Matvei
Bronstein and quantum gravity:
70th anniversary of the unsolved problem
// Physics-Uspekhi
2005, vol 48, no 10, pp. 1039-1053 (pdf)]
Gennady Gorelik,
Center for Philosophy and History of Science, Boston
University
http://people.bu.edu/gorelik/
[Russian original: Матвей
Бронштейн
и квантовая гравитация. К 70-летию нерешенной проблемы //
УФН 2005, №10 (pdf)]
Contents
2. Quantum gravity before 1935
3. Semiconductors or quantum gravity?
4. The problem of ch-measurability. Is
the uncertainty principle too certain?
5. Gravity and microphysics in the
1930s
7. Expansion of the universe in 1937
9. How to attain inner perfection
without external confirmation
Abstract. Matvei Bronstein's 1935 work on quantum gravity, the first in-depth study of the problem, is analyzed in the context of the history of physics and the scientist's career. Bronstein's analysis of field measurability revealed "an essential difference between quantum electrodynamics and the quantum theory of the gravitational field" and showed that general relativity and quantum theory are fundamentally difficult to unify. Featured in the story are Plank, Einstein, Heisenberg, Pauli, Rosenfeld, Landau, and Bohr. The methodological uniqueness of the quantum gravity problem is discussed.
1. Introduction
The subtitle of this
article may perplex the reader. Indeed, what on earth happened
in 1935? Had no one combined the words 'quantum' and 'gravity'
before or written a formula containing all three fundamental
constants: c, G, and h (the speed of light,
the gravitational constant, and the Planck constant)?
Certainly, all this had been done, and the latter even
preceded the former. However, it was in 1935 that the problem
of quantum gravity was first comprehended in its depth.
It was Matvei Bronstein who made this breakthrough in his
doctoral dissertation defended at the Leningrad
Physico-Technical Institute (LPTI) in November 1935; the
results were published in two articles in 1936 [1, 2]
(republished in part in [3]).
Today, seventy years
later, the real crux of the problem is especially evident
since it is still unsolved and remains probably the most
'cursed' question of fundamental physics.
To better see the
path that brought Bronstein to his work of 1935 and understand
its meaning, let us start with an overview of the historical
background1.
1
More details can be found in [4].
2.
Quantum gravity
before 1935
The simplest and
most tangible synonym of quantum gravity, the so-called Planck
scales, first emerged in Planck's article that dates
back to 1900; it has no relation to quantum gravity, however.
Nobody realized at that time that a new, quantum, era was
about to begin in physics. Planck hoped that the newly
proposed constant h (then denoted by the letter b)
would be possible to integrate into the edifice of
classical physics. He suggested new 'natural units of measure'
with the sole
'practical' purpose that they 'retain their significance for
all times and all cultures, even extraterrestrial and
extrahuman ones" [5a]. Such an exotic suggestion was based on
a solid philosophy of the first pure theoretical physicist in
which the ideal of classical physics is readily perceived. In
Planck's view, a fundamental goal of physics was to liberate
the physical world picture from the individuality of the
creative mind, from any anthropomorphic element [5b].
Planck's strange
quantities met with little sympathy. In 1922, they were
disapproved by the famous experimental physicist P. Bridgman
(in his book Dimensional Analysis [6]) whose
philosophy of operationalism was distilled from the practice
of physical measurements. He entered the history of physics by
expanding the confines of the practically accessible pressure
range from thousands to hundreds of thousands of atmospheres,
and that was still one hundred orders of magnitude below the
Planck scales. It is easy to understand a 'convert to
[Bridgman's] somewhat materialistic exposition' who would say
there was no place for such values in physics; no wonder that
Planck's 'natural units' looked ridiculous in the eyes of
Bridgman. A length unit of 10-33 cm seemed so
non-operational that he did not care much about
argumentation.
Such an impressive
philosophical gap, which in addition has a quantitative scale,
is a remarkable characteristic of the quantum gravity problem,
even if neither Planck nor Bridgman talked about the theory
of gravity as such.
Meanwhile, by the
time Bridgman's book came out, the theory of gravity had
undergone historic metamorphosis into the relativistic theory
of gravity or general relativity (GR). Just a few months after
GR had been published, Einstein emphasized the necessity of
unifying the new concept of gravity and the quantum theory.
Having obtained the formula for the intensity of gravitational
waves, he remarked: " to the intra-atomic movement
of electrons, atoms would have to radiate not only
electromagnetic but also gravitational energy, if only in tiny
amounts. As this is hardly true in nature, it appears that
quantum theory would have to modify not only Maxwellian
electrodynamics, but also the new theory of gravitation" [7]2.
2
Einstein repeated this argument in [8].
This short remark
contains three important points. First, Einstein assigned a
leading role to the quantum idea. Second, he implied
parallelism between electrodynamics and gravity (in the 1920s,
he turned this concept into the conviction that the two forces
were closely related and set out on the path of the unified
field theory, which led him nowhere). Finally, this remark
shows that Einstein was a theorist of no less exalted thought
than Planck; surely, his wording "in minute amounts" sounds
too weak in this case.
Einstein made no
quantitative estimates but evidently had in mind famous
problem of 'classical' Rutherford's atom collapse due to the
electrons orbiting the nucleus should radiate and fall into
the nucleus. The loss of electromagnetic energy calculated by
the formulas of Maxwell electrodynamics takes the extremely
short time of ~ 10-10 s to occur, whereas
gravitational out-radiation calculated by Einstein's newly
derived formulas would last ~ 1030 years. Even the
age of the universe, ~ 1010 years, is insignificant
compared with this time, although in 1916 the phrase 'age of
the universe' made no sense in physics. Einstein's "this
cannot be the case in nature" in fact related to the universe
rather than to the atom. In the next year of 1917, Einstein
demonstrated a way to treat the universe as a physical object.
He paid no attention to the magnitude of the effect as if it
was to be rejected as contradicting his cosmological
prerequisite, i.e., the static picture of the universe. In the
eternal static universe, instability of atoms is unacceptable
regardless of the magnitude of the effect.
After the discovery
Hubble made in 1929, physicists for the first time obtained
grounds on which to talk about the age of the universe as an
experimentally measurable quantity. They could reject
Einstein's static prerequisite for 'operational-measuring'
reasons, but the thought of theorists flew higher than that.
By historical coincidence, the article by Heisenberg and Pauli
published in 1929, where the general scheme for quantizing
electromagnetic field was developed, optimistically stated
that "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 fully
analogous to that applied here" [9, p. 3]. They referred to
Einstein's aforementioned remark of 1916 and O Klein's
statement of 1927 on the necessity of a unified description of
gravitational and electromagnetic waves taking into
consideration Planck's constant h. In other words, the analogy
between gravity and electromagnetism was again implied.
Heisenberg and
Pauli's optimistic confidence was apparently based on the
idea that quantization should be applied to the equations of
the weak gravitational field, or linearized equations of GR
obtained by Einstein in 1916. Such an approach was employed in
1930 by Leon Rosenfeld (who worked under Pauli) to answer the
question raised by Heisenberg [10]. The question was
metaphysical rather than physical, that is, whether
self-energy in quantum electrodynamics (QED) is infinite even
in the absence of charges if the gravitational field of light
is taken into account. Rosenfeld confirmed Heisenberg's
supposition by showing the corresponding gravitational energy
to be infinitely large, whence 'a new difficulty for the
Heisenberg-Pauli quantum theory of wave fields emerged'.
However, Rosenfeld did not explain why one should trust this
new infinity inferred from the weak field assumption.
So weakish was the
state of quantum gravity by the time Bronstein got to the
problem. The general mood might be described as sluggish
optimism and summarized in the following way: gravity
should be quantized by the same means as electromagnetism
but these means need to be properly developed to get rid of
infinities. While the quantum theory of the
electromagnetic field was indispensable to understand real
phenomena in atomic and nuclear physics, the reasons for
creating the quantum theory of gravity were merely some
'high-brow general considerations' not necessarily of interest
to practical-minded physicists.
3.
Semiconductors or quantum gravity?
Among the motives
that led Bronstein to work on his dissertation on the quantum
theory of gravity, one was quite practical and down-to-earth:
there was no such thing as a dissertations for a scientific
degree in the USSR before 1934. The proletarian power
abolished the old tsarist tables of ranks, including
scientific ones. However, after the revolutionary fervor was
pacified in the course of building Stalinism, the government
decided to introduce the scientific degrees of Candidate and
Doctor of Science (within two years starting from January
1934) "in order to stimulate research work and raise the
skills of scientific and educational cadres." To make the new
machinery workable, a certain number of degrees were conferred
without defending the theses.
In this manner,
Bronstein was given a candidate degree by the Scientific
Council of LPTI (June 1935) for his work in astrophysics and
invited to submit a doctoral dissertation on 'the theory of
semiconductors'. Ya Frenkel, head of the Theoretical
Department, wrote: "By now, he [Bronstein] has actually
written his doctoral dissertation (on electronic
semiconductors) and will defend it in the near future" [11].
