arXiv:gr-qc/9712099v1 29 Dec 1997 bevr ol s odsrb hsclraiymade reality physical a the describe that to language mere use a would the of observers elements to the relegated of be role be should sujective would by which it energy multiply case, latter and or the momentum divide In we squared. space-time Planck-length whether a treated the on as be or depending to object, object, prepared energy-momentum an is as zero, both being Hamiltonian which, the by the constraint given dy- is the of cosmology moreover, of structure content formulation, namical canonical scale and its being large distances In as the between universe. at characterize relations looked of that be set times can the by reality cosmology physical described relativistic Here, is experi- development all itself. which other for scale, The resolution Planck maximum ments. the finite at fundamen- the possibly nonzero determine time, realization still and but the length small, is tal very them exist of developments there One that fundamental physics. two gravitational approach from in an allowed time such be and Adopting fundamental would space reality. the of as of them parts bits taking constituent and objectivizing man- matter, of that than reality consists rather physical It an- the exists at still ner. look of There to task pieces. approach the minutest other along its into reality it physical dividing analyse to way matter. ticular of bit observed which an and called objective observer is the the part of with identified part is which system theory physical to the difference of no predictions makes fundamental it the the and as relativity, take of and must blocks one observer building that the matter, -namely of is bits theory it the objetivized the Actually, the by of bits. allowed kinds matter two entities only of the when laws set between reality relation a the the objective as describe the interpreted lan- to is of a this behaviour use of the may elements govern observer the that of any role that subjective guage the to time and betvzn iso atri eeteesjs par- a just nevertheless is matter of bits Objectivizing space relegates relativity that fact well-known a is It ASnme() 33.p 36.a 42.v 04.60.Ds 04.20.Cv; 03.65.Ca; 03.30.+p; theory. number(s): quantum PACS canonical the of operators togethe the l which, method for quantization field ordering Dirac’s dynamical the where new scenario canonical a e leaving The equivalence, described. equivalently an be in can formulation laws alternative physical an inertial encompass t extending also requires to principl It postulates foundational formulation. suitable canonical a the on set based to attempt an review We ersnain frltvt,qatmgaiyadcosmol and gravity quantum relativity, of Representations .INTRODUCTION I. osj ueird netgcoe Cient´ıficas, Serrano Investigaciones de Superior Consejo etod ´sc Mge aa´n,Isiuod Matem´at de Catal´an”, F´ısica Instituto “Miguel de Centro er .Gonz´alez-D´ıaz F. Pedro Dcme 0 1997) 20, (December 1 w eaiitcrpeettossol eepce obe to above expected the be Since should quantities. representations subjective relativistic of two role en- the and to momentum alter- ergy relegating the approach from follow relativistic would nate formu- which mechanics could wave one a late then limit, only nonrelativistic is well-defined mechanics wave currently corre- the that able the to view adhere of we accepted velocity if most Thus, of the and [4]. velocity with particle wave, group coincides sponding and the packet that wave particle proof a Broglie’s of relativistic de correspondence his without without the obscure by of becomes formulated idea as rule But, the Broglie de Duane, before foundation. Du- rule relativistic though, momentum light without crystals the of in [3] theory diffraction postulated corpuscular x-ray ane the explain of to means order by Broglie founda- In relativistic de a [2]. course, on tion based Of relation known [1]. his theory formulated derived relativity consistently requires be it to whether to refers mechanics physical the of elements objective formal- as reality. a such taken of be time quantities should two and ism latter space the and though well, energy, as and transfor- momentum that same for the to mations reproduce identical and should relativity formally they spacetime formalism for language, relativistic rele- subjective be a a to enter of energy elements and to It momentum gated for meters. and that clocks then with appears measure may distances observer space an and that durations the time transformations, of changes Lorentz relativistic through time. or gives, space systems of tial bit is observed part an which of role and the physical observer play the the to taken of with part identified which is the of representation system predictions this the expect to in difference would theory no between One makes relation again bits. it the that objectivized being build- such now the and theory with observers the time, of and space blocks of ing bits objectivized of up nteohrhn,ataiinldbt bu wave about debate traditional a hand, other the On iner- for relativity Einstein of kinematics the Clearly, rmrltvsi ocpsadhsa nprinciple in an has and concepts relativistic from tnint oieta rmsbek such breaks frames noninertial to xtension esaeiedsrpino h relativistic the of description spacetime he ast ossetdfiiin fhermitian of definitions consistent to eads oetmeeg otnu hr the where continuum momentum-energy o ossetqatzto fgravity, of quantization consistent for e ihgaiy lost osrc a construct to allows gravity, with r 2,206Mdi (SPAIN) Madrid 28006 121, csyF´ısica Fundamental, y icas ogy deriv- equivalent while keeping within the inertial framework, function in p-representation Ψ(p) can also be written as one would also expect the two resulting formulations of a function of the wavelength, namely Ψ(λ), by using the h wave mechanics to be equivalent for inertial relativistic de Broglie relation p = λ , which also holds in the non- frames. relativistic limit. Therefore, Ψ(p) = Ψ(λ) can always The idea that we shall explore in this paper is: when be interpreted as the probability amplitude for the pre- we generalize to noninertial frames, besides the same en- supposed particle-like system to behave as a wave with ergy, momentum, time and space intervals, these rela- wavelength λ. At the same time, in x-representation, tivistic representations should give rise to two generally Ψ(x) is regarded as the probability amplitude for the distinct inequivalent dynamical field quantities -namely presupposed point-like particle to be localized in space the usual gravitational field and a new field which we at x. However, if one would presuppose the system to interpret as describing cosmological interactions without be a wave with wavelength λ, whereas Ψ(p) could equiv- nonrelativistic counterpart. Clearly, Einstein’s general alently be regarded as the probability amplitude for the relativity has proved successful in dealing with the cos- wave to propagate with a momentum “localized” at the mological problem under the special restrictions about single value p, there is no known fundamental quantum spatial homogeneity and isotropy of matter in the uni- relation allowing the spatial distance x = R in Ψ(x) (with verse implied by the cosmological and Weyl principles. R being the objectivized bit of spatial distance character- Our claim is that the spacetime relativistic description of izing the system in wave representation) to be discretized cosmology is only valid in the approximation of large uni- so that this wave function can be re-written as a function verses, but as one goes back to the earliest stages of the of a corresponding particle property (which we take to be cosmological evolution it is the cosmological field which the mass), namely Ψ(m), interpretable as the probabil- dominates the dynamics of the evolution. In turn, when ity amplitude for the wave-like system to behave like a the universe becomes large enough, the cosmological in- particle with mass m. teractions considered here are by themselves alone not In the relativistic formalism, x could still be discretized adequate to describe the universe. Actually, what would in terms of a relativistic Compton wavelength of the sys- h exactly describe the evolution of the universe at any stage tem, R λc = mc , i.e. in terms of the spatial scale is the combined effect of the Einstein equations and the at which≡ the system undergoes purely relativistic inter- equations for the introduced cosmological field. A conse- actions with effects such as the fine-structure originat- quence from this point of view is that neither the gravi- ing from its spin. However, this relation would be lost tational nor the cosmological field can be quantized sepa- in the limit c , where R 0, and in any case, rately in a fully consistent way, being the above combined cannot be considered→ ∞ as a fundamental→ quantum relation effect of the two fields which admits a full quantization that could be regarded to be at the same footing as the free from the usual problems of the canonical formalism de Broglie formula, in this case relating a measurable of quantum gravity. bit of objectivized space distance to mass. Moreover, In Sec. II we consider in detail the momentum-energy even at the relativistic level, there exists no known quan- formulation of spacial relativity. It is shown how this and tum relation whatsoever which would link a discretized Einstein relativity can be both derived from a generalized bit of objectivized time, T (characterizing the system), abstract relativistic formalism where action coordinates to the mass of that system, leading to a transformation are used. The wave-mechanical implications from such Ψ(T ) Ψ(m), analogous to as the Einstein-de Broglie a relativistic approach is also dealt with in this section. relation→E = hν does with energy and frequency to allow An extended canonical formalism for noninertial frames the transformation Ψ(E) Ψ(ν). is discussed in Sec. III, where we see how the gravi- Although, given the mass→ of the electron, it is our tational and cosmological fields can be combined in a choice whether to measure its position or momentum, unique picture. Sec. IV deals with the quantization of and this is still enough to describe objective reality in the the generalized canonical formalism. A proof is given inertial approximation, the alluded inequivalence appears that the resulting quantum approach does not show any to be detrimental to the beauty of the underlying theory problem with the hermitian order of operators. and leads, in fact, to the known difficulties encountered in any attempt to quantize gravity (see Subsec. II-D and Secs. III and IV). The electron has a mass, but e.g. II. REPRESENTATIONS OF RELATIVITY in experiments where its interaction with the Coulomb field of the hydrogen nucleus is measured, it also shows There is one sense in which quantum-mechanical po- another element of its objective reality which, like mass, sition and momentum representations are not formally only depends on relative velocity (through the relativis- equivalent if wave-particle duality is, as usual, meant to tic factor): the spatial domain given by the Compton imply equal contributions from the two pictures (wave wavelength where the Darwin interaction takes place, or and particle) to that duality. When one presupposes a equivalently, the time interval that a train a light waves system to be an elementary particle with mass m, the would take in traversing that spatial domain. particle is being assumed to be point-like and its wave The lack of a fundamental quantum relation between R and m and between T and m leading to the above for-

