To our present knowledge (but there is little doubt about) both Gravity and Quantumness are fundamental and general phenomena, meaning that their laws govern anything & everything. (The third such is Relativity, and as far as we know any further is at best such a general philosophy as e.g. Causality.) But Gravity and Quantumness, as well in their old forms as in the present best ones contradict each other.

          But General Laws of Physics must not contradict each other. So of they contradict, they are incorrect. (At least one of them; but most probably both.) Very probably the way out would be Unification. Gravity & Relativity contradicted each other until 1916; then Einstein unified them, and the unified theory, General Relativity, is free of at least self-contradiction. Similarly, Relativity and Quantumness contradicted each other, but then they were unified c. 1947 as Quantum Field Theory. The unification was gradual, with such names as Dirac, Tomonaga &c., and the present status is a matter of argumentation. QFT does give infinities; but in some cases they are renormalisable, i.e. the infinities "can be removed". Also a lot of Field Theories, quite respectable in nonrelativistic limit, are unrenormalisable. Now this may mean that QFT is an erroneous Unification; but also that Relativity rules out the unrenormalisable Field Theories. This latter is the opinion of the overwhelming majority of QFT experts; and it may quite be correct. Anyways, Relativity rules out an acceleration to FTL velocities, and this "prediction" is generally accepted (with some sad feeling) by physicists.

          Quantumness and Gravity are also in contradiction in their original forms (as I told), but up to now the unification is not ready. The need of it and the reasons why it has not been made are the topics of this series (which will be being unfold or in Latin simply evolving gradually in several steps. The main reason to write these sequences is Advertisement; either of my own or of my viewpoint about. My feeling is that we are not far from the unification (see e.g. Refs. [1]-[7]), but surely we are not yet ready.

          I must state that (Unified) Quantum Gravity is/will be not the (Final?) Theory of Everything. The latter (sought by many, claimed by few and made by none so far) would be a Unification of Gravity, Quantumness & Relativity, a theory containing 3 fundamental constants, G, h & c, in a theory free of self-contradiction. I am less ambitious (more realistic?) for looking the third and last dual Unification, a Theory of G & h. You will see that it is a task enough in itself: and it would have quite enough new predictions in itself.

          Now let us proceed. The sixth part happens in Hungary from 1985 to 1994.

 

GRAVITY VERSUS/AND QUANTUMNESS, PART 6

 

FALSE DAWN, HUNGARY, OTL, 1985

 

B. Lukács

 

CRIP RMKI, Theory Dept., & President of Matter Evolution Subcommittee of HAS

 

H-1525 Bp. 114. Pf. 49, Budapest Hungary

 

lukacs@rmki.kfki.hu

 

 

 

ABSTRACT

            Successes in the second half of the 80's. Newtonian Quantum Gravity is at last formulated even in OTL.

 

0. ON FALSE DAWNS

            False dawn is a meteorological phenomenon which seems as if it were dawn, maybe it is dawn, but it is not followed by morning; or at least not for a long time. I can mention 3 kinds of well known false dawns.

            The first is a peculiar light of sky at the end of some nights. It is not dawn; not yet. But it is always followed by dawn; sooner or later. At least, always was followed so far.

            The second kind is the False Dawn on Mercury [1]. Mercury's rotational and orbital periods are in 2:3 resonance. For first approximation this would mean simply that 1 day = 2 years. However the orbit is highly eccentric. Therefore there are latitudes where Sun first starts to peep in at the horizon, but then he turns back. This is a false dawn. After some time he again starts; and then goes up and there will be daylight for a Mercury year.

            The third kind happens at specific Terrestrial points. Maurus Jókai, famous although now forgotten (abroad) Hungarian novellist depicts the usual scenario at the industrial town Torockó (Transylvania) in a novel about the 1848 revolt [2]. From the viewpoint of some parts of the town the Eastern horizon is partially obscured by a rocky wall. Therefore there are two dawns. Sun comes up, but is immediately going behind the wall; and one has to wait one more hour for permanent morning.

