**TRIALITY IN THE DEPTH OF PHYSICS? ON THE FUNDAMENTAL
UNIFICATION**

*From: Triality in Evolution, eds. B. Lukács & al.,
KFKI-1995-21/C*

B. Lukács

Central Research Institute for Physics RMKI, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

**ABSTRACT**

It
seems as if physics suggested 3 fundamental phenomena, accompanied by 3
fundamental constants known as G, c and ~~h~~.
If so, then a contradiction-free description would need a trialistic
unification. Dual unifications are known as e.g. General Relativity, but the
trial unification is not yet available. Here the reasons suggesting a trial
unification, the possible phenomena described by it and the difficulties are
discussed.

**1. INTRODUCTION**

There
are, to our present knowledge, at least 3 fundamental and generally valid
phenomena in physics. Here these we call symbolically *Gravity*, *Relativity* and *Quantumness*.
We do not know exactly, what is behind these phenomena, we know only some
consequences of them, and some theories which have been made to describe them.
Still it is easy to define the meaning of the above 3 keywords in such a way
that all physicists can agree.

*Gravity* is a phenomenon by which
any pair of physical entities disturb each other through "empty
space", although in degrees decreasing with distance. There are many ways
by which mutual disturbances are possible; but only gravity is
"general". E.g. electromagnetism mimics gravity quite well, but acts
only on electrically charged bodies. As we know, gravity acts on everything.

*Relativity* is a phenomenon showing
that only relative motion means anything, and 3-space and time, as we know
them, do not mean too much, only the space-time is "objective", but
causality is strict.

*Quantumness* (a term whose
clumsiness reflects how late the phenomenon was recognised) is behind a lot of
correlated violations of basic concepts of the physics from Aristotle to Mach.
It means that the point particle is a nonexistent idealisation; that there may
be two bodies in the same time at the same place. Maybe its most direct manifestation
is the existence of "uncertainty relations", inequalities stating the
nonexistence of states sharply defined synchronously in two quantities
"canonically conjugate" to each other.

Interestingly
enough, each such *Phenomenon*
possesses its *Fundamental constant*.
For *Gravity*, there is the *Cavendish constant*: G = 6.67*10^{-8}
cm^{3}/gs^{2}. For *Relativity*,
space and time scales are converted to each other via the *light velocity*: c = 3.00*10^{10}
cm/s. Finally, in *Quantumness*
elemental and indivisible portions of energy, angular momentum, uncertainties,
&c. are all defined by the *Planck
constant* ~~h~~ = 1.05*10^{-27} gcm^{2}/s.

Obviously, for a phenomenon at least one scale parameter is needed, and if the phenomenon is of general validity, the constant is such as well. We do not know what would happen if two scale constants were to belong to one fundamental phenomenon; no such case is reported. One may believe that further fundamental and generally valid physical phenomena exist as well: up to now they have not been reported beyond doubt. One may question the general validity of any of the 3 phenomena mentioned above; but the loss of general validity would result in anomalies.

Now,
fundamental and generally valid *phenomena* cannot be contradictory, but *theories*
can and are. Until it is so, physics does not work, in principle in any case,
in practice in some ones. So some answers are impossible, ambiguous or
arbitrary. So some "final" questions cannot be answered. It seems
that this (not necessarily only this) is behind some unsolved questions in
cosmology, biology or psychology.

The standard way of solution is Unification. If two theories are unified, the new unified theory is not self-contradictory (a self-contradictory theory is not a correctly made theory, so is not a real theory but rather an educated guesswork). The problem is that in unification the general structures of the theories must change (being different from each other even at the beginning). But then nothing leads the hand of the unifier: great inspiration and intuition is needed, and even if a unification has been made, there is no guarancy that the result is correct.

We not yet have the unified theory for all the 3 Phenomena. But the overview of successes and failures will be edifying.

*2. GRAVITY
IN SICH:*** THE NEWTONIAN
THEORY OF MUTUAL ATTRACTION**

In ancient ages the whole problem did not exist at all. Our weight, and in some rare cases the orbital motion of Moon, was explained by Universal Weight, the universal tendency of bodies to approach an exceptional point of the Universe. The most consequent theories put this point into Earth's center, because then the formation of the spherical Earth got a simple explanation, and experience was absent at places far from this point. Asimov points out [1] that this would have been impossible if instead of Earth Venus or Mars had possessed Moon; similarly the Galilean moons of Jupiter are not too far from visibility by naked eyes. In both cases the early astronomers, many millenia ago, would have established at least the multiplicity of centers of attraction, but (by accidents at the formation of the Solar System?) this was really possible only at the discovery of telescopes. Then in happened in one year. The logical consequence was the theory that gravity is mutual between bodies.

The
first mathematical formalism of mutual gravitational attraction was made by Newton [2]. He formulated gravity as a mutual *force*
causing acceleration. From the data of parabolic (really elliptic) paths of
ballistics on terrestrial grounds, from the orbital velocity and radius of
Moon, and from the radius of Earth (all known with some accuracies in his time)
he found that if an inverse square law in distance holds (which is rather
natural from geometric viewpoint and very simple mathematically), then the same
force can explain the motion of the cannon ball and that of Moon, a description
conform to Occam's razor *Entia non sunt multiplicanda praeter necessitatem*,
expressing the principles of all economic theories. So he stated

ma =
F = -GMm/r^{2} (2.1)

Such a formula predicts that the accelerations of all falling bodies are equal at the same place, but this had become then an established fact up to several percents of accuracy from Galileo's experiments on the Pisa Tower. Later the fact was confirmed by Bessel to 4 digits from astronomical observations and in this century to 9 digits in laboratory [3].

After
the formulation of the force law of gravity a science called *celestial
mechanics* came into existence and sooner or later all but two observed
details of the motions of celestial bodies of the Solar System became explained
in details up to observational errors by the Newtonian Theory of Gravity. The
two exceptions were the secular deceleration of the angular velocity of Moon
and the anomalous perihelion advance of Mercury. Both effects are very minute:
the rate of lunar angular deceleration is some 10^{-10}/year, and
Mercury's perihelion advances 5.75"/y instead of the 5.32"/y
calculated as the perturbation of all known planets (while in a year Mercury
itself moves some 1500°, some 10^{7} times more than the anomalous
advance).