2 1 mal inequivalence appears to be related to the fact that 3 2 wave mechanics originated from a relativistic mechanics ds(q) = (dq0)2 (dqi)2 . (2.1) − where one just objetivizes bits of matter relative to an " i=1 # X observer, but leaves spacetime to always play the role of An inertial reference system for action coordinates qα coordinates labeling events that occur through the emer- 0 1 2 3 gence in spacetime of such bits of matter. It is the au- will then be an orthonormal frame, q , q , q , q , charac- terized by a constant value of the dimensionless quantity thor’s contention that, relative to an observer, one would dq also need a relativistic theory of momentum-energy itself dq0 . We assume (2.1) to be relativistically invariant in in order to objetivize bits of the spacetime -i.e. bits of any of such action reference frames. Note however that space distances and time durations, and hence derive the since they do not correspond to visualizable objetivized missing relations between R and m and between T and elements of the physical reality, the values of these in- m, following steps paralell to de Broglie’s. On the other tervals cannot be measured by any experimental devices. hand, a priori presupposing that a microscopic system This abstract action interval should follow an action line is a particle or a wave would require some appropriate of the universe which at every point has a tangent whose physical conditions to be satisfied by the system. direction in action space is defined by a vector with unit In Einstein relativity space and time are relegated to length given by play the subjective role of elements of a language that is dqα used by the observer to describe his environment, and it uα , (q) = (q) (2.2) is the relation of the bit of matter (with its own space- ds time trajectory) with the observer what makes the ob- q with uα u( ) = 1. jective reality out of which the world is constructed [5]. (q) α What would be new in a relativistic formalism described We regard the appropriate physical conditions that al- in terms of a momentum-energy continuum is the ex- low an abstract wave-particle entity to be objetivized so plicit renouncement to presuppose the Einsteinian ob- that it contains a bit of either space and time (wave pic- jective relation between the observer and bits of matter ture) or matter (particle picture) as being described by a mapping of the action coordinates onto coordinates of, re- as a necessarily establised and unique element of the pos- α α sible physical reality. Instead, we take all three notions, spectively, 4-momentum, dq T0cdp , and 4-position, dqα m cdxα, where T and→ m are objetivized bits space, time and matter -when considered independent of → 0 0 0 the observer- as a priori being merely the elements of of time and matter, and c is the velocity of light. In a subjective language. The observer can then get re- the first case, we allow the system to accommodate null lated to either bits of matter or bits of space and time by rays (null geodesics) along which repetitive, reliable mea- some introspective process that leads to either a distinct, surements of its “objetive” spacetime characteristics are enabled, while the resulting unobjetivized 4-momentum purely theoretical world picture, or to the design of re- α lated experiments and observations, so that, depending components dp are kept as coordinates that label events on the very nature of the system and the predisposition with the above objetivized spacetime characteristics. In of the observer toward it, either the bits of matter or the second case, the mapping allows the system to evolve dp null the bits of space and time become objectivized relative along lines with constant values of dE (which we call to the observer, while space-time or momentum-energy cosmodesics) and this permits repetitive, reliable mea- remains respectively relegated to play the subjective role surements of “objective” particle-like characteristics of the system, while the resulting spacetime components of coordinates. α On the other hand, in order to presuppose “noth- dx are kept as coordinates that are used to label events ing” about the system an abstract relativistic formalism with the above objetivized particle-like characteristics. should be established in which the coordinate labeling The allowance of null cosmodesics to probe the evo- events do not imply any objetivization either of matter lution of the system makes then the action line element or of spacetime. Consistently imposing then the appro- (2.1) and the action velocity vector (2.2) to transform as priate physical conditions on this formalism would finally ds(q) m cds(x) (2.3) result in usual spacetime relativity or the alternate de- → 0 scription in terms of momentum-energy relativity for ob- jetivized bits of, respectively, matter or space and time. α (q)α dx (x)α Quantities that one may take to play the role of the co- u = u , (2.4) → ds(x) ordinates labeling events in the generalized, abstract for- malism are the components of some unobjetivized action where ds(x) is the usual line element of spacetime Ein- qα, α =0, 1, ..., 3. Note that one can make these coordi- stein relativity and u(x)α the corresponding velocity of nates simple dimensionless numbers by using the Planck the universe. If we allow the system to accommodate constant, thus showing the abstract character of them. null geodesics in order to probe its evolution, then it is The usual line element of Einstein relativity would then instead obtained generalize to an action element ds(q) T cds(p) (2.5) → 0

3 dpα u(q)α = u(p)α, (2.6) along different values of the energy. We can then intro- → ds(p) duce momentum-energy reference systems evolving uni- formly relative to each other with relative constant rates with ds(p) the line element in momentum-energy coor- v 2 , so as an extended principle of relativity according to dinates and u(p)α the velocity of the universe defined on c (p) which all the laws of nature are identical in all “inertial” them. The invariance of the interval ds would give rise momentum-energy reference systems, if the equations ex- to a formulation of relativity which is formally equivalent pressing the laws and the events that take place in such to that of Einstein spacetime relativity for inertial frames. reference systems are all described in terms of momenta and energies. Such laws must then be invariant with re- spect to transformations of momenta and energies from A. Special relativity in momentum-energy one momentum-energy reference system to another. A differential interval defined in one of such reference In what follows I will formulate a momentum-energy systems can be given by representation for relativity. In order for the resulting theory to be self-consistent, such a formulation should de2 ds(p)2 = dp2 dp2 dp2. (2.7) satisfy the following requirements. c2 − x − y − z (i) The kinematics of special relativity (i.e. the re- lations between coordinate labels) in the momentum- The principle of relativity for momentum-energy contin- (p) energy representation must satisfy all mechanical Ein- uum implies that ds will be the same in all inertial stein four-momentum transformations, and its associated momentum-energy systems, and leads to the definition mechanics (i.e. the quantities derived from an action of a proper energy given by principle) must in turn obey the usual Lorentz transfor- 1 mations. de′ v2 2 de = , γ = 1 . (2.8) (ii) Whereas description of a given system in space- γ − c2 time implies that such a system occupies just a space-   time part (often just a point) from a necessarily larger Let us consider two inertial momentum-energy refer- system where at least an external observer is also in- ence systems independently evolving with a relative rate v cluded, its description in momentum-energy continuum c2 . From the above discussion it follows that if the en- requires considering the system and the observer as lo- ergy origin is chosen at the point where both systems cated at distinct particular values of momentum and en- coincide, and such systems evolve so that their px-axes ergy intervals on the same frame, so that no evolution of always coincide, then we will have in the limit c → ∞ a system independent of the observer is possible. ′ ′ ′ ′ (iii) The nonrelativistic limit c of the resulting px = px, py = py, pz = pz, e = e + pxv. (2.9) mechanical relations between time→ durations ∞ and space c distances should produce either known or rather trivial On the other hand, if is kept finite, it is easy to see that results, or not exist at all. The nonrelativistic limit of the the transformations that leave invariant the interval are kinematical transformations of momentum and energy p′ + v e′ p = x c2 , p = p′ , must predict values of the energy which depend on the x γ y y chosen reference system, and values of the momentum such that this behaved as an absolute quantity. The latter requirement needs some further explana- p′ v + e′ p = p′ , e = x , (2.10) tion. Consider a system S which evolves uniformly (i.e. z z γ dp at a constant rate de ) in the vacuum momentum-energy continuum. Since, after requirement (i), its evolution which, in turn, coincide with the transformation formu- dp v las for momentum-energy 4-vector of Einstein relativistic rate is de = c2 , we can see why the components of mo- mentum must become absolute quantities in the nonrel- mechanics. Equations (2.10) lead to expressions for the ativistic limit, where energy will still depend on the bare transformations of velocities, general 4-vectors, and unit velocity v. In such a limit, one would not expect the sys- 4-velocities, which exactly coincide with those of Einstein relativistic kinematics, and reduce to (2.9) as c . tem S with energy e1 to interact with itseft with a differ- → ∞ ent energy e2 because, then, the maximum rate of signal Thus, the transformations (2.10) do satisfy the kinemat- 1 ical parts of the requirements in (i) and (iii). propagation in momentum-energy, c , becomes zero. Passing to the domain where c is finite, we see that In order to formulate the relativistic mechanics in the maximum rate of signal propagation in momentum- momentum-energy representation, let us consider a free energy is no longer zero and, therefore, the momentum system evolving in the momentum-energy continuum. components become no longer absolute quantities. This For such a system there should exist a certain inte- will give rise to the emergence of a purely relativistic gral (the counterpart to action of Einstein relativity interaction of the system S with itseft when it evolves in momentum-energy continuum) which has the mini- mum value for actual evolution of the system in the

