First Steps of Quantum Gravity

and the Planck Values,

by Gennady GorelikStudies in the history of general relativity. [Einstein Studies. Vol.3].

Eds. Jean Eisenstaedt, A.J. Kox.

Boston: Birkhaeuser, 1992. P.364-379.

1.
The Birth of the Planck Values

2.
Gravitation and
Quanta

3.
Bronstein and His Way
to Quantum Gravity

4. *"...
an
essential difference between quantum electrodynamics and
the quantum theory of
the gravitational field."* Quantum Limits of General
Relativity

5. The
Planck Values and
Quantum Gravity

6.
Conclusion

Every contemporary discussion of quantum
gravity inevitably
includes the so-called Planck values, which are combinations of
three
fundamental constants: *c *(the velocity of light), *G
*(the gravitational
constant), and *h *(Planck's constant):

*x _{pl} *=

where the exponents *x, y, z *are rational numbers.
Depending upon the
choice of *x, y, z *the Planck values can have any
dimensionality—length,
time, density, and so forth.

Today, the Planck values are connected with the quantum limits of general relativity (GR). Such an interpretation of these values is reached in different ways—from arguments within the framework of tentative variants of quantum-gravitational theory to simple dimensional considerations. Because the latter approach does not require any complex theoretical constructions, one can surmise that the quantum-gravitational role of the Planck values may have been known to Planck himself.

In reality, however, Planck introduced his *cGh *values
in 1899,
without any connection to quantum gravity. Quantum limits to the
applicability
of general relativity (and, implicitly, their Planck scale) were
first
discovered in 1935 by the Soviet theorist Matvey P. Bronstein
(1906-1938). It
was not until the 1950s that the explicitly
quantum-gravitational significance
of the Planck values was pointed out almost simultaneously by
several
physicists. In the following, the history of the Planck values
will be
reviewed. The topic is an important one, because the Planck
scale is almost the
only element certain to be a part of a future synthesis of
general relativity
and quantum theory (Misner, Thorne, Wheeler 1973).

**1. The Birth of the Planck Values**

In the fifth installment of his continuing study of
irreversible radiation
processes (Planck 1899), Max Planck introduced two new universal
physical
constants, *a *and *b, *and calculated their
values from experimental
data. The following year, he redesignated the constant *b *by
the famous
letter *h *(and in place of *a, *he introduced *k
= b/a, *the
Boltzmann constant).

In 1899, the constant *b *(that is, *h) *did not
yet have any
quantum theoretical significance, having been introduced merely
in order to
derive Wien's formula for the energy distribution in the
black-body spectrum.
However, Planck had previously described this constant as
universal. During the
six years of his efforts to solve the problem of the equilibrium
between matter
and radiation, he clearly understood the fundamental, universal
character of
the sought-for spectral distribution.

It was perhaps this universal character of the new constant
that stimulated
Planck, in that same paper of 1899, to consider a question that
was not directly
connected with the paper's main theme. The last section of the
paper is
entitled "Natural Units of Measure" ["Natürliche
Maasseinheiten"]. Planck noted that in all ordinary systems of
units, the
choice of the basic units is made not from a general point of
view "*necessary
for all places and times*," but is determined solely by "*the
special
needs of our terrestrial culture*" (Planck 1899, p. 479).
Then,
basing himself upon the new constant *h *and also upon c
and *G, *Planck
suggested the establishment of

*units of length,
mass, time, and
temperature that would, independently of special bodies and
substances,
necessarily retain their significance for all times and all
cultures, even
extraterrestrial and extrahuman ones, and which may therefore
be designated as
natural units of measure.* (Planck 1899, pp. 479-480)

These new units are chosen in such a way that
in the new
system of units every mentioned constant (c, *G, h) *is
equal to one.
Thus, Planck introduced the following units:

*l _{pl} =
*(

*m _{pl} = (hc/G)^{1/2}
= *10

*t _{pl} *= (

(where modern notation and values are used). In cosmology, a "natural unit" of density is also now very popular:

*r _{pl} = c^{5}/hG^{2}
*= 10

While this last section of the paper bears only a slight
connection to the
rest, it is strongly connected with Planck's broader philosophy
of physics, in
particular, with his attention to the absolute elements of the
scientific world
picture and his opposition to anthropomorphism in physics
(Goldberg 1976).
Eddington (1918) seems to be the first who attempted to uncover
a theoretical
significance in the Planck length, a significance that may be
the clue to some
essential structure of gravitation. But, both Planck's and
Eddington's
suppositions were ignored or even ridiculed (see, for example,
Bridgman 1922).