The lunar anomaly was explained by G. H. Darwin as a consequence of tidal frictions [4]. Moon generates a tidal protuberance on the oceans just below herself, but friction with the other oceanic waters carries forward this protuberance. In consequence gravitational attraction between the protuberance and Moon pulls forward Moon and backwards some water. So Moon gets some extra (orbital) angular momentum at the price of the rotational angular momentum of Earth. Therefore Moon spirals outwards with increasing angular momentum but decreasing angular velocity. While the actual rate of angular momentum transfer could not be calculated from first principles (friction with seabed is a complicated phenomenon), the rates did not contradict to anything known. As for the anomalous perihelion advance of Mercury, the simplest solution was an unseen planet, called Vulcan, within Mercury's orbit. One time the discovery of Vulcan was reported [5], but without calculated orbit, and the result was irreproducible.

**3. RELATIVITY
IN SICH: EINSTEIN'S SPECIAL THEORY OF RELATIVITY**

The
relativity of motions come from Galileo, but there it was true only for
"mechanical" motions [6]. It came from common sense. In Newton's mechanics it turned out to be true if all forces
depend only on *relative* distances and velocities, an obvious assumption.
However it had no further consequence, because the general belief was that
optics was not subject of this relativity principle. Namely the propagation of
the transversally oscillating light wave was explained via a very rigid medium,
the aether, of course permeating the space up to at least the visible stars.
The (local) rest frame of this medium defines a natural and physically
preferred coordinate system and by combinations of mechanical and optical
experiments the velocities can be measured in this system. While this rest
frame may be not absolute in philosophical sense, it is quite universal in
physics.

However,
serious attempts to measure Earth's velocity in this K_{o} frame
failed, e.e. they gave 0 velocity with a margin much below Earth's orbital
velocity around Sun [7]. For a while ad hoc explanations existed to explain
particular failures, but finally Einstein invented a general explanation [8],
appended soon by a geometric interpretation of Minkowski. Now this appended
version is known as Einstein's theory of Special Relativity.

The fundaments of this theory go as follows. The fundamental manifold of physics is not Space, but Space-Time, a four-dimensional manifold whose points are physical events. An event is labelled by its spatial coordinates x and its time t.

By
using four coordinates (x,y,z,ct) one gets a 4 dimensional *pseudo*-Euclidean
space with the invariant elementary distance

ds^{2}
= dx^{2} + dy^{2} + dz^{2} -(dct)^{2} (3.1)

same for any observer, where first c is a scale factor of velocity dimension.

Any linear transformation of the coordinates, preserving this invariant form, gives another permitted coordinate system, related to each other via unaccelerated motions, none of them is preferred. Only such formulae are permitted, which transform covariantly together with the coordinate transformations. Therefore when comparing observations of two observers of relative velocity v on some noninvariant, 3 dimensional quantity (as lengths of rods, time intervals between events &c.), the ratio v/c appears in the relation of the two observations.

If, in addition, one postulates that real physical motion can connect events only if

ds^{2}
µ 0 (3.2)

then events happen
in a strict causal order for any observer (time evolution goes *forward*),
as always seen.

Since the whole theory was based on tendentional failures of optical measurements for absolute velocity, c turned out to be the velocity of light, but it became an universal terminal velocity (on 3 dimensional language)

**4. QUANTUMNESS
IN SICH: QUANTUM (WAVE) MECHANICS**

At the end of the last century there were problems in explaining the frequency distribution of the blackbody or cavity radiation. It turned out that sufficiently "black" bodies emit similar radiations when heated at the same temperature. The radiation was, of course, proportional to the surface and depended on temperature T and its distribution changed with the actual frequency x, but otherwise the formula was universal:

dE ~ Ff(x;T)dx (4.1)

with f~x^{2}
at low x's and f~e^{-Âx} at high x's. The frequency of the maximum
obeyed

x_{m}
= T/ß (4.2)

where ß~10^{-27}
ergs.

The explanation might have been anything about the microscopic structure of the matter. However, experimenters showed that the same radiation emerged from empty containers of reflecting or reradiating inner walls if a negligible hole was opened. Since no particular matter was inside, a fundamental thermodynamical explanation was needed, of course with a possibility that c appears too via Maxwell's equations.

With combined electromagnetism and thermodynamics Wien arrived at a scaling

f(x;T)
= c^{-3}x^{3}g(x/T) (4.3)

and there became stuck. The experiments suggested

lim y‑>0 g(y) ~ 1/y

lim
y‑>∞ g(y) ~ e^{-ßy} (4.4)

y ≡ x/T

and, by choosing any smooth function for g between the asymptotics, eq. (4.2) follows with a number factor of order 1. However the new constant ß seemed to belong to vacuum, which is strange.

In 1900 Planck was able to get a function g(x/T), conform to observations, from the postulate that all emissions and absorptions of energy in atomic systems happen in portions

E = hx (4.5)

[9]. Then the principles of thermodynamics will result in

g(y)
= 8πh(e^{-hy} - 1)^{-1} (4.6)

where ~~h~~ ≡
h/2π was a completely new constant, to be taken from measurement. With ~~h~~
= 1.054*10^{-27} ergs this Planck spectrum agrees with all
measurements.

In the
next years the new constant appeared in various rôles, e.g. in the photoelectric
electron radiation from alkali metals, in atomic spectra and in the specific
heat of crystallic matter. The synthesis was reached in 1925-6, when Heisenberg
formulated the matrix mechanics [10] and Schrödinger the wave mechanics [11].
They turned out to be equivalent, and Dirac formulated the fundamental
postulate of *Quantum* *Mechanics* [12] as follows.

The
physical quantity is replaced by an operator. The commutator of two operators
is i~~h~~*(their classical Poisson bracket). The operator acts on the
physical state vector or function, and the measurable values of a physical
quantity are their eigenvalues according to the particular operator. Then
uncertainty relations appear if two operators do not commute, and the
uncertainties are proportional with ~~h~~. Later von Neumann appended the
theory with the axiomatic description of measurement [13]. According to him, if
we measure the system (anything it means, e.g. an interaction with a macrosystem,
us or the apparatus), then it jumps into an eigenstate of the operator of the
measured quantity, with a probability calculable from the actual state.

