Subject: New Look at the Large Numbers
From: <Saved by Microsoft Internet Explorer 5>
Date: Thu, 6 May 2004 12:15:08 -0500

International Journal of Theoretical Physics, vol. 25, No. 8, 1986

New Look at the Large Numbers

Thomas Görnitz'

Received January 31, 1986

A new interpretation for the large number hypothesis is given, referring to
the close connection between the Bekenstein-Hawking entropy and Weizsäckers ur
theory.

1. INTRODUCTION

Eddington (1931, 1936) points out that the empirical values of dimensionless
numbers appearing in physics may crave and admit a theoretical explanation. Two
of his examples, the inverse fine structure constant hc/e2=137 and the mass ratio of proton and electron,
mp/me=1840, are today subsumed
among the unsolved problems of elementary particle physics. Two others, known as
the "large numbers," belong to gravitation and cosmology. They are

E1 = e2/Gmemp =
FCoul/Fgrav

the relation between the Coulomb and gravitational forces in the hydrogen
atom, and

E2 = number of nucleons in the universe

Empirically, E1 is of the order 1040 and E2
of the order lO80 (if we are permitted to use a closed world
model); hence

E2 = E12

(1)

Dirac (1937, 1938) proposes an explanation of the two large numbers in terms
of a third one, the age of the universe tU, the epoch, measured in
units of a typical atomic time e2/mc3. Using the present
age of the universe, it turns out that

def

tU

E3

=

––––––

= E1

(2)

e2/mc3

Dirac's idea is that this relation, too, should hold as a law of nature: "Any
two of the large dimensionless numbers occurring in nature are connected by a
simple mathematical relation, in which the coefficients are of order of
magnitude unity." From (2) and (1) it would follow that E1 and
E2 are dependent on the epoch. Hence, Dirac postulates, the number of
nucleons in the universe must increase with tU2, and the
gravitational "constant" G must decrease with tU-1. Dirac
also mentions the possibility that hc/e2 and/or
mp/me might vary proportionally to the logarithm of
tU.

The theory of a variable G is further extended by Jordan and his co-workers
(Jordan, 1955; Jordan er al, 1964; Ludwig, 1951). A review of all theories with
variable G is given by Wesson (1980). In these theories, G is taken as the only
natural ,,constant" varying in time.

Dirac's approach can be said to suffer from two weaknesses, a theoretical and
an empirical one.

Theoretically, in our present understanding of physical theory there is no
logical necessity for an explanation of the large numbers by some law of nature.
They might describe contingent initial conditions of the universe or of coupling
constants. The striking numerical relations (1) and (2) might either be chance
coincidences or they might even indicate those values of the constants at which
there is a possibility of there being living organisms in the world able to
observe it [see, eg., Misner et al. (1973)].

Empirically, no sign for a variation of G with tU has so far been
found. Hellings et al. (1983), analyzing the Viking lander data, have shown
strong, while not yet compelling, empirical arguments against it.

In this paper we shall, however, propose a further study of the theoretical
meaning of the large numbers. In the present observational situation it seems
likely that a final empirical test may presuppose a better understanding of the
theoretical foundation and implications of Dirac's hypothesis.

2. BLACK HOLES

The theory of black holes has given rise to a further type of large number,
described by Bekenstein (1973, 1981) and Hawking (1975) as black hole entropy.
For simplicity of expression we shall describe the theory in Planck-Wheeler
units. For Dirac, a natural system of units consisted of c, h, and the mass of
some elementary particle. In a black hole, elementary particles are no
recognisable; hence G is then a more natural unit. Planck (1899) found that h,
c, G, and the Boltzmann constant k constitute a system of natural units which
"holds as long as the laws of gravitation, of propagation of light in vacuum,
and the principles of thermodynamics remain valid." Our units of time, length
and mass are of the order

t0= 1043
sec

l0=1033cm

(3)

m0 = l05g

The idea of black hole entropy had its origin in the remark that black hole
formation is an irreversible process. It was found that in any process involving
a black hole its surface cannot decrease; the same holds for processes involving
several black holes for the sum total of their surfaces. The surface of a black
hole is proportional to the square of its mass Mbh, and it was found
that the statistical entropy Sbh of a Schwarzschild black hole, being
a dimensionless number, has the value