            Anything may happen. A white dwarf is a dying star. Now, a brown dwarf is a star born already dying.

 

1. A QUEST FOR NEWTONIAN QUANTUM GRAVITY

            Not too much happened between 1974 & 1985 about Quantum Gravity in Hungary. The most important step was Ref. [3]: we started to evaluate Károlyházy's theory for experimental situations. A new kind of Brownian motion is obtained from the stochastic Reductions, and the effect is temptingly near to be observable; indeed, it would be, but every kind of noises would frustrate the experiments. In the subsequent more than 2 decades no experiment was performed.

            An Italian group started something fundamentally similar to Jánossy's idea (see Part 5) but much better elaborated. I mention here the names Ghirardi, Rimini, Weber & Grassi, and give a reference from amongst many as an example [4].

            And then the Relativity Group of CRIP obtained some money by which it was able to organise the first Hungarian Relativity Workshop (now we are at the sixth), called then the Balatonszéplak Relativity Workshop. It was small, but really international; and it was a chance for poor Hungarian physicists to propagate their ideas. (Hungarian currency was not convertible to any other currencies, which was a barrier against Conference participations. Interestingly, Hungarian currency was nearest to free conversion in the Second World; then already some banks in USA, Austria & Italy bought Hungarian Forints on a bad parity.)

            In order to make use of the occasion, three Hungarian lectures were made about three different Quantum Gravity approaches, as follows here:

            Károlyházy. A brief recapitulation of the model mentioned in Part 5, General Relativity + Quantum Mechanics, plus the newest evolution of the theory (that was [3]). This lecture is [5].

            Diósi, Lukács, Martinás & Paál. This was General Relativity + Quantum Field Theory, attacked from thermodynamic viewpoint. You take the Universe. Its geometry is time-dependent, so you cannot do anything else than to solve QED on a changing manifold, so a Hawking radiation of second kind appears [6]. However this radiation does not seem objectively existing, e.g. it does not show Doppler shift between observers of relative velocities. This may be simply a contradiction between the partial unifications (G+c) and (h+c), but may be "real" as well, since it may be the influence of the changing geometry on the observer, and then it is not an external radiation, so why to be Doppler-shifted? With some caution (you are using simultaneously two incompatible partial unifications!) we were able to get a Universe not geodetically complete in the past (so being have been created; sorry for the inevitable clumsiness of Indo-European languages, where only Active, Passive and sometimes Medial voices are possible; I wonder what would be the best translation from Magyar to Basque; the Universe may have created itself, which is not Passive Voice in Magyar). The Universe appears with natural initial conditions: extension of Planck length and temperature of Planck energy [7]. While this is interesting, the group produced one more publication [8], and then the collaboration broke down.

            L. Diósi & B. Lukács. Now, this Ref. [9] was a pure (G+h) approach, in the way told in Part 2. What would happen if we deliberately forget about Relativity?

            The next Chapter summarizes this approach.

 

2. QUANTUM GRAVITY WITHOUT RELATIVITY

            Now I continue where I chopped the story at Padraig O'Gallaghan in ATL 1932. But this is our OTL. I of course remember a lot, and I have our conference lecture from 1985 [9]; but I would rather not ask my coauthor, and the whole thing happened 20 years ago, so in minor details I may remember erroneously. However surely, I knew Károlyházy's theory well enough to calculate in it, and Diósi was interested in Measurement problem in Quantum Mechanics. Now, if Gravity Reduces the wave function, then She Reduces only macro objects, which is good.

            As you see, I calculated real experimental situations in the Károlyházy model; Diósi did it too, but they were not published. (As far as I remember, he calculated the noises from microearthquakes, and you can imagine how vague the results might have been.) But he knew the model well enough. In addition he had the opinion that the Measurement means a stochastic process acting on the wave function [10]. Then Micro is a system (having a coherent Ψ) where this new stochastic term is negligible, and Macro is where it is not.

