of scalar or spinorfields (see, e.g., Ref. [1]). However, when we consider the evolution of the universe,we usually talk about positions of objects, e.g., we say that galaxies are recedingfrom us, that there is a black hole in the center of a galaxy, etc. The works suchas those considered, e.g., in Refs. [2–7], inspired by the implications of the Wheeler-DeWitt equation, consider models in which the wave function of the universe is givenin terms of positions of particles. Here we provide an explicit demonstration for the1rst time of how a wave functional of, e.g., a scalar field and a gravitational field isrelated to a wave functional of many particle positions and a gravitational field. Inthe literature on quantum gravity the space of configurations of a scalar or whateverfield is called configuration space, but the same name ‘configuration space’ is usedin other branches of physics for the configuration space of one, two three, or manyparticle positions.First we analyze the conventional field theory of a scalar field in the Schr¨odingerfunctional representation in which quantum fields are c -numbers, whilst the corre-sponding canonically conjugated variables are functional derivatives. A generic state,represented as a functional of the scalar field, can be expanded over the basis of theFock space. In distinction to the usual approaches, in which a generic state as asuperposition of multiparticle states in momentum space, we consider multiparticlestates in position space. The scalar field QFT state functional can then be generalizedto include a fixed (background) metric field q ij ( x ) on a space like 3D hypersurface Σin spacetime.Next, the theory is formulated within a more general framework in which anunfixed, dynamical 4D metric is taken into account and is split according to the ADMprescription. The classical action contains a matter part, I m , plus the gravitationalpart I G [ g µν ]. For the matter part it is customary to take a functional of some fields,such as a scalar, spinor or Maxwell field, etc. In the case of a scalar field, ϕ , theaction is thus I m [ ϕ, g ].In this paper we explore an alternative procedure in which we start directly from aclassical action I m [ X µ , g µν ] for a system of “point particles” coupled to a gravitationalfield. We point out that such a procedure makes sense, if the coordinates X µ arenot assumed to be associated with true point particles, but with effective positionsof extended particles. In the Gupta-Bleuler quantization the classical constraints,obtained by variation of the Lagrange multipliers, become operator constraints onthe quantum states. As usual, the Lagrange multipliers in I m [ ϕ, g ] are the lapseand shift functions N , N i , i = 1 , ,
3, whereas in I m [ X µ , g ] we also have the Lagrangemultiplier λ , associated with the τ -reparametrization invariance, which gives the massshell constraint, and, after quantization, the Klein-Gordon equation.In the procedure, discussed in this paper, in which we start from the classicalaction I m [ X µ , g ] + I G [ g µν ], upon quantization the time X does not disappear fromthe equations. On the contrary, in the usual procedure that starts from the action I m [ ϕ, g ] + I G [ g µν ], there is manifestly no time in the quantum equations, which is thenotorious “problem of time”.The paper is organized as follows. In Sec. 2 we first briefly review some basic factsabout the Schr¨odinger functional representation for a scalar field and show how afixed gravitational field can be included into the description. In Sec. 3 we considerthe dynamical gravity as well, first with the novel and then with the traditional2pproach to the matter term, and show how those distinct procedures are related toeach other. In Sec. 4 we resume our findings, and discuss a larger framework intowhich they can be embedded. In particular, we point out, that the classical matter,consisting of particles, can be extended to superparticles or spinning particles thatare described not only by the commuting coordinates X µ , but also by anticommutingcoordinates, in the literature often denoted as ξ µ or θ µ . Coupling of such a systemto gravity and quantizing it along the lines discussed in this paper, would then bringspinor fields into the description. To make our presentation self-consistent we will first review some known facts fromquantum field theory, and then show a novel way of introducing the gravitational fieldinto the description of quantum states. But first let us recall how in flat spacetimewe arrive at a quantum field theory from a relativistic point particle. It is describedby a “minimal” length action that can be cast into the following equivalent form: I [ X µ ( τ ) = 12 Z d τ ˙ X µ ˙ Xµλ + λm ! , (1)where λ is a Lagrange multiplier, variation of which gives the constraint( p µ p µ − m ) = 0. Upon quantization it becomes the Klein-Gordon equation (cid:0) ∂ µ ∂ µ + m ) ϕ ( x ) (cid:1) = 0 . (2)The latter can be derived from the action for a scalar field, I [[ ϕ ( x )] = 12 Z d x (cid:0) ∂ µ ϕ∂ µ ϕ − m ϕ . (cid:1) , (3)Here x ≡ x µ = ( t, x ), µ = 0 , , , x ≡ x i , i = 1 , ,
3, are spacetime coordinates. Thecanonically conjugated variables, ϕ ( t, x ) and Π( t, x ) = ˙ ϕ ( t, x ), where dot denotes thederivative with respect to the time t , become upon quantization the operators ˆ ϕ ( t, x ),ˆΠ( t, x ), satisfying the equal time commutation relations[ ˆ ϕ ( t, x ) , ˆΠ( t, x )] = iδ ( x − x ′ ) , (4)[ ˆ ϕ ( t, x ) , ˆ ϕ ( t, x )] = 0 , [ ˆΠ( t, x ) , ˆΠ( t, x )] = 0 . (5)A general solution of the Klein-Gordon equation (2) can be expanded accordingto ˆ ϕ ( t, x ) = Z d p p (2 π ) ω p (cid:0) a ( p )e − ipx + a † ( p )e ipx (cid:1) , (6) We use the normalization as adopted, e.g., in the textbook by Peskin [8] ω p = p m + p and px ≡ k µ x µ . The operators a ( p ), a † ( p ) are consistent with(4), (5) if they satisfy [ a ( p ) , a † ( p ′ )] = δ ( p − p ′ ) , (7)[ a ( p ) , a ( p ′ )] = 0 , [ a † ( p ) , a † ( p ′ )] = 0 . (8)In the ϕ ( x ) representation the field operator is represented by a c -number, timeindependent, field ϕ ( x ), whereas the conjugate momentum operator is representedby the functional derivative [9, 10]:ˆ ϕ ( x ) −→ ϕ ( x ) , ˆΠ( x ) −→ − i δδϕ ( x ) . (9)A state | Ψ i is represented by a time dependent wave functional Ψ[ t, ϕ ( x )] ≡h ϕ ( x ) | Ψ i , which satisfies the Schr¨odinger functional equation i ∂ Ψ[ t, ϕ ( x )] ∂t = H Ψ[ t, ϕ ( x )] . (10)The Hamilton operator is (see e.g.,Ref. [10]) H = Z d x (cid:16) ˆΠ ˙ˆ ϕ − L (cid:17) = 12 Z d x (cid:18) − δ δϕ ( x ) + ϕ ( m − ∇ ) ϕ (cid:19) (11)One could object that the second order functional derivatives give singularity. Andyet, this is precisely what happens in quantum field theory in whatever representation.Namely, stacionary solution of Eq. (11) satisfy H Ψ[ ϕ ( x )] = E Ψ[ ϕ ( x )] . (12)An example of a solution is the vacuum functionalΨ [ ϕ ( x ) = η exp (cid:20) − Z d x ϕ ( x ) √ m − ∇ ϕ ( x ) (cid:21) , (13)which, when inserted into the stacionary Schr¨odinger equation (12), gives12 Z d x (cid:16) √ m − ∇ δ (0) (cid:17) Ψ = E Ψ . (14)This corresponds to the singular (“zero point”) energy of the vacuum.However, in Ref. [11, 12] (see also [13]) the following generalization of the Hamil-ton operator (11) was considered: H = 12 (cid:16) − ρ a ( x ) b ( x ′ ) ∂ a ( x ) ∂ b ( x ′ ) + ϕ a ( x ) ω a ( x ) b ( x ′ ) ϕ b ( x ′ ) (cid:17) , (15)4here ρ a ( x ) b ( x ′ ) is a metric in the space of fields ϕ a ( x ) ≡ ϕ a ( x ). Then, in general,no singularity arises, despite that the expression contains the second order functionalderivatives ∂ a ( x ) ≡ ∂∂ϕ a ( x ) . As to the ordering ambiguity, one can extend the procedureof Ref. [14] by introducing the basis vectors h a ( x ) , satisfying the Clifford algebrarelations h a ( x ) · h b ( x ′ ) ≡ (cid:0) h a ( x ) h b ( x ′ ) + h b ( x ′ ) h a ( x ) (cid:1) = ρ a ( x ) b ( x ) , (16)and, instead of (15), define the following Hamilton operator, covariant in functionspace: H = − (cid:16)(cid:0) h a ( x ) ∂ a ( x ) (cid:1) (cid:16) h b ( x ′ ) ∂ b ( x ′ ) (cid:17) + ϕ a ( x ) ω a ( x ) b ( x ′ ) ϕ b ( x ′ ) (cid:17) . (17)Here ω a ( x ) b ( x ′ ) is a coupling between the fields that generalizes √ m − ∇ , The mo-mentum vector operator Π = h a ( x ) Π a ( x ) = − ih a ( x ) ∂ a ( x ) is then Hermitian [14].We are not interested here into the mathematical intricaces [15] concerning her-miticity of the momentum operator ˆΠ( x ) that must be taken into account in the casesof generic function spaces. For us it is important that the definition (9) works wellin the relevant physical calculations as nicely shown in Ref. [10]. The expectationvalue is h ˆΠ( x ) i = Z D ϕ ( x )Ψ ∗ [ ϕ ( x )]( − i ) δδϕ ( x ) Ψ[ ϕ ( x )] , (18)where for the measure we take D ϕ ( x ) = Y x d ϕ ( x ) , (19)which holds in the function space with the metric ρ ( x , x ′ ) = δ ( x − x ′ ) , (20)so that the squared distance element in the function space is Z d x d x ′ ρ ( x , x ′ )d ϕ ( x )d ϕ ( x ′ ) = Z d x d ϕ ( x ) . (21)Taking the complex conjugate, we have h ˆΠ( x ) i ∗ = Z D ϕ ( x )Ψ[ ϕ ( x )] i δδϕ ( x )Ψ ∗ [ ϕ ( x )]= Z D ϕ ( x ) (cid:20) Ψ ∗ [ ϕ ( x )]( − i ) δδϕ ( x ) Ψ[ ϕ ( x )] − i δδϕ ( x ) (Ψ ∗ Ψ) (cid:21) (22)Let us assume that Ψ ∗ [ ϕ ( x )]Ψ[ ϕ ( x )] → ϕ → ∞ , which means that thestate is localized around a fixed, expected field configuratiion Then the expecta-tion value is real and the momentum operator is Hermitian. If we also assume that5 D ϕ ( x ) Ψ ∗ [ ϕ ( x )]Ψ[ ϕ ( x )] = 1, then Ψ[ ϕ ( x )] can be interpreted as the probabilityamplitude, and Ψ ∗ [ ϕ ( x )]Ψ[ ϕ ( x )] as the probability density for the field configuration ϕ ( x ), whose expectation value is h ϕ ( x ) i = Z D ϕ ( x )Ψ ∗ [ ϕ ( x )] ϕ ( x )Ψ[ ϕ ( x )] . (23)Using (9), the operators a ( p ), a † ( p ) can be represented as follows [9]: a † ( p ) = Z d x e − i px q (2 π ) ω p ϕ ( x ) − p (2 π ) ω p δδϕ ( x ) ! , (24) a ( p ) = Z d x e i px q (2 π ) ω p ϕ ( x ) + 1 p (2 π ) ω p δδϕ ( x ) ! , (25)If we take the Fourier transform of the operators a † ( p ), a ( p ), namely a † ( x ) = 1 p (2 π ) Z d p e i px a † ( p ) (26) a ( x ) = 1 p (2 π ) Z d p e − i px a ( p ) (27)we obtain [ a ( x ) , a † ( x ′ )] = δ ( x − x ′ ) , (28)[ a ( x ) , a ( x ′ )] = 0 , [ a † ( x ) , a † ( x ′ )] = 0 . (29)These are the commutation relations for creation (annihilation) operators that create(annihilate) a particle at x . From (28),(29), by using (24),(25), we find the followingexplicit expressions: a † ( x ) = p (2 π ) m − ∇ ) / ϕ ( x ) − p (2 π ) m − ∇ ) / δδϕ ( x ) (30) a ( x ) = p (2 π ) m − ∇ ) / ϕ ( x ) + 1 p (2 π ) m − ∇ ) / δδϕ ( x ) (31)From the latter equations we obtain ϕ ( x ) = 1 p (2 π ) m − ∇ ) / ( a ( x ) + a † ( x )) , (32) δδϕ ( x ) = p (2 π ) m − ∇ ) / ( a ( x ) − a † ( x )) . (33)6sing (32),(33), we can express the Hamiltonian (11) in terms of a ( x ) and a † ( x ): H = 12 Z d x (cid:16) a † ( x ) √ m − ∇ a ( x ) + (cid:16) √ m − ∇ a ( x ) (cid:17) a † ( x ) (cid:17) . (34)If expressed in terms of a ( p ), a † ( p ), the Hamiltonian has the usual form, H = 12 Z d p ω p (cid:0) a † ( p ) a ( p ) + a ( p ) a † ( p ) (cid:1) . (35)A generic state is a superposition of many particle states:Ψ[ t, ϕ ( x )] = ∞ X r =0 Z d p d p ... d p r φ ( t, p , p , ..., p r ) a † ( p ) a † ( p ) ...a † ( p r )Ψ [ ϕ ( p )] , (36)where φ ( t, p , p , ..., p r ) is a complex valued wave packet profile for an r -particle statein momentum representationIn terms of the Fourier transformed creation operators (26),(27), the same statereads:Ψ[ t, ϕ ( x )] = ∞ X r =0 Z d x d x ... d x r ψ ( t, x , x , ..., x r ) a † ( x ) a † ( x ) ...a † ( x r )Ψ [ ϕ ( x )] , (37)where ψ ( t, x , x , ..., x r ) is a complex valued wave packet profile for an r -particle statein position representation. Its absolute square | ψ | = ψ ∗ ψ gives the probability den-sity of observing at a time t a multi particle configuration at positions x , x , ..., x r ,and as discussed in the next paragraph and the references cited therein, the apparentnon covariance of | ψ ( x ) | is not problematic.Within relativistic quantum field theories, the wave packet profiles in momentumrepresentation are occasionally used in the literature when considering the S -matrix.In the textbook by Peskin and Schr¨oder [8], a single particle wave packet is consid-ered, and it is mentioned that φ ( p ) is the Fourier transform of the spatial wavefunc-tion. In general, a wave packet depends on time. In a given Lorentz reference frameone can thus consider either φ ( t, p ) or ψ ( t, x ) (and their multiparticle extensions),the corresponding creation operators being a † ( p ) or a † ( x ). Although asymptoticstates are usually taken to have definite momenta , there are situations, when theyform wave packet profiles, in which case it is possible to measure the detection time,the time of flight, etc. Such wavepacket profiles, of course, can be observed fromdifferent Lorentz reference frames. When observed from another Lorentz frame, thewave packet, and hence the probability