Sbh = 4 (pi)
Mbh2

(4)

if Mbh is measured in the "natural" units (3). For a black hole of
solar mass Mbh = 2 * 1033g = 1038m0, we
get

Sbh =
1077

(5)

a large number indeed.

For a fundamental theory of black holes, star masses are as contingent as
elementary particle masses; star masses are in fact defined by stability
conditions which depend on atomic constants. From the point of view of pure
gravitation, the "natural" limiting values of black hole entropy would be the
entropy of the smallest and of the largest possible black hole. These are given
by

Mbh(min) = m0

Mbh(max) = MU

=

M(mass_of_the_universe)

(6)

For Mbh(min), Sbh is of the order of unity; for
Mbh(max), we get

E 4 = SU
=

10120 =

E13

(7)

If the universe is spatially compact, we may indeed consider it as something
like a huge black hole in whose inside we have the good luck to live.
Then E4 would be the entropy of the universe. What does such a
statement mean?

Entropy is a subconcept of the general concept of information. In Boltzmann's
language, the entropy of a closed system is the logarithm of the number of its
possible microstates contained in its macrostate. The information content
Imac of knowing the macrostate alone is the actual information we
possess about the system in phenomenological thermodynamics. The information
content of a microstate Imic is the virtual information, i.e.,
the information we would possess if we knew the microstate.

The quantity

S

= Imic

- Imac

(8)

is the information on the system that we lack by only knowing the macrostate;
this is what we mean by the system's entropy. Sbh is thus the
information an outside observer loses by the process of black hole formation; it
is hence the information an inside observer of the black hole might maximally
gain by describing its state as it can be known to that observer observing the
full microstate. Now, we are inside observers of the universe. Hence
SU would mean the information content of the universe.

3. UR THEORY

How can we define a microscopically observable information content of the
universe that might be compared with this number? For this purpose we use the
concepts of Weizsäcker’s (1971) theory of ur alternatives, as presented in 1968;
for further information on the theory cf. Weizsäcker (1971b, 1985), Castell el
al. (1975, 1977, 1979, 1981, 1983, 1985), and Drieschner (1973). Basically, the
theory is a consistent interpretation of quantum theory in terms of information.
In a separable Hilbert space a basis can be given by eigenvectors of a
self-adjoint operator with a discrete eigenvalue spectrum. Such an operator is
interpreted as an observable, i.e., as an empirically decidable, infinite-valued
alternative. In fact, only finite alternatives can be empirically decided.
Logically, every discrete alternative can be decided by a sequence of binary
alternatives. Corresponding to that, every separable Hilbert space can be
described as a tensor product of two-dimensional spaces, or as a subspace of
such a tensor product. If we are permitted to describe the possible quantum
states of the whole universe in a separable Hilbert space, it is mathematically
trivial that the states of the universe can be built up from states of smallest,
binary "subjects." These are called ,,urs" by Weizsäcker (from German
Ur-Alternativen = original alternatives).

The state space of a single ur is C2. It is symmetry group
is SU(2) *U(1). Assuming
that the theory of a composite object is invariant under the same group as that
of its smallest equal parts, one concludes that any object in the universe, and
the universe itself, ought to be have at least SU(2) as its symmetry group. A
natural description of the quantum states of an object with a given symmetry
group is given by the square integrable functions on a homogeneous space of this
group. SU(2) is locally isomorphic to SO(3). Weizsäcker considered this as the
reason why all physical objects can be naturally described in a
three-dimensional real position space. Thus, the ur theory does not presuppose
the existence of the space-time continuum; on the contrary, it deduces the
position space from the mathematical structure of abstract quantum theory.