Since we will refer back to this point later, let us see the so called Schrödinger equation, describing the evolution of a simple point system in Quantum Mechanics in wave language. We assume that only potential and kinetic energies exist, the potential is external and depends only on the coordinates as V(x). Then we need the operators of coordinates, momenta, energy and time derivative. First, let us choose the correspondence

x^{i}
‑> x^{i}· (4.7)

Being the [x,p] Poisson bracket 1, now one gets

p_{k}
‑> -i~~h~~(∂/∂x^{k}) (4.8)

Via eq. (4.7)

V(x) ‑> V(x)· (4.9)

Finally, Hamiltonian mechanics + analogy with eqs. (4.7-8) suggest that

i~~h~~(∂/∂t)
<‑ E
(4.10)

Then, being E=p^{2}/2m+V
in classical mechanics, the evolution of the wave function Ň between two
measurements is described by the equation

ih∂ψ/∂t
= -(~~h~~^{2}/2m)(∂^{2}/∂x^{2})ψ + Vψ
(4.11)

This equation is fully deterministic for ψ; the deterministic evolution is interrupted by measurement and there a stochastic step appears, but not between measurements.

Quantum
Mechanics states that the measurable physical quantities are all operators.
Then in the measurement of 2 quantities always commutators enter, which, in
turn, contain h. In addition, the evolution equations always contain ~~h~~
too. So *Quantumness* is an
universal phenomenon, and ~~h~~ is its fundamental constant.

**5. ON
CONTRADICTIONS**

The above 3 fundamental theories have been proven correct in themselves, i.e. their predictions were proven correct in measurements innumerable times, if nothing else disturbed the results. However, the 3 fundamental theories contradict each other pairwise.

Special Relativity Theory predicts the light to propagate with a constant velocity c. On the other hand, mass m drops out from eq. (2.1), so anything moving with v=c at infinity, would travel with something else near a mass. A ball thrown towards a mass M with almost v=c would exceed c nearer to the center but this is impossible in Special Relativity. Both Newtonian Gravity and Special Relativity claimed universality but they cannot be synchronously true.

Quantum Mechanics predicts an inevitable spread of momentum (and so of velocity) of a point mass closed into a box of linear size L as

Δv
≥ ~~h~~/2mL (5.1)

If L<~~h~~/2mc,
then the point mass would have probabilities for v>c, prohibited by the also
universal Special Relativity Theory.

The
potential or force field of universal Newtonian Gravity is measured from
ballistic paths of test bodies. In the theory the gravitational potential is a
c-number. On the other hand, Quantum Mechanics claims universal validity too,
so the test bodies have inevitable spatial and momentum spreads. Then there is an
universal limit when measuring V_{gr}(x) (see in Sect. 8). So the sharp
potential of Newtonian gravity cannot be measured in any way, i.e. does not
exist.

Since everything contradicts everything, at least two of the 3 "fundamental theories" must be wrong, maybe all 3. Now we turn to pairwise unifications.

**6. GRAVITY +
RELATIVITY: GENERAL RELATIVITY**

In 1916 Einstein was able to formulate the common theory of Gravity and relativity [14]. In it the manifold is a 4 dimensional pseudo-Riemannian one, with the infinitesimal distance

ds^{2}
= g_{rs}(x)dx^{r}dx^{s} (6.1)

(there is a
summation for indices occurring pairwise, above and below). The metric tensor g_{ik} is the solution of the Einstein equation

R_{ik}(g_{lm})
- ˝g_{ik}R_{rs}g^{rs} = -8πGc^{-4}T_{ik} (6.2)

where R_{ik}
is the Ricci tensor, formed from g_{ik} and its first and second
derivatives, while T_{ik} is the energy-momentum tensor of the matter
present. Motions happen between events ds^{2}>0, and force-free
motions do on straight(est) lines called geodesics. Gravitational force does
not exist, but a central body changes the space-time geometry, and the other
orbits it on straightest lines of the influenced geometry, as if it were
attracted. Of course these geodesics are independent of the orbiting body, so
the measured acceleration is matter-independent up to any digits.

This is really a unified theory. For small masses the Minkowski spacetime recovers, so special relativity. For slow motions far from the gravity center the motion mimics the Newtonian motion under the force (2.1), which will not be shown here. However the formalism differs from those of both precursor theories.

Mercury, the planet closest to Sun, does not move very slowly at a very weak curvature. So deviations are expected from the Newtonian orbit, and they are obtained by calculation as 0.43"/year. The exact value of the observed perihelion anomaly is obtained without further assumptions.

In the unified theory both previous constants appear. For an example where both appear in a physical situation, consider the horizon distance

r_{h}
= 2GM/c^{2} (6.3)

around a point mass M. Below this shell even light is trapped.

**7. RELATIVITY
+ QUANTUMNESS: QUANTUM FIELD THEORY**

The first attempt to unify Quantum Mechanics and Special Relativity was made by Dirac [15], in a straightforward way. Starting with relativistic mechanics, and then again replacing physical quantities with operators as old in Section 4, one arrives at relativistically covariant quantum mechanical equations. However this formalism does not define a self-consistent theory.

Namely, when starting in Quantum Mechanics, one must fix the number of particles, since the wave function Ň depends on the particle coordinates. On the other hand, putting more energy into the system than the double of the rest energy of the particle considered, a particle-antiparticle pair may appear. Since uncertainty principles permit energy fluctuation for a short time, additional pairs are always present with a small probability, so the structure of Quantum Mechanics loses its validity if Relativity is included.

The solution was the so called "second quantization" in Quantum Field Theories [16]. Instead of the wave function Ň another operator appears, producing the actual, changing state from a fixed one. The actual physical field, considered, must be defined in advance.

There are as many independent Quantum Field Theories as independent interactions (only one in Grand Unification). Two fundamental constants, c and h, are always present and there appear the coupling constants of the fields too, but they are dimensionless.