The semiconductor studies carried out by Bronstein were
equally highly valued by A Ioffe, director of LPTI [12,13].
In such
circumstances, it was not at all trivial to choose quite a
different subject for the dissertation. Still less trivial was
the new subject. As Bronstein explained to his colleague I
Kikoin, a doctoral dissertation should contain 'long
unintelligible formulas' and, in this respect, gravity
obviously has an advantage compared with semiconductor
physics. Both physicists did have a strong sense of humour.
Bronstein appears to
have been writing his dissertation during the summer months of
1935; his first article on quantum gravity dates from August
[1]. The session at which he defended his thesis was held on
November 22, 1935, with I Tamm and V Fock as the official
reviewers. The surviving shorthand records of the session and
personal reminiscences show that Bronstein just reported his
recent work and attacked rather than defended when he
disagreed with the arguments of the reviewers [14].
There is no archival
evidence on how much the colleagues of Bronstein were
surprised by the drastic thematic change of his research, from
semiconductors to quantum gravity. In those days, the gap
between these subjects was no smaller than it is today. In the
mid-1930s, the theory of gravity was concerned only with
celestial mechanics and cosmology. All hopes to have a
generalized theory of gravity or Unified Field Theory for
earthly microphysics were in the past, even though a few
enthusiasts still remained, including Einstein. The Soviet
physics showed itself to be maturely independent in that no
prominent theorist in the country shared Einstein's passion of
that time despite the huge respect to the great physicist and
the admiration of the work he had done in the first quarter of
the century.
Certainly, the area
of theoretical physics was much narrower then. In the
mid-1930s, Lev Landau explained that "theoretical physics,
unlike experimental physics, is a small science open to
perception in its entirety by any theorist" [15]. Insofar as
mastering theoretical physics meant active work rather than
mere passive understanding, it was as small for Bronstein as
for Landau.
Bronstein started on
the path to science in the circle of physics lovers at Kiev
University under the leadership of P Tartakovskii. In January
1925, the then 18 year-old Bronstein submitted an article 'On
a Consequence of the Light Quanta Hypothesis' to the Journal
of the Russian Physical and Chemical Society (the
forerunner of the Journal of Experimental and Theoretical
Physics, JETP). Assuming the photon structure of X-rays, he
obtained the dependence of the boundary of a continuous X-ray
spectrum on the radiation angle and came to the conclusion
that the discovery of this effect added another argument in
favor of the quantum theory of light; "otherwise, some light
will be shed on the applicability limits of the quantum theory
to the X-ray range." It is worthwhile to remind ourselves that
the very idea of photons was at that time rejected by Bohr
himself, who changed his view only after the 1925 Bothe-Geiger
experiments. So, young Bronstein plunged just into the
troubled midst of physical discussions. In the same 1925,
Bronstein published another article on the subject in the then
most reputable German journal Zeitschrift fur Physik [16].
In 1926, Bronstein
entered Leningrad University and soon joined the so-called
'jazz-band', a cheerful group of gifted young physicists. The
core of the group were 'the three musketeers': George Gamow,
Dmitry Ivanenko, and Lev Landau. So, Bronstein had to play
D'Artagnan. Life separated the three musketeers much farther
than Alexandre Dumas could have imagined, but it was only
death that cut short the friendship of Landau and Bronstein
[17].
In his student years
Bronstein made an important contribution to the theory of
stellar atmospheres in the form of the so-called
Hopf-Bronstein relation [18]. E Milne, one of the founders of
the field, recommended Bronstein's paper for publication in
the Monthly Notices of the Royal Astronomical Society [19].
In 1931, Uspekhi
Fizicheskikh Nauk published a detailed survey by
Bronstein entitled "The Modern State of Relativistic
Cosmology". It was the state after the Hubble law, the first
observational fact of physical cosmology was established in
1929 [19]. And it was the first review of cosmology in the
USSR.
The young theorist felt at home on different floors of the physics building. This feeling emerges in Bronstein's reviews of conferences he wrote for scientific journals and popular science magazines [20].
|
|
The early 1930s. Bronstein and some participants in the 1933 All-Union Conference on Nuclear Physics (drawings by N Mamontov for Bronstein's account of the conference). Bronstein is said to have had the picture of a frog on his arm-band that he wore as a secretary for the conference. The picture was apparently prompted by a German phrase then popular with theorists: "Jetzt kommt der Moment, wo der Frosch ins Wasser springt" (Here comes the moment when the frog jumps into the water). Physicists of that time anticipated some radically new idea jumping out of the troubled water to help in comprehending the microworld. |
|
|
G Gamow, P Dirac, A Ioffe, V Fock, Ya Frenkel |
Of special relevance
was the conference on theoretical physics held in Kharkov in
May 1934 that showed that Soviet theoretical physics as a
whole and its Kharkov branch in particular held an important
place in world physics. The participants in the conference
included, besides leading theorists from Moscow and Leningrad,
Niels Bohr (for whom it was the first visit to the USSR) and
his close associate Leon Rosenfeld. In Bronstein's words, "the
conference was a kind of business meeting rather than a
congress to demonstrate achievements"; he limited his account
to ideas that could be of interest not only for theorists but
also to physicists working in other fields [21, p. 516].
The participants
discussed various 'business problems' of importance for
physics at that time. The most dramatic reports were made by I
Tamm. One of his works (in co-authorship with S Al'tshuler)
predicted the neutron magnetic moment and was challenged by
Bohr who believed it incompatible with the zero electric
charge of the neutron. Here, the great Bohr was wrong.
As regards his other
work on the hypothesis of pair forces in the nucleus, Tamm
knew himself that he was 'wrong' but nevertheless reported the
negative result of his calculations. Today, we understand that
this work was an important step to the Yukawa meson and that
Tamm regarded his 'wrong' idea as his strongest one. Here is
how Bronstein described this dramatic episode: "Tamm told how,
based on the Fermi theory of beta-decay, one can calculate the
interaction between a proton and a neutron. It is called an
exchange interaction during which a proton and a neutron
switch roles as they exchange electron and neutrino or
positron and neutrino. In his calculations, Tamm assumes that
both the proton and the neutron are stable. As a result, he
comes to the conclusion that the interaction is too weak to
explain the binding of proton and neutron in the nucleus.
Tamm's paper provoked an animated discussion. His methods of
calculation were criticized by Landau; opinions on the issue
differed" [21, p.518]3.
3 Thus, it is clear that the term "Tamm-Ivanenko forces" does not reflect historical reality, contrary to the opinion of S Gershtein [22] and in accordance with E Feinberg [23] (see [24] for details).
Bronstein combined
interest in topical problems and a broad view on the general
architecture of the edifice of physics then under
construction. It was he who introduced the currently
well-known cGh-plan of this edifice or "Relations of
Physical Theories to Each Other and to the Cosmological
Theory" as he titled a section in his 1933 article [25] where
he schematically ranked the existing and anticipated theories
according to their applicability taking into account the
fundamental constants c, G, and h. At that
time, physics was waiting for a 'relativistic quantum theory'
or ch-theory. But Bronstein looked farther than that:
"After the relativistic quantum theory is created, the task
will be to develop the next part of our scheme, that is to
unify quantum theory (with its constant h), special
relativity (with constant c), and the theory of
gravitation (with its G) into a single theory."
It those days,
astrophysics already had a focus of its own for the ch-theory:
white dwarfs. There was also a vague hope first expressed by
Bohr in the late 1920s that the relativistic quantum theory
would be able to account for the source of stellar energy. For
all that, gravity remained an external factor, like the walls
of a container. Bronstein realized the need for the cGh-theory
in astrophysics and explained it in a simple way: if the sun
were compressed to nuclear density, its radius would be
comparable with the gravitational radius [26].
In Bronstein's view,
however, cosmology should be the main task for the cGh-theory:
"...a solution to the cosmological problem requires first
to create a unified theory of electromagnetism, gravity,
and quanta." [25, p. 28]. The addition of fundamental
forces unknown in 1933 to electromagnetism would make quite a
modern, even if pretty banal, statement. But in 1933 such an
understanding of the cosmological problem was new.
Since Bronstein made
calculations in both astrophysics and cosmology, these were
not merely 'general considerations' for him but were still too
general for a man with imagination and enthusiasm to be
absorbed in writing 'long unintelligible formulas' of quantum
gravity, be it for the sake of his own dissertation or for
world science.
Indeed, Bronstein as
a theorist had a more specific reason for investigating not
only in the breadth but also in the depth of the problem.
History preserves some evidence, e.g., a photo in the
newspaper Khar'kovskii rabochii [Kharkov worker] of
May 20, 1934 published to illustrate information about the
aforementioned conference on theoretical physics; the photo
features Landau, Bohr, Rosenfeld, and Bronstein sitting at a
round table and conversing.
The photo
published in the newspaper Khar'kovskii rabochii
(Kharkov worker) on May 20, 1934 among materials on the Kharkov
conference on theoretical physics. Left to right: Landau,
Bohr, Rosenfeld, and Bronstein.