We can choose the parameter space of SU(2) itself as its homogeneous space.
It is an S3. It is the simplest and thus the most natural
assumption that this S3 is cosmic space. The
wave function for a single ur can then be represented as a function over the
S3 possessing only one knot line. Hence an ur is to be
understood as an extremely nonlocal object. Localized states in the universe are
obtained by superposition of many urs. The state space of any object has to be a
subspace in the Fock space of all urs which itself is isomorphic to a subspace
of the infinite tensor product of C2. Castell (1975) has
provided a method to construct the state space of a massless particle from urs.
Models for massive particles have also been found (Weizsäcker, 1985; Görnitz and
Weizsäcker, 1985).

What would be the information content of the universe in this picture? The
decision of one ur alternative provides one bit of information. Hence the
information content of the universe may be considered as equal to the number N
of urs in it. Weizsäcker (1973) gave an estimate of this number prior to the
theory of black hole entropy. He subdivides the universe into elementary volumes
equal to ln3, where ln is the Compton wavelength
of the nucleon. For every elementary volume a binary alternative is defined by
asking whether it contains a nucleon or not. Nucleons are chosen since they form
the bulk of ponderable matter, and ponderable matter is needed for actual
experiments. Since the radius R of the universe is 1040 ln, the result
is

N=
ln 3

=10120

(9)

in accordance with (7).

Of course, this estimate makes use of a basis of the Hilbert space of the
universe which is different from the direct counting of urs. The ur is nonlocal,
while here we start from localized volumes. We assume, however, that the two
bases give the same number of dimensions to the Hilbert space
(d=2N), and hence the same information content. In
any case, the Hilbert space of this model of the universe is finite-dimensional.
The universe is not Lorentz-invariant. Only in a locally tangential Minkowski
space will infinite-dimensional Hilbert spaces be defined.

In order to localize a nucleon in a universe of radius R down to an accuracy
comparable to ln, we need a superposition of 3R/ln urs, R/ln for every space
dimension. Now

R/ln = E3

= 1040

(10)

This will permit us to subdivide the N urs of the universe into
N/E3 nucleons, and we get

E2= N/E3

= 1080

(11)

in accordance with (1). Generally speaking, ur theory requires the equation
with

N = E4

= E2 * E3

(12)

E2= N2/3

E3 = N1/3

(13)

The result (10) on the information content of the nucleon also has been
derived from the theory of black hole entropy. For a particle of mass m falling
into a black hole, the entropy rises by an amount

Sdiff

=

4(Pi)

[(Mbh+m)2

-Mbh2)

=

8(Pi)

Mbh*m

This is the information lost for an outside observer due to the particle
falling into the black hole. The authors were surprised also to find
Sdiff to be exceedingly large. The maximum possible value of
Sdiff is reached for Mbh = MU. In our units (3)
is

mn=
10-20m0

and

MU =
1060m0

Hence

Sdiff(max) =
l040

(14)

The ur theory explains this value. A nucleon "is" l040 urs.

So far, our description refers only to the momentary state of the universe.
Within this limit it turns out to be a trivial mathematical consequence of
abstract quantum theory if we add only the postulate that the information
content of the universe as presently accessible to our theory is finite. For a
theory of particles and fields as well as for the history of the universe we
need a law of dynamics. We leave that to later papers (cf. Görnitz, 1985) and
confine ourselves to a qualitative remark. If a law of dynamics admits a
noncompact symmetry group, its unitary representations need an
infinite-dimensional Hilbert space. Time with an open future would mean a group
of time translations in R1. This law of dynamics will not keep the
number of urs constant. This is the ur-theoretical expression of the idea that
in an open future there is no upper bound to the degree of decidable
alternatives; they are finite at any moment, but unbounded with proceeding time.
If, in such a dynamics, we begin with a finite number N(t0) of
urs, it is to be expected statistically that N(t) will on the average
increase monotonically with t. If, at any time, N1/3= R/ln , and if ln is supposed to be the
unit of length used in ponderable measuring instruments and hence defined as
constant through time, this means an expansion of the universe.