density, looks different, it is subjected to an A state with definite momentum, a † ( p ) | i ≡ | p i , is an approximation. In reality, there is alwayssome spreading. A sharp momentum state is not in Hilbert space, it belongs to generalized states. a ( x )Ψ [ ϕ ( x )] = 0 . (38)Using Eq. (31) in the latter equation, we find for the solution the vacuum functional(13) that was obtained directly from the stacionary Schr¨odinger equation (12).Usually authors do not work in the coordinate ( x ), but in the momentum ( p )representation, which in many respects is more practical. In particular, in momen-tum representation it is straightforward to calculate the normalization constant forthe vacuum functional. But for the purposes of our paper, the coordinate represen-tation is useful, because once the expressions (30),(31) and (13) are obtained, wecan generalize them to include the metric field q ij ( x ) on a 3D hypersurface Σ by re-placing ∇ with the covariant operator q ij D i D j , which when acting on a scalar fieldgives √ q ∂ i ( √ q∂ i ϕ ( x )), where q = det q ij , i.e., the determinant of the 3-metric. Toour knowledge such procedure has not yet been considered in the literature, and inthe following we will show its usefulness in introducing a gravitational field into thedescription.If we make the replacement ∇ → q ij D i D j = D i D i ≡ D , (39)then the creation and annihilation operators (30),(31) become functionals of q ij ( x ),and so does the vacuum functional (13): a ( x ) → a [ x , q ij ( x )] , (40) a † ( x ) → a † [ x , q ij ( x )] , (41)Ψ [ ϕ ( x )] → Ψ [ ϕ ( x ) , q ij ( x )] . (42)It turns out that they satisfy the same commutation relation (28),(29)The so modified operator a † [ x , q ij ( x )] creates a particle at position x in the grav-itational field q ij ( x ). One finds that the modified operator a [ x , q ij ( x )] annihilates themodified vacuum functional Ψ [ ϕ ( x ) , q ij ( x )]. A generic state can then be representedby the following functional:Ψ[ t, ϕ ( t, x ) , q ij ( x )] = ∞ X r =1 Z d x ... d x r ψ [ t, x , ..., x r , q ij ( x )] × a † [ x , q ij ( x )] ...a † [ x r , q ij ( x )]Ψ [ ϕ ( x ) , q ij ( x )] . (43) It is usually stated that the notion of particle depends on the metric, and by “particle” it isunderstood an excitation with definite momentum. Here by “particles” we mean just the excitations,created or annihilated by the modified position dependent operators (40),(41). ψ [ t, x , ..., x r , q ij ( x )] is the amplitude for the probability that we will find matterparticles at positions x , x , ..., x r in a gravitational field q ij ( x ). This is thus a wavefunctional that depends on particle positions and on the corresponding gravitationalfield at those positions. In the case when only one particle is created, its wavefunctional is ψ [ x , q ij ( x )]. The state (43) is then given byΨ[ t, ϕ ( x ) , q ij ( x )] = Z d x ψ [ t, x , q ij ( x )] a † [ x , q ij ( x )]Ψ [ ϕ ( x ) , q ij ( x )] . (44)In Ref. [17] the wave functional of the form ψ [ X µ , q ij ( x )], which depends on a parti-cle position and a gravitational field, was considered, and found to satisfy the Klein-Gordon and the Wheeler-DeWitt equation. This was an alternative to the usuallyconsidered state functional Ψ[ ϕ ( x ) , q ij ( x )] that depends on a scalar field ϕ ( x ) insteadon x . In Eq. (44) we have a translation between those two possible descriptions, andin Eq. (43) we have a translation for the general case of many particles. In onedescription we have a functional Ψ[ t, ϕ ( x ) , q ij ( x )], and in the other description wehave a set of functionals ψ [ t, x , q ij ( x )], ψ [ t, x , x , q ij ( x )], ... , ψ [ t, x , ..., x N , q ij ( x )],which are components of the expansion of the state functional in terms of the ba-sis states a † [ x , q ij ( x )] ...a † [ x r , q ij ( x )]Ψ [ ϕ ( x ) , q ij ( x )], r = 0 , , , ..., N , where N isarbitrary and can go to infinity. Schr¨odinger equation
If we substitute the expression (37) for the state functional and (34) for theHamiltonian into the Schr¨odinger equation (10), we obtain a set of equations for themultiparticle wavefunctions: i ∂ψ (( t, x , ..., x r ) ∂t = r X ℓ =1 q m − ∇ x ℓ ψ ( t, x , ..., x r ) , r = 1 , , ..., ∞ , (45)where ∇ ℓ ≡ − ∂ ∂x iℓ ∂x iℓ , i = 1 , , , (46)In Eq. (45) we have omitted the infinite zero point energy, because it cancels out inthe expressions containing the probability density ψ ∗ ψ .If we take into account also the gravitational field q ij ( x ), the Schr¨odinger equation(10) generalizes so that instead of Ψ[ t, ϕ ( x )] we have Ψ[ t, ϕ ( x ) , q ij ( x )] and in theHamiltonian (34) the operator ∇ generalizes according to (39): i ∂ Ψ[ t, ϕ ( x ) , q ij ( x )] ∂t = H Ψ[ t, ϕ ( x ) , q ij ( x )] . (47)Then, instead of (45) we obtain i ∂ψ [ t, x , ..., x r , q ij ( x )] ∂t = r X ℓ =1 q m − D x ℓ ψ [ t, x , ..., x r , q ij ( x )] , r = 1 , , ..., ∞ . (48)9quation (47), or equivalently, (48), describes the evolution of a many par-ticle state in the presence of a bakground gravitational field q ij ( x ). At a cer-tain time t , the probability density | ψ [ t, x , ..., x r , q ij ( x )] | is centered around aconfiguration x , ..., x r and a 3-metric q ij ( x ). At another time t , it is centeredaround a different configuration and a different intrinsic metric. In other words, R Ω | ψ [ t, x , ..., x r , q ij ( x )] | d x d x ... d x r D q ij ( x ) is the probability that at time t we find r particles within a domain Ω around the positions x , x , ..., x r and in thegravitational field q ij ( x ). Because in the right hand side of Eq. (48) there is no op-erator term acting on q ij ( x ), the probability density remains at all times t centeredaround the same 3-metric q ij ( x ). In particular, it can be q ij ( x ) = δ ij , but in generalit is a position dependent metric with a non vanishing 3-curvature. Equation (48)then describes evolution of a wave function in a fixed non trivial gravitational field.There is no dynamics of q ij ( x ) itself in such a formalism In the previous section we started from a relativistic point particle in flat spacetimeand arrived upon first quantization at the Klein-Gordon equation for a scalar field ϕ ( x ). Then we quantized the field ϕ ( x ) as well, and arrived at quantum field theory inwhich a generic state can be represented as a functional Ψ[ t, ϕ ( x )], expanded in termsof many particle states with wave packet profiles (wave functions) φ ( t, p , p , ..., p r ),or equivalently, ψ ( t, x , x , ..., x r ), satisfying the Schr¨odinger equation (45). We thendiscussed how such wavefunctions can be generalized to include a fixed gravitationalfield. We have thus arrived at the quantum evolution of a many particle wave function(48) in a fixed 3-metric field q ij ( x ) on a simultaneity hypersurface Σ. In order toinclude a dynamics of the gravitational field g µν ( x ), x ≡ x µ , µ = 0 , , ,