They did have a
common topic for the conversation, it being the subject of
their articles. Nothing is said about this subject in
Bronstein's review of the conference published in Uspekhi
Fizicheskikh Nauk in 1934 because his aim was to dwell
on matters interesting 'not only for theorists'. Meanwhile,
the 'common topic' of the four researchers, the coming relativistic
quantum theory, was so theoretical that it could be of
interest only to a very few. In modern vocabulary, the term
should be substituted by 'quantum electrodynamics', but such a
substitution would not give the feeling of the dramatic
changes in the mentality of microphysics theorists
experienced in the early 1930s.
The quantum theory
of the electromagnetic field was regarded as an important
component of the relativistic quantum theory, but not the sole
one. In the late 1920s, nobody thought about forces of the
microworld other than electromagnetism, and what was known
about electromagnetism could not explain how the nucleus
confined its positive charge. In that pre-neutron epoch,
nuclei were believed to be composed of protons and
'intranuclear' electrons. The uncertainty relation and the
small size of the nucleus suggested a high relativistic speed
of 'intranuclear' electrons. At the same time, before the
positron was discovered, Dirac's ch-equation was
considered to be burdened with a most serious 'plus-minus'
problem. Therefore, theorists hoped that the coming
relativistic quantum theory would solve a cluster of puzzling
problems, such as infinities, nuclear spins, and continuous
spectra of beta-decay.
They awaited the
revolutionary reconstruction of physics comparable with
relativistic and quantum physics. Niels Bohr, the chief
inspirer of the revolutionary mood, was even prepared to
sacrifice the law of conservation of energy for the sake of
successful reconstruction. This attitude was shared by Landau,
who had met Bohr in 1930 and at once adopted him as his sole
teacher.
Landau soon made a
step from general hope to specific calculations. In January
1931, he and R. Peierls arrived at a revolutionary conclusion,
which is that the most natural problem of the 'relativistic
quanta theory' — the quantum theory of the electromagnetic
field — is unsolvable because of the defectiveness of the
basic notion 'field at a point'. It was the beginning of the
story, the development of which should have been discussed by
the four theorists gathered at the round table in Kharkov in
May 1934.
4.
The problem of ch-measurability. Is the uncertainty
principle too certain?
Quantum mechanics
and its uncertainty principle (1927) brought some limitations
on the applicability of concepts inherited from classical
physics. These 'h-limitations' concerned joint
measurability of certain pairs of variables, such as
coordinate and momentum: DxDp > h, but at the same time
left open the possibility of obtaining an arbitrarily accurate
value of either variable. This gave reason to apply these
variables in the h-theory.
Soon after the
meaning of h-limitations was understood, the question
arose as to the character of quantum constraints imposed when
relativity was taken into account as well, or ch-limitations.
Thought
experiments (such as the 'Heisenberg microscope') provided
arbitrarily accurate results only if the c-theory was
ignored. However, a most important physical subject, the
electromagnetic field, was relativistic even before the theory
of relativity was created, since Maxwell equations contained
the constant c.
An article published
by Landau and Peierls in 1931 was entitled "Extension of the
uncertainty principle to the relativistic quantum theory".
After having considered thought experiments in the ch-domain,
the authors arrived at the conclusion that not only were
combined pair uncertainties inevitable but so were individual
ones. The physics of the new limitation was related to the
fact that measurement of 'the field at a point' required
maximally accurate measurement of the position of the test
charge possible only at a sufficiently large momentum
(therefore, small wave length) of the measuring particle. In
this case, however, the recoil momentum of the test charge
produced an additional electromagnetic field that distorted
the field being measured. Hence, the conclusion that the
notion of 'field at a point' is undefinable. Based on this
inference, the authors questioned the then accepted approach
to quantization of the electromagnetic field and predicted
that "the correct relativistic quantum theory to come will
contain neither physical quantities nor measurements in the
sense of wave mechanics." [27].
This paper written
in Zurich (in January 1931) manifested the great influence of
Bohr by referring to his articles and oral discussions in
Copenhagen. Evidently, the authors were sure they were
developing Bohr's ideas in particular by theoretically
substantiating his hypothesis about energy non-conservation in
ch-physics. However, when Landau and Peierls came to
Copenhagen to see Bohr in February 1931, he rejected their
conclusion. The situation is depicted in a drawing by G Gamow
and in recollections by Leon Rosenfeld, then Bohr's assistant:
"When I arrived at the institute on the last day of February 1931, for my annual stay, the first person I saw was Gamow. As I asked him about the news, he replied in his own picturesque way by showing me a neat pen drawing he had just made. It represented Landau, tightly bound to a chair and gagged, while Bohr, standing before him with upraised forefinger, was saying 'Bitte, bitte. Landau, muss ich nur ein Wort sagen!' ('Please, please, Landau, may I just say a word?') I learned that Landau and Peierls had just come a few days before with some new paper of theirs which they wanted to show Bohr, 'but' (Gamow added airily) 'he does not seem to agree — and this is the kind of discussion which has been going on all the time.' Peierls had left the day before, 'in a state of complete exhaustion,' Gamow said. Landau stayed for a few weeks longer, and I had the opportunity of ascertaining that Gamow's representation of the situation was only exaggerated to the extent usually conceded to artistic fantasy." [28].
Landau and Bohr discussing measurability of
field, 1931.
Nevertheless, Landau
held to his opinion and the article was published.
For two years,
highbrow theorists regarded this paper as very important,
although it closed the old direction of thought rather than
opened a new one — various paradoxical problems in
'paranuclear' physics turned out to have a common deep root.
Bronstein saw it this way. While in his very first article he
mentioned the possibility that experiment would demonstrate
"applicability limits of the theory," here, the applicability
limits came from 'theoretical experiments'.
Considerations of
observability and measurability played an important role in
the analysis of the simultaneity notion in the theory of
relativity. In quantum mechanics such consideration had
become an ordinary tool and even commonplace. In his 1931
review of Dirac's book4, Bronstein reproached the
author for the underestimation of quantum-relativistic
problems and quoted witty Pauli's definition: "Die Observable
ist eine Groesse, die man nicht messen kann" (The observable
is a variable that is unmeasurable) and suggested that "The
uncertainty principle of ordinary quantum mechanics is too
certain for the relativistic quantum theory" [30].
4
The book was published in Russian in 1932 [29].
Meanwhile, Bohr
worked together with Rosenfeld to transform his oral
objections to Landau into a well-grounded text to defend
quantum uncertainty from the 'relativistic threat'. The work
took two years to complete and resulted in a lengthy article,
"famously obscure and difficult" in the words of the
well-known physicist and historian of science S Schweber [31].
Indeed, this super-theoretical article is frightening both in
its volume (more than 60 pages) and the abundance of
laboratory terminology used to describe thought experiments,
such as test bodies of arbitrary mass and charge able to
penetrate each other, countless small mirrors at every part of
the test body, rigid bindings to a hard frame, flexible
magnetic threads, etc. [32, 33, pp. 139-142].
However, the main
idea of the defense is clearly formulated at the very first
pages of the article, indicating the weak point of Landau -
Peierls's reasoning: to measure the field, they used
point-like charges as test bodies, the idealization taken
from the quantum mechanics of atomic phenomena. But the
notion of the point-like charge is illegitimate in classical
field theory. On the other hand, classical physics allows
measuring an average field in a finite space region with any
desired accuracy. If such measurement is impossible for some ch-reason,
a certain characteristic length should exist that limits the
size of the space region where measurement is still possible.
However, the quantum theory of the electromagnetic field is
based only on two universal constants, c and h, that
could produce no characteristic length. Values of charges and
masses of elementary particles are merely external
characteristics not integrated into the edifice of the theory
[33, p. 121].
For all the power of
dimensionality considerations, they are no more than
'theoretical physics for the poor (experimenters)', they can
yield a result but can not account for it. Meanwhile, Bohr
sought to obtain a comprehensive explanation, and the greater
part of the paper was to realize the idea that a measuring
instrument must be macroscopic (i.e., classical) in principle.
The physical idea underlying his laboratory technique can be
described as follows: if a field is to be measured with a
desired accuracy, the test body must be chosen such as to have
a relatively large mass in order that the recoil momentum does
not produce too large a field. Thus: "...as regards the
measurability problem, the quantum field theory is a
controversy-free idealization insofar as it permits
abstraction from all constraints imposed by the atomistic
structure of field sources and measuring devices" [33, p.
162].
If Bohr's intention
was to make Landau change his mind he did not succeed because
Landau never recognized that his work with Peierls was flawed.
As to Bronstein, not only he understood and accepted Bohr and
Rosenfeld's result but seemed to comprehend it even better
than the authors. It follows from Bronstein's short note
submitted to the Doklady Akademii Nauk in
January 1934 [34]. In this three-page presentation, instead of
the sixty pages written by Bohr and Rosenfeld, Bronstein
elucidated the logic of their thought experiments and brought
out the physical essence of Bohr's conclusion of the
'non-fatal' nature of ch-limitations for
electrodynamics: a thought experimenter needed unlimited
freedom to choose the charge and the mass of the test body.