            Now, Jánossy's idea was that there is a new Law of Nature, reducing when Ψ extends too big. Say, everything is reduced whose Ψ extends on a region >R_red; then we can collect experimental bounds for R_red, e.g. highly excited H atoms are inflated enough, still Schrödinger equation reproduces the measured spectrum well, so the new term does not yet operate [11]. And so on. Similarly, in the Ghirardi-Rimini-Weber scenario some extra laws determine Reduction; such theories improve Quantum Mechanics, without unifying it with something.

            As for the stochastic description Diósi was up to date with models, Markovian processes and whatnot. And we were already working together in 3 problems, one I mentioned in the previous Chapter, another was the Riemannian structure of the thermodynamic state space, and the third the very early Universe at the spontaneous breakdown of the SU(5) symmetry of GUT. So why not a fourth; he presented me a “philosophical” problem.

            It was something similar to the problem of Copernicus (I mean, for structure, not for importance). Copernicus started from the observation (known for every astronomer after Claudius Ptolemy, to be sure) that planets move on independent epicycles, still the directional vectors of the first epicyclic cycles are all parallel. Why? You cannot expect this from the Ptolemaic model, and such an accidental coincidence is contrary to Occam's Razor. It must have a fundamental reason; he suggested that these first epicycles are just the mirror images of Earth's cycle.

            Now, in Károlyházy's theory light velocity c plays important and fundamental role, but c cancels in the formula separating Micro/Macro behaviours. Why?

            Then I suggested the scenario "If Einstein got encephalitis...". The result became [9]. We got exactly Károlyházy's Micro/Macro border even if light velocity appeared nowhere at all in the theory. The recapitulation of the essence of [9] deserves a separate Chapter; unfortunately 1985 is practically pre-PC Age in Hungary, so the paper never existed in file. My earliest extant file is an MS-DOS Word 4.0 one from the middle of 1987.

 

3. SHARP GRAVITATIONAL POTENTIALS ARE MEANINGLESS CONSTRUCTIONS

            There are only G and h as fundamental constants of universal phenomena. (No Einstein, no Relativity.) There are two fundamental theories, closed, excellent, but both of them neglect the other. The first is Newton's Gravity Theory, checked for many digits in Celestial Mechanics, with the only shortcoming at Mercury's perihelion advance; and nobody would refuse an argument that the immediate solar neighbourhood is not the best place to check fundamental theories. The second is the Quantum Mechanics of Schrödinger & Heisenberg, checked on atomic spectra many times. However, if we believe in the universality of both, the true theory should contain both constants.

            Namely, let us assume that we want to measure the gravity potential Φ(r). The simplest way is to hurl compact pebbles, and observe their paths by telescopes. The equation of motion is

              md²r/dt² = -mgrad Φ(r)                                                                                                          (1)

Now, m drops out (which in itself is interesting, but is a fact). So observing the path (also in time), the gradient of Φ is measured, and thence Φ can be integrated, up to a constant, which is simply the freedom in the zero point of energy.

            However the stones we hurl obey Quantum Mechanics. Take the simplest possible stones without excitable internal structure and such. The stones have, of course, masses M and sizes R. Their wave functions extend on a length L. It is not important if the stones are all alike or not.

            Obviously we cannot measure Φ(r) on scales <L, because we do not know where the stone "is". Surely we can prepare arbitrarily sharp Ψ, so there is no lower limit to L. Also, we may restrict ourselves to situations L<R. The observation goes until time T. Then we in fact measure a space and time average of acceleration g (which is -grad Φ). So:

              <g>_V,T = (1/VT)∫g((x+ξ,t+τ)d³ξdτ

              ξ < L, V = 4πL³/3                                                                                                                  (2)

              -T/2 < τ < T/2

The measurement goes for time T, during which the change of the momentum is p. Then

              <g> = p/MT                                                                                                                           (3)

If we can measure <g> for arbitrarily small T and L, then we are ready.