These last considerations lead us back to Dirac's hypothesis, but with a
so-far-undetermined function R(t). A more precise description of dynamics will
be needed for its elaboration.

ACKNOWLEDGMENTS

I thank Prof. C. F. v. Weizsäcker for numerous discussions about all aspects
of the ur theory and for his continuous support. I also thank Prof. A. O. Barut
for helpful discussions. I am grateful to the Deutsche Forschungsgemeinschaft
for financial support.

REFERENCES

Bekenstein, J. D. (1973). Physical Review D, 7, 2333.

Bekenstein, J. D. (1981). Physical Review D, 23, 287.

Castell, L. (1975). In Quantum Theory and the Structures of Time and
Space, L. Castell, M. Drieschner, and C. F. v. Weizsäcker,
eds, Vol. 1, Hanser, Munich.

Castell, L., Drieschner, M., and Weizsäcker, C. F. v., eds. (1975, 1977,
1979.1981, 1983, 1985). Quantum Theory and the Structures of Time and Space,
Vols. 1-6. Hanser. Munich.

Dirac, P. A. M. (1937). Nature, 139, 323.

Dirac, P. A. M. (1938). Proceedings of the Royal Society, A
165, 199.

Drieschner, M. (1973). Voraussage - Wahrscheinlichkeit - Objekt ,
(Lecture Notes in Physics. vol. 99), Springer, Berlin.

Eddington, A. S. (1931). Proceedings of the Cambridge Philosophical
Society, 27, 15.

Eddington, A. S. (1936). Relativity theory of protons and Electrons,
Cambridge University Press, Cambridge.

Görnitz, Th. (1985). On the Connection of Abstract Quantum Theory and General
Relativity. Part 1. The Cosmological Model, Internal Report, Starnberg.

Görnitz, Th., and Weizsäcker. C. F. v. (1985). De-Sitter representations and
the particle concept in an ur-theoretical cosmological model, In Proceedings
oft the 1985 Symposium on Conformal Groups and Structures, Clausthal, tou
appear.

Hawking, S. W. (1975). Communications in Mathematical Physics,
43, 199.

Hellings, R. W., Adams, P. J., Anderson, J. D., Keesey, M. S., Lau, E. L.,
Standish, E. M., Canuto, V. M. and Goldman, I. (1983). Physical Review
Letters, 51, 1609.

Jordan, P. (1955). Schwerkraft und Weltall, 2nd ed., Vieweg,
Braunschweig.

Jordan, P., Ehlers, J., and Kundt, W. (1964). Zeitschrift für Physik,
178, 501.

Ludwig, G. (1951). Fortschritte der projektiven Relativitätstheorie,
Vieweg, Braunschweig.

Misner. C. W., Thorne, K. S., and Wheeler. J. A. (1973). Gravitation,
p. 1216, Freeman, San Francisco.

Planck, M. (1899). Sitzungsberichte der Deutschen Akademie der
Wissenschaften Berlin, Klasse für Mathematik, Physik und Technik,
1899, 440.

Weizsäcker, C. F. v. (1971a). The unity of physics, in Quantum Theory and
Beyond, T. Bastin, ed., University Press. Cambridge.

Weizsäcker, C F. v. (1971b) Die Einheit der Natur, Hanser, Munich;
English translation: The Unity of Nature, Farrar Straus Giroux, New York
(1980)]

Weizsäcker C. F. v. (1973) A comment to Dirac's paper, in The
Physicist's Conception of Nature, J. Mehra, ed., Reidel, Dordrecht

Weizsäcker, C. F. v. (1985) Der Aufbau der Physik, Hanser,
Munich

Wesson, P. S. (1980). Gravity, Particles and Astrophysics, Reidel
Dordrecht