The general conclusion remained the same but Bronstein
emphasized that potentialities of any theory must correspond
to those of nature. "The impossibility, in principle, to
measure, with an arbitrary accuracy, a field in the coming
relativistic quantum theory will be essentially a consequence
of the atomism of matter, i.e. the impossibility, in
principle, to infinitely increase [charge density])" [34, p.
389].
Bronstein's note had
already been published when a newsman took photo of the four
physicists at the round table in Kharkov in May 1934. It is
very likely that the ch-topic was not central to their
discussion. The situation radically changed after 1931 when
Landau got to the ch-problem. There was no longer a
need to cut the Gordian knot of quantum-relativistic problems.
Most of them had been solved by that time by experimenters.
The neutron, positron, and neutrino were within a few months
integrated into the physical world picture; as a result, a
number of former problems turned out to be a triumphant
confirmation of theoretical propositions. In light of present
knowledge, the solution to a number of puzzling problems
achieved at those times may seem rather prosaic but physicists
of that period thought differently. For them, the picture of
the microworld changed drastically; suffice it to say that
they had to do with four times the number of elementary
particles and antiparticles than they had before (eight
instead of two). In that situation, gravity appeared to have
little relevance to microphysics. But, strange as it may seem,
it did have something to do with the history of microphysics.
5.
Gravity and microphysics in the 1930s
The neutrino had the
most ambiguous status of all the newly obtained particles,
with its direct experimental observation being a matter of the
remote future. For a few years Bohr's hypothesis of
non-conservation of energy in the relativistic quantum theory
successfully competed against the neutrino hypothesis
suggested by Pauli to explain the continuous spectrum of
beta-electrons. The work by Landau on the mass limit of a star
composed of a Fermi-gas (1932) which is now considered in the
context of the theory of white dwarfs and black holes was
viewed differently at those times. Landau himself believed
that he substantiated the existence of 'pathological' regions
in stars that required the ch-theory to be described
and, in accordance with Bohr's idea, generated stellar
radiation energy from 'nothing'. "Following the beautiful idea
of professor N Bohr, one may think that stellar radiation is
due to a mere violation of the law of energy conservation that
does not hold, as was first noticed by Bohr, in the
relativistic quantum theory where the laws of ordinary quantum
mechanics fail (as confirmed by experiments on the continuous
spectrum of electrons in beta-decay and ensues from
theoretical considerations [here Landau refers to his and
Peierls’ article [27] — GEG]). We expect all this to
be manifest when matter density comes to be so large that
atomic nuclei get in close contact resulting in a single giant
nucleus" [35].
In the same frame of
mind, Bronstein suggested in his paper "On the Expanding
Universe" (1933) a cosmological model with which to realize
Bohr's hypothesis; the non-conservation of energy was
effectively taken into consideration in equations of GR in
the form of a time-dependent cosmological lambda-term.
Einstein's theory of gravity was thus brought in touch with
microphysics and actually invalidated the Bohr's hypothesis.
The supplementary note to the proof of Bronstein's paper
dated 13 January 1933 read as follows: "Landau drew my
attention to the fact that the validity of Einstein's
gravitational equations for empty space surrounding a material
body is incompatible with the non-conservation of its mass.
This inference is strictly verified for the solution of
Schwartzschild (spherical symmetry); physically, it is related
to the fact that Einstein's equations of gravity allow only
transverse but not longitudinal gravitational waves..." [36].
In other words, no
matter how exotic the physics of the nucleus (or the
'pathological region' of a star) might be, the laws of GR (far
away from any exotic regions) forbid the variability of
mass-energy.
As soon as Bohr came
to know this simple consideration (from Gamow's letter), he
replied to the effect that `so much the worse for gravity,' with: "I fully agree
that a renunciation of energy conservation will bring with it
equally sweeping consequences for Einstein's theory of
gravitation as a possible renunciation of conservation of
charge would have for Maxwell's theory." And he blurted out
right away his own quantum-relativistic news: "In the course
of the autumn, Rosenfeld and I have succeeded ... in verifying
the complete correspondence between the basis of the formalism
of quantum electrodynamics and the measurability of the
electromagnetic field quantities. I hope it will be a comfort
for Landau and Peierls that the stupidities they have
committed in this respect are no worse than those which we
all, including Heisenberg and Pauli, have been guilty of in
this controversial subject" [37].
In 1934, in
connection with the same remark from Landau Bohr still
desperately asked: "Shall we necessarily demand that all these
gravitational effects be as closely associated with atomic
particles as electrical charges are with electrons ?" [38]. By
this time, the idea of the neutrino was widely accepted in
physics, supported by both experimental findings and Fermi's
theory of beta-decay. An evidence is I Tamm's work on pair
nuclear forces. One week before the conference in Kharkov
opened, Tamm wrote to Dirac: "What do you think of this Fermi
theory? I have some distaste for the idea of the neutrino, but
at present I see no other way to overcome the difficulties.
Enclosed please find a short note on some corollaries from
Fermi's theory. Would you kindly submit it to Nature if you
find it interesting enough" [39].
A modern physicist
may feel awkward that the great Bohr so insistently tried to
discredit the law of energy conservation or be disappointed at
the 'childish' argument of the great Landau (because of
ridiculously small gravity effects in microphysics). But the
embarrassment turns to sympathy when one comes to know that
the great Pauli (who never believed in the non-conservation of
energy and instead invented ad hoc a new neutral
particle) described Landau's gravitational argument as an
important achievement in a lecture delivered during his stay
in the USSR at the end of 1937 (in the published lecture this
achievement was attributed to Einstein, not Landau, probably
because Landau had been arrested by the time of publication)
[38].
To sum up, the score
in the match between Bohr and Landau was 1 to 1 to the benefit
of science after Bohr had neutralized the radicalism of
Landau's inference with respect to the ch-theory and
Landau 'rendered harmless' Bohr's radical theory of
non-conservation of energy with the aid of the cG-theory
or non-quantum theory of gravity.5
5
Landau most likely believed that the score was actually 1.5: to
0.5 in his favor; he never disproved Bohr's reasoning but
considered his thought measurements unrealizable in practice.
Peierls was of the same opinion (see [41]).
While Bronstein could be motivated to address the quantum gravity problem by the outcome of the first round of this match, important was the fact that gravitation was in Bronstein's field of vision, as it was in Landau's, as it manifested in the second round.
6.
"...An essential difference between quantum electrodynamics
and the quantum theory of the gravitational field"
It seems reasonable
to associate the origin of the problem with Bronstein’s note
of 1934 where he showed that to measure electromagnetic field,
Bohr's thought experimenter had to be able to set the
arbitrary charge and mass densities of the test body.
Bronstein could notice that gravity gives no such freedom for
two reasons. First, gravitational charge and mass are the
same. Second, when arbitrarily increasing the density of such
a body, the observer would inevitably encounter the
gravitational radius and would therefore lose the sight of the
test body. Hence, the logic of the Bohr-Rosenfeld defense
fails.
The limitation of
the Bohr-Rosenfeld argumentation is even more apparent if
their idea that the universal constants of quantum
electrodynamics, c and h, produce no
characteristic length is extended to gravity. The theory of
gravity deals with three constants, c, G, and h,
whose combination lPl = (hG/c3)1/2
= 10-33 cm, gives the Planck length. However, there
is no evidence in Bronstein's writings that he was aware of
this simple argument, nor did other theorists appear to have
mentioned Planck values until the mid-1950s. (By strange
coincidence, the book by Bridgman [42] translated into
Russian and published in 1934 where the Planck values were
mentioned — and renounced — was edited by S I Vavilov,
director of the Physical Institute where Bronstein worked as a
researcher.)
True, the
dimensional argument does not provide as strong a motivation
to pose the cGh-problem as the difference between the
charge freedom in electrodynamics and gravity, the source of
the real problem.
Before addressing
this problem, Bronstein developed a quantum theory of the weak
gravitational field to solve two problems being natural in
this approximation and required by the principle of
correspondence: emission of gravitational waves and Newton's
law of gravity. Representing gravitational interaction of
material bodies via "an intermediate agent — gravitational
quanta", Bronstein obtained, from the cGh-theory of
the weak field, Einstein's cG-formula of
gravitational radiation in the non-quantum limit and Newton's
G-law of universal gravitation in the classical limit.
The solution of
these problems occupied the major part of Bronstein's theses,
and the results, even if expected, were absolutely necessary
to seriously consider the very possibility to quantize
gravity. In connection with this part of the dissertation, V A
Fock, who spoke at the meeting where it was presented, said:
"This work of Matvei Petrovich is the first one devoted to
quantization of gravitational waves in which final physical
results have been obtained. Rosenfeld, who worked out the same
problem, reported only general mathematical results... The
approximation considered by Matvei Petrovich (weak field
approximation — G EG) raises no doubt. The result
would be the same even if Einstein's theory turned out to be
wrong" [43, p. 317].