4 momentum-energy continuum. This integral must have usual transformation formulas for 4-vectors of Einstein the form relativity. It is now inmediately seen that by substitut- ing (2.14) into such formulas, one obtains usual Lorentz b β e2 e2 P = β ds(p) = deγ = Lde,˜ (2.11) transformations. This completes fullfilment of require- − − c ment (i). Za Ze1 Ze1 We also note that the formula for T in (2.13) has no b where is an integral along a momentum-energy world nonrelativistic counterpart. In fact, in the limit c , a → ∞ line of the system between two particular events charac- we obtain from (2.13) R terizing the momentum of the system when it has ener- 2 gies e and e , and β is some constant that characterizes T0v 1 2 T T0 + T0, R T0v, (2.15) the system. The coefficient L˜ of de plays the role of a La- ≈ 2c2 ≈ ≈ grangian and has the physical dimensions of a time. For i.e. the nonrelativistic limit of T and R reduces, respec- P to have the dimensions of an action, unlike Einstein tively, to the rest time and a distance-velocity law which relativity where each system is characterized by its rest 2 may be trivially interpreted as the customary definition energy mc , here each system should be characterized by of velocity. the complementary quantity to its rest energy, that is its On the other hand, it also follows from (2.13) rest time T0. We take therefore β = cT0, and hence the integral P for a free temporal system becomes R = T v (2.16)

e2 P = T0 deγ, (2.12) 2 2 2 2 2 − T c = R + T0 c . (2.17) Ze1 Expression (2.16) should now correspond to the relativis- with L˜ = T γ. − 0 tic expression for the definition of velocity of the object. Instead of a momentum and an energy, the mechanical We finally note that (2.17) must correspond to the ana- system will now be described by a space distance R and logue of the usual relativistic Hamiltonian in our com- a time duration T . Assuming the momentum-energy co- plementary momentum-energy formalism for relativity. ordinate space to be homogeneous, so that the properties If we express time T in terms of the distance R, then we of the system remain invariant under infinitesimal paral- v have a complementary relativistic ”Hamiltonian” lel displacements of rate c2 and energy e, the quantities 1 R and T would be conserved and can be obtained using 1 2 2 2 2 T HT = R + T c , (2.18) the same Lagrangian principles as in classical mechanics, ≡ c 0 but in our complementary representation, i.e.
which has the physical dimension of a time. Law (2.18) ∂L˜ T v v T must correspond to the Minkowskian function F which R 0 ,T R L˜ 0 . = v = = 2 = (2.13) is the conjugate counterpart to Hamiltonian and whose ∂( 2 ) γ c − γ c existence has been recently suggested [6]. It describes the We have to check that the relativistic mechanics ex- way in which objectivized bits of space distance,R, and pressed by (2.11)-(2.13) is consistent with the full rel- time interval, T , are related to each other. ativistic picture, i.e. we have to check that by substi- We still have to check that our mechanical relation tuting space distance and time duration given in (2.13), (2.17) satisfies requirement (iii). Unlike the conventional expressed as a 4-vector in terms of the corresponding 4- Hamiltonian of Einstein relativity, which in the limit velocity, in the transformation formulas for a general 4- c produces the known nonrelativistic Hamiltonian p2→ ∞ vector, one obtains usual Lorentz transformations. That 2m plus the rest energy, the relation (2.18) gives only this is indeed the case can be readily seen by using the the rest time T0 in that limit where, therefore, it in- (p) α 1 principle of least action, δP = 0, and ds = (dpαdp ) 2 , duces no mechanical effects. Of course, for high-velocity 0 e 1 2 3 experiments one would expect time T to increase with with p = c , p = px, p = py, p = pz. We then obtain (p) (p) dpα (x) (x) velocity v and T , such as it is also predicted by Einstein δP = T u δpα, where u = = u = u , u 0 0 α α ds(p) α α α relativity and verified many times in laboratory experi- being− the Einstein unit 4-velocity (see the next subsec- ments. When suitably generalized to noninertial frames tion). It follows that so that it becomes applicable to the whole universe, this ∂P law will describe the cosmological evolution in the vac- xα = = (cT,R)= T0uα (2.14) uum momentum-energy continuum (see Sec. IIIB). −∂pα is the distance 4-vector. It turns out that the square of 0 2 3 i 2 B. The R-m and T-m relations the length of momentum 4-vector, (p ) i=1(p ) , is invariant under transformations (2.10).− Generalizing to any 4-vector Aα which transforms like theP components We note now that the velocities of the universe u(x)α, of the momentum 4-vector under (2.10), we recover the u(p)α and u(q)α are all the same; i.e.:

5 v (x) (p) (q) α where P ′ = P , P ′′ = P , and similarly for Q′ and Q′′. uα = uα = uα = uα = (2.19) m0c T0c cγ On the other hand, rays of the universe will be deter- This invariance would be a particular example of an in- mined by the Fermat principle [2], i.e. variance notion which refers to quantities that preserve Q their values in all the above three types of coordinate sys- δ dϕ =0, (2.24) tems. Of course, all dimensionless quantities that can be ZP formed in the theory should respect this kind of invari- for the representation-invariant phase dϕ ance which we hereafter refer to as representation invari- ance (q) α (x) α (p) α . Thus, the de Broglie theorem of phase harmony dϕ =2πOα dq =2πOα dx =2πOα dp , (2.25) [2] can be regarded to be a consequence from this in- variance. We can in fact visualize any microscopic entity where the Oα’s are the wave vector of the universe [2] on as evolving along lines of the universe on three distinct the respective representation. sheets. Evolution on the action sheet would describe If null cosmodesics are allowed to occur and be used an unobjetivized wave-particle entity propagating with as probes to follow the evolution of the system, then dq q rate 0 and having a pure action phase ϕ . On the dq (x) action line of the universe the entity would carry no defi- (q) Oα Oα = , (2.26) nite observable energy or characteristic time. The action m0c sheet can be unfolded by the above-mentioned mappings into the usual spacetime sheet and a momentum-energy and if, alternatively, null geodesics are permitted to probe sheet, each with the corresponding line of the universe the evolution of the system, we obtain projected on it. Along the spatial line of the universe (p) on the spacetime sheet, the entity would manifest as a (q) Oα Oα = . (2.27) bit of energy propagating on that sheet with given ve- T0c locity v, and along the momentum line of the universe The de Broglie’s extension of the quantum relation [2] on the momentum-energy sheet, it manifested like a bit dp v generalizes then to read of time “propagating” with corresponding rate de = c2 in momentum-energy, or like the phase wave with phase (q) hOα = uα, (2.28) (x) c2 ϕ and velocity v , relative to the spacetime sheet. Likewise, projected on the spacetime sheet, the entity with h the Planck constant. Thus, whereas (2.28) yields would manifest like the phase wave with phase ϕ(p) and the known Einstein-de Broglie relations between momen- 1 propagation rate v , relative to the momentum-energy tum and energy and, respectively, wavelength and fre- sheet. Since the phase is dimensionless, we must then quency whenever null cosmodesics are allowed to occur have and be used to follow the evolution of the system, as far as null geodesics are used to do that, (2.28) gives rise to ϕ(q) = ϕ(x) = ϕ(p) = ϕ. (2.20) the new fundamental quantum relations