            For this, however, we should start with very small L. But

              δ₁<g> ~ h/MTL                                                                                                                      (4)

since

              Lδp ~ h                                                                                                                                  (5)

That is, we cannot start with too small L. Or, we can, but only if M is big enough. Namely, L doubles in time

              t₂ ~ ML²/h                                                                                                                              (6)

Good; can then the sharp value of the acceleration g be measured by means of pebbles of infinitely large masses?

            No. The pebble is also the source of the potential. Were we to know the exact position of the (center of mass of the) pebble, we could extract its contribution from g; but we have a position uncertainty L. The acceleration uncertainty hence is

              δ₂<g> ~ GM/L²                                                                                                                      (7)

(My proof here is somewhat simplified; I have assumed that the averaging goes just to the interval L. We could average on areas >L too. They give even worse results.)

            The total uncertainty

              δ<g> ~ {(h/MTL)² + (GM/L²)²}^1/2                                                                             (8)

This diverges if T->0; if T is not small, we cannot measure sharp instantaneous values. Also, it diverges if L->0. The smallest uncertainty is achieved with pebbles

              M_opt ~ (hR/GT)^1/2, L ~ R                                                                                                        (9)

and then

              δ_opt<g> ~ (hG/VT)^1/2                                                                                                  (10)

            This uncertainty is inevitable; there is no way around. We can measure very sharply averages for large domains and times; but there is no sharp instantaneous measurement for the acceleration or for the gravity potential.

            In Quantum Mechanics we accepted in the 20's that the fact that p and x cannot be measured simultaneously sharply means that p and x are no c-numbers, but operators; and only their averages or expectation values can be expressed via c-numbers. Now the inevitable lower bound for the inaccuracy of the gravity potential must mean something analogous. We shall see what, in the next Chapter.

 

4. STOCHASTIC POTENTIAL

            Let us now write simply

              Φ = <Φ> + δΦ                                                                                                                       (11)

where <Φ> is simply the old c-number solution of Newton's Gravity equation. Then:

              <δΦ> = 0                                                                                                                               (12)

              <δ²Φ> = hG/VT                                                                                                                     (13)

            Obviously one cannot go beyond (12-13) without at least a model; results (12-13) state that Newtonian Gravity and Quantum Mechanics hurt each other and that how much they are inconsistent. Obviously for Measurement Problem Ψ should be stochastic [10]; if the energy operator in Schrödinger's Equation is stochastic, then Ψ is stochastic via that equation; and the energy operator is stochastic if Φ is stochastic. So let us write

              δΦ = Φ_sto                                                                                                                               (14)

However from (12-13) we do not yet know the full stochastic behaviour. Still, the simplest choice, that Φ_sto is a white noise, works [12], and then

              <Φ_sto(x,t)Φ_sto(X',t')> = ChGδ⁽³⁾(x-x')δ(t-t')                                                                              (15)

where C is a number constant of order 1, simply collecting all the neglected 2's and π's in the above calculations. I am too lazy to evaluate them, so I write henceforth C=1.

            But then we can determine, where is the transition zone between Micro and Macro physices. Namely, let us rewrite the Schrödinger equation as

              ihdΨ/dt = E_kin + E_sto = (-h²/2m)ΔΨ + MΦ_stoΨ (16)

Now, order of magnitude estimations give for a wave packet of width L~R:

              E_kin ~ h²/MR², E_sto ~ M(hG/RT)^1/2                                                                              (17)

For L>R the body is practically point-like; for L<R we must somehow average up for the body, which is possible but will be ignored here.

            If the kinetic term dominates, the body approximately evolves as its center of mass; this is the case of Classical Mechanics. However, if the stochastic term dominates, then the behaviour cannot even approximately be represented by means of unstochastic expectation values. This is Quantum Mechanics. If none dominates, then we arrive at unknown behaviours; at a region where Physics cannot be guessed from the two partial theories known until now.