Quantum
Field Theories tend to give infinite results via divergences, but for a class
of theories these divergences can be removed by renormalisation. If the
coupling constant is not <<1, technical problems may appear. However for
Quantum Electrodynamics e^{2}/~~h~~c~1/137, the theory can be
evaluated via perturbations, and the results (e.g. small corrections in the
spectrum of the H atom) are correct.

**8. GRAVITY +
QUANTUMNESS: NEWTONIAN QUANTUM GRAVITY**

This pairwise
unification is not yet ready, mainly because nobody tried to formulate it for
decades. Namely, we have seen that in 1925, when Quantum Mechanics appeared,
and any contradiction between Quantum Mechanics and Newtonian Gravity could
show up, Newtonian Gravity was no more the contemporary theory of *Gravity*, General Relativity being
available.

However this was a historical accident, not an inherent onthologic fact. So let us assume for a moment that Quantum Mechanics has preceeded General, moreover even Special Relativity; which contradiction appears then between Newtonian Gravity and Quantum Mechanics, and how to resolve it?

In Newtonian Gravity there is a well-defined sharp gravity potential V, giving a force, acting on bodies. In Quantum Mechanics bodies have wave functions, so their positions have uncertainties. We shall see that these two statements contradict each other. For the details see [17].

Assume
first a sharp (c-number) gravity potential V(r). We want to measure it. The
proper way is to throw test bodies through the region affected by V; the
accelerations will show the local gradients of V, and then V can be
reconstructed (up to a zero point constant always appearing in potentials).
Without *Quantumness* this
is possible in the limit of infinite many test bodies. With *Quantumness* the process does not
converge. Namely, take a test body of mass M which is point-like (but of
course, its wave function is not). We want to measure the average of the
potential or the gravitational acceleration g in a volume R^{3} and a
time interval T. Obviously the wave function should be concentrated in the
volume at least during T. Starting with a very concentrated wave packet it
spreads too rapidly. The optimal strategy is to choose an initial spread below
but in the order of R in such a way that at t=T it grows to just R. Only order
of magnitude formulae are given here; choose a spread R/2 and wait until it
becomes R. This time will be T if

T ~
MR^{2}/~~h~~ (8.1)

which is a lower limit for the mass of the test particle. In addition the momentum P must be 0 at the middle of the measurement, otherwise the particle would leave the volume prematurely. Now, the particle picks up a momentum

P ~ MTg (8.2)

with a quantum uncertainty

_P ~
~~h~~/R (8.3)

Then the sensitivity of the measurement is

σ(g)
~ ~~h~~/MRT (8.4)

So the accuracy of the measurement is increasing with increasing M. However with a high enough M the test body is no more a test body because it disturbs the gravity to be measured, from an unknown point. Therefore

σ^{2}(g)
~ (~~h~~/MRT)^{2} + (GM/R^{2})^{2} (8.5)

Now, our only free parameter is M, because R and T were fixed at the beginning. Therefore σ(g) has its minimum at

M_{opt}
~ (~~h~~R/GT)^{1/2} (8.6)

whence

σ^{2}(g)
~ ~~h~~G/R^{3}T (8.7)

This is the final accuracy until which the gravity acceleration is observable in a domain of size R during a time T. Eq. (8.7) is a special kind of uncertainty relations.

Now let us make one more step by postulating that physical theories should not contain unobservable quantities. Then eq. (8.7) indicates that the gravity potential is not a c-number; it contains inherent "smearing". The simplest way to it is a stochastic description. The gravitational acceleration g (gradient of the potential) has a deterministic and an indeterministic part as

g =
g_{cl} + g_{st} (8.8)

and just so V is built up. We get eq. (8.7) if

<g_{st}(x,t)>
= 0 (8.9)

<g_{st}(x,t)g_{st}(x',t) = **1**~~h~~Gδ^{3}(x-x')δ(t-t')
(8.10)

which is just a white noise type fluctuation.

Of course, there is a back reaction to Quantum Mechanics. The Schrödinger equation automatically gets the form

i~~h~~∂ψ/∂t
= -(~~h~~^{2}/2m)(∂^{2}/∂x^{2})ψ +
(V_{cl}+V_{st})ψ (8.11)

So quantum objects
are subjects of a general stochastic force, determined by the fundamental
constants ~~h~~ and G. This would lead to an "anomalous" Brownian
motion, or, for microparticles, simply to the breakdown of quantum superpositions.

Such effects are, in principle, observable. Some efforts have been made to find hopeful scenarios, since some 30 years ago Károlyházy proposed a theory which belongs to Sect. 10, but in which the Quantum Mechanics was also disturbed via a stochastic gravity [18]. The results are up to now that the anomalous Brownian motion is on the verge of observability if circumstances are lucky; but they are probably not [19], [20].

On the other hand, there is a phenomenon without good explanation, whose characteristic data are conform to eq. (8.11); it is rather hard to calculate the theoretical predictions in this moment. The phenomenon is called spurious scattering. Spurious scattering was observed when evaluating particle tracks in photoemulsions. In the 30's cosmic radiation was measured by balloons carrying photoemulsion, but it was impossible to establish the standard electric and magnetic fields on board the baloon to determine the particle energies. However via Coulomb scattering the track is winding in a way whose statistics is definite and a parameter is energy- and emulsion-dependent. Calibrating the same emulsion down one can determine then the particle energy from the winding of the track. However it turned out that besides the measurement errors and Coulomb scattering, a third effect exists too, mimicking Coulomb scattering but with a different statistics similar to that of random walk. It was necessary to determine its amplitude too when determining the energy of the particle. There is no generally accepted explanation.

Being
a mere side effect, publications on spurious scattering were rare; an exception
is Ref. [21]. L. Jánossy, involved into cosmic radiation measurements in that
time, later believed that there is some limitation of the quantum superposition
behind it [22], and proposed a methodical measurement of it [23]. The result is
that many observations point to a characteristic length in the "random
walk" in the order of 10^{-8} cm; such a characteristic length can
be obtained by formulae containing only ~~h~~, G and a mass M from a mass
~10^{-13} g; and this is the characteristic grain size in emulsions
[24]. If anyone is interested further, the details of the argumentation are in
Ref. 24. In this time no new data are expected because of changes in detection
techniques in particle physics.