The quantum-gravitational meaning of the Planck values could be revealed only after a relativistic theory of gravitation had been developed. As soon as that was done, Einstein pointed out the necessity of unifying the new theory of gravitation with quantum theory. In 1916, having obtained the formula for the intensity of gravitational waves, he remarked:

*Because of the
intra-atomic
movement of electrons, the atom must radiate not only
electromagnetic but also
gravitational energy, if only in minute amounts. Since, in
reality, this cannot
be the case in nature, then it appears that the quantum theory
must modify not
only Maxwell's electrodynamics but also the new theory of
gravitation.* (Einstein
1916, p. 696).

(For a similar comment, see Einstein 1918.)
Einstein
obviously had in mind the problem of the instability of the
atom. But his
conclusion was hardly based on quantitative estimates. Atomic
radiation,
calculated according to classical electrodynamics, results in
the collapse of
the atom in a characteristic time interval on the order of 10^{-10}
s
(in stark contradiction with reality), whereas atomic
gravitational radiation,
calculated according to Einstein's formula, is characterized by
an enormous
time collapse on the order of 10^{37} s. Thus, there was
no reason to
worry that the relativistic theory of gravitation would
contradict empirical
data, but the analogy with electrodynamics determined the
direction of
Einstein's thinking.

The "cosmological" value of the time of atomic gravitational out-radiation reminds one that during these years Einstein was also thinking about cosmological problems. Even though the effect of gravitational radiation is small, Einstein had to reject its possibility, apparently because of his prerequisites for a cosmology. In a static, eternal universe, any instability of the atom is inadmissible, wholly independently of its magnitude. It is interesting to compare this attitude, quite natural for that time, with the current, more tolerant attitude toward the instability of the proton (Salam 1979). The change is explained by the acceptance of an evolutionary picture of the universe.

For two decades after Einstein pointed out the necessity of a quantum-gravitational theory in 1916, only a few remarks about this subject appeared. There were too many other more pressing theoretical problems (including quantum mechanics, quantum electrodynamics, and nuclear theory). And, the remarks that were made were too superficial, which is to say that they assumed too strong an analogy between gravity and electromagnetism. For example, after discussing a general scheme for field quantization in their famous 1929 paper, Heisenberg and Pauli wrote:

*One should
mention that a
quantization of the gravitational field, which appears to be
necessary for
physical reasons, may be carried out without any new
difficulties by means of a
formalism wholly analogous to that applied here. *(Heisenberg
and Pauli
1929, p. 3)

They grounded the necessity of a quantum
theory of
gravitation on Einstein's mentioned remark of 1916 and on Oskar
Klein's remarks
in an article of 1927 in which he pointed out the necessity of a
unified
description of gravitational and electromagnetic waves, one
taking into account
Planck's constant *h.*

Heisenberg and Pauli obviously intended that quantization techniques be applied to the linearized equations of the (weak) gravitational field (obtained by Einstein in 1916). Being clearly approximative, this approach allows one to hope for an analogy with electromagnetism, but it also allows one to disregard some of the distinguishing properties of gravitation—its geometrical essence and its nonlinearity. Just such an approach was employed by Leon Rosenfeld, who considered a system of quantized electromagnetic and weak gravitational fields (Rosenfeld 1930), studying the mutual transformations of light and "gravitational quanta" (a term that he was the first to use).

The first really profound investigation of the quantization of
the
gravitational field was undertaken by Matvey P. Bronstein. The
essential
results of his 1935 dissertation, entitled "The Quantization of
Gravitational Waves," were contained in two papers published in
1936. The
dissertation was mainly devoted to the case of the weak
gravitational field,
where it is possible to ignore the geometrical character of
gravitation, that
is, the curvature of space-time. However, Bronstein's work also
contained an
important analysis revealing the essential difference between
quantum
electrodynamics and a quantum theory of gravity not thus
restricted to weak
fields and "nongeometricness." This analysis demonstrated that
the
ordinary scheme of quantum field theory and the ordinary
concepts of Riemannian
geometry are not sufficient for the formulation of a consistent
theory of
quantum gravity. At the same time, Bronstein's analysis led to
the limits of
quantum-gravitational physics (and to Planck's *cGh*-values).