3, andconsequently of the induced metric q ij ( x ) on Σ, let us consider the classical actionfor a many particle system coupled to gravity: I [ X µn , g µν ( x ) , λ n ] = I m [ X µn , g µν ( x ) , λ n ] + I G [ g µν ( x )] , (49)where I m [ X µn , g µν ( x ) , λ n ] = 12 N X n =1 Z d τ ˙ X µn ˙ X νn g µν λ n + λ n m n ! , (50)and I G = κ Z d x √− g R, κ = (16 πG ) − . (51) The parameter τ is arbitrary and in principle different on each worldline. To simplify thenotation we write τ instead of τ n , X µn ( τ ) of an arbitrary monotonically increasing parameter τ de-scribe the worldlines associated with positions of particles, e. g., their centers of mass.Realistic particles are not point like, they are extended beyond their Schwarzschildradii , but in our description we take into account only the particle’s center of mass.Despite that the definition of the center of mass is a subject of controversy [18], forour purpose here it is important that each of the objects whose motion is governedby the action (50) is extended, not point like, and that can be described by fourcoordinates only, so that the infinitely many degrees of freedom of an extended objectare neglected. How this works for an extended object confined within a narrow tube inspacetime is shown in the derivation of the Papapetrou equation [19]. The derivationis based on the moments of the stress-energy tensor around a chosen worldline withinthe tube. If only the first moment (“monopole”) is taken into account, one obtainsthe geodesic equation for such a worldline. In such sense one should also considerthe wordlines occurring in the action (50). For further support of our argument, seeAppendix A.In the continuum limit of many densely packed worldlines the action (50) becomesthe dust action. Coupling of dust to gravity was considered by Brown and Kuchar [20]in order to resolve the problem of time in quantum gravity. In their approach dust issupposed to be present everywhere and its degrees of freedom incorporate time. Wewill show that instead of dust one can as well employ a system of “point” particlesthat even need not be present everywhere. It comes out that the generator of timetranslations is directly associated with the stress-energy tensor of such system ofparticles.If one performs the ADM split of spacetime, then the action (49) can be writtenas a functional of the 3-metric q ij , and the lapse and shift functions, N and N i , i, j = 1 , ,
3. Rewriting the matter part of the action, (50), into the phase spaceform, I m [ X µn , p nµ , λ n , g µν ] = X n Z d τ (cid:18) p nµ ˙ X µn − λ n g µν p µn p νn − m n ) (cid:19) , (52)and performing the ADM split, we obtain [17] I m [ X µn , p nµ , λ n , q ij , N, N i ] = X n Z d τ (cid:16) p nµ ˙ X µn − λ n (cid:20) N ( p n − N i p ni )( p n − N j p nj ) − q ij p ni p nj − m n (cid:21)(cid:19) (53) The action (50) can also describe a system of (mini) black holes with their positions beingparametrized with X µn and tracing the worldlines X µn ( τ ). Such a system, instead of being consideredwithin a complicated detailed description involving the mutual dynamics of black holes’ gravitationalfield, could be as well approximately described in terms of their positions X µn . The very fact thatwe talk about a black hole in the center of our galaxy, or in the center of another galaxy, meansthat we ascribe to a black hole a position, and parametrize it by a set of coordinates.
11n the above action, momenta are the quantities, variation of which gives the relation p µn = ˙ X µn /λ n .In order to have the matter action on the same footing as the gravitational action,we must insert 1 = R √− g d x δ ( x − X n ( τ )) / √− g = R d x δ ( x − X n ( τ )). But ourparticle is not exactly point like, it is extended, e.g., a ball, whose worldvolume isdescribed by X µ ( τ, σ ), σ ≡ σ a = ( R, ϑ, ϕ ), 0 ≤ R ≤ R , 0 ≤ ϑ ≤ π , 0 ≤ ϕ ≤ π , where R is greater than the Schwarzschild radius. Therefore, δ ( x − X n ( τ ))should be considered as an approximation to R d σ p − ¯ f δ ( x − X n ( τ, σ )), where¯ f ≡ det ¯ f ab , ∂ a ≡ ∂/∂σ a (see also [21], where the so called “good” δ -function isdefined). Despite that such a δ -function associated with a ball is not invariant , itis covariant . In Appendix A we consider the dynamics of a ball, modelled by aspace filling brane, described by the action that is covariant with respect to generalcoordinate transformation of x µ and ξ A = ( τ, σ a ).For the gravitational part of the action we obtain [22] I G [ q ij , π ij , N, N i ] = Z d t d x (cid:2) π ij ˙ q ij − N H G ( q ij , π ij ) − N i H G i ( q ij , π ij ) (cid:3) , (54)where H G = − κ G ij kℓ π ij π kℓ + κ √ qR (3) , (55) H i G = − j π ij , (56)and G ij kℓ = 12 √ q ( q ik q jℓ + q iℓ q jk − q ij q kℓ ) . (57)is the Wheeler-DeWitt metric.Variation of I = I m + I G with respect to λ n , N , and N i , which have the role ofLagrange multipliers, gives the constraints [17] that involve both the gravitational For instance, if we consider the case of
Minkowski space , then in another Lorentz reference framethe δ -function, corresponding to a ball-like extended object, looks different (no longer associatedwith a ball at rest, but a moving ellipsoid). However, in a new Lorentz frame one can have withrespect to the new simultaneity 3-surface a ball-like extended object, described by X n ( τ, σ ), thecorresponding delta-function being exactly the same as our “original” delta-function. The conceptof the delta-function, though not invariant, is covariant . δλ n : 1 N ( p n − N i p ni )( p n − N j p nj ) − q ij p ni p nj − m n , (58) δN : H G = − X n Z d τ λ n N δ ( x − X n ( τ ))( p n − N i p ni )( p n − N j p nj )= − X n δ ( x − X n ) 1 N ( p n − N i p ni ) (59) δN i : H G i = − X n Z d τ λ n N δ ( x − X n ( τ )) p ni ( p n − N j p nj ) , = − X n δ ( x − X n ) p ni (60)In Eq. (59) we used p n = g ν p nν = (1 /N )( p n − N i p ni ), replaced ( p n ) with ˙ X n /λ n ,and then proceeded as follows: Z d τ δ ( x − X n ( τ )) δ ( x − X n ( τ )) 1 N ( p n − N i p ni ) ˙ X n = Z d τ | ˙ X n | δ ( x − X n ( τ )) 1 N ( p n − N i p ni ) ˙ X n (cid:12)(cid:12)(cid:12) τ = τ c = δ ( x − X n ) 1 N ( p n − N i p ni ) , (61)where τ c is the solution of the equation x = X n ( τ ), i.e., τ c = ( X n ) − ( x ), and wherewe have taken positive ˙ X n . Since this equation is valid at any x , we omit in the laststep the subscript | τ = τ c . Similarly we proceeded in Eq. (60).Algebraic structure of the constraints (58)–(60) is independent of foliation ofspacetime, determined by N , N i . In any foliation, the form of the equations re-main the same. The constraints (58)–(60) are thus covariant under diffeomorphisms.By taking a linear combination of those constraints, we obtain a Hamiltonian: H = Z d x (cid:0) N H + N i H i (cid:1) = 0 . (62)Here H = H G + H m , H i = H Gi + H mi , where H m , H mi , H G , and H Gi are given inEqs. (59),(60), (55)–(57). The terms with N i p ni cancel out, and so we obtain H = Z d x ( N H G + N i H G i ) + X n p n = 0 . (63)In the latter equation, which holds for arbitrary N , N i , the matter Hamiltonian isjust the sum of particle’s momenta p n , i.e., their energies.Eq. (63) can also be obtained directly from the Einstein equations integrated overa space like hypersurface according to κ Z dΣ ν √− g G ν = − Z dΣ ν √− g T ν = − P . (64)13f the stress-energy tensor is concentrated so that it effectively forms a discrete set ofparticles in the sense as previously described, then the total energy P in Eq. (64) isjust the sum of particle energies, occurring in Eqs. (63).Instead of Eq. (64) we can take the corresponding covariant form κ Z dΣ ν √− g G µν n µ = − Z dΣ ν √− g T µν n µ = − P µ n µ , (65)where n µ is a unit vector field. If T µν = ρu µ u ν , i.e., a dust stress-energy tensor and u µ the 4-velocity, then it is convenient to take n µ = u µ . A discrete version of thedust stress energy tensor is the one obtained from the system of ”point particles”,described by the action (52). Then for n µ we can take a vector field that at thelocations of particles coincides with their 4-velocities u µn = p µn /m n .Because the constraints (59)–(60) are in fact just the the ADM split of Einstein’sequations, we see that as a consequence of the Bianchi identities they are conservedin time, or along any time like curve whose tangent is u ν : u ν D ν ( G µν + 8 πT µν ) = 0 . (66)The constraint (58), namely, g µν p µn p µn − m n = 0, also is conserved, u α D α ( g µν p µn p νn − m n ) = 2 g µν u α p µn D α p νn = 2 p nν u α D α p νn = 0 , (67)where we now assume that u α is tangent along a geodesic, so that p αn = m n u α . Then u α D α u ν = 0, and the equality (67) is satisfied.When quantizing the theory, one has to fix a gauge. In our case this is achievedby choice of N , N i . Because N and N i behave as Lagrange multipliers, they canbe arbitrary functions of spacetime coordinates x µ . We will use the choice N = 1, N i = 0. It is straightforward to show that such gauge fixing imposes no secondclass constraints . Upon quantization, in the Schr¨odinger representation in which X µn and q ij are diagonal, the momentum operators are the covariant generalizationsof ˆ p nµ = − i∂/∂X µn and ˆ π ij = − iδ/δq ij . If acting on a functional only once, thenˆ p nµ behaves as partial derivative, and ˆ π ij as functional derivative, otherwise ˆ p nµ iscovariant derivative with respect to g µν , and ˆ π ij is covariant functional derivative withrespect to the metric G ijkℓ δ ( x − x ′ ). The constraints (58)–(60) become conditionson a state represented as ˜ ϕ [ T , T , ...T N , X i , X i , ...X iN , q ij ( x )] ≡ ˜ ϕ [ T n , X in , q ij ( x )], n =1 , , ..., ∞ , where T n ≡ X µn : (cid:0) ˆ p n − q ij ˆ p ni ˆ p nj − m n (cid:1) ˜ ϕ [ T n , X in , q ij ( x )] = 0 , (68) H G − X n δ ( x − X n ) i ∂∂T n ! ˜ ϕ [ T n , X in , q ij ( x )] = 0 , (69) See explanation after Eq. (92). H G i − X n δ ( x − X n ) i ∂∂X in ! ˜ ϕ [ T n , X in , q ij ( x )] = 0 . (70)Integrating the last two equations over x , we obtain H G − X n i ∂∂T n ! ˜ ϕ [ T n , X in , q ij ( x )] = 0 , (71) H G i − X n i ∂∂X in ! ˜ ϕ [ T n , X in , q ij ( x )] = 0 , (72)where H G = Z d x H G = Z d x (cid:18) κ G ij kℓ δ δq ij δq kℓ + κ √ qR (3) (cid:19) , (73) H G i = Z d x H G i = − Z d x ( − i )2D j δq ij ( x ) . (74)At this point let us recall that, as shown by Moncrief and Teitelboim [23], themomentum constraint (60) is a consequence of the conservation of the Hamiltonconstraint (59). Therefore, it is sufficient if we consider the Hamilton constraint (59)and its quantum versions (69) or (71) only. We then have( H G + H m ) ˜ ϕ [ T n , X in , q ij ( x )] = 0 , (75)where H m = X n ˆ p n = − i X n ∂∂T n . (76)In this description a quantum state is represented by ˜ ϕ [ T n , X in , q ij ( x )], which isa function of the particles’ spacetime coordinates, X µn = ( T n , X in ), and a functionalof the dynamical variables of gravity, q ij ( x ). In the absence of horizons, which isthe situation that we consider here, we can choose coordinates X µn so that all timecoordinates X n ≡ T n on a given 3-surface Σ are equal: T = T = ... = T N = T . Thenwe can write ˜ ϕ [ T n , X in , q ij ( x )] as a function of a single time coordinate T :˜ ϕ [ T n , X in , q ij ( x )] = φ [ T, X in , q ij ( x )] , n = 1 , , ..., r, (77)and d ˜ ϕ d T = X n ∂ ˜ ϕ∂T n ∂T n ∂T = X n ∂ ˜ ϕ∂T n = ∂φ∂T . (78) The ordering issue can be settled by replacing δ/δq ij with covariant functional derivatives withrespect to the Wheeler-DeWitt metric, and proceeding `a la Ref. [14]. A further development of thisimportant point is beyond the scope of the present paper. As in so many other papers, also here δ/δq ij has only a symbolic meaning, unless considered as the covariant functional derivative. H G φ [ T, X in , q ij ( x )] = i ∂φ [ T, X in , q ij ( x )] ∂T . (79)We see that if one starts from the classical action (49) in which the matter part isexpressed in terms of the worldlines of individual particles, then upon quantization wearrive at the Wheeler-DeWitt equation (79) which contain time. The wave functionaldepends on coordinates of particles, X µn = ( T n , X in ), and the induced metric on ahypersurface Σ, defined by X n = T = constant (on which all particles have the sametime coordinates T n = T ).Besides the time dependent equation (79), the wave functional also satisfies theKlein-Gordon equation (68). Assuming real ˜ ϕ , the second order Klein-Gordon equa-tion can be cast into the form of a first order equation for a complex wave func-tion [24, 25] ψ = ψ R + iψ I , (80)where ψ R = φ ( T, X , X , ..., X r ) = ˜ ϕ ( T , T , ..., T r , X , X , ..., X r ) , (81)and ψ I = i Ω − ˙ φ ( T, X , X , ..., X r ) . (82)So instead of (68) we obtain the equivalent equation (see Appendix B) i ∂ψ∂T = Ω ψ. (83)Here Ω is a matrix, by means of which the above equation is just a compact formof Eq. (48) that we derived in Sec. 2 after expanding the Schr¨odinger wave functionalΦ[ t, ϕ ( x )] in terms of multiparticle wave functions according to Eq. (37), and aftergeneralizing it so to include the 3-surface induced metric q ij as well.The function ψ , occurring in Eq. (83), is the true, complex valued, wave function(related to the probability density), whilst the function ˜ ϕ satisfying the Klein-Gordonequation (68), is just a real field.By using the expression (80) and Eq. (79), it is straightforward to derive that inaddition to Eq. (79), valid for the real field ˜ ϕ , we also have the similar equation forthe complex wave function ψ : H G ψ [ T, X in , q ij ( x )] = i ∂ψ [ T, X in , q ij ( x )] ∂T . (84)In the above treatment we started from the classical action (49) and the cor-responding constraints which upon quantization became the conditions on states(68)–(70) that can be represented either as a wave functional ˜ ϕ [ T n , X in , q ij ( x )] or16 [ T, X in , q ij ( x )]. No further, i.e., a “second” quantization is performed here. Thecomplex functionals ψ [ T, X in , q ij ( x )], n = 1 , , , ... , are obtained according to theprescriptions (80)–(82). The matter Hamiltonian H m plus the gravitational Hamil-tonian H G acting together on ψ [ T, X in , q ij ( x )] give zero. Time automatically appearsin the equations, such as (83) or (84). In the previous subsections we discussed a novel approach in which we started from aclassical matter represented as a multi particle system. In the usual treatments matteraction is not I m [ X µn , g µν ( x )], or equivalently, I m [ X µn , p nµ , λ n , g µν ] (Eqs. (52),(53)), butis a functional of fields, for instance scalar fields: I m [ ϕ a ( x ) , g µν ( x )] = 12 Z d x √− g (cid:0) g µν ∂ µ ϕ a ∂ ν ϕ a − m ϕ a ϕ a (cid:1) , (85)which after the ADM split reads: I m [ ϕ a , q ij , N, N i ] = 12 Z d t d x N √ q (cid:20) N ( ˙ ϕ a − N i ∂ i ϕ a )( ˙ ϕ a − N j ∂ j ϕ a ) − q ij ∂ i ϕ a ∂ j ϕ a − m ϕ a ϕ a (cid:21) . (86)Variation of the total action I = I m [ ϕ a , q ij , N, N i ] + I G [ q ij , N, N i ] (87)with respect to N and N i gives the constraints H = H G + H m and H i = H G i + H m i .Here the gravitational part of the constraints, H G , H G i , are given in Eqs. (55),(56),whilst the matter parts H m , H m i are now H m = √ q (cid:18) Π a Π a q + q ij ∂ i ϕ a ∂ j ϕ a + m ϕ a ϕ a (cid:19) (88) H m i = ∂ i ϕ a Π a , (89)where Π a = ∂ L ∂ ˙ ϕ a = √ qN ( ˙ ϕ a − N i ∂ i ϕ a ) . (90)The Hamiltonian is a linear combination of the constraints H = Z d x ( N H + N i H i ) . (91)17he Lagrange multipliers N and N i are arbitrary, and, as before, we choose N = 1, N i = 0, so that H = Z d x H = Z d x ( H G + H m ) = H G + H m . (92)Such a choice brings no second class constraints, because it involves only Lagrangemultipliers and their conjugate momenta π , π i , and no other phase space variables.Namely, setting φ = N − φ = π , φ i = N i , φ i = π i , and calculating the timederivative of the constraints according to ˙ φ = { φ, H } , we obtain ˙ φ = 0, ˙ φ = − H ,˙ φ i = 0, ˙ φ i = H i , which are the original constraints.Upon quantization we have ( H G + H m ) | Φ i = 0 , (93)which in the Schr¨odinger functional representation gives h ϕ a ( x ) , q ij ( x ) | ( H G + H m ) | Φ i = 0 . (94)Explicitly, the latter equations reads Z d x (cid:20) κ G ijkℓ δ δq ij δq kℓ + √ qR (3) + √ q (cid:18) − δ δϕ a δϕ a + ϕ a ( − D i D i + m ) ϕ a (cid:19)(cid:21) Φ[ ϕ a ( x ) , q ij ( x )] = 0 , (95)where H G , H m are represented as matrices in the space of fields ϕ a ( x ), q ij ( x ), i.e., asfunctional differential operators. Concerning the factor ordering of the operators ˆ π ij we can adopt and generalize the procedure of Ref. [14].The functional Φ[ ϕ a ( x ) , q ij ( x )] is “first quantized” with respect to the metric q ij ( x ), and “second quantized” with respect to particle position. This is a “hybrid”procedure, and there is manifestly no time in Eq. (95).The matter Hamiltonian in Eq. (95), H m = Z d x √ q (cid:18) − δ δϕ a ϕ a + ϕ a (cid:0) − D i D i + m (cid:1) ϕ a (cid:19) , (96)can be expressed in terms of the operators a [ x , q ij ( x )], a † [ x , q ij ( x )], defined in Sec. 2,so we have H m = Z d x (cid:16) a † [ x , q ij ( x )] p m − D i D i a [ x , q ij ( x )] + z . p . (cid:17) . (97)This is an exact expression for H m . 18et us now take the following Ansatz:Φ[ ϕ a ( x ) , q ij ( x )] = X n d x ... d x n ψ [ x , ..., x n , q ij ( x )] ˜Φ n [ ϕ ( x ) , q ij ( x )] , (98)where ˜Φ n [ ϕ ( x ) , q ij ( x )] = a † [ x , q ij ( x ) ...a † [ x n , q ij ( x ) ˜Φ [ ϕ, q ij ( x )] (99)is obtained by the action of the creation operators on the vacuum ˜Φ [ ϕ, q ij ( x )], whichis now not the vacuum (42), valid in the case of a fixed background metric, but a suit-ably generalized expression accounting for the fact that now the metric is dynamical.Formally, let us set˜Φ [ ϕ, q ij ( x )] ∝ exp (cid:20) − Z d x √ qϕ ( x ) p m − D i D i ϕ + Q [ q ij ] (cid:21) , (100)which contains in the exponential an additional functional Q [ q ij ], that could contain,among others, the expressions such as R d x √ qR (3) .Using the commutation relations [ a [ x , q ij ( x )] , a † [ x ′ , q ij ( x )]] = δ ( x − x ′ ) , (101)[ a [ x , q ij ( x )] , a [ x ′ , q ij ( x )]] = 0 , [ a † [ x , q ij ( x )] , a † [ x ′ , q ij ( x )]] = 0 , (102)the expansion (98), and the matter Hamiltonian (96), we find H m Φ = ∞ X r =1 Z d x d x ... d x n r X n =1 q m n − D n ψ [ x , x , ..., x r , q ij ( x )] × a † [ x , q ij ( x )] ...a † [ x r , q ij ( x )]Φ [ ϕ ( x ) , q ij ( x )] , (103)where D n ≡ q ij D ni D nj , D ni ≡ DD X ni . (104)Equation (103) is just like Eq. (84) (see also (48)). If we postulate that the quan-tizations based on the actions (49) and (85) are equivalent, then the wave functional ψ [ x , ..., x n , q ij ( x )] is the same one as considered in Eqs. (68)–(72), depends on time T and satisfies (84), now for positive sign only. Here we have renamed X in ≡ X n ,occurring in Eqs. (68)–(72), into x n , n = 1 , , ..., N = ∞ . If time T occurs in ψ , thenit automatically also occurs in Φ.Taking now into account Eq. (84), we obtain H m Φ = i ∂ Φ ∂T , (105) As we pointed in Sec. 2, the metric and its determinant cancel out, so that they do not appearin the r.h.s. of the commutation relation.