            Now, this region of new behaviour is then around

              M³R ~ h²/G                                                                                                                            (18)

In principle we cannot yet know the behaviour there, because we do not yet have the Unified but Newtonian Quantum Gravity. Still, a first model is available: Stochastic Gravity and Continuous Measurement. Diósi elaborated a handy formalism for Continuous Measurement, and then everything can be evaluated in the model. Unfortunately, he made a tactical error.

            Eq. (18) is the same as in Károlyházy's theory. (Cannot be anything else; as I told above, we started from the observation that Károlyházy's Micro/Macro border formula does not contain c at all. From h and G (18) is the only possible result; I do not prove it here.) However some other results differed between Károlyházy and us; and we would have known, why. (I was in an interesting position. I was coauthor in [3] and [13], and we observed the differences in formulae derived there. So the results of Károlyházy, Frenkel & Lukács differed from those of Diósi & Lukács, even in the nonrelativistic limit where the formalisms should have coincided (say, c->∞). Where is the difference; and which one is in error? Maybe both; but surely both cannot be correct.)

            In 1989 the situation was as follows. Károlyházy was a Doctor of Science since 1974, full professor, and what you like; at the Roland Eötvös University. Diósi & I were at CRIP, and I just became Doctor of Science. When I got the invitation to write my Theses, I did not chose General Relativity, deliberately, but tried Heavy Ion Physics. I did not have enemies in Heavy Ion Physics, e.g. HIP community was not interested in Quantum Gravity honours.

            Then we looked after the difference between Diósi & Lukács and the nonrelativistic limit of the Károlyházy model; and we found it. It is explicitly and very diplomatically written in [14]; we definitely did not write that one condition for reduction is rather strange there; we only wrote that then the spectrum of the stochastic fluctuations is not that of white noise, and a white noise spectrum is more "attractive". But even this was too much.

            Namely in 1990 Diósi got the idea to write Theses too. He might have chosen Thermodynamics, and then the history I narrate here would have been OK. But he chose Continuous Measurement.

            Now, the process includes 3 opponents who must not work at the same institution as the candidate. In this case that meant practically opponents from the Roland Eötvös University, and one opponent was from the department of the opposite theory (I mean, where Károlyházy worked). Consequently Diósi was refused on a tragicomically unjust process (the opponent simply announced that he did not get answer for any of his questions). Of course the voting process and the argumentation was secret, but it took 21/2 hours instead of the usual 1/2, so you can imagine what happened. I was angry and Diósi was angry & hurt. The honour of the opponent theory was defended in a definitely feudal manner.

            Then we threw discretion to the winds, and evaluated two realistic situations in the Károlyházy model. They are [15] & [16]; the titles are explicit enough, I only add that in [15] we showed that the improbable reduction condition would result also in strange celestial mechanics and in [16] that lethal X-ray radiation would emerge from laboratory lead bricks. So the condition cannot be correct. QED.

            By the way, there was some symbolism in the choice of journals too. [12] was sent to Annalen der Physik, because Einstein's famous papers were published there, and [15] was sent to Nuovo Cimento, because Károlyházy's very first Quantum Gravity paper was published there.

            Diósi wrote a string of papers about Measurement, density matrices, Markovian processes and Itoh formalism, sent some letters to foreign colleagues, got more citations, and defended his Theses in 1998. However then already political climate did not favour our collaboration; if you do not understand this understatement, read again Part 3 for analogy.

            However we are really still at the beginning of the 1990's. Two things happened, important to Newtonian Quantum Gravity. First, in 1990 multiparty democracy returned to Hungary, therefore the intellectuels in weeks formed two solid phalanges and started to hurl slanders to each other. We in CRIP were slightly immune to this; but the hate grew, and Diósi & I were not in the same group. Second, my colleague Ágnes Holba told some details of her measurements for the Jánossy theory (I narrated them already in Part 5) while some of us drinking coffee together in CRIP. This was the time when I got the information that she did not publish the results after Jánossy's untimely death because she did not know THE THEORY. But I knew it!