As told above, the simplest way to satisfy the uncertainty relation (8.7) is to introduce a stochastic element into Quantum Mechanics and Newtonian Gravity (both completely deterministic in their own spheres). The unified theory (Newtonian Quantum Gravity) is not yet ready, but proper stochastic terms have been introduced into Quantum Mechanics; see Ref. 25 and citations therein.

We note that in NQG a borderline can be derived in the form

~~h~~^{2}/G
~ M^{3}R
(8.12)

Well above this
line the body is moving approximately on a trajectory of NG and we can forget
about wave functions, well below its evolution is almost exactly as in QM, with
superpositions and so. So this dividing line is the border between Macroscopy
and Microscopy. Objects near to the line would need full NQG, not yet ready.
For terrestrial densities ~ 1 g/cm^{3} grains of R~10^{-5} cm
and M~10^{-14} g are on the borderline. There can one expect new,
unexplained behaviour.

Now, these characteristic data hold for colloid grains. What is more, they hold also for the smallest known true living organisms below bacteria: Mycoplasmatales and Rickettsia. In addition, they hold for the relevant parts of neurons too. Now the extra stochastic nature of NQG means that all the argumentation about determinism and indeterminism up to now is invalid for objects near to the dividing line, and the unsolved problem of "free will" is obviously intimately connected with neural networks, whose elements are on the dividing line.

**9. FURTHER
CONTRADICTIONS AND LIMITATIONS**

Now we have 3 unified theories (one is being still in construction, but never mind), but they are unified pairwise. So they cannot be the unified theory. One can see this, if limitations of the theories are detected or if two unified theories still contradict.

Limitations are ample. For example, in a wide class of material sources GR cosmologies start or end with a singularity, where density is infinite [26]. One may or may not believe in physical infinities, but, without doubt, histories cannot be continued through infinities. Quantum Field Theories are able to describe some interactions, although in strict mathematical sense they do not exist [27]; but without doubt they cannot give the masses of the elementary particles appearing in them.

Let us
clarify this statement. Consider a QFT. It describes an interaction, mediated
by a vector boson of mass m, with a coupling constant g^{2}, between
various particles of masses M_{i}. In addition, of course, ~~h~~ and
c appear.

First
question: can ~~h~~, c and g^{2} determine m? The answer is *no*,
because the dimension of g^{2} is that of ~~h~~c. One cannot get
anything of dimension of m from ~~h~~, c and g^{2}.

Second
question: may ~~h~~, c and g^{2} determine m? The answer is no, from
the structure of QFT's, since QFT's with massive vector bosons are not renormalisable,
so they give infinities, not removable, as results. Non-zero masses must be
generated by a way called spontaneous symmetry breaking. We will not go into
its details, but it needs a scalar boson (Higgs), with a quartic potential.
Then an effective mass will be generated, determined by ~~h~~, c, g^{2}
and the coefficients in the quartic potential. As we have seen, ~~h~~, c and
g^{2} are insufficient, and M_{i} must not appear, because m is
unique and M_{i} are various. So the coefficients of the quartic
polynomial must set the mass m; however this is a description, not an
explanation.

In
addition, m is generally in the same order of magnitude as the M_{i}'s.
This suggests a common particle mass scale ~1 GeV. However this characteristic
mass cannot be obtained from QFT's, only can be introduced into them from
outside.

As for contradictions, we mention only one and half, from confronting GR and QFT.

First consider the Einstein equation (6.2) of GR. The Einstein equation is necessary to determine the metric of the space-time, but it does not exist outside of pure GR. Namely, the left hand side is a c-tensor, but the right hand side is a q-one, so they cannot be equal.

Second, try with some ideas that e.g. the Einstein equation holds e.g. in expectation value. Then we have a generally non-Minkowski space-time with Quantum Fields in it. Now, vacua of QFT's are Lorentz-invariant, but not invariant for general transformations of GR. So no covariant answer exists if we ask for the states of the quantum fields. Even if somehow there is a "most natural" coordinate system, strange results are obtained. Here only one will be mentioned.

Consider a Universe solution. Then there is a "natural" coordinate system, whose time coordinate lines are trajectories orthogonal to the 3-space of constant curvature [26]. Let us perform the QFT calculations in this coordinate system. Since the geometry is generally time-dependent, the fields become excited, the vector bosons are created with some rates. This is a kind of Hawking radiations [28].

Now, there is no problem with the existence of such a radiation. The problem is that, when it has been calculated in a de Sitter Universe, the result has become rather strange. A de Sitter Universe is a metric of form

ds^{2}
= dt^{2} - e^{αt}(dx^{2} + dy^{2} + dz^{2}) (9.1)

and it has 10 Killing vectors of space-time symmetry [29]. Now let us calculate the Hawking radiation in the special coordinate system of an observer moving with a "cosmologic" velocity, a velocity field orthogonal to the 6 spatial symmetries. A more or less thermal energy distribution is obtained. Such particles may have been created by the expanding Universe.

However consider another observer moving with a relative velocity to the first, but still orthogonal to 6 spatial symmetries. It is possible because there are 4 timelike symmetries too, and linear combinations of Killing vectors by constant coefficients are Killing vectors too. He will, of course, see the same distribution from symmetry reasons. But for a real radiation, external to both observers, a Doppler shift would be expected due to the relative motion, which does not appear now [28]. So there are here onthologic problems with the radiation. That is only philosophy; but then should we write the energy-momentum tensor of this radiation into the Einstein equation for determining the expansion or should not.

Without doubt, the 3 pairwise united theories cannot be the final words of natural sciences.

**10. TRIALISTIC
UNIFICATIONS**

Contradictions would automatically vanish in a trially unified theory if it were free of self-contradictions. Here we note first that no such theory, describing in addition correctly the known phenomena, has been found up to now. We mention some examples, without too much details or completeness.