These equalities would in fact represent a generalization h µ = mc = ,T = hΩ, (2.29) of the de Broglie theorem of phase harmony. R Let us consider any two points P and Q along the action line of the universe on the action sheet. We can where Eqn. (2.14) has been used, and Ω is the energy then form the integral frequency in momentum-energy continuum. The first of relations (2.29) provides us with the wanted Q Q relation between a discretized R and mass m. It al- ds(q) = u(q)dqα, (2.21) lows the interpretation of the wave function Ψ(x) in x- − − α ZP ZP representation as the probability amplitude for a micro- which should have a stationary value. It is then possible scopic system to be a particle with mass m when one pre- to introduce a general vector of the universe supposes the system to be a wave. One could say that a particle is not but just a wave propagating in momentum- Jα = uα, (2.22) energy continuum with characteristic “wavemomentum” µ. The relation Rµ = h promotes the definition of the rel- and a principle of least action such that ativistic Compton wavelength to the same fundamental status as that played by the de Broglie relation pλ = h. Q α This new fundamental relation already has been therefore δ Jαdq P tested in all those atomic-physics experiments aiming at Z e.g. measuring the relativistic interaction between the

′ ′′ electron and the Coulomb field produced by the hydro- Q Q α α gen nucleus, corresponding to the fine-structure Darwin = m0cδ Jαdx = T0cδ Jαdp =0, (2.23) ′ ′′ term. In particular, the first of equations (2.29) would ZP ZP

6 predict that an electron undergoing Darwin interaction as labels for the events but at the same time they also would be sensible to the ensemble of values taken by the inform us through the Lorentz transformations about ac- Coulomb field within a spatial domain which would de- tual time durations and space distances, measurable with crease as the electron is excited to upper energy levels. clocks and meters. Moreover, although in Einstein rela- Equivalently, the fourth-component relation T = hΩ tivity momentum components and energy can never be provides us with the missing relation between time and taken to label real events, they can be nevertheless ob- a particle-like property, and discretizes an objectivized tained as actual quantities from the associated relativistic time T for the system which corresponds to the time scale mechanics where mass is introduced as an objetivized bit that light waves (null geodesics) would last in traversing of matter. Likewise, in the momentum-energy represen- the spatial domain R. According to it, the objectivized tation of special relativity one would expect the coordi- time appears to be quantized in discrete portions, each nates energy p0 = e and momentum pi to have also a carrying the total energy of the system. This entails no double function: serving as labels of events characterized violation of energy conservation as the time portions are by objetivized bits of spatial sizes and time durations, independent of each other. Both relations in (2.29) have and informing us about the actual values of the energy no counterpart in the nonrelativistic limit c . and momentum of the system. Such values should be the → ∞ same as those predicted by Einstein relativistic mechan- ics. In momentum-energy representation of relativity, C. Wave mechanics in momentum-energy one would also obtain the same transformation formulas for time durations and space distances as in spacetime A quantum-mechanical wave equation can also be de- relativity, though in this case these quantities are given ˆ δ rived from (2.18) by introducing the operators T = i¯h δe as mechanic rather than kinematic quantities. and Rˆ = i¯h δ . Using a wave function Ψ Ψ(p,e), we Since the Klein-Gordon relativistic wave equation gives δp ≡ the eigenenergies of the system in terms of mass eigen- obtain 2 values, en = mnc , and time periods are related to the 2 2 2 ∂ Ψ 1 2 ∂ corresponding wavelengths by an explicit relation, one ¯h = ¯h + T 2c2 + V (p) Ψ, (2.30) − ∂e2 c2 − ∂p2 0 should expect the quantum theory derived from relativ-   ity in momentum-energy coordinates to be formulated in where we have introduced a generic potential V (p). This terms of wave functions which admits a completely equiv- is the counterpart in momentum-energy to the Klein- alent interpretation to that of the wave functions of the Gordon equation. If, as it is the case for the whole uni- quantum theory derived from Einstein relativity, when verse, the system is closed, then one would expect a dis- both are applied to inertial systems. Therefore, the non- crete T -spectrum which would associate with an infinite gravitational quantum theory formulated above must be set of universes ”frozen” at the given eigenvalues of T . completely equivalent to that derived from Einstein spe- This spectrum would only tend to become continuous in cial relativity. the classical region that corresponded to very large val- ues of T . We finally note that the quantum description of systems that show time asymmetry could only be ac- D. The cosmological field counted for whenever we assume a haft-integer intrinsic angular momentum for the whole system, so that, instead The conclusion obtained in the precedent subsection is of (2.30), one would have a Dirac-like wave equation no longer valid for noninertial frames. In general rel- ∂ ativity we must distinguish between coordinate labels γα + cT + V (p) Ψ(p)=0, (2.31) ∂pi 0 from proper intervals, entering at the two totally dif-   ferent levels that correspond, respectively, to differential α with γ the 4 4 Dirac matrices, which is invariant under topology and metric geometry. In spacetime general rela- × e e, but not under T T . Indeed, just as for an- tivity showings of a physical clock are predicted not only →− →− timatter in momentum representation, the negative time by the labels that distinguish events, but also by the states could not be physically ignored, since there is noth- metric, and the change of the metric respect to space- ing to prevent a system from making a transition from a time coordinates describes at the same time a dynamical state of positive time to a state of negative time. Equiv- quantity: the gravitational field. In general theory of alence between the two relativistic quantum-mechanical relativity formulated in terms of momentum-energy co- representations manifests here in the sense that states ordinates, besides mechanical time durations and space with negative time in momentum representation should distances, one would likewise expect the emergence of a be equivalent to states with negative energy in position new quantity: the metric of the momentum-energy con- ι representation as far as an antiparticle moving forward in tinuum, fαβ fαβ(p ), which would help, together with time is equivalent to the corresponding particle moving the pι-coordinate≡ labels, to construct actual momentum backward in time. and energy intervals. Actually, in Einstein relativity the Minkowskian coor- Although as far as it describes the geometry of the dinates x0 = t and xi have a double function: they serve

7 momentum-energy continuum, the dimensionless metric where only the second and third terms in the r.h.s. of (p)2 tensor fαβ would be the same as the usual tensor gαβ by the expression for ds above have been disregarded. representation invariance, its variations with respect to On the other hand, by analogy with the Einstein equa- momentum-energy coordinates should, at the same time, tions, the field equations in momentum-energy represen- describe an independent quantity with “dynamical” con- tation can be written tent by itself; i.e.: a new field which would generally 1 differ from gravity and only coincides with this under R(f,p)αβ =4πK Sαβ fαβS , (2.35) − 2 particular, limiting conditions. These two fields would   in general induce different behaviours in systems acted where R(f,p)αβ is the Ricci tensor espressed in terms upon by them. γ It is in this sense that curved spacetime and curved of tensor fαβ and coordinates p , and K is the coupling momentum-energy are not equivalent representations of constant for the new field. Since this constant has the a unique general-relativity theory. I will give now some dimension of a conventional force, one can regard it as arguments in support of the interpretation that the vari- the universal force exerted upon the system by a universal ι constant field other than ρ, which is defined in spacetime. ations of the momentum-energy metric fαβ(p ) with re- spect to the momentum-energy coordinates must de- The sole field which appears to be able to generate such scribe cosmological interactions. a force is gravity and since this is attractive, K must be negative: K = K . Finally, the tensor Sαβ is a space- (i) Because the dimensionless quantities fαβ, as regarded − | | as the components of a metrical tensor, are representa- time 4-tensor, the counterpart of the momentum-energy 4-tensor of Einstein equations in momentum-energy rel- tion invariant, we should have fαβ = gαβ and therefore, ativity. When the involved velocities are small compared if fαβ is taken to describe a cosmological field, appropri- ate solutions of the usual Einstein equations satisfying to the velocity of light, we have Weyl and cosmological principles are also cosmological β β S uαu τ , (2.36) solutions. α ≃ 0 (ii) For the reasons discussed in the precedent subsec- so that the dominating term in this tensor becomes tions, one would not expect a cosmological field to have 0 T S τ0. The parameter τ0 = , where Vp is the 3- nonrelativistic counterpart. This must actually be the 0 ≃ Vp volume in momentum space, accounts for the time that case for the interactions described by the field equations characterizes the system in momeentum-energy contin- derived from fαβ. Nevertheless, one can consider the v2 uum in a unit momentum volume. Then, Eqn. (2.35) limit of very small but yet nonzero values of c2 , where reduces to a very weak but still nonzero cosmological field with po- 0 tential ρ is present. Assuming this field to be described R(f,p) 4π K τ0. (2.37) ι 0 from the metric fαβ, ρ ρ(p ), the time-Lagrangian in ≃− | | ≡ momentum space could be written as We note that the terms in (2.37) which contain deriva- tives of the affine connections in momentum-energy, T v2 α e L˜ = T + Tc2ρ, (2.32) Γ(f,p)βγ, with respect to c involve extra power c and − 2c2 − therefore are large as compared to the derivative with respect to the momenta pi, i = 1, 2, 3. Hence, we can where, similarly to as the nonrelativistic gravitational po- dr 2 approximate tential goes like ( dt ) , i.e. like a squared velocity, the dp 2 dr 2 potential ρ should go like ( de ) = ( c2dt ) , i.e. like the in- 1 ik ∂e f ∂efik 4π K τ . (2.38) verse of a squared velocity. Hence, in the nonrelativistic 2 ≃ | | 0 limit v 0, ρ will in fact strictly vanish. The situation   c → we shall nevertheless consider is one where From (2.34) and (2.38) we obtain