19e have thus reproduced the functional representation of the time dependentSchr¨odinger equation for the scalar field in the presence of a 3-metric field q ij ( x ).Returning to Eq. (94) and using the latter result, namely that H m Φ = i∂ Φ /∂T ,we obtain H G Φ = − i ∂ Φ ∂T . (106)We have thus obtained a time dependent Schr¨odinger equation for the gravitationalpart of the Hamilton operator. Both equations, (105) and (106), together give theWheeler-DeWitt equation (94). It thus turns out that though time does not mani-festly take place in the Wheeler-DeWitt equation (94), it is hidden in the wave func-tional Φ, if the quantization procedures discussed in Secs. (3.1)and (3.2) are equivalentin the sense that they lead to the same physics. When investigating the universe and behaviour of objects in it, we normally havetheir positions in mind. In quantum gravity the entire universe is envisaged to bedescribable by a wave function. However, in practical calculations only few degreesof freedom of the universe are taken into account, but we assume that in realitythere exists a wave function(al) of the universe that comprises all degrees of freedom,including those of observers. We base our discussion of the universe on canonicalgravity which we modified in the part that relates to matter. Usually matter isrepresented by fields, such as a scalar, spinor, gauge field, etc. In the model consideredin this paper we started directly from the classical action for a multi particle systemcoupled to gravity, and arrived after quantization to the Schr¨odinger equation for aset of multi-particle wave functionals ψ [ t, x , x , ..., x r , q ij ( x )], r = 1 , , ... . . Thus,in our approach a state satisfying the Wheeler-DeWitt equation is represented as afunctional of gravity and matter degrees of freedom, the latter being given in termsof multi particle configurations. It then turns out that time does not disappear fromthe quantum equations, it is given in terms of the time like coordinates of particles,associated with the clocks situated on particle’s worldlines. We have thus set upa theoretical framework for a quantum gravity description of the universe, whichinvolves positions of particles, a concept in many respect closer to our intuition andobservations than the concept of scalar field.In this paper we have confined us to discussing the Wheeler-DeWitt equationwhose semiclassical solutions are well-known to contain singularities. How to avoidsingularities has been discussed, e.g., by Claus Kiefer and Barbara Sandhofer [26](see also refs [27–31] cited therein) who conclude: ”Upon discussing the Wheeler-DeWitt equation, one finds that all normalizable solutions lead to a wave function20hat vanishes at the point of the classical singularity; this we interpret as singular-ity avoidance. An analogous situation of singularity avoidance is found in the loopquantum cosmology of this model [32].” Our classical action (49) of point particlescoupled to gravity, if taken literally, would be impossible, because at the positions X µn of point particles there would be black hole singularities. We avoided singularityby postulating that the particles are extended and that X µn are coordinates of theireffective positions (analogous to the center of mass of a non relativistic particle).Our procedure could as well be upgraded into a promising direction: namely, byextending the classical point particle to the superparticle, by including, besides thecommuting coordinates X µ , also the Grassmann anticommuting coordinates. andcoupling such system to gravity. Thus, because of the presence of the additional,anticommuting coordinates, ξ µ , the problem of a point particle coupled to a gravi-tational field would be avoided, because the extra coordinates would bring into thegame in an elegant way the particle’s effective extension. Upon quantization of sucha model in which a superparticle is coupled to the metric (which now depends on X µ and ξ µ ), spinor fields would occur in the description. For the time being, in our cur-rent paper we describe a classical particle by four coordinates X µ only, and assumethat it is extended and held from collapsing into a black hole by the forces that arenot included into the description. In our opinion this is legitimate, because a physical model necessarily involves a restricted set of variables, fields, etc., and neglects therest.However, we have to bear in mind that the action of Einstein’s gravity containingonly the first order term of the curvature scalar, R , leads upon quantization to di-vergences that cannot be renormalized. In the present paper we have not addressedthis problem. One possibility is to include into the gravity part of the action (49)the quadratic, R , and higher order/derivative terms as well, and re-run the proce-dure considered in this paper, either with the same matter action consisting of pointparticles, or generalizing them to superparticles, as mentioned above. Inclusion ofhigher derivative terms would then solve the problem of renormalizability. But, aswidely recognized, we would then have problems with ghosts and instabilities. Thereis a vast literature on how higher derivative gravity theories can be made physicallyviable (see, e.g., [38], and references therein.)A higher derivative gravity action arises as an effective action of string theory.Another promising theory is loop quantum gravity. Both those theories have sat-isfactorily addressed the issues of quantum divergences. We anticipate that thosedifferent theoretical structures, such as a higher derivative canonical gravity with su-perparticles as sources (that can be extended to superstrings), string theory, and loopquantum gravity, will at the end turn out to be revelling different aspects/corners ofa fundamental theory yet to be discovered.21 cknowledgement This work has been partly supported by the Slovenian Research Agency.
Appenix A: Coupling of “point-like” sources to the gravi-tational field
Because classical gravity contains black hole solutions, point particle sources areproblematic . However, if a particle is extended beyond its Schwarzschild radius, inprinciple there is no problem. When describing motion of an extended object, we mayneglect its internal dynamics and consider only the motion of an effective worldline,a “center of mass”. In such a case the coupling of the object with the gravitationalfield may be given by Eq. (53), with understanding that X µ ( τ ) are center of masscoordinates, and that the range of the considered spacetime coordinates is outsidethe Schwarzschild radius.As a model of extended object let us consider an open space filling brane, de-scribed by coordinates X µ ( ξ A ), µ = 0 , , , ..., D − A = 0 , , , ...p , where in ourcase of a space filling brane, p = D −
