symme-trized extension of Wheeler-DeWitt equation and the symmetrized quantum potential for gravity
If at the fundamental level of quantum processes, non local corre-lations are due to a background space which acts as a direct informa-tion medium between the particles under considerainforma-tion, it is legitimate consider the possibility that also in the quantum gravity domain space functions as a direct information medium. In this regard, in this chapter we want to focus the attention of the reader on a mathematical model of quantum gravity (recently developed by the authors) in which the idea of stage of processes as a direct, immediate information medium can be embedded. The crucial idea that is at the basis of this model is to build a symmetrized version of the bohmian approach of Wheeler-DeWitt equa-tion and thus to introduce the consideraequa-tions made in chapter 5 inside Wheeler-DeWitt equation.
As we know, in quantum gravity and cosmology universe can be described by a wave-functional Ψ which satisfies the Wheeler-DeWitt (WDW) equation (here we have made the position~=c= 1):
(8πG)Gabcdpabpcd+ 1 16πG
√g
2Λ−(3)R
Ψ = 0 (25) whereGabcd= 12√
g(gacgbd+gadgbc−gabgcd)is the supermetric,pabare the momentum operators related to the 3-metric gab, g = detgij, (3)R is the 3-dimensional curvature scalar,Λis the cosmological constant,G is the gravitational constant. In the bohmian approach, by decomposing the wave-functional Ψ in polar form Ψ = ReiS/~ the WDW equation becomes
(8πG)Gabcd
δS δgab
δS δgcd
− 1 16πG
√g
2Λ−(3)R
+QG= 0 (26) where
QG=~2N gGabcd
1 R
δ2R
δgabδgcd (27) N being the lapse function. The term QG can be defined “quantum potential for gravity”. Moreover, in the bohmian approach, Einstein’s equations – in absence of source of matter-energy - assume the following
forms :
Rµν−1
2gµνR=−1 N
δR
QGd3x
δgij (28)
for the dynamical parts, and R0ν−1
2g0νR= QG 2√
−gg0ν (29)
for the non-dynamical part. (The reader can find some interesting de-velopments as regards the bohmian approach to WDW equation, for example, in the references [41, 42, 43, 44, 45, 46]).
Equations (26), (28) and (29) show that the termQGis responsible of the behaviour of the universe intended as a whole. On the basis of equa-tions (26), (27), (28) and (29) regarding the bohmian approach to WDW equation, one can say that universe presents a sort of aggregate, com-prehensive “order” which guides it : this order is just determined by the
“quantum potential for gravity” (27). The quantum potential for gravity (27) can be thus considered as a sort of generalization of the bohmian quantum potential to the universe as a whole. Just like in Bohm’s pilot wave theory the quantum potential (9) guides the motion of a subatomic particle in the regions where the wave function of that particle is more intense, in analogous way the quantum potential for gravity (27) can be considered the crucial element that guides the behaviour of the universe.
Moreover, just like for the quantum potential (9) of the original Bohm’s pilot wave theory, also for the quantum potential for gravity (27) we can make important considerations which derive from its ma-thematical expression. In fact, if the quantum potential for subatomic particles (9) has a like-space, instantaneous action, in analogous way also the quantum potential of gravity (27) has an instantaneous, like-space action. As a consequence, if the original Bohm’s quantum potential (9) can be associated – in the context of quantum non-locality in the subato-mic world - with the idea of space as an immediate information medium between subatomic particles, in analogous way the quantum potential for gravity (27) - just in virtue of its non-local instantaneous action – can be associated with the idea of space in the quantum gravity domain as an immediate information medium. The quantum potential for gravity (27) can be thus considered an appropriate candidate to represent the special state of space in the quantum gravity domain as an immediate, direct information medium.