However, Bronstein
was perfectly aware that the main physical problems requiring
quantum gravity -- the final states of stars and the initial
state of the Universe -- equally needed strong field
treatment. The only way to somehow get in touch with strong
field was an analysis of measurability. Indeed, Landau and
Peierls suggested this method to deal with the formal problem
of infinity in the ch-theory in a physically
meaningful way. Bohr and Rosenfeld further developed and
modified it along the same physical line. Bronstein applied
this approach to the problem of cGh-theory.
M P
Bronstein reading a lecture on the theory of gravity and quantum
theory.
Bronstein considered
the measurability problem in a separate paragraph ("Let us
make some thought experimentation!") in the first of the two
papers on quantum gravity (August 1935 [1]). In the second one
(December 1935), he went on with the analysis and carried it
through to achieve a definitive conclusion.
'A device' for
measuring gravity with the field strength represented by the
Christoffel symbol [00,1] (or Г100 in
modern notation) is governed by equations of GR in the weak
field approximation (gmn = emn
+ hmn,
hmn < 1). The equation of
motion for the test-body has the form
Following Bohr and
Rosenfeld, to measure the field strength Г averaged on volume
V and time T , one uses a test body of volume
V (and mass rV) whose momentum is
measured in the beginning and in the end of the time interval
T. If the duration of measurement is Dt (< T) and
Dx is the coordinate
uncertainty, then the uncertainty Dp is the sum of the
usual quantum-mechanical uncertainty h/Dx and the gravity
field uncertainty created by the recoil of the test-body
during the measurement (the recoil field being given by
Einstein's equation of gravitation DDh01 = Grvx).
By adding
constraints on the parameters of the measuring procedure, Dx < V1/3
(because of measuring the average over V) and Dx < cDt (relativistic
constraint), Bronstein obtained two boundaries from below for
the uncertainty of the field being measured, DГ:
He concluded: "Of
these two boundaries for the case of light test-bodies (rV < (hc/G)l/2,
i.e.
smaller than 0.01 mg), the former is the only essential one.
The latter boundary is essential for heavier test-bodies.
Evidently, a heavy test-body should be recommended for the
most accurate possible measurement [00,1]; this means that
theoretically only the second boundary is of importance.
Finally, we have
Thus, it is clear
that in a region where all hmn are small
compared with 1 (this is what is meant by 'weak' in the title
of this work), the accuracy of gravitational measurements can
be made arbitrarily high: since approximate linearized
equations are applicable in this region and the principle of
superposition is valid, there is always a possibility to have
a test-body of arbitrarily large density r. We therefore
conclude that it is possible to construct a consistent quantum
theory of gravity in the framework of the special theory of
relativity (i.e. when the space-time continuum is
'Euclidean'); such an attempt is made in this work. Matters
are different, however, in the realm of the theory of general
relativity, where deviations from 'Euclidean conditions' may
be arbitrarily large. The thing is, the gravitational radius
of the test-body (GrV/c2) used in the
measurements should by no means be larger than its linear
dimensions (V1/3); hence, the upper boundary
for its density (r < c2/G V2/3).
Thus, the possibilities for measurements in this region are
even more restricted than follows from the quantum-mechanical
commutation relations. It appears hardly possible to extend
the quantum theory of gravity to this region without profound
modification of classical notions" [1, pp. 149, 150].
Friendly
jest showing how Bronstein saw the socialist planning of science
(all-union conferences were held to discuss the topic): "Any
plan is a forecast". However, he predicted the theory of quantum
gravity without making use of tarot cards, by sheer force of
scientific logic.
The denominator in
the formula for DГ contains volume V
and time t, but they can be used to reduce
minimum uncertainty only by increasing V and T, that
is, by expanding the region of averaging, i.e., by
disregarding localization of the measurement, which should be
point-like in the limit.
Note the
'accidental' appearance of the Planck mass scale, "(hc/G)l/2,
i.e., smaller than 0.01 mg" and the cautious general
conclusion: "hardly possible."
Three months after
the paper had been written, Bronstein defended his thesis.
The unanimous verdict of Fock and Tamm, the official
reviewers, was "deserves" (the scientific degree). More
interesting, however, was the dispute between them and the
dissertator briefly (and probably incompletely) recorded in
the minutes of the meeting [43, p. 317).
Fock: "Of interest
here is the analogy between gravitational and electromagnetic
waves. This analogy, interesting in itself from the physical
standpoint, made it possible to use the apparatus of
electrodynamics. Einstein's equations are nonlinear ones...
There is yet no non-linear theory in electrodynamics, and the
generalization concerning it is still in the bud. The work of
Matvei Petrovich [Bronstein] is of value in that it may shed
light on the relationship between linear and non-linear
theories. As regards measurability of the gravitational
field, there is again an analogy with the situation in
electrodynamics. Therefore, the introduction of the
gravitational radius raises the same objections as in
electrodynamics. The results obtained by Matvei Petrovich are
beyond any doubt, and I have nothing to do but wind up."
However, Bronstein
challenged this view: "For me, the analogy between the
non-linear theory of gravity and nonlinear electrodynamics
such as the Born-Infeld theory is debatable. Non-linear
electrodynamics is actually unitary while the general theory
of relativity is not. I don't think any important corollaries
can be deduced from the comparison of the present theory and
the general theory of relativity."
In the vocabulary of
those times, the term 'unitary' was applied to a field theory
in which a 'particle' was a specific field configuration and
its mass stood for the energy of this field. The standard
electrodynamics was regarded as dualistic because its notions
of field and particle were independent (see, for instance,
[44]). It was hoped that non-linear electrodynamics would be
instrumental in the solution to the problem of the electron's
infinite self-energy, and its concrete variant, the
Born-Infeld theory, was based on the Lagrangian that was not
derived from any profound physical considerations but was
'hand-made' so as to avoid infinities even at the classical
level: lbi = e-1 (1 + e
LM)
, where LM is the usual Maxwellian Lagrangian and e
is a small
constant (needed to obtain classical electrodynamics in the
linear approximation).
It is therefore
difficult to agree with Fock's analogy between the electron
radius in the Born-Infeld theory (as a characteristic distance
at which the field behavior begins to deviate from the Coulomb
one) and the gravitational radius associated with the
fundamental physical fact of equality between gravitational
and inert masses or with the theoretical expression of this
fact, the principle of equivalence.
Although the very
notion of gravitational radius was introduced to physics only
in conjunction with the Schwartzschild solution and Riemannian
geometry of GR, the phenomenon of the black hole was
known even to Laplace as early as 1798; therefore, it was even
then possible to speak about a gravitational radius to which a
body should be compressed to prevent its light from escaping,
i.e., to have the escape velocity equal the speed of light
(see, for instance, [45]). The physics of this phenomenon is
determined by Newton's law of gravitation (it would be
impossible to 'run away' to infinity if the force of gravity
decreased, say, according to the law 1/r). The forerunner of
this universal phenomenon, the fact that the escape velocity
is independent of the mass of a 'spacecraft', was discovered
by Galileo, who found out that the motion of a body under
gravity is independent of the mass of the (test) body. This
fact is certainly very far from the notion of the 'event
horizon' and other geometric subtleties, but the physical
essence of Einstein's geometrization of gravity is rooted in
Galileo's discovery.
Bronstein displayed
the common sense of a theorist in the concluding comments on
the criticism of his theses. Frenkel, speaking about "the
brilliant work" of the dissertator, remarked that "when
constructing a quantum theory of gravity, one should describe
and establish the relationship between this theory and
electrodynamics. Matvei Petrovich did not take this into
account although it would be desirable to do so in a work like
this." Bronstein's opinion was that "this advice is
dangerously misleading. As is known, Einstein wallowed in
failure trying to establish the relationship between these
theories." The last question was asked by V K Frederiks: "What
influence can the physical effect of gravitational wave
emission have?" The theoretical experimenter answered that
this effect would change the rotation of a binary star.
It follows from the
above that Bronstein's colleagues all agreed on the analogy
between gravity and electromagnetism and did not appreciate
the fundamental difference between the two indicated by the
dissertator. This probably made him emphasize his view in the
second (more comprehensive) paper published in the Zhurnal
Eksperimental'noy i Teoreticheskoy Fiziki (Journal of
Experimental and Theoretical Physics), dated December
14, 1935:
"Until now, all
considerations have on the whole paralleled those in quantum
electrodynamics; but here we should take into
account a circumstance that reveals the fundamental
distinction between quantum electrodynamics and the quantum
theory of the gravitational field.. The difference
is due to the absence of any limitations to the increase in
charge density r
in formal
quantum electrodynamics that ignores
the
structure of the elementary charge. Components of the
electric field can be measured with an arbitrarily high
accuracy provided the charge density is large enough. In nature, there
are probably some fundamental limits on the electric charge
density (no more than one elementary charge per volume of
linear dimensions comparable with the length of the
classical electron radius), but formal quantum
electrodynamics does not takes these limitations into
account. Due to this, we may consider measurements of
electrodynamic quantities as 'predictable' without falling
into contradiction. Matters are different in the quantum
theory of gravitational field. It has to take into
consideration the limitation arising because the
gravitational radius of the test-body... can not be larger
than its real linear dimension... whence
D
[00, 1] >
h2/3G2/3/сTV4/9
.