**3. Bronstein and His Way to Quantum Gravity**

In the 1930s, the name of the Leningrad theorist Matvey
Petrovich Bronstein
(1906-1938) was well known in Soviet physics, and this in spite
of Bronstein's
youth. He was a friend and colleague of the bright young
physicists George
Gamow, Lev Landau, and Dmitriy Ivanenko, as well as the
astronomers Victor
Ambarzumian and Nikolai Kozyrev. Bronstein's papers were
published in the main
physical journals. In 1932, Dirac's *Principles of Quantum
Mechanics *was
published in his translation and with many of his footnotes. At
Leningrad
University and the Leningrad Physical-Technical Institute, the
students and
faculty appreciated Bronstein's participation in the journal *Physikalische
Dummheiten,
*which circulated in a single copy. Many more readers knew
Bronstein's popular scientific articles and books and his three
wonderful books
for children. The latter are being reprinted today, fifty years
after the death
of their author. In 1937, Bronstein was arrested and then killed
by Stalinism.
For two decades, his name was banned as the name of an "enemy of
the
people."

Bronstein had broad scientific interests—his papers dealt with astrophysics, semiconductors, cosmology, and nuclear physics. But, the investigation of quantized gravity became his most important achievement (see Gorelik 1983; Gorelik and Frenkel 1985, 1990).

One of his first papers of 1926 contained evidence of his acquaintance with general relativity, but the full measure of his knowledge of the topic was in an extensive and profound survey article on relativistic cosmology that he published in 1931. The appearance of this review, which contains sections on the foundations of general relativity, on the formulation of the cosmological problem, and on all of the then-known cosmological models, was stimulated by Hubble's discovery of 1929.

In this article, one finds no doubts about the possibility of
constructing a
cosmological theory based solely on general relativity. In
Bronstein's
subsequent papers concerning cosmology, such doubts are
characteristic, and
they are based on his general conception of physics. Bronstein
looked at
physics through the "magic *cGh *cube." He drew a chart
describing the structure of theoretical physics based on the
role played by the
three fundamental constants *c, G, *and *h, *and
on the corresponding
limits of applicability of different theories (ignoring some of
these
constants). His next paper on the topic (Bronstein 1933b)
contains a chapter
entitled "*Relations of Physical Theories to Each Other and to
Cosmological Theory*," the basic analysis of which was
illustrated by
the following scheme:

Here, as Bronstein wrote, "*solid lines correspond to
existing
theories, dotted ones to problems not yet solved*"
(Bronstein 1933b, p.
15). Relativistic quantum theory, that is, *ch *theory,
was then the
center of attention. But, as a rule, gravity was ignored.

Bronstein also wrote about the quantum limits of general
relativity in other
papers, in the context of astrophysics and cosmology (see, for
example,
Bronstein 1933a). One phrase from Bronstein's popular article of
1929 (which
concerned Einstein's attempt to unify gravitation and
electromagnetism) reveals
his general attitude to the problem of the quantum
generalization of general
relativity: "*The construction of a space-time geometry that
could
result not only in laws of gravitation and electromagnetism
but also in quantum
laws is the greatest task ever to confront physics*"
(Bronstein 1929,
p. 25). Thus, it is no surprise that Bronstein chose the
quantization of
gravitation as the topic of his dissertation (when the system of
scientific
degrees was introduced in the USSR). However, this choice was
rather surprising
in the contemporary scientific environment, because in the
1930s, fundamental
theorists were concentrating on quantum electrodynamics and
nuclear physics.

Bronstein defended his thesis in November 1935. His examiners,
the prominent
Soviet theorists Vladimir Fock and Igor Tamm, praised the thesis
highly as
"*the first work on the quantization of gravitation resulting
in a
physical outcome*" (Gorelik and Frenkel 1985, p. 317).
Bronstein's
dissertation and two corresponding publications (Bronstein
1936a, 1936b) were
mainly devoted to the quantization of weak gravitational fields,
but one of the
most important results is an analysis of the compatibility of
quantum concepts
and classical general relativity in the general case (this
latter topic will be
examined separately).

Bronstein considered gravitation in the weak-field approximation (where it is described by a tensor field in Minkowski space) in accordance with Heisenberg and Pauli's general scheme of field quantization (Heisenberg and Pauli 1929). From quantized weak gravitation, he deduced two consequences: (1) a quantum formula for the intensity of gravitational radiation coinciding in the classical limit with Einstein's formula, and (2) the Newtonian law of gravitation as a consequence of quantum-gravitational interactions.

One might think that what we see here is merely an expression of the simple requirement of the correspondence principle. In fact, Bronstein's results had real significance, because the peculiar character of the gravitational field (namely, its identification with the space-time metric) gave rise to doubts about the possibility of synthesizing quantum concepts and general relativity. For example, Yakov I. Frenkel (the head of the theoretical department at the Leningrad Physical-Technical Institute, where Bronstein worked) was very sceptical about the possibility of quantizing gravitation, because he, like some other physicists, considered gravitation as a macroscopic, effective property of matter. It should be mentioned that even in the 1960s, Leon Rosenfeld supposed that the quantization of gravitation might be meaningless since the gravitational field probably has only a classical, macroscopic nature (Rosenfeld 1963)—and Rosenfeld was the first to consider a formula for the expression of quantized gravitation. On the other hand, Einstein's attitude was also well known. He believed that the correct full theory was separated from general relativity by a much smaller distance, so to speak, than from quantum theory.