However, just like it happens in the original Bohm’s pilot wave theory, also the original Bohm’s approach to quantum gravity cannot be consi-dered completely convincing from the point of view of the mathema-tical symmetry in time. The standard quantum laws regarding WDW equation (and thus also the Bohm’s approach to WDW equation which derives from it) are not time-symmetric and therefore if one inverts the sign of time, the filming of a process in the quantum gravity domain could not correspond to what physically happens. In particular, in the Bohm’s approach, the quantum potential for gravity (27) - although has a like-space, an instantaneous action - just because it comes from WDW equation which is not time-symmetric, cannot be considered completely satisfactory from the mathematical point of view, in the sense that it can meet problems by inverting the sign of time.
On the basis of these considerations, in order to have a more appro-priate candidate for the state of space in the quantum gravity domain (and thus for the universe as a whole), which could assure a mathema-tical symmetry in the exchange of t in –t, it is necessary to consider a symmetrized version of the quantum potential for gravity. In this re-gard, before all, the problem of time-symmetry in the standard inter-pretation of WDW equation can be resolved in analogous manner to the considerations made by Wharton in his attempt to develop a time-symmetric formulation of standard quantum mechanics, by considering a time-symmetric extension of WDW equation of the form
H 0 0 −H
C= 0 (30)
where
H =
(8πG)Gabcdpabpcd+ 1 16πG
√g
2Λ−(3)R
(31) and C=
Ψ Φ
, Ψis the solution of the standard WDW equation,Φ is the solution of the time-reversed WDW equation.
The second step is to build a time-symmetric reformulation of the bohmian approach to WDW equation in the light of the symmetrized WDW equation (29). This can be obtained in analogous way to the program followed in chapter 5 regarding time-symmetric extension of Bohm’s pilot wave theory. The crucial point is to decompose the time-symmetric WDW equation (30) into two real equations, by expressing
the wave-functionalsΨandΦin polar form :
Ψ =R1eiS1 (32)
Φ =R2eiS2 (33)
whereR1andR2are real amplitude functionals andS1 andS2 are real phase functionals. Inserting (32) and (33) into (30) and separating into real and imaginary parts we obtain the following quantum Hamilton-Jacobi equation for quantum general relativity
(8πG)Gabcd In this way a symmetrized extension of bohmian version of WDW equa-tion emerges. This approach is characterized by a symmetrized quantum potential for gravity at two components
QG = ~2N gGabcdR1 The first component (35) coincides with the original “quantum potential for gravity” (27) : it can explain the forward-time process of the space-like, instantaneous action of quantum gravity. The second component (36) must be introduced in order to reproduce also the time-reverse of a process in the quantum gravity domain through the instantaneous action (this second component assures that one could see what really happens if one would imagine to film the process by going backwards in time). As a consequence, just because it allows us to recover a symmetry in time, the symmetrized quantum potential for gravity can be considered a good mathematical candidate for the state of space in the quantum gravity domain that expresses a direct, immediate information medium (just like the symmetrized quantum potential (24) can be considered the star-ting point to develop mathematically the idea of space as an immediate
information medium between elementary particles). The symmetrized quantum potential for gravity implies that also in the quantum gravity domain a fundamental stage of physical processes exists which acts as an immediate information medium [47].
On the basis of the argumentations provided in chapters 5 and 6 as regards a symmetrized Bohm’s version of non-relativistic quantum mechanics and a symmetrized Bohm’s version of WDW equation, we can therefore conclude that both the wave functions of subatomic particles and the wave-functionals of the gravitational field in the quantum gravity domain determine a space medium, a special state of physical reality (represented, respectively, by the symmetrized quantum potential and the symmetrized quantum potential for gravity) which acts as a direct, immediate information medium in its respective domain.
7 Conclusions
On the basis of the treatment made in this article, time that is mea-sured with clocks is not an independent dimension that flows on its own but is a numerical order of the material change i.e. motion that runs in a timeless space. This understanding introduces interesting perspectives in the interpretation of the instantaneous communication between subato-mic particles in an EPR-type experiment. Information transfer between subatomic particles is instantaneous because space functions as a di-rect, as an immediate information medium. The symmetrized quantum potential appears as the most adequate candidate to represent mathema-tically the idea of space as a direct information medium. In the quantum domain, space assumes the special state represented by the symmetri-zed quantum potential which produces an instantaneous communication between the particles under consideration and allows us to interpret in the correct way both the forward-time and the time-reverse of the same physical process. Forward-time and time-reverse of a physical process are a matter of numerical order of material changes in a timeless space.
Moreover, as regards the idea of a background timeless space as an imme-diate medium of information, the symmetrized quantum potential seems to occupy a fundamental role also in the quantum gravity domain. In a complete physical theory the possibility is opened that a fundamen-tal arena in which space functions as an immediate information medium and which is associated with a symmetrized quantum potential should assume a crucial role and all the objects of physics might emerge from it as special states.
References
[1] P. Yourgrau, A World Without Time : The Forgotten Legacy of Gödel And Einstein, New York : Basic Books, 2006.
[2] J. F. Woodward, “Killing time”, Found. Phys. Lett.9(1996).
[3] C. Rovelli, “Time in quantum gravity : an hypothesis”, Phys. Rev.D43, 2, 442 (1990).
[4] C. Rovelli, “Analysis of the different meaning of the concept of time in different physical theories”, ll Nuovo Cimento110B, 81 (1995).
[5] C. Rovelli,Quantum gravity. Cambridge : Cambridge University Press, 2004.
[6] J. Barbour, “The Nature of Time”, http ://arxiv.org/abs/0903.3489 (2009).
[7] T. N. Palmer, “The Invariant Set Hypothesis : A New Geometric Frame-work for the Foundations of Quantum Theory and the Role Played by Gravity”, http ://arxiv.org/abs/0812.1148 (2009).
[8] F. Girelli, S. Liberati, L. Sindoni, “Is the notion of time really fundamen-tal ?”, http ://arxiv.org/abs/0903.4876 (2009).
[9] E. Prati, “The nature of time : from a timeless hamiltonian framework to clock time metrology”, arXiv :0907.1707v1 (2009).
[10] P. Eckle et al., “Attosecond Ionization and Tunneling Delay Time Mea-surements in Helium”, Science322, 5907, 1525 – 1529 (2008).
[11] Pirandola S.et al., “Quantum direct communication with continuous va-riables”, A Lett. J. Explor. Front. Phys. (2008).
[12] G. C. Hegerfeldt, “Causality problems for Fermi’s two-atom system”, Phys. Rev. Lett.72, 596 - 599 (1994).
[13] D. Fiscaletti, A. S. Sorli, “About absolute physical phenomena and rela-tive physical phenomena in a timeless space”, IUP Journal of Physics4, 1 (2011).
[14] J. H. Field, “Quantum electrodynamics and experiment demonstrate the nonretarded nature of electrodynamical force fields”, Phys. Part. and Nucl. Lett.6, 4, 320–324 (2009).
[15] A. L. Kholmetskii et al., “Experimental test on the applicability of the standard retardation condition to boun magnetic fields”, J. Appl. Phys.
101, 023532 (2007).
[16] C. Rovelli, “Quantum spacetime : what do we know ?, e-print arXiv gr-qc/9903045 (1999), printed inPhysics meets philosophy at the Planck scale, C. Callender and N. Huggett eds., Cambridge : Cambridge University Press, 2001.
[17] A. Sorli, D. Fiscaletti and D. Klinar, “Time is a measuring system derived from light speed”, Phys. Ess.23, 2, 330-332 (2010).
[18] C. Rovelli, “Loop Quantum Gravity”,Living Reviews in Relativity, http ://relativity.livingreviews.org/Articles/lrr-1998-1/ (1997).
[19] I. Licata, “Minkowski Space-Time and Dirac Vacuum as Ultrareferential Reference Frame”, Hadronic Journal14, 3 (1991).
[20] F. Selleri (2000), “Space and time should be preferred to spacetime – 1”, International workshop Physics for the 21stcentury, 5-9 June 2000.
[21] F. Selleri (2000), “Space and time should be preferred to spacetime – 2”, International workshop Physics for the 21stcentury, 5-9 June 2000.
[22] R. Manaresi and F. Selleri (2004), “The international atomic time and the velocity of light”, Found. Phys. Lett.17, 65.