Thus, the quantity
on the right-hand side of this inequality represents the
absolute uncertainty minimum in the measurement of the
strength of the gravity that can not be exceeded by
introducing an adequately chosen measuring device. This
absolute limit is a result of very rough calculations
because deviations from the superposition principle are
likely to interfere when the measuring device has a
sufficiently large mass (here, we consider a case of
gravitational waves, that is, approximatelyassume the
equations of gravity to be linear; this approximation ceases
to be valid near the surface of a heavy body whose
gravitational radius tends to equal its real dimensions).
One may think, however, that a similar result will be
preserved in a more precise theory because it does not in
itself follow from the principle of superposition and only
corresponds to the fact that the general theory of
relativity does not admit bodies of a given volume to have
an arbitrarily large mass. There is no analogy to this fact
in electrodynamics (just because it obeys the principle of
superposition); that is why quantum electrodynamics can be
free from internal controversy. In contrast, it is
impossible to overcome the internal controversy in the
theory of gravitational waves; measurements of gravity field
values may be regarded as 'predictable' only if the
consideration is restricted to sufficiently large volumes
and time intervals. 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 Thaler." [2, pp. 217, 218] (see also [3, pp. 441,442]).
The conclusion is
formulated resolutely and shows that the author was fully
aware of its radicalism. It is further emphasized by the
German phrase standing for an exclamation mark: "He who does
not believe it owes one thaler."6 In 1936, this
radical forecast brought to memory Landau and Pieirls's
argument stated five years earlier and thereafter refuted by
Bohr and Rosenfeld; therefore, Bronstein’s enthusiasm for the
prediction had to be moderated and at the same time
accentuated.
6
This phrase concludes the story of the incredible adventures
in the Brothers Grimm fairy tale "The Valiant Little Tailor"
7.
Expansion of the universe in 1937
Bronstein defended
his thesis a few days before his 29th birthday. He had only
one and a half years to live till August 1937. But he managed
to do much within his very short lifespan. He combined his
work for the Leningrad Physico-Technical Institute and
lecturing at Leningrad University (A B Migdal was his
postgraduate student in 1937 and Ya A Smorodinskii one of his
undergraduates). He translated Electron Theory by R
Becker and The Principles of Quantum Mechanics by P
Dirac (and also edited its second Russian edition). He wrote a
few articles for the encyclopaedic Physical Dictionary and
worked on a textbook on statistical physics. Most
surprisingly, Bronstein wrote three popular science books: Solar
Matter, X-Rays, and Inventors of the Radiotelegraph
edited by his wife, Lidiya Korneevna Chukovskaya. The
fourth book in the same line was to be about Galileo. He was
attracted to children's literature by S Ya Marshak
Bronstein's last two
papers were published in the Zhurnal Eksperimental'noy
i Teoreticheskoy Fizikim March 1937. One of them,
presenting nuclear-physical calculations requested by I V
Kurchatov, testifies to the author's interest in 'earthly'
scientific problems [46]. The other, a larger and
'extraterrestrial' article, was "On the Possibility of the
Spontaneous Splitting of Photons" [47] (an excerpt is
published in [3, pp. 283-290]), the first paper dealing with
'cosmomicrophysics' (using modern terminology) and the first
real result of 'cooperation' between the physics of elementary
particles and cosmology. The result was so elegant that Ya B
Zeldovich and I D Novikov included it in their famous
monograph on cosmology, even though it was "no more" than the
substantiation of the expansion of the universe [48].
In 1937, the
situation looked much more dramatic. Hubble's red shift in the
spectra of remote galaxies interpreted as the Doppler effect
in the expanding universe gave evidence that its age had to be
some 2 billion years in the then accepted astronomical time
scale, in tremendous contrast to the results of isotopic
dating in accordance with which the geological history of the
Earth spanned several billion years. Bronstein pointed out
this discrepancy in his review published in Uspekhi
Fizicheskikh Nauk in 1931 [49].
As early as 1929,
the well-known astrophysicist F Zwicky proposed a simpler
explanation for the Hubble law describing the grandiose
picture of receding galaxies. According to Zwicky, the
galaxies did not recede; rather, photons emanating from them
had enough time "to become reddish" during their long journey:
the longer they travelled, the greater the red shift. This
astronomical hypothesis was quite unexpectedly supported by a
new physics, the theory of electron-positron vacuum, when
Halpern hypothesized in 1933 that the red shift was due to the
interaction of photons with this vacuum and Heitler expounded
this hypothesis in 1936 in the well-known monograph on quantum
electrodynamics [50].
Bronstein disproved
this hypothesis by showing that, regardless of the mechanism
of "spontaneous splitting of a photon," the relativity
principle results in the frequency dependence of the
probability of photon decay. In other words, had the
corresponding red shift effect actually existed it should have
been different in different regions of the spectrum, rather
than homogeneous as in the Hubble law and Doppler effect.
Thus, the only observable fact of cosmological character known
then was given a microphysical substantiation.
In addition,
Bronstein directly calculated a hypothetical process in the
framework of QED of that time (a small effect could be
superimposed on the Doppler shift) and obtained a zero
probability of spontaneous photon decay (the calculation
occupied the main part of the paper). This agreement was
essential for the physics of those days because, as Bronstein
put it, "At present, there is no complete theory of vacuum
polarization."
However, he was not
to participate in the further development of such a theory. It
is not known whether he managed to submit more papers for
publication. If he did, their fate could be illustrated by a
miraculously preserved trace of his destroyed article entitled
"Quantum Statistics" in the second volume of the Physical
Dictionary published in 1937. One of the 14,000 copies of the
volume retains the concluding part of this article with the
name of Bronstein under it, along with another version having
the same page number but signed by a different author; in
other words, the entry begins as one and ends as two,
evidently due to the careless printer who failed to tear out
one page.
Matvei Petrovich
Bronstein was arrested on the night of August 6, 1937. He was
then thirty. They demanded that he turn over his arms and
poisons — he gave a laugh as his response. Bronstein was
executed in a Leningrad prison in February 1938.
The
last photograph of Bronstein
8.
Decades later
In the 1930s, the
problem of quantum gravity was not topical; most researchers
were concentrated on a variety of vital tasks posed by the
physics of the nucleus, molecules, and the condensed state.
Only one, the French physicist Jacques Solomon (1908-1942),
undertook to develop Bronstein's idea and appreciated the
issue of unmeasurability of the strong gravitational field.
However, he also was not to continue the work either — in 1942
he was arrested as an activist of the French Resistance and
executed by the German Gestapo [51].
For all that,
Bronstein's studies were not forgotten in his own country ten
years after his death. Fock wrote in the official review of a
work nominated for the Stalin prize in 1948: "The work of
Ivanenko and Sokolov is entitled 'Quantum Theory of Gravity'.
This title is at variance with the contents and should be
replaced by a less ambitious one, e.g. "Simplified Exposition
of the Quantum Theory of Gravity". The thing is, the true
quantum theory of gravity was created by a Leningrad physicist
M P Bronstein and expounded in his paper "Quantization of
Gravitational Waves" (Zhurnal Eksperimental'noy i
Teoreticheskoy Fiziki, Vol. 6, pp. 195-236) published
in 1936. Ivanenko and Sokolov use Bronstein's results but make
no reference to his work in their text... Regardless of the
reasons that may have led the authors to abstain from
mentioning the contribution made by Bronstein, their work can
by no means be regarded as the creation of the quantum theory
of gravity because such a theory was constructed by Bronstein
11 years before" (cited from [52]).
The work nominated
for the Stalin prize contained the concept of weak field
approximation but no new physics whatsoever, even though
Ivanenko spoke about the conversion of gravitons to other
particles as an indication of the conversion of space-time to
matter. The coefficient ~ 10-40 characterizing the
relationship between the forces of gravity and
electromagnetism in the microworld efficiently protected any
calculation from experimental verification. (It should be
noted that Bronstein did not use the term 'graviton' even
though the word had already been in use as early as 1934
[53].)
Twenty years later,
several physicists almost simultaneously, using different
approaches, reopened the theme of quantum gravitation
discovered by Bronstein.
Landau pointed out
the gravitational boundary of QED at which two fundamental
interactions are equalized and beyond which QED can not be
regarded as a closed theory due to the necessity of taking
into account gravitation [54]. O Klein discovered the
gravitational boundary of the relativistic quantum theory
[55]. Finally, J Wheeler discovered the quantum boundary of
GR [56]. As a result, the triple physical sense of Planck
scales was exposed even though none of the researchers
mentioned Planck's name. The now universally accepted notion
of 'Planck values' was introduced by Wheeler two years later
[57].7
7
In a letter to the author of the present publication, Wheeler
wrote that in 1955 he did not know of Planck's 'natural units'.