The
first such example is the so called K-model [18]. It was not intended to be a
trial unification, but formally it is, containing G, c and ~~h~~ as
fundamental constants. It is a space-time, approximately Minkowskian, but
"gravity waves" are travelling in it. They are stochastic with such
amplitudes that some combined "uncertainty relations" hold. So in any
prediction ~~h~~, G and c appear together. Unfortunately the theory gives
infinities, and even with physically reasonable cutoffs gives unphysical
results [30], [31], [32].

There is a clear kinship between the K-model and the Newtonian Quantum Gravity (Sect. 8), but c does not appear in the latter. The structural similarities and differences are discussed in Ref. 33. Indeed, NQG can be extended to give a reasonable near-Minkowskian space-time. However that is a first approximation, not a trial unification. Quite recently Ref. 32 suggested to use space-times conformal to Minkowski (or those of maximal symmetry) and to put fluctuations into the conformal factor. This is a good idea, e.g. causality (the lightcone structure) would not fluctuate, but such a theory does not contain QFT's, so it is not a trial unification.

Another
approach, but slightly related, was taken by Lánczos [34]. He generalised the
Einstein equation (6.2) to a fourth order equation via an elegant variational
principle. All vacuum solutions of the old equation remain valid but there are
new ones. He showed that there are solutions periodic in all 4 directions. Now
let us assume that (for a reason unknown for us) such a solution came into
existence in the early Universe, with a period length l~10^{-33} cm,
same in all four directions. This periodicity is of course directly
unobservable in macroscopy, being averaged out. Still, some fluctuations remain,
and their characteristic parameter (we have seen in Sect. 4 that a parameter of
dimension gcm^{2}/s is needed) contains, of course, l and the two
fundamental constants of the (modified) GR. Then the only possibility is

H ~
c^{3}l^{2}/G ~ 10^{-27} gcm^{2}/s
(10.1)

in the order of
the Planck constant. So it is possible that *Quantumness* is not fundamental, but is a consequence of
the microstructure of our space-time. It is possible, but then first one should
(approximately) derive QFT's from this structure, and, second, what did produce
this particular structure?

For another approach we must see first something about multidimensional space-times. The topics will be treated in another paper of this Volume [35], but something is necessary to be repeated here. The 4-dimensionality of the spacetime is an observed fact, but only in macroscopy. Extra timelike dimensions would disturb causality, but extra spacelike ones closed (compact) at microscopic sizes do not disturb anything fundamental. Their existence or nonexistence is a matter of fact. Therefore one may invent spacetimes with N spatial and 1 temporal coordinates. Still the arguments leading to GR remain valid, so some lightcone structures (SR) and Einstein-type equations are advisable.

After the pioneering work of Kaluza [36] particle physicists are eager to generate interactions from extra dimensions. The problem is that electromagnetic or stronger interactions can be four-dimensional projections of particles moving "too fast" in the extra dimensions, so that their full N+1 dimensional velocity would be spatial [37]. Then either 4 dimensional causality remains unexplained, which is an unpleasant idea, or electrodynamics, and strong and weak interactions do not come from the extra dimensions. Now we can continue with supergravity.

There
is a theory in Minkowski space-time called *supersymmetry* (see e.g. Ref.
38 and citations therein). The idea is that there are operators of space-time
transformations as e.g. rotation and translation, with some commutators. There
are operators transforming quantum numbers (connecting for example different
quarks, &c.), with some other commutators (e.g. SU(3)). Now, it is
unnatural to have non-simple groups for fundamental operators, so the commutators
between the two sets of operators should be nontrivial too. The new theory,
e.g., predicts new particles, boson pairs to each fermion and vice versa. They
are not seen, but they may be very massive. Now we come to a trial unification,
if the "external" and "internal" operators act in the
macroscopic 4 and the microscopic N-3 dimensions, and we apply the formalism of
GR on all the N+1 dimensions. Automatically, ~~h~~ appears in the commutators.
That is the theory of *supergravity* [39]. Unfortunately it seems unrenormalisable,
so if it gave infinities, they could not be removed; in addition, as told
above, one may have serious doubts in deriving the stronger interactions from
extra dimensions.

Another
attempt is using *superstrings* [40], fundamental objects whose first
excited quantum states are at 10^{19} GeV (10^{16} erg). This
quantity contains also h, similarly to the Lánczos theory. If the macroscopic spacetime
structure is derived from the actions of superstrings, then there is a hope
that such a construction *might* explain all these pairwisely unified
theories, but until now no great success has been reported.

We stop here with the list of attempts. The problem is not hopeless, but obviously the trially unified theory is not yet in hand. However in the next Section it will be shown that some, very general, features of the unified theory can be guessed even now.

**11. THE
FUNDAMENTAL SCALES OF THE UNIFIED THEORY**

A trially
unified theory contains the 3 fundamental constants of *Gravity*, *Relativity* and *Quantumness*
(or their independent combinations); and the minimal theory does not contain
more. Then the characteristic data or its fundamental objects or phenomena can
be determined for orders of magnitude, because, with 3 and only 3 fundamental
constants the length, time and mass scales of the unified theory can be
uniquely calculated. Namely, besides G, c and ~~h~~ everything other results
of the calculations must be *numbers*, and, without postulating very large
or very small number constants in the theory (unnatural) or very special
formulae (also unnatural) the number factors would be in the
"neighbourhood" of the order of unity.

Consider
the set (G,c,h). All of them have different dimensions. Imagine an arbitrary
function of them. The only function which can have a *dimensional*
argument is the power function; all the others must have dimensionless
arguments. Let us see only one example. Consider the exponential function; it
can be expanded into a Taylor series as

e^{x}
= ∑_{0}_{∞} x^{n}/(n!)
(11.1)

Now if x is not dimensionless, then the sum does not exist at all, quantities of different dimensions being unsummable.