0

8 depend on p0. In this case, we obtain from (2.33) and same values of field ρ. Thus, if a ray of light is emitted at (2.37) a point where the potential is ρ1 and the energy frequency is Ω, then upon arriving at a point where the potential 2 2 c p ρ 4π K τ0, (2.41) 1+c ρ1 is ρ2 it will have an energy frequency Ω 2 . For an △ ≃− | | 1+c ρ2 ∂2 observer at the arrival point the energy frequency would where p = i . Eqn. (2.41) is formally the same as 2 ∂pi∂p c (ρ1−ρ2) △ then be shifted by an amount Ω=Ω 2 that the Newtonian Poisson equation of nonrelativistic grav- △ 1+c ρ2 ity, except for the sign in the r.h.s. The latter feature corresponds to a proper-energy shift given by shows the essentially repulsive character of field ρ which, E Ec2 (ρ ρ ) , (2.46) therefore, could be a good candidate to describe cosmo- △ ≃ 2 − 1 logical interactions. The analogy between (2.41) and the where E is the proper energy at the emission point where Poisson equation for gravity allows one to solve (2.41) in the potential is ρ . a way which parallels the solution of the Coulomb law. 1 If we assume that the system is our universe and that In the simplest situation of a single system with charac- every point considered represents a galaxy of approxi- teristic time T , we have mately the same size and luminosity, then the light com- τ dVp K T ing to our galaxy from the inner regions of any other ρ ρ(pi) K 0 = | | . (2.42) ≡ ≃| | c2P c2P galaxy would be produced in a physical environment sim- Z ilar to our own. In this case, ρ1 ρ2 and hence E 0. We define the p-force between two sources T and T ′ as However, as the light source separates≃ from the△ core≃ and

′ enters outer regions of the emitting galaxy where the mo- 2 ′ ∂ρ K TT ρ >ρ E < Fp = c T | | . (2.43) menta become smaller, 1 2 and from (2.46) 0. − ∂P ≃ P 2 Although the approximation used in (2.46) breaks△ down as ρ increases, the above discussion appears to point out It is worth noticing that the p-force Fp has the dimension 2 of the inverse of a conventional force, that is the dimen- that, rather than attributing this actually observed effect GN [7] to the presence of some sort of dark matter, it would sion of the Newton constant 4 . Therefore, we can re- c instead be atributted to the noninvariance of proper en- gard GN as the quantity that characterizes the constant ergy under propagation in curved momentum-energy. ρ-field in our universe. Since Fp is repulsive GN is then positive. On the other hand, since ρ does not depend on In what follows, the above results will be taken to im- xi ρ ply that the field derived from variations of the metric the position , it must take on the same value ( 0 say) ι at any two distant spatial points in a system, provided tensor fαβ with respect to coordinates p essentially de- these points are characterized by the same momenta and scribes cosmological interactions. We shall therefore refer times. Assuming then that the total mass of the system to this field as the cosmological field. is M and taking l = cT , V = P , we obtain from (2.42) 0 M0 K III. EXTENDED GEOMETRODYNAMICS V l Hl. = | | 3 = (2.44) ρ0M0c   Let us introduce an arbitrary system of coordinates Xι Thus, H can be interpreted as a Hubble constant and in a Riemannian spacetime, and an arbitrary system of (2.44) as a cosmological law. coordinates P ι in a Riemannian momentum-energy. De- (iii) For a constant ρ-field, p0 should be related to the scribe then a hypersurface in spacetime and a hypersur- 0 ι i proper energy e by e = √f00p . In the case of weak face in momentum-energy by giving four functions X (q ) 0 2 i ι i fαβ-fields, e cp (1 + c ρ). Let us consider then the of three action coordinates q and four functions P (q ) ≃ propagation of a light ray in momentum-energy contin- of the same action coordinates qi, respectively; i.e.: uum when a weak constant ρ-field is present. The light ray will be characterized by an energy frequency Ω which Xι = Xι(qi), P ι = P ι(qi), (3.1) would be given by the derivative with respect to the energy coordinate of the phase-eikonal in momentum- with ι = 0, 1, 2, 3 and i = 1, 2, 3; the 0-component in α energy, η = rαp + φ (with φ an arbitrary constant), momentum-energy corresponds to energy. These two hy- for a “plane− wave” eiη in momentum-energy. As ex- persurfaces are thus labeled hypersurfaces [8], i.e.: in this pressed in terms of∼ the energy p0, the energy frequency case two hypersurfaces together with a common intrinsic ∂η i becomes cΩ0 = 0 , and if we express it in terms of the action coordinate system q for them. Expressions (3.1) − ∂p ι ι proper energy e, we have tell us that the point of the X (P )-hypersurface carrying the intrinsic label qi is located in spacetime (momentum- ∂η 1 ∂η Ω c c c 0 . energy) at the point carrying the spacetime (momentum- Ω= = 0 2 (2.45) ι ι − ∂e −√f00 ∂p ≃ 1+ c ρ energy) label X (P ). This implements the unfolding discussed in Sec. II in the geometrodynamical formalism We lift then the above restriction that spatial points of (see Fig. 1). a system have all the same local momenta and hence the