Bronstein's attitude was near the golden mean, since he was
sure that one
should look for the synthesis of both great theories—general
relativity and the
quantum theory. His investigation brought to light deep
connections between the
classical and quantum ways of describing gravitation and so gave
testimony to
the effect that the quantum generalization of general relativity
is both
possible and necessary.

**4. " ... an essential difference between quantum
electrodynamics
and the quantum theory of the gravitational field."
Quantum Limits of
General Relativity**

Before obtaining the two mentioned results and just after deducing commutation relations, Bronstein analyzed the measurability of the gravitational field, taking into account quantum restrictions.

The question of measurability had occupied an important place
in fundamental
physics since Heisenberg had discovered the restrictions on
measurability
resulting from the indeterminacy relations (restrictions on the
measurability
of conjugate parameters). As physicists looked toward the
development of *ch*-theory
(relativistic quantum theory), the question of measurability
attracted great
attention, especially after Landau and Peierls rejected in their
1931 paper the
concept of an "electromagnetic field at a point," based upon its
immeasurability within the *ch*-framework. This paper led
to the detailed
and careful analysis of the situation that was undertaken in
1933 by Bohr and
Rosenfeld. They saved a local field description in quantum
electrodynamics, but
at the high price of assuming arbitrarily high densities of mass
and electrical
charge.

Bronstein had an eye for this subject, and just after Bohr and
Rosenfeld's
paper, he summed up the situation in a short but clear note
(Bronstein 1934).
So it was quite natural that in considering quantum gravity
Bronstein decided
to analyze the problem of the measurability of the gravitational
field within
the *cGh*-framework.

He proceeded from the weak-field approximation, where the
metric *g _{ik}
= *

where *T _{ik}, *is the
energy-momentum tensor,
and

In the weak-field approximation, the equation of a geodesic

can be written as *(x ^{1} *=

To measure Ã_{1,00} averaged over the
volume *V *and
the time interval *T, *one should measure the component
of momentum *p _{x}
*of a test body with volume

where r* *is the density of the test
body. If the
measurement of the momentum has indeterminacy ~*Dp _{x}*,
then

The indeterminacy *Dp _{x}* is the sum of two
parts: the ordinary
quantum mechanical term (

If a separate measurement of momentum requires a time interval *Dt*
(<< *T), *then the indeterminacy in *ho\, *connected
with the
indeterminacy in the recoil velocity *v _{x} *~

and, in accordance with Eq. (2), *D*Ã_{1,00}
~ *krDx. *So
the indeterminacy in the momentum that is connected with the
gravitational
field of the test body would be

Two conditions determine the lower bound on the duration At of
the
measurement of momentum. First, it is necessary that *Dt >
Dx/c* in
order for the recoil velocity to be less than *c. *Hence,
from Eq. (5), we
get

Second, from the very meaning of the measurement of the field in
volume *V, *it
follows that the indeterminacy *Dx < V ^{1/3}*.
Hence,

Having obtained these two lower limits, Bronstein remarked that
the ratio

"*depends on the mass of the test body, being quite
insignificant in the
case of the electron and becoming of order 1 in the case of a
dust particle
weighing one hundredth of a milligram*" (Bronstein 1936b,
p. 216).
There are then two limits for *D*Ã, respectively

As we have seen, to measure F in a certain volume *V *as
precisely as possible,
one should use a test body with as large a mass (density) as
possible. Hence,
only the first limit is essential.

Bronstein wrote that the preceding considerations are analogous to those in quantum electrodynamics. But there arises here the essential difference between quantum electrodynamics and quantum gravity, because

*in formal quantum
electrodynamics, which does not take into consideration the
structure of the
elementary charge, there is no consideration limiting the
increase of density
p. With sufficiently high charge density in the test body, the
measurement of
the electrical field may be arbitrarily precise. In nature,
there are probably
limits to the density of the electrical charge... but formal
quantum electrodynamics
does not take these limits into account.... The quantum theory
of gravitation
represents a quite different case: it has to take into account
the fact that
the gravitational radius of the test body (krV) must be less
than its linear
dimensions*

(Bronstein 1936b, p. 217)

It follows that Eq. (10) gives "*the
absolute minimum
for the indeterminacy*":

Bronstein understood that "*the absolute
limit is
calculated roughly*" (in the weak-field framework), but he
believed
that "*an analogous result will be valid also in a more exact
theory*."
He formulated the fundamental conclusion as follows:

*The elimination
of the logical
inconsistencies connected with this requires a radical
reconstruction of the
theory, and in particular, the rejection of a Riemannian
geometry dealing, as
we have seen here, with values unobservable in principle, and
perhaps also the
rejection of our ordinary concepts of space and time,
replacing them by some
much deeper and nonevident concepts. Wer's nicht glaubt,
bezahlt einen Taler. *(Bronstein
1936b, p. 218)

(The same German phrase concludes one of the Grimm brothers' very improbable fairy tales).