But
then *all* lengths derived from the united theory must contain *powers*
of the 3 constants, so the result must be αG^{A}c^{B}~~h~~^{C},
where α is a number, and, as told above, generally one expects it not too
far from unity. The exponents must be so arranged that (cm^{3}/gs^{2})^{A}(cm/s)^{B}(gcm^{2}/s)^{C}=cm.
This is 3 equations for the 3 exponents, so the result is unique. In this way
we get the scales, traditionally called Planck scales, as

L_{PL}
= (~~h~~G/c^{3})^{1/2} ~ 10^{-33} cm
(11.2)

T_{Pl}
= (~~h~~G/c^{5})^{1/2} ~ 10^{-44} s
(11.3)

M_{Pl}
= (~~h~~c/G)^{1/2} ~ 10^{-5} g
(11.4)

The first 2 scales
are completely strange and inaccessible for us; the third is familiar, being
the mass of a dust grain, for example. However the fundamental
"particle" or fluctuation of the unified theory would be of ~L_{Pl}
size and ~M_{Pl} mass. Therefore it would be of density ~10^{93}
g/cm^{3}, rather high. All other scales, if needed, can be combined
from eqs. (11.2-4).

This means that for our present techniques genuine phenomena of the trially unified theory cannot be directly investigated. However from any proposed unified theory some side effects can be calculated within the reach of our possibilities. An example comes from supergravity. It predicts fermionic pairs of bosons, say the photino (gravitino is doubtful, because Gravitation is not an interaction in General Relativity, and supergravity is a multidimensional generalisation of GR). Now, the photino is not seen, but the possible mass was guessed just above the present accelerators.

In a
unified theory the expansion of the Universe would not start from singularity
(infinite density) but from a state of density ~M_{Pl}/L_{Pl}^{3},
and such a state may spontaneously occur for a time TPl as quantum fluctuation.
Such a result can be seen even from a very crude approximation, by using the
Einstein equation (6.2) with a Universe symmetry, and writing the Hawking
radiation, calculated from QFT on that geometry, on the right hand side [41].
Then one gets a solution incomplete in past time direction, i.e. starting from
something else than singularity. The initial curvature radius is, for order of
magnitude, L_{Pl} and the initial radius is ~M_{Pl}/L_{Pl}^{3}
[41], but it cannot be anything else, of course.

As seen a trially unified theory could explain some "final" questions. Unfortunately it is not ready.

**12. DOUBTS AND
CONCLUSIONS**

Unfortunately
there are some doubts about the finality of the trially unified theory to be
looked for. One such serious signal exists, the particle masses. As told above,
the general scale for particle masses is M_{0}~1 GeV. Let us see the
generality of this statement.

In
pre-quark particle physics all non-masless elementary particles have masses
from 0.105 GeV (µ^{±}) upwards to several GeV, with the only
exception of the e^{±}, possessing 0.000511 GeV. There was some hope to
explain the ratio m_{p}/m_{e}=1836 and tricky mathematical
formulae were invented for it.

In the
present Grand Unification the elementary particles are leptons, quarks, Higgs
bosons and vector bosons. The masses of the last two groups are derived
effective masses, and for the first two ones (except again the electrons) if
not zero then are in the order of GeV magnitude. Two mass scales seem to appear
in the effective masses: one of ~100 GeV of the electroweak sector, hopefully
derivable somehow from the GeV scale, and one of 1015 GeV of the X and Y bosons
(see proton decay) [42], maybe derivable from M_{Pl}c^{2}~10^{19}
GeV.

So if
we are very lucky, then only the GeV scale needs explanation. But a very
exceptional formula would be needed to derive 10^{0} Gev from 10^{19}
GeV. It seems as if there were a fourth fundamental constant.

But
this fourth constant does not appear anywhere in *Gravity*, *Relativity*
or *Quantumness*. It seems
as Quantum Field Theories would be the (still incomplete) pairwise unification
of the individual theories of *Relativity*
and *Quantumness*, not
containing any M_{0}. Until the trial unification this idea was
possible, because some external parameters may get explanations from the third;
but it seems as if the trial unification could not be successful at this point.

There
are various possibilities, and no one can choose among them at the present
stage of art. It is possible that there is a fourth fundamental phenomenon,
independent of *Quantumness*,
behind the non-massless elementary particles; but what is it, and in what else
is it manifested? For ninety years everybody expected the quantization of
particle masses from Quantum Theories. It is possible that somehow elementary
particles are "accidental": they dropped out somehow from the unity
of the original Oneness in the early stage of expansion of the Universe and
thinning of the matter; but then why the particle masses are so uniform in the
whole observable Universe? No mass difference is seen even in far spectra, and
at least small inhomogeneities would be expected in a decaying Oneness. Or, it
is marginally possible that finally someone will be possible to explain a
number 10^{-19} between Planck and particle mass scales. It is not
impossible by an *exponential* formula: e^{-N}~10^{-19} if
N=44 and why could a number in an exponent not be 44?

It
could; only such dirty trick never happened up to now in the history of
physics. So we stop here, with an example given by Eddington. As known from
observations, the present average particle density of the Universe is cca. 1
particle in a cubic meter, i.e. ~10^{-6} cm^{-3}. The
observable part of the Universe is some 10 billion lightyears, or ~10^{28}
cm, and it is still unknown if the Universe is closed or not. It is still
possible that it is a surface of a hypersphere of radius not too far above this
size. If so, then the total particle number is ~10^{78}.

Now,
this is a dimensionless number and very large. By no means, the number of all
the conserved particles in a closed Universe (finite) is a fundamental number;
let us denote it by ~~N~~ and try to guess how to explain it._{0}

Eddington, as told, guessed that

~~N~~
~ 10_{0}^{78} >> 1
(12.1)

and found a useful
formula in a 16^{th} century theology book which calculated the exact
number of independent graces of the Holy Virgin (gratia plena). The author got
an exponential formula

N_{G} = 2^{Θ} where Θ≡2^{Ξ} and Ξ≡2^{3}
(12.2)

(exponential formulae are natural in classification counting independent possibilities). Now

N_{G}
= 1.158*10^{77} ~ 0.1~~N~~
(12.3)_{0}

so if ~~N~~
has a fundamental explanation at all, it probably contains the construction
(12.2). We cannot derive the formula (12.2) from today's physics. Now observe
that_{0}

e^{2}/GM_{p}M_{e}
~ 10^{39} ~ ~~N~~_{0}^{1/2}
(12.4)