9 Ω A. Deformations and relabelings N (qi)= N (qi)+ N (qi), (3.3) X T ν E

Changes in a labeled hypersurface on a given projected in which we have used ∂NT = Ω , with Ω as given by the sheet (spacetime or momentum-energy) will generally in- ∂NE ν second of expressions (2.29) and ν is the usual frequency duce changes in the labeled hypersurface on the other defined by the Einstein-de Broglie relation E = hν. Had projected sheet. These labeled hypersurfaces are changed we deformed sheet Π , instead of Π , then we had ob- either by leaving both fixed in the respective embedding T E tained the infinitesimal deformation spaces (spacetime and momentum-energy) but relabeling uniquely their points, or by deforming both hypersur- ̺ [ε(q)]P ι(qi)= P ι(qi)+ mι(qi)δN (qi), (3.4) faces into other pair of hypersurfaces, while leaving their E P labeling fixed. Any arbitrary change of a pair of such hy- where mι(qi) is the unit normal to the hypersurface persurfaces may be decomposed into these two changes P ι(qi), and [8]. ν The first kind of changes represents a pure deforma- N (qi)= N (qi)+ N (qi). tion of the hypersurface in the Riemannian space with- P E Ω T out changing of labeling. It can be carried out as fol- ι i ι i Similarly, relabeling is the operation (which we denote lows. Start from hypersurfaces X (q ) and P (q ). Draw i k k ι i by ̺[¯q (q )]) that takes the label q from fixed space- geodesics perpendicular to X (q ) and cosmodesics per- ι ι ι i time and momentum-energy points X and P and re- pendicular to P (q ). Move then along the geodesic and ι ι i attaches it to the points X¯ and P¯ which originally had cosmodesic that start from the point q , eventually meet- i k ing a point of the deformed hypersurface X¯ι and a point the labelq ¯ (q ). Here deformations of the spacetime (or on the deformed hypersurface P¯ι, respectively. Attach momentum-energy) sheet are again necessary. We have to these points the same label qi as that of the starting ̺ [¯qi(qk)]Xι(qk) points, and describe the displacement of X¯ι with respect E to Xι by giving the proper time τ(qi) measured along the geodesic, and that of P¯ι with respect to P ι by the proper i k i k ι k ι k = ̺T [δNT (q ),δNE(q )]X (q )= X (q ) energy ε(qi) measured along the cosmodesic. Repeating this operation at each point of the two original hypersur- faces will give rise to two single functions τ(qi) and ε(qi) ∂N i +Xι(qk) (qk) δN i (qk)+ T (qk)δN i (qk) that describe the operations of pure deformation of the ,i T i E ∂NE two surfaces; i.e.: ̺[τ(qi)] and ̺[ε(qi)] (see Fig. 2).     Since in the curvilinear formalism the cosmodesic does ι k ι k i k not match the respective geodesic, the label of the end = X (q )+ X,i(q )δNX (q ), (3.5) point on X¯ι(qi) will not coincide with the corresponding ′ label of the end point on P¯ι(qi ), and therefore the sheet with the subscript ,i meaning the derivative with respect i ΠE (or the sheet ΠT ) of Fig. 2 should be deformed in to q , and we have used a generalized shift function which an amount that allows these two final labels to exactly is given by coincide. But deforming e.g. the sheet ΠE induces an ad- ι i i k i k λ i k ditional deformation of hypersurface X (q ) itself. Hence, NX (q )= NT (q )+ NE(q ), (3.6) the action of an infinitesimal deformation of hypersurface µ Xι(qi) will be given by where use of the de Broglie relation and the first of ex- ι i i i ι i pressions (2.29) has been made. ̺T [τ(q)]X (q ) ̺E[δNT (q ),δNE(q )]X (q ) ≡ It is also obtained

i k ι k ι ι k i k = Xι(qi) ̺E[¯q (q )]P (q )= P + P,i(q )δNP (q ), (3.7) in which

ι i i ∂NT i i i k i k µ i k +n (q ) δNT (q )+ (q )δNE(q ) NP (q )= NE(q )+ NT (q ). ∂NE λ    

Note that δNX , as defined from (3.3), will give the ac- ι i ι i i i = X (q )+ n (q )δNX (q ), (3.2) tual proper time separation, T (q ) say, between any two hypersurfaces and is generally different from τ(qi). The where δNT and δNE account for the proper time and the set of deformations of hypersurfaces turns out to be an proper energy, respectively, nι(qi) is the unit normal to infinitely dimensional set [8] whose elements are charac- ι i k i k the hypersurface X , and NX (q ) is a generalized lapse terized by functions T (q ),q ¯ (q ). One can define gen- function having the dimension of a time and is given by erators for the relabeling. Let us use the notation such

10 k i k iq ι that e.g. i(q ) iq, NX (q ) NX , etc. Then if the the change of n produced by the displacement of hy- action of groupH on≡ the H function space≡ is expressed as persurface Xκ induced by displacing P κ by an amount κ ′ ′ ′′ δP (q). In (3.15) the Greek indices are raised and low- iq ιq ¯ ιq κq iq ̺E[NX ]X = X [X ,NX ], (3.8) ered by gαβ (first term) and fαβ (second term), and the Latin indices by, respectively, the metric tensors then the generators can be identified through the in- 4 ι κ 4 ι κ finitesimal transformation gik = gικXi Xk , fik = fικPi Pk , (3.16) ′ iq ιq ι ι ι ι ̺E(δNX )X where Xi X,i and Pi P,i. The terms (3.15) then contribute≡ by an amount ≡

′ ′′ iq ι λ λ ′ ¯ ιq κq ιq δX [X ,NX ] iq δ⊥n (q) iι δXi (q) λ δPi (q) = X + δN , (3.9) = X (q).nλ(q) + δN iq X δXκ(q′) − δXκ(q′) µ δP κ(q′) X N iq =0   X

iq in the neighborhood of the identity N = 0. = Xiι(q)n (q) δλδ (q, q′)+ δλδ (q, q′) X ′ λ κ ,i κ ,i iq iq − If we denote the coefficient for δN in (3.9) by ξ , X iq
the operators (q, q′) (3.17) ′ − iq δ Xiq = ξ ′ (3.10) iq δXιq iι ′ ′ X (q)nκ(q)δ,i(q, q ) (q, q ), (3.18) ′ ≡− − iq will be the generators of the relabelings. The vectors ξiq in which the indices in the first term of (3.17) are raised are obtained by comparing (3.9) with (3.5). It follows and lowered by gαβ, gik and those of the second term in

′ the same equation and in (3.18) by fαβ, fik and also gαβ, iq ι ′ ′ ξiq = X,i(q )δ(q, q ), (3.11) gik. Substituting (3.18) in (3.14), we obtain so that the generators of relabeling are [Xq,Xq′ ]

ι δ Xiq = X,i(q) . (3.12) δXι(q) κ ′ iι ′ δ ′ = n (q )nκ(q)X (q)δ,i(q, q ) (q, q ). (3.19) δXι(q) − Proceeding similarly, we can also identify the generators of pure deformations. They are: By employing then the usual procedure [8], we finally get commutators with exactly the same formal structure as ι δ in conventional geometrodynamics, but with the indices Xq = n (q) ι . (3.13) δX (q) in the r.h.s. being raised and lowered by gik and also by f , which are defined in (3.16). The structure constants of the infinitely dimensional ik group corresponding to relabelings and deformations are determined from the commutation relations of their gen- B. Hamiltonian formalism erators (3.12) and (3.13). Of most interest is the com- mutator between two generators (3.13) A minimal representation of this extended formulation ι of geometrodynamics should use as canonical variables κ ′ δn (q) δ ′ [Xq,Xq′ ]= n (q ) + (q, q ), (3.14) − δXκ(q′) δXι(q) both the metric tensor gik and the metric tensor fik as well as their respective conjugate momenta πik and ωik. where (q, q′) means the same expression with q and q′ Our task now is to find the superhamiltonian and the H interchanged. This antisymmetrization kills all terms supermomentum i which should be constructed out of δnκ(q) the above metric tensorsH and momenta, while respecting in δXι(q′) which are proportional to the delta function κ the commutation relations of geometrodynamics, with an ′ δn (q) δ(q, q ) and, therefore, only the “tilting” term of δXι(q′) action functional remains to contribute [8]. Such a tilting term has in this 4 ik ik ˙ case the form S = d q(πq g˙qik + ωq fqik Z iι ι λ ι X nι δX,i + δP,i , (3.15) − µ i   NXq q N iq), (3.20) − H − XqH where the first term gives the change of nι when the hy- κ δ i persurface X is displaced directly by pure X-deforming where ˙ 0 and NX and NX are given by (3.3) and ≡ δq by an amount δXκ(q), and the second term accounts for (3.6), respectively. The Hamiltonian that corresponds H

11 kl kl to this action functional determines then the change, π xi ω pi πkl = | , ωkl = | (3.24) δF , of any arbitrary function F of the geometrodynamic |i µ |i λ ik ik variables (gik,fik, π ,ω ) induced by the deformation 0 i i 0 δNX = NX δq , δNX = NX δq of the two hypersurfaces. It then follows Under such a deformation kl µ kl ′ ′ π iq = 2 qikπ i + ω i q lq H − |x λ |p δF = [F, q′ δN + lq′ δN ], (3.21) H X H X   ′ ′ q q where N and N are given by (3.3) and (3.6). Spe- T µ E X Xl = i (x)+ i (p) iX (3.25) cializing to pure relabeling (δNX = 0), H λ H ≡ H