Meanwhile, nobody
reproduced Bronstein's 'renunciation of Riemannian geometry'.
On the contrary, attempts to quantize gravity continued by
means of crossing Riemannian geometry of GR and quantum
theory. This unrestrained optimism was probably encouraged by
the success of QED, the construction of which was so advanced
by the late 1940s as to transform it into the most exact
physical theory, in compliance with the optimistic prediction
Bohr and Rosenfeld made in 1933.
Thirty years later,
a notable event was Rosenfeld's hypothesis that quantization
of a gravity might make no sense because such a field has a
purely classical macroscopic nature [58]. It should be
recalled that Rosenfeld was the author of the first work
concerning quantum gravity published in 1930 implying that
"quantization of the gravitational field…may be carried out
without any new difficulties by means of a formalism wholly
analogous to" electrodynamics.
Bronstein thought
otherwise after he found an essential difference between
quantum electrodynamics and quantum gravity and predicted "the
rejection of our ordinary concepts of space and time,
replacing them by some much deeper and nonevident concepts."
7 In a letter to the
author of the present publication, Wheeler wrote that he did
not know of Planck's 'natural units' in 1955.
The very first
physical model of this kind ('gravity as an elasticity of a
quantum vacuum') was suggested by A D Sakharov in 1967 [59].
This model was enthusiastically welcomed by Wheeler, a pioneer
of quantun gravity [60].
By another
coincidence, in 1967 the name of Bronstein was almost
officially mentioned in the gala volume of Oktyabr' i
nauchny progress (October [Revolution] and Scientific
Progress) where Tamm summarized the achievements of Soviet
theoretical physics. He wrote: "Some exceptionally bright and
promising physicists of this generation passed away
prematurely: M P Bronstein, S P Shubin, A A Vitt" [61] (these
physicists of the first generation educated in the Soviet
Union were arrested in 1937 and all died in 1938, even though
one was sentenced to be shot and the two others were given
eight and five-year labor camp sentences respectively).
Forty years later,
after the discovery of Hawking's effect (evaporation of black
holes), quantum gravity became a respectable area of research
[62]. Since then, around 60 books having the word combination
'quantum gravity' in their titles have been published and
dozens of conferences held (a bit too many for a still
non-existent theory).
The centenary of
Einstein's birth in 1979 was commemorated by the publication
of collected original works "that have made an important
contribution to the development of the theory of gravitation."
This volume included a section entitled "The general theory of
relativity and the physics of the microworld" that contained
papers by Bronstein, Ivanenko-Sokolov, and Fock (recall that
the last author compared the works of the first two in his
review of 1948) [3].
In the 1980s,
Bronstein’s contribution became the subject for history of
science research, and in the 1990s his results were introduced
to western science [4, 63].8
8
John Stachel was the first western author to expound on the
works of Bronstein [64]. He is a prominent historian of science,
founding editor of the series Einstein Studies and The
collected papers of Albert Einstein (Princeton
University Press).
The seventy years of
developments were marked by two monographs published in 2004
under the same title, 'Quantum Gravity' [65]. Both authors
acknowledged that the problem remains wide open but quite
differently presented the work of Bronstein. One quotes the
measurability considerations of Landau and Bohr and alludes to
Bronstein's paper only to renounce both his inference and
basic logic of his analysis. The other sympathetically cites
the 'rejective' conclusion of Bronstein but does not explain
his argumentation of (un)measurability.
This discrepancy
perhaps most adequately reflects the current state of the 70
year-long problem.
9.
How to attain inner perfection without external confirmation
In his
'Autobiographical Notes', Einstein pointed out the two main
criteria for the assessment of a theory: its 'external
confirmation' as an agreement with empirical facts and its
'inner perfection' as a naturalness and logical simplicity
[66].
These criteria seem
sound and even trivial for the entire realm of physics ...
barring the problem of quantum gravity.
Its 'external
confirmation' is counteracted by the astronomical number 1040.
This limitation manifested itself in the very first argument
of Einstein in favor of gravity quantization (gravitational
collapse of the atom). Theoretically, it would be possible to
deal with numbers of such astronomical scale by transition
from physical experiments to astronomical observations, but
no practical way to get to real cGh-objects of
observation is thus far available. The difficulty of such
transition provoked Landau to say about astrophysicists that
they are often in error but never in doubt (true, this
aphorism was coined before the new astrophysical era began in
the 1960s).
It is a bit awkward
to talk about 'inner perfection' in relation to the attempts
to quantize gravity when one looks through a wide variety of
unworkable theoretical constructions and sees the authors'
vain zeal that was never realized into anything of immortal
value. The graveyard of theoretical constructions reminds one
of the abandoned graveyards of perpetual motion projects or
hydrodynamic theories of the ether. Flushes of pioneer
optimism are easier to explain by the 'semicriterion' of
external attractiveness of the next candidate theory. Added to
this seems to be a popular - among physics students - belief
that 'mathematics is more clever than man', i.e. that careful
calculations sooner or later have to produce a physical
result.
The analysis of
field measurability of the 1930s may be called, in addition to
Einstein's criteria, 'inner justification' of
the theory. In fact, it was the analysis of theory’s
applicability limits carried out from the inside. Certainly,
such an analysis can not be absolutely strict and does not
lead to physical statements amenable to direct verification in
experiment. Nonetheless, the analysis of field measurability
is a physical one.
The disagreement of
Landau, the initiator of the analysis, with the results of
Bohr’s extended version, looks strange but had its reasons.
The freedom of thought experiment that Bohr accepted because
it was not forbidden by the known laws of nature was too much
freedom for Landau, since he saw no means to realize this
freedom in real experiment. Indeed, how was it possible to
consider test-bodies of an arbitrary mass and charge in the
framework of microphysics when only a few elementary particles
were really known?
Nevertheless, the
guarantee given by Bohr and Rosenfeld to the constructors of
QED in 1933 proved valid 15 years later, when the most
accurate physical theory was created.
Bronstein's 1935
prediction with respect to quantum gravity was of a
prohibitive character: it forbade the solution of the problem
'at a small price' while conserving the Riemannian geometry of
GR. In itself, this by no means compromises the prediction.
Some great physical laws are of such character as exemplified
by the prohibition on the existence of perpetual motion
machines of the first and second kinds. The special theory of
relativity could be formulated as a prohibition or
impossibility of finding the speed of a light source from
measurements of the velocity of light.
The problem is how
to deduce physically meaningful result from Bronstein's
prohibition. By way of a modest example, I shall try to defend
quantum gravity from one of the founders of QED.
F Dyson has recently
suggested that "quantum gravity is physically meaningless."
Therefore, the 70 year-old history of research endeavors has
come to an end and they should be stopped for the lack of a
subject to study. He argues as follows:
"The essence of any
theory of quantum gravity is that there exists a particle
called the graviton which is a quantum of gravity, just like
the photon which is a quantum of light. … It is easy to detect
individual photons, as Einstein showed, by observing the
behaviour of electrons kicked out of metal surfaces by light
incident on the metal. The difference between photons and
gravitons is that gravitational interactions are enormously
weaker than electromagnetic interactions. If you try to
detect individual gravitons by observing electrons kicked out
of a metal surface by incident gravitational waves, you find
that you have to wait longer than the age of the universe
before you are likely to see a graviton. … If individual
gravitons cannot be observed in any conceivable experiment,
then they have no physical reality and we might as well
consider them non-existent. They are like the ether, the
elastic solid medium which nineteenth-century physicists
imagined filling space. … Einstein built his theory of
relativity without the ether, and showed that the ether would
be unobservable if it existed. He was happy to get rid of the
ether, and I feel the same way about gravitons. According to
my hypothesis, the gravitational field described by Einstein's
theory of general relativity is a purely classical field
without any quantum behaviour" [67].
This reasoning has a
weak point in the very beginning. No matter how common the
analogy between the photon and the graviton is and for all the
similarity of the Coulomb law and the Newton law of
gravitation, the two interactions are different in essence as
was emphasized by Bronstein. The difference undermines the
notion of 'graviton' as standing on an equal footing with the
notion of the 'photon'. Bronstein actually discovered that the
usual notion of 'field quantum', as applied to gravitation, is
essentially approximate like many other important 'working'
physical notions, such as absolute space, simultaneity, ray of
light, temperature, etc., that are also approximate or rather
have limited applicability. It can be said that Bohr and
Rosenfeld justified the notion of the photon in the framework
of electrodynamics and Bronstein showed defectiveness
(approximate nature) of the graviton notion in the gravity
theory. This essential difference arises from the equality of
inert and gravitational masses, an experimental fact that is
sometimes referred to as the first great discovery of modern
science and provides the basis for one of the greatest
theories, GR.