            Then we wrote the paper; it is [17]. I stop here; the Spurious Scattering deserves a new Part of its own. I close this part with recording that in 1994, when our project financially ended, we (I mean, Diósi & I) organised from the remaining money a very moderate Workshop where everybody participated who was working in Quantum Gravity, excepting Penrose. There was a Spurious Scattering lecture, [18]. And nobody defended [19] against [15] & [16].

 

6. OUTLOOK

            It still looked like a True Dawn.

 

REFERENCES

 [1]       I. Asimov: The Solar System and Back. Discus, Avon Books, New York, 1972

 [2]       M. Jókai: Egy az Isten. (The title means: God Is One; that is the slogan of Unitarians, sometimes called Modern Arians, who refuse Trinity. I do not know if the novel had English edition at all.)

 [3]       F. Károlyházy, A. Frenkel & B. Lukács: On the Possibility of Observing the Eventual Breakdown of the Superposition Principle. In: Physics As Natural Philosophy, ed. by A. Shimony & H. Feshbach, MIT Press, Cambridge Mass. 1982, p. 204

 [4]       G. C. Ghirardi, A. Rimini & T. Weber: Unified Dynamics for Microscopic and Macroscopic Systems. Phys. Rev. D34, 470 (1986)

 [5]       F. Károlyházy: Gravitation and the Breakdown of the Superposition Principle. In: Proc. Balatonszéplak Relativity Workshop, 1985, ed. B. Lukács, p. 63

 [6]       G. W. Gibbons & S. Hawking: Cosmological Event Horizons, Thermodynamics and Particle Creation. Phys. Rev. D15, 2738 (1977)

 [7]       L. Diósi, B. Lukács, K. Martinás & G. Paál: Thermodynamic Analysis of the Vacuum. In: Proc. Balatonszéplak Relativity Workshop, 1985, ed. B. Lukács, p. 73

 [8]       L. Diósi, B. Lukács, K. Martinás & G. Paál: Thermodynamics of the Vacuum. Astroph. Space Sci. 122, 371 (1986)

 [9]       L. Diósi & B. Lukács: Newtonian Quantum Gravity. In: Proc. Balatonszéplak Relativity Workshop, 1985, ed. B. Lukács, p. 95

[10]      L. Diósi: Gravitation and the Quantummechanical Localisation of Macroobjects. Phys. Lett. 112A, 288 (1985)

[11]      L. Jánossy: The Physical Aspects of the Wave-Particle Problem. Acta Phys. Hung. 1, 423 (1952)

[12]      L. Diósi & B. Lukács: In Favor of a Newtonian Quantum Gravity. Annln. Phys. 44, 488 (1987)

[13]      F. Károlyházy, A. Frenkel & B. Lukács: On the Possible Role of Gravity in the Reduction of the Wave Function.  In: Quantum Concepts in Space and Time, ed. by R. Penrose & C. J. Isham, Clarendon Press, Oxford, 1986, p. 109

[14]      L. Diósi & B. Lukács: On the Minimum Uncertainty of Space-Time Geodesics. Phys. Lett. A142, 331 (1989)

[15]      L. Diósi & B. Lukács: Károlyházy's Quantum Space-Time Generates Neutron Star Density in Vacuum. Nuovo Cim. 108B, 1419 (1993)

[16]      L. Diósi & B. Lukács: Calculations of X-Ray Signals from Károlyházy's Hazy Space-Time. Phys. Lett. A181, 366 (1993)

[17]      Ágnes Holba & B. Lukács: Is the Anomalous Brownian Motion Seen in Emulsions? Acta Phys. Hung. 70, 121 (1991)

[18]      Ágnes Holba & B. Lukács: Is the Spurious Scattering a Quantum Gravity Phenomenon?. in Stochastic Evolution of Quantum States in Open Systems and in Measurement Problems, eds. L. Diósi & B. Lukács. World Scientific, Singapore, 1994, p. 69

[19]      F. Károlyházy: Gravitáció és makroszkópikus testek kvantummechanikája. Magy. Fiz. Foly. XXII, 23 (1974)