′ lq Using the same ansatz as in usual geometrodynamics δF = [F, lq′ δN ], (3.22) H X [8], we can similarly obtain the superHamiltonian and taking into account that both gik and fik tranform ik lm q = Gqiklmπ π (√gR) like tensors and both πik and ωik do like tensor densi- H q q − q ties of weight 1 under relabeling, so that the respective changes are given by the Lie derivatives of a tensor and ik lm + Fqiklmωq ωq fC , (3.26) a tensor density, we obtain a set of equalities, i.e. − q   ′ ′ p lq δ lq′ lq where [gikq, lq′ δN ]= H δN H X δπikq X Gqiklm l l l = gik,lδNX + gilδNX,k + glkδNX,i, 1 = (g g + g g g g ) (3.27) 2 g ilq kmq imq klq ikq lmq ′ ′ √ − lq δ lq′ lq ′ [fikq, lq δNX ]= Hikq δNX H δω is the metric on usual superspace and Fqiklm, which is given by the same expression as (3.26), but with the g f = f δN l + f δN l + f δN l , ’s replaced for the corresponding ’s, is the metric on ik,l X il X,k lk X,i the equivalent superspace constructed from momentum- energy coordinates. Finally, R and C are the scalar cur- ′ ′ lq δ lq′ lq vatures in the respective 3-space. ikq ′ [π , lq δNX ]= H δNX The superHamiltonian will correspond to the op- H δgikq q δ H eration q 0 . Depending on which of the two sub- H ≡ δq spaces it is projected onto, q can be written either as ik l il k lk i H = (π δNX ),l π δNX,l π δNX,l, − − δ q =Ω =Ω X (3.28) H δτ H ′ ′ lq δ lq′ lq ikq ′ [ω , lq δNX ]= H δNX or as H δfikq δ q = ν = ν P . (3.29) ik l il k lk i H δε H = (ω δNX ),l ω δNX,l ω δNX,l, − − Therefore, whose unique solution reads: ik lm q =Ω GqiklmπX πX (√gR)X kl kl H − iq = 2 gikπ + fikω , (3.23) H − |i |i
  ik lm where the subscript i means the corresponding covariant +ν FqiklmωP ωP fC − P derivative. All derivatives| in (3.23) are taken with respect  p   to the action-like coordinates qi. These coordinates were however defined such that hypersurface Xι(qi) would cor- T ν E Ω (x)+ (p) =Ω X . (3.30) respond to a constant value of q0. Due to the mutual ≡ H ΩH H complementary character of X and P , qi may either be   Using (3.2), (3.5), (3.24) and (3.29) in action (3.20) we given by qi = µxi, when it is projected onto spacetime, obtain or by qi = λpi if is is projected onto momentum-energy. Therefore, the covariant derivatives in (3.23) can be writ- 4 ik SX d x(π g˙Xik ten ∝ X Z

12 ik i + ω f˙P ik NX X N Xi), (3.31) IV. QUANTIZATION P − H − X H δ g δ where ˙= δτ when it is over and˙= δε when it is over All of the essential steps that we shall adopt in what f. From δSX we obtain the new Hamiltonian constraint δNX follows are not but hints and guesses as they concern the quantization of the canonical formalism developed in Sec. ν T + E =0, (3.32) III. To my knowledge, there is no other way to proceed H ΩH with the quantization of any field, not even for inertial with T and E the supeHamiltonians of geometro- systems. We start with the action functional obtained in dynamicsH and cosmodynamicsH which, separately, are no the previous section for a gravitating system, i.e. longer zero in the present formalism. 4 ik Of course, one could re-formulate the above canonical SX d x(π g˙Xik ∝ X formalism in terms of the cosmological field rather than Z the gravitational field. We would then derive an action functional ik i + ω f˙P ik NX X N Xi), (4.1) P − H − X H 4 ik SP d p(π g˙Xik which has the same form as that of parametrized field ∝ X Z theories, but contains the additional (second) term, and differs in the specific form of the superHamiltonian and ik i supermomentum which, in (4.1), read + ω f˙P ik NP P N P i), (3.33) P − H − P H ν i T E where N and N are as given in Sec. IIIA, and X = (x)+ (p), P P H H ΩH

E Ω T HP = (p)+ (x) (3.34) ν T µ E H H iX = (x)+ (p), (4.2) H Hi λ Hi

E λ T both being equal to zero. HiP = (p)+ (x). (3.35) Hi µHi Instead of (4.1), one could use the action functional relative to the momentum-energy sheet, SP , which is These are the basic equations for the canonical formula- given in terms of the superquantities P and iP . In tion of the cosmological field which we may call cosmo- the form given by (4.1) and (4.2), our actionH is preparedH dynamics. From δSP , we would then obtain again (after δNP to be quantized just on spacetime. Spacetime quantiza- multiplying by ν and dividing by Ω the resulting expres- tion would proceed by turning into operators the met- ik sion) the constraint (3.32). ric gik(x), the momentum π (x) and, as a consequence Clearly, physical systems that show observable gravita- from the fact that they are given in terms of gik(x)’s ik T T tional effects are usually of large size (even astrophysical and π (x)’s, the quantities and i , as well as the E E H H black holes are remarkably large). Such systems will then quantities and i by themselves. The resulting op- be characterized by small values of ν and rather huge val- erators areH assumedH to satisfy the commutation relations ues of Ω. Hence, using the constraint T = 0 for them (in what follows we set ¯h = c = G = K = 1) becomes an excellent approximation.H However, for pri- 1 mordial black holes or in the very early universe, one [g (x), πlm(x′)] = i δlδm + δmδl δ(x, x′) (4.3) would expect the quantum characteristics of the systems ik 2 i k i k to be exactly the opposite -i.e. such systems would have
large ν and small Ω. In this case, it would be the cos- ′ ik lm ′ mological Hamiltonian which became approximately con- [gik(x),glm(x )] = [π (x), π (x )] = 0 (4.4) strained so that E 0. Therefore, one would also ex- pect this constraintH rather≃ than the usual one to contain T E ′ Ω br ′ [ (x), (p )] = i g (x)δ(x),bδ(x, x ),r (4.5) almost all the relevant dynamical information required H H − ν to describe the latest stages of black-hole evaporation or the earliest stages of the evolution of the universe. T Ei ′ Finally, we note that by independently varying any of [ j (x), (p )] the two above action functionals with respect to either H H metric gik or metric fik, we would respectively obtain [9] Einstein equations and the cosmological field equations λ br ′ i = i g (x)δ(x),bδ(x, x ),rδ , (4.6) (2.35). − µ j where the subscript ,l means derivation with respect to xl.

13 2 In order to proceed with the quantization of the com- T δ Ψ ˆ Ψ Giklm(x) plementary momentum-energy canonical formalism, we H ≡− δgik(x)δglm(x) would start with (3.33)-(3.35) and similarly turn into op- ik E E erators fik(p), ω (p) and hence the quantities , i , δΨ as well as the quantities T and T by themselves.H H The + g(x)R(x)Ψ = i (4.14) i δ E p resulting operators wouldH then satisfyH the commutation − ( ) p H relations ˆT δΨ δΨ lm ′ 1 l m m l ′ i Ψ 2i = i E . (4.15) [fik(p),ω (p )] = i δiδk + δi δk δ(p,p ) (4.7) H ≡ δgik(x) − δ (p) 2  |k Hi
These equations should always be different of zero, un- ′ ik lm ′ less for systems of infinite size. We have therefore decon- [fik(p),flm(p )] = [ω (p),ω (p )] = 0 (4.8) strained our wave equations, leaving them in a manifest Schrdinger-like form. As in the parametrized field the- E T ′ ν br ′ ories, equation (4.15) implies that the state functional [ (p), (x )] = i f (p)δ(p),bδ(p,p ),r (4.9) H H − Ω is unchanged under relabeling of the hypersurfaces. In- deed, by a relabeling of the hypersurface the metric must change into [ E(p), T j (x′)] Hi H gik g¯ik = gik δN δN → − Xi|k − Xk|i µ br ′ i E = i f (p)δ(p),bδ(p,p ),rδ , (4.10) while, since has the dimension of a spacetime dis- − λ j i tance, it undergoesH the transformation where the subscript ,l denotes now derivation with re- E E E ¯ = + δNXi. spect to pl, instead of xl. Hi → Hi Hi For the state functional to be kept unchanged, one should then have A. Spacetime quantization 3 δΨ δΨ d x 2 δNXi|k P δNXi =0. Here, we shall restrict ourselves to explicitly deal with δgik − δ Z  Hi  quantization in spacetime. We shall adopt the metric representation in which the state functional Ψ will be- By integrating by parts the first of these integrals and taking into account the arbitrariness of δNXi, we recover come a functional of the 3-metric gik(x) and the quanti- ties E(p) and E(p), in such a way that in fact the supermomentum wave equation (4.15). The H Hi state functional thus depends on the spatial geometry S G E E and physical distances , but not on the particular met- Ψ Ψ gik, , (4.11) ≡ H Hi ric and position chosenD to represent it. Likewise, one can can be interpreted as containing the information about show [8] the invariance of (4.14) under pure deformations the showings of clocks and meters among its arguments. of the hypersurfaces, so that now the wave functional will also depend on a generic time , but not on any of the Then [10]: T (i) the 3-momentum πik(x) is replaced by the varia- particular moments that may be chosen to represent it. tional derivative with respect to the metric gik(x) Thus, S δ Ψ Ψ , , . (4.16) πˆik(x)= i , (4.12) ≡ G D T − δgik(x) It follows that the proper domain of the state func-