In other words, the
graviton is not intrinsic in the anticipated theory of quantum
gravity as much as the photon is in QED. Only a formal
approach can associate any wave with a certain quantum. There
is hardly anyone who would associate a wave on the sea surface
with a 'surfon' particle in order to study the behavior of
such a wave.
Moreover, Dyson did
not propose what to do with two physical phenomena of primary
importance, the earliest stage of cosmological expansion and
the final stage of stellar collapse. How they could be
explained without quantum gravity? In either case, the need of
a new theory is determined by the Planck scale. Bronstein was
the first to show that this quantitative characteristic
reflects physical nature rather than mere dimensional
considerations.
It is not necessary
to use an advanced apparatus to obtain this quantitative
borderline. Suffice it to resort to the simplest gravity and
quantum physics, i.e., Newton's law of gravity and quantum
postulate suggested by Bohr in 1913 (and the reminder that the
black hole phenomenon was predicted in classical physics).
Following the example of Bohr and Bronstein in taking full
quantitative freedom, let us consider a simple physical
system of a 'double star' (or ‘gravitium’ molecule in the
language of the microworld), that is, two identical point-like
masses m bound by gravitational interaction and
moving in a circular orbit with radius r. Submitting
this system to classical mechanics and the Bohr quantum
postulate makes it possible to determine the values of system
parameters at which its description must take into account
both quantum and 'black hole' effects; this procedure yields
Planck values of m and r. Before the era of
quantum mechanics, such a derivation would not have been more
'illegal' than Bronstein's reasoning in 1935 or Klein and
Wheeler's in the 1950s.
Bronstein's papers
contain no exact directions on whether it is indeed necessary
to reject "our ordinary concepts of space and time,
replacing them by some much deeper and nonevident concepts"
and, if 'yes', what these new concepts must be. By way of
example, it can be imagined that such a substitution would
entail the replacement of the formula for the Planck length, lPl
= (hG/с3)1/2, by the formula for
the classical gravitational constant, like G = l2c3/h,
where l is a certain
constant of the theory of quantum gravity to come.
The history of
quantum gravity makes one think that the majority of
publications would have never appeared had their authors known
and seriously apprehended the analysis of the problem
undertaken by Bronstein. At the very least, this would have
saved a large amount of paper and working time.
Could Matvei
Bronstein have promoted the development of the theory of
quantum gravity if Russian history had not killed him at the
age of 30? Unfortunately, a historian of science can not
answer questions like this. He can only offer his historic
thaler to the one who can.
I am thankful to L P
Pitaevskiy for discussions on the subject of this paper, and
to B L Altshuler and the reviewers for their helpful comments.
References
1. Bronstein M
"Quantentheorie schwacher Gravitationsfelder" Phys. Z. Sowjetunion 9 140-157 (1936)
2. Bronstein M P "Kvantovanie gravitatsionnykh voln" (Quantization of Gravitational Waves) Zh. Eksp. Tear. Fiz. 6 195 (1936)
3. Albert Einshtein i teoriya gravitatsii (Albert Einstein and the Theory of Gravty) (Ed. E S Kuranskii) (Moscow: Mir, 1979)
4. Gorelik G "First steps of quantum gravity and the Planck values", in Studies in the History of General Relativity (Einstein Studies, Vol. 3, Eds J Eisenstaedt, A J Kox) (Boston: Birkhauser, 1992) p. 364 - 379; Gorelik G E, Frenkel V Ya Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties (Boston: Birkhauser, 1994)
5a. Planck M. Uber
irreversible Strahlungsvorgange Ann. Phys. 4(1), S.122,1900.
5b. Planck M Wissenschaftliche Selbstbiographie. (Leipzig: Barth, 1970)
6. Bridgman P W Dimensional
Analysis (New Haven, Conn.: Yale Univ. Press, 1922) p. 101
7. Einstein A "Naherungsweise
Integration der Feldgleichungen der Gravitation" Sitzungsber.
Kbniglich
Preufi. Akad. Wiss. (Berlin) 688 (1916)
["Approximative integration of the field equations of
gravitation" (22 June 1916), in The Collected Papers of
Albert Einstein Vol. 6 The Berlin Years: Writings,
1914-1917 (Princeton, NJ: Princeton Univ. Press, 1997) p. 201]
8. Einstein A "Uber
Gravitationswellen" Sitzungsber. Koniglich Preuss. Akad. Wiss. (Berlin) 154 (1918)
["On Gravitational Waves" (31 January 1918), in The
Collected Papers of Albert Einstein Vol. 7 The
Berlin Years: Writings, 1918-1921 (Princeton, NJ:
Princeton Univ. Press, 2002) p. 9]
9. Heisenberg W, Pauli W
"Zur Quantendynamik der Wellenfelder. Zeitschrift fuer Physik
56:1-61 (1929).
10. Rosenfeld L "Uber die Gravitationswirkungen des Lichtes." Z. Phys. 65 589 -599 (1930)
11. Gorelik G, Frenkel' V M P Bronshtein i kvantovaya teoriya gravitatsii (M P Bronshtein and the Quantum Theory of Gravitation.), in Einshteinovskii Sbornik 1980-1981 (Ed. I Kobzarev) (Moscow: Nauka, 1985) p. 322
12. Bronstein M P "On the theory of electronic semiconductors" Phys. Z. Sowjetunion 2 28 (1932); Bronstein M P "On the conductivity of semiconductors in magnetic field" Phys. Z. Sowjetunion 3 140 (1933)
13. Ioffe A F Elektronnye poluprovodniki (Electronic Semiconductors) (Lenigrad-Moscow: ONTI, 1933)
14. Gorelik G E, Frenkel' V Ya "M.P. Bronstein i ego roP v stanovlenii kvantovoy teorii gravitatsii" (M P Bronstein and His Role in the Development of Quantum Theory of Gravity), in Einshteinovskii sbornik, 1980-1981 (Ed. I Kobzarev) (Moscow: Nauka, 1985) pp. 317-320
15. Shpinel' V S, Letter to G. Goreliku, 17 March 1998
16. Bronstein M P
"Ob odnom sledstvii gipotezy svetovykh kvantov" (About One
Corollary of the Hypothesis of Light Quanta) Zh. russk.
fiz.-khim. obshch. 57 321-325 (1925); Bronstein M P "Zur Theorie des
kontinuierlischen Rontgenspektrums" Z. Phys. 32 881
(1925)
17. Gorelik G E, Frenkel' V Ya
Matvey Petrovitch Bronstein: 1906-1938 (Moscow:
Nauka, 1990)
18. Bronstein M P l"Zum
Strahlungsgleichgewichtsproblem von Milne" Z. Phys. 58
696-699 (1929); "Uber das Verhaltnis des effektiven Temperatur
der Sterne zur Temperatur ihrer Oberflache" Z. Phys. 59144-148(1929)
19. Bronstein M P "Note on the temperature distribution in the deeplayers of stellar atmospheres" Man. Not. R. Astron. Soc. 91 133(1930)
20. Bronstein M P "Convegno di Fisica Nucleare" COPEHA (1) 176-177 (1933); "Konferentsiya po tverdym nemetallitcheskim telam" (Conference on Solid Non-Metallic Bodies) Priroda (1) 73-74 (1933); "Vsesoyuznaya yadernaya konferentsiya" ("All-Union Nuclear Conference") Usp. Fiz. Nauk 13 768 (1933); SORENA (9) 155-165(1933)
21. Bronstein M P "O konferentsii po teoretitcheskoy fizike" (About Conference on Theoretical Physics) Usp. Fiz. Nauk 14 (4) (1934)
22. Gershtein S S "Na zare yadernoy fiziki" (At the Dawn of Nuclear Physics) Priroda (8) 62 (2004)
23. Feynberg E L Epoha i litchnost'. Fiziki. Otcherki i vospominaniya (Epoch and Personality. Sketches and Reminiscences) (Moscow: Fizmatlit, 2003) pp. 80-81
24. Gorelik G E "Neelementarnyy vopros fiziki elementarnyh tchas-tits" (Non-Elementary Problem of Physics of Elementary Particles), http://ggorelik.narod.ru/Dau/ Tamm_Ivanenko_1934_Priroda_2004.htm
25. Bronstein M P "K voprosu o vozmozhnoy teorii mira kak tselogo" ("On the Question of a Possible Theory of the World as a Whole"), in Uspekhiastronomitcheskihnauk. Sbornik,No. 3 (Moscow: ONTI, 1933) p. 3-30
26. Bronstein M P
"Vnutrennee stroenie zvezd i istotchniki zvezdnoy energii"
(Inner Structure of Stars and Sources of Stellar Energy), in Uspehi
astronomitcheskih nauk. Sbornik, No. 2 (M.: ONTI,
1933) p. 102
27. Landau L, Peierls R
"Erweiterung des Unbestimmtheitsprinzips fur die
relativistische Quantentheorie" Z. Phys. 69 56-70
(1931)
28. Rosenfeld L "On quantum electrodynamics", in Niels Bohr and the Development of Physics (Ed. W Pauli) (London