Eddigton interpreted the left hand side as the ratio of strengths of fundamental interactions and guessed that the total number of existing particles might govern the strengths of interactions. This is obviously impossible in the present construction of physics without instantaneous long-range connections; it might or might not come naturally in a Machian world description. And observe that

M_{Pl}/M_{0}
~ ~~N~~_{0}^{1/4}
(12.5)

But it
is quite possible that ~~N~~ is 0 (or 1 or such). Namely in
Grand Unification the number of conserved charges is only 2: Z and B-L [42].
Now, for the electric charge Z we have local density data, and the local
electric charge density is 0 (equal numbers of protons and neutrons). As for
the barion and lepton charges we could make an incomplete count. In our
neighbourhood we mainly have protons, electrons, neutrons, e-neutrinos and
e-antineutrinos. Then, for densities,_{0}

(B-L)/V ~ n_{proton} - n_{electron} + n_{neutron} - (n_{neutrino}
- n_{antineutrino}) (12.6)

Now, the first two terms cancel, the third is some 15% of the first, and we do not know anything for the bracketed difference. So with a slight luck both conserved charges may be 0, which is a natural initial condition.

But
then the present N~10^{78} of particles conserved in the
SU(3)*SU(2)*U(1) Standard Theory, having emerged somewhere at the breakdown of
Grand Unified symmetries about 10^{28} K temperature and 10^{-34}
s from the Beginning is a cosmologic accident. Then M_{Pl}/M_{0}~N^{1/4}
can be another accident or the consequence of the same accident, and the trial
unification can be complete. But how to explain a very old accident?

**ACKNOWLEDGEMENT**

Partly supported by OTKA T/01822.

**REFERENCES**

[1] I. Asimov: The Tragedy of the Moon. Doubleday & Co., New York, 1973

[2] I. Newton: Principia Mathematica Philosophiae Naturalis.London, 1687

[3] L. Eötvös,
D. Pekár & E. Fekete: Annln. Phys. **68**, 11 (1922)

[4] Sir G.
H. Darwin: Phil. Trans. Roy. Soc. **172**, 491 (1881)

[5] M. Lescarbault
d'Orgčres: Cosmos **16**, 22 (1860)

[6] G. Galileo: Dialogo dei due massimi sistemi del mondo. Landini, Florence, 1632

[7] A. A.
Michelson & E. W. Morley: Amer. J. Sci. **31**, 377 (1886)

[8] A.
Einstein: Annln. Phys. **17**, 891 (1905)

[9] M.
Planck: Lecture at the German Physical Society (Berlin),
14^{th} Dec., 1900

[10] W.
Heisenberg: Z. Phys. **33**, 879 (1925)

[11] E.
Schrödinger: Annln. Phys. **79**, 361 (1926)

[12] P. A. M.
Dirac: Proc. Roy. Soc. London **109**, 642 (1925)

[13] J. v.
Neumann: Gött. Nachr. **1**, 1 (1927)

[14] A.
Einstein: Annln. Phys. **49**, 898 (1916)

[15] P. A. M.
Dirac: Proc. Roy. Soc. Lond. **117**, 610 (1928)

[16] S. Tomonaga:
Prog. Theor. Phys. **1**, 27 (1946)

[17] L. Diósi
& B. Lukács: Annln. Phys. **44**, 488 (1987)

[18] F. Károlyházy:
Nuovo Cim. **42**, 390 (1966)

[19] F. Károlyházy, A. Frenkel & B. Lukács: in Physics as Natural Philosophy, eds. A. Shimony & H. Feshbach, MIT Press, Cambridge, Mass. 1972, p. 204

[20] F. Károlyházy, A. Frenkel & B. Lukács: Quantum Concepts in Space and Time, eds. R. Penrose & C. J Isham, Clarendon Press, Oxford, 1986, p. 109

[21] P. J. Lavakare
& E. C. G. Sudarshan: Nuovo Cim. Suppl. **XXVI**, 251 (1962)

[22] L. Jánossy:
Lecture given at his 60^{th} birthday at CRIP, Budapest, May, 1971, unpublished

[23] Ágnes Holba
& B. Lukács: Acta Phys. Hung. **70**, 121 (1991)

[24] Ágnes Holba & B. Lukács: in Stochastic Evolution of Quantum States in Open Systems and in Measurement Pocesses, eds. L. Diósi & B. Lukács, World Scientific, Singapore, 1994, p. 69

[25] L. Diósi:
Europh. Lett. **22**, 1 (1993)

[26] S. Hawking & G. F. R. Ellis: The Large-Scale Structure of Space-Time. Cambridge University Press, Cambridge, 1973

[27] M. Banai:
J. Math. Phys. **28**, 193 (1987)

[28] G. W.
Gibbons & S. Hawking: Phys. Rev. **D15**, 2738 (1977)

[29] H. P. Robertson & T. W. Noonan: Relativity and Cosmology. Saunders, New York, 1969

[30] L. Diósi,
B. Lukács: Nuovo Cim. **108B**, 1419 (1993)

[31] L. Diósi,
B. Lukács: Phys. Lett. **A181**, 366 (1993)

[32] J. L.
Rosales & J. L. Sanchez-Gomez: Phys. Lett. **A199**, 320 (1995)

[33] L. Diósi,
B. Lukács: Phys. Lett. **A142** 331, (1989)

[34] C. Lánczos:
Found. Phys. **2**, 271 (1972)

[35] B. Lukács: in this Volume, p. 2

[36] Th. Kaluza:
Sitzungber. Preuss. Akad. Wiss. Phys. Math. Kl. **LIV**, 966 (1921)

[37] B. Lukács
& T. Pacher: Phys. Lett. Phys. Lett. **113A**, 200 (1985)

[38] J. Ellis: Proc. Neutrino '82 Balatonfüred, p. 304

[39] J. Wess
& B. Zumino: Nucl. Phys. **B70**, 39 (1974)

[40] M. Green: Sci. Amer. 1986/9

[41] L. Diósi
& al.: Astroph. Space Sci. **122**, 371 (1986)

[42] P. Langacker:
Phys. Rep. **72C**, 185 (1981)

_

**My HomePage, with some other studies**, if you are curious.