[23] A. Sorli, D. Klinar and D. Fiscaletti, “New insights into the special theory of relativity”, Phys. Ess.24, 2 (2011).
[24] A. S. Sorli, Einstein’s timeless universe, Saarbrücken : LAP Lambert Academic Publishing, 2010.
[25] D. Bohm,Quantum theory, New York : Prentice-Hall, 1951.
[26] A. Sorli A I.K. Sorli, “From space-time to a-temporal physical space”, Frontier Perspectives14, 1, 38-40 (2005).
[27] D. Fiscaletti, “A-temporal physical space and quantum nonlocality”, Electr. Jour. Theor. Phys.3, 2, 15-20 (2005).
[28] D. Bohm, “A suggested interpretation of quantum theory in terms of hidden variables”, Phys. Rev.85, 166-193 (1952).
D. Bohm, Phys. Rev.89(1953).
[29] L. de Broglie, J. Physique,6, 225 (1927)
L. de Broglie,Conseil de Physique Solvay, octobre 1927, Bruxelles ; Rap-ports et discussions, Gauthier-Villars, Paris 1928, 253-256.
L. de Broglie, Ann. Fond. L. de Broglie,12, 1 (1987).
[30] P. R. Holland,The quantum theory of motion, Cambridge : Cambridge University Press, 1993.
[31] D. Bohm,An ontological foundation for the quantum theory. Symposium on the foundation of modern physics – 1987, P. Lahti e P. Mittelstaedt eds., Singapore : World Scientific Publishing Co, 1988.
[32] J. S. Bell,Speakable and unspeakable in quantum mechanics, Cambridge : Cambridge University Press, 1987.
[33] D. Bohm, B. J. Hiley,The undivided universe : an ontological interpreta-tion of quantum theory, London : Routledge, 1993.
[34] G. Chew, Sci. Progr.51, 529-539 (1960).
[35] B. J. Hiley, “Non-commutative geometry, the Bohm interpretation and the mind-matter relationship”, paper presented at Proc. CASYS’2000, Liege, Belgium, Aug. 7-12 (2000).
[36] D. Fiscaletti, A. Sorli, “Toward an a-temporal interpretation of quantum potential”, Frontier Perspectives14, 2, 43-54 (2005).
[37] K. B. Wharton, “Time-symmetric quantum mechanics”, Found. Phys.37, 1, 159-168 (2007).
[38] D. Fiscaletti, A. Sorli, “Non-locality and the symmetryzed quantum po-tential”, Phys. Ess.21, 4, 245-251 (2008).
[39] C. Philippidis, C. Dewdney and B. Hiley, Il Nuovo Cimento 52B, 15 (1979).
[40] C. Dewdney, “Calculations in the causal interpretation of quantum me-chanics”, inQuantum Uncertainties – Recent and Future Experiments and Interpretations, New York : Plenum Press, 1987, pp. 19-40.
[41] J. Kowalski-Glikman, “Quantum potential and quantum gravity”, arXiv :gr-qc/9511014 v1 (1995).
[42] F. Shojai and M. Golshani, “On the geometrization of bohmian mecha-nics : a new approach to quantum gravity”, Int. J. Mod. Phys.A13, 677 (1998).
[43] F. Shojai, A. Shojai and M. Golshani, “Conformal transformations and quantum gravity”, Mod. Phys. Lett.A13, 2725 (1998).
[44] F. Shojai and A. Shojai, “Pure quantum solutions of bohmian quantum gravity”, J. High En. Phys.5, 037 (2001).
[45] A. Shojai, F. Shojai, “Constraints Algebra and Equations of Motion in Bohmian Interpretation of Quantum Gravity”, arXiv.gr-qc/0311076 v1 (2003).
[46] A. Shojai and F. Shojai, “Bohmian Quantum Gravity in the Linear Field Approximation”, Physica Scripta68, 4, 207 (2003).
[47] D. Fiscaletti and A. Sorli, “Perspectives towards the interpretation of physical space as a medium of immediate quantum information transfer”, Prespacetime Journal1, 6, 883-898 (2010).
(Manuscrit reçu le 27 mai 2011)