1. The action for such a system is I m = µ B Z d p +1 ξ ( − det ∂ A X µ ∂ B X µ ) / , f AB ≡ ∂ A X µ ∂ B X µ , (107)where µ B is the brane tension. When considered as a matter source of gravity theabove action has to be rewritte so to include a δ -function: I m = µ B Z d p +1 ξ ( − det( ∂ A X µ ∂ B X µ ) / δ D ( x − X ( ξ ))d D x. (108)This action is invariant under reparametrizations of the worldsheet parameters ξ A ,and under general coordinate transformations of spacetime coordinates x µ . Rewrittenin terms of new ξ A and x µ , it retains the same form (108), but with new functions X µ ( ξ A ), representing the same worldsheet.Let us now split the derivative according to [34, 35] ∂ A = n A ∂ + ¯ ∂ A , (109)where n A is a vector in the direction of the hypersurface element dΣ A = dΣ n A , and ∂ is the derivative in the direction of n A (normal derivative), whilst ¯ ∂ A is the tangentialderivative, orthogonal to n A ∂ . Then we have f AB = ∂ A X µ ∂ B X µ = n A n B ∂X µ ∂X µ + ¯ ∂ A X µ ¯ ∂ B X µ , (110) See the paper [33], where those problems are thoroughly analysed not only for point particles,but also for branes. This means that the brane does not fill the entire space, but only the volume inside a sphere,so that our brane is in fact a ball-filling brane. n A n A = 1 ∂X µ ∂X µ , (111)Inserting (110) into the definition of f = det f AB , we obtain [34, 35] f = ˜ fn , ˜ f = 1 p ! ǫ A ...A p ǫ B ...B p ¯ f A B ... ¯ f A p B p , ¯ f AB ≡ ¯ ∂ A X µ ¯ ∂ B X µ . (112)In a gauge in which n A = (1 , , , ... ), using the procedure of Ref. [35], we have ξ A = ( τ, σ a ), a = 1 , , ..., p , ∂X µ = ∂/∂τ ≡ ˙ X µ , ˜ f = det f ab ≡ ¯ f , ∂ a = ∂/∂σ a , anddΣ = d p σ .Using (109)–(112), the action (108) then reads I m = µ B Z d τ d p σ q − ˜ f p ∂X µ ∂X µ δ D ( x − X ( τ, σ ))d D x, (113)which is a covariant expression, because it does not change its form under the trans-formations ξ A → ξ ′ A = χ A ( ξ ) and x µ → x ′ µ = F µ ( x ). In a particular gauge (choiceof parameters ξ A ), considered above, the action becomes I m = µ B Z d τ d p σ q − ¯ f q ˙ X µ ˙ X µ δ D ( x − X ( τ, σ ))d D x, (114)Let us now choose a line X µ T ( τ ) and write X µ ( τ, σ ) = X µ T ( τ ) + w µ ( τ, σ ) . (115)Then I m = µ B Z d τ d p σ q − ¯ f q ˙ X µ T ˙ X T µ δ D ( x − X T ( τ ) − w ( τ, σ ))d D x, (116)where we have now ¯ f = det ∂ a w µ ∂ b w µ .Next, let us use the expansion , δ ( x − X T − w ( τ, σ )) = δ ( x − X T ( τ ))+ Z d p σw µ ( τ, σ ) δ δ ( x − X T ( τ ) − w ( τ, σ )) δw µ ( τ, σ ) (cid:12)(cid:12)(cid:12)(cid:12) w ( τ,σ )=0 + ..., (117) We can verify on a simpler example that such expansion indeed works: Z d x F ( x ) δ ( x − a ) = F ( − a ) Z d x F ( x ) (cid:18) δ ( x ) + ∂δ ( x − a ) ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 a + ... (cid:19) = F (0) − F ′ (0) a + ... = F ( − a ). f does notdepend on τ , the quantity m = µ B R d p σ p − ¯ f can be factored out. Assuming thatthe size of the brane is small and neglecting the terms with the powers of w µ , weobtain I m = m Z d τ q ˙ X µ T ˙ X T µ d D x √− g δ D ( x − X T ( τ ) √− g , x / ∈ Ω . (118)In the latter equation we have effectively approximated X µ ( τ, σ ) with X T ( τ ), andtaken only the region x / ∈ Ω of the spacetime outside the ball.The action (118) implies that X T ( τ ) is a geodesic. This is consistent with thebrane equations of motions corresponding to the action (107), D A D A X µ = 0, whichcan be split as D τ D τ X T + D τ D τ w + D a D a w µ = 0. The w -terms in the latter equationare due to the extension of the object and represent a deviation from the geodesicequation, like in the Papapetrou equation. If we neglect them, then we have thegeodesic equation.The above example of a ball, modelled `a la space filling brane, shows how anextended object can be approximately described as a point particle coupled to thegravitational field, with understanding that only the region outside the horizon istaken into account. The region inside horizon is not taken into account, becausethe object is actually not point-like, but extended. With the ball, we do not have asmeared δ -function, but a “true” δ function, namely, δ D ( x − X ( τ, σ )) / √− g , i.e., anobject which is covariant under general coordinate transformations of x µ , and alsounder reparametrization of ξ A = ( τ, σ a ).The procedure explained above can be straightforwardly adapted to hold not onlyfor a brane, filling a ball, but also for any brane, for instance, for a closed 2-brane,considered by Dirac as a model of electron.In our procedure we in fact avoided the problem of defining the center of mass inspecial an general relativity. Namely, a far away observer cannot distinguish amongarbitrarily chosen lines within the extended object whose size is negligible in com-parison with the considered distances. Let us now nevertheless demonstrate how thecenter of mass could be defined, first in flat spacetime and then in a curved one.Choosing a unit time like direction n µ in Minkowski space, let us define the centerof mass coordinates for a system of point particles according to X µ T = P k p ρk n ρ X αk N αµ P k p σk n σ , (119)where N αµ = δ αµ − n α n µ is the projector onto the hypersurface Σ µ , orthogonal to n µ .The center of mass coordinates can thus be interpreted as being defined with respectto a chosen simultaneity surface Σ µ , associated with an observer.The Poisson brackets between so defined center of mass coordinates and the total24omentum P ν = P k P νk are { X µ T , X ν T } = ∂X µ T ∂X βk ∂X ν T ∂p kβ − ∂X ν T ∂X βk ∂X µ T ∂p kβ = 0 , (120) { X µ T , P ν } = N µν = η µν − n µ n ν . (121)These equations are Lorentz covariant. In the particular case of n µ = (1 , , , { X , P } = η − , (122) { X r T , P s } = η rs , r, s = 1 , , { X , P s } = η s − n n s = 0 , (124) X µ T = P k p k ( X µk − X k n µ ) P k p k = ( µ = 0 , P k p k X rk P k p k if µ = r. (125)We see that these are correct Poisson bracket relations.For a generic stress-energy tensor we have X µ T = R dΣ ν T νρ n ρ X α N αµ R dΣ ν T νρ n ρ . (126)How to define the center of mass in curved spacetime is much debated, with nounique generally accepted solution. Our tentative proposal is first to consider x α asa vector field which in given coordinates [13] is a α ( x ) = x α . Then we can generalize(126) to X µ T = R [ x ] dΣ ν √− g T νρ n ρ a ( x ) α N αµ R dΣ ν √− g T νρ n ρ . (127)where R [ x ] denotes the covariant integral over a vector field, which in the abovecase is A µ ( x ) = a α N αµ . This means that the vectors A µ ( x ) at different points x areparalelly transported along a geodesic from the point x to a chosen point x (the“origin”), where they are summed (integrated). How precisely this works is shown inRefs. [36, 37] and [14].For the purpose of the procedure adopted in this paper it is sufficient that thecenter of mass, or any other point that samples the motion of a finite size particle,does exist. The precise location of such point within the particle is not important forthe validity of our procedure. 25 ppenix B: The connection between the multiparticleSchr¨odinger and Klein-Gordon equation The multiparticle Schr¨odinger equation (48) can be cast into the following systemof equation for the real and the imaginary part of the complex wave function ψ = ψ R + iψ I : ∂ψ R ∂t = r X k =1 ω x k ψ I ( t, x , x , ..., x r ) , (128) ∂ψ I ∂t = − r X k =1 ω x k ψ R ( t, x , x , ..., x r ) , (129)Introducing the compact notation x , x , ..., x r ≡ ψ ( X r ) , ψ R ( x , x , ..., x r ) ≡ ψ ( X r ) R , (130)Ω ( X r )( X s ) = r X k =1 ω x k δ rs δ ( X r − X ′ s ) , ω x k = p m − D x k . (131)we can rewrite Eqs. (128),(129) as˙ ψ ( X r ) R = Ω ( X r )( X s ) ψ ( X s ) I , (132)˙ ψ ( X r ) I = − Ω ( X r )( X s ) ψ ( X s ) R , (133)Expressing ψ ( X s ) I in Eq. (132) in terms of ˙ ψ ( X r ) R , ψ ( X s ) I = Ω − X r )( X s ) ˙ ψ ( X r ) R , (134)and inserting it into Eq. (133), we obtain the following second order equation:¨ ψ ( X r ) R + Ω ( X r )( X s ) Ω ( X s )( X k ) ψ ( X r ) R . (135)Taking into account the explicit form (131) of the matrix Ω ( X r )( X s ) , the latter equationbecomes¨ ψ R ( t, X , X , ..., X r ) + r X m =1 r X n =1 ω x m ω x n ψ R ( t, X , X , ..., X r ) = 0 . (136)Introducing now the notation φ ( t, X , X , ..., X r ) ≡ ψ R ( t, X , X , ..., X r ) ≡ ˜ ϕ ( t , t , ..., t r , X , X , ..., X r ) and using the relation (78), which impliesd φ d t = X m,n ∂ ˜ ϕ∂t m ∂t n , (137)26q. (136) can be written in the form X M,n (cid:18) ∂ ˜ ϕ∂t m ∂t n + ω x m ω x n ˜ ϕ (cid:19) = 0 . (138)In the latter equation is embraced the multiparticle Klein-Gordon equation (68) ∂ ˜ ϕ∂t n + ω x n ˜ ϕ = 0 . (139)For illustration let us now consider a flat space solution of the above equation,˜ ϕ ( t , ..., t r , x , ..., x r )= Z d p ... d p r (cid:0) c ( p , ..., p r )e − P k p k x k + c ∗ ( p , ..., p r )e P k p k x k (cid:1) . 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