In such a way, the quantum limits of general relativity were
revealed for
the first time. The only thing lacking for the modern theorist
is the Planck
scale of these limits. Of course the Planck values were
implicitly present,
because all three constants *c, G, *and *h, *were
involved in
Bronstein's analysis. One such value, the Planck mass *(hc/G) ^{1/2},
*appeared
in Bronstein's text, "

It is not difficult to supplement Bronstein's reasoning in order to demonstrate the Planck scale of quantum gravity, if only one tries to measure the gravitational field as precisely as possible in a volume as small as possible (Gorelik 1983). Then, one can obtain, for example, the indeterminacy of the metric

*Dg = l _{pl }/cT.*

The true value of Bronstein's fundamental result concerning the quantum limits of general relativity was not recognized by his contemporaries. For example, Vladimir Fock had, on the whole, a very high opinion of Bronstein's dissertation, but in a summary written for an abstract journal, Fock gave only the following vague characterization of this important result: "a few observations on the measurability of the field quantities are appended" (Fock 1936, p. 88).

In the 1930s, it was much easier to unite the constants *c *and
*h *with
*e, m _{e}, *and

**5. The Planck Values and Quantum Gravity**

For two decades after Bronstein's work, there was calm in the
field of
quantum gravity. Only in the mid-1950s did the length *l _{0}
= (Gh/c^{3})^{1/2
}*appear almost simultaneously in a few different
forms in a few
papers. For example, in 1954, Landau pointed out that the length

In a report at the Bern relativity jubilee conference in 1955,
Oskar Klein
remarked that *l _{0} *corresponds to

The most famous quantum-gravitational interpretation of the
length *l _{0
}*was developed in John Archibald Wheeler's 1955 paper
entitled
"Geons." The main subjects of the paper are hypothetical
gravitational-electromagnetic pure field configurations (geons).
In addition,
Wheeler evaluated the manner in which quantum fluctuations of
the metric

The term "Planck values," which is now generally accepted, was
introduced later (Misner and Wheeler 1957). According to
Wheeler, he did not
know in 1955 about Planck's "natural units" (private
communication).

A history of the Planck values provides interesting material
for reflections
on timely and premature discoveries in the history of science.
Today, the
Planck values are more a part of physics itself than of its
history. They are
mentioned in connection with the cosmology of the early universe
as well as in
connection with particle physics. In considering certain
problems associated
with a unified theory (including the question of the stability
of the proton),
theorists discovered a characteristic mass ~ 10^{16}*m _{p}
*(

*This is known as
the Planck
mass, after Max Planck, who noted in 1900 that some such mass
would appear naturally
in any attempt to combine his quantum theory with the theory
of gravitation.
The Planck mass is roughly the energy at which the
gravitational force between
particles becomes stronger than the electroweak or the strong
forces. In order
to avoid an inconsistency between quantum mechanics and
general relativity,
some new features must enter physics at some energy at or
below 10** ^{19}
proton masses.* (Weinberg 1981, p.
71).

The fact that Weinberg takes such liberties
with history in
this quotation is evidence of the need to describe the real
historical
circumstances in which the Planck mass arose. As we saw, when
Planck introduced
the mass (*ch/G*)^{1/2}^{ }(~ 10^{19}*m _{p}*)
in 1899, he did not intend to combine the theory of gravitation
with quantum
theory; he did not even suppose that his new constant would
result in a new
physical theory. The first "

Theoretical physicists are now confident that the role of the
Planck values
in quantum gravity, cosmology, and elementary particle theory
will emerge from
a unified theory of all fundamental interactions and that the
Planck scales
characterize the region in which the intensities of all
fundamental
interactions become comparable. If these expectations come true,
the present
report might become useful as the historical introduction for
the book that it
is currently impossible to write, *The Small-Scale Structure
of Space-Time.*

*Acknowledgment: *The author is thankful to Jean
Eisenstaedt and Don
Howard for their help in preparing this report

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———
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[——— (1994) [Matvei
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