T T tional is an extended superspace which, besides on the and (ii) the quantities (x) and i (x) are replaced H H E 3-geometry, depends also on suitable distance and time by the functional derivatives with respect to (p) and concepts. The specific mathematical characteristics of E(p), respectively, i.e.: H Hj such an extended superspace will be considered in a fu- ture publication. We have in this way succeeded in sepa- δ δ ˆT (x)= i , ˆT (x)= i . (4.13) rating suitably defined space and time concepts from the H − δ E(p) Hj − δ jE(p) H H dynamical variables. Following this procedure, we substitute these opera- tors into the superHamiltonian and supermomentum in B. Momentum-energy quantization spacetime representation, and impose the general con- straints (4.2) as restrictions on the state functional, that is By following a completely parallel procedure, we finally obtain in the case of the cosmological field

14 δ ωˆik(p)= i , (4.17) Hamiltonian constraints. For the ordering chosen, in the − δfik(p) present case one can find

′ rs T 2i[ X (x), X (x, )] = δ(x, x ),r g (x) (x) δ δ H H Hs ˆE(p)= i , ˆE(p)= i . (4.18) H − δ T (x) Hj − δ jT (x)  H H + T (x)grs(x)+ grs(x′) T (x′) Hs Hs δ2Φ ˆEΦ F (p) iklm δf x δf x ν2 H ≡− ik( ) lm( ) + T (x′)grs(x′) + (x, p; g,f; T , E) Hs Ω2 H H  δΦ + f(p)C(p)Φ = i (4.19) 2ν − δ T (x) + [ T (x), E (p′)] + [ E (p), T (x′)] . (4.22) p H Ω H H H H
δΦ δΦ We can readily check that the troublesome terms (those ˆE rs rs i Φ 2i = i T , (4.20) that have factors g or f occurring to the right of the H ≡ δfik(p) |k − δ i (x) T E   H s or s [12] in the second and third lines of (4.22)) Hare all canceledH by the commutators mixing Hamiltonian with in x with that in p in the last line of (4.22). Using then M ′ Ω ′ Φ Φ , , , (4.21) δ,r(p,p )= ν δ,r(x, x ) and (4.5) and (4.9), we finally ob- ≡ G M E tain where the subscript |k now means covariant derivative ′ 1 rs T M [ X (x), X (x )] = i g (x) (x) with respect to the metric of momentum-energy, de- H H −2 Hs notes the geometry of a 3-momentum superspace,G and and some concepts of generic momentum and en- ergy,M definedE parallely to as for generic space and time rs ′ T ′ ν T E ′ +g (x ) s (x )+ (x, p; g,f; s , s ) δ,r(x, x ) concepts in the case of spacetime quantization. Eqns. H Ω H H (4.17)-(4.21) form up the essentials of the formulation of  what we may call quantum cosmodynamics, with the first 1 ′ r r ′ = iδ,r(x, x ) ( (x)+ (x )) . (4.23) two ones being different of zero always unless for systems −2 HX HX of zero size. Thus, (4.23) must vanish weakly. Since in the covariant T E form i or i : (1) the interchange of momenta and H H ′ ′ C. Consistent operator-ordering coordinates only leads to terms with δ,i(x, x ) or δ,i(p,p ) which can be put equal to zero, and (2) the commutators Let us now see how the operator-ordering problem T E ′ λ ′ which appears in conventional geometrodynamics can be [ j (x), i (p )] = i δ(x),j δ(x, x ),i (4.24) worked out in our extended formalism. The problem can H H − µ be expressed by using the hermitian ordering that corre- sponds to the quantum operators proposed by Anderson E T ′ µ ′ [ (p), (x )] = i δ(p),iδ(p,p ),j (4.25) [11]. In our extended formalism of geometrodynamics, Hi Hj − λ Anderson’s ordering translates into will also give terms with derivatives of δ-functions, the or- 1 kl kl µ kl kl der of factors in supermomentum operators does not lead iX = gikπ + π gik + fikω + ω fik H 2 |l |l λ |l |l to any factor-ordering problem. Hence, one can have a h  i closed algebra of the generalized constraints also in the ik quantized theory, and therefore sX Ψ = 0 and X Ψ=0 π lm H H X = (gilgml + gimgkl gikglm) π √gR can be satisfied simultaneously [10]. The same conclusion H √g − − can also be obtained in the quantum-mechanical descrip- tion of cosmodynamics. Thus, the quantization of the ν extended formalism of both geometrodynamics and cos- + (g,f; π,ω; R, C), Ω modynamics leads to no problem with a hermitian order of operators. The issue of quantizing the gravitational where (g,f; π,ω; R, C) denotes the same expression as in field may then be persued without restricting to domains all the explicited terms but with the g’s, π’s and R re- where the factor-ordering problem is circumvented or re- placed for, respectively, the f’s, ω’s and C. The ordering placing the dynamical content of Eqns. (4.19) and (4.20) problem is manifested through the commutator between for a cosmological constant.

15 ACKNOWLEDGMENTS Legends for figures

This work was supported by DGICYT under Research Fig. 1: Relation of points of the two hypersurfaces which Project No PB94-0107-A. carry the same intrinsic label and their location in space- time and momentum-energy sheets.

Fig. 2: Related changes of the normals to the two hyper- surfaces when each of these hypersurfaces is displaced an infinitesimal amount.

[1] A. Land´e, ”Foundations of Quantum Theory”, Yale Uni- versity Press, New Haven, 1955; ”From Dualism to Unity in Quantum Physics”, Cambridge university Press, Lon- don, 1960; ”New Foundations of Quantum Mechanics”, Cambridge Universy Press, London, 1965; E. MacKin- non, Am. J. Phys. 44, 1047 (1976). [2] L. de Broglie, ”Recherches sur la Th´eorie des Quanta”, Ph. D. Thesis, University of Paris, 1924 (Re-edited by Masson et Cie, Paris, 1963). [3] W. Duane, Proc. Natl. Acad. Sci. U.S. 9, 158 (1923). [4] M. Born and W. Biem, Phys. Today, August 1968, pp. 51-56. [5] M. Sachs, Phys. Today, February 1969, pp. 51-60. [6] P.F. Gonz´alez-D´ıaz, Phys. Lett. B317, 36 (1993). [7] P.J.E. Peebles, ”Principles of Physical Cosmology”, princeton University Press, Princeton, 1993. [8] J.A. Wheeler, in: ”Relativity, Groups and Topology”, ed. B.S. DeWitt and C.M. DeWitt, Gordon and Breach, New York, 1964; K. Kucha˘r, in: ”Relativity, Astrophysics and Cosmology”, ed. W. Israel, Reidel Publishing Company, Dordrecht-Holland, 1973. [9] D.R. Brill and R.H. Gowdy, Prog. Theor. Phys. 33, 413 (1970) and references therein. [10] P.A.M. Dirac, ”Lectures on Quantum Mechanics”, Belfer Graduate School of Science, Yeshiva University, New York, 1964. [11] J.L. Anderson, Phys. Rev. 114, 1182 (1959). [12] J.L. Anderson, in: ”Eastern Theoretical Physics Confer- ence”, ed. M.E. Rose, Gordon and Breach, New York, 1962.

16 This figure "fig1-1.png" is available in "png" format from:

http://arxiv.org/ps/gr-qc/9712099v1 This figure "fig1-2.png" is available in "png" format from:

http://arxiv.org/ps/gr-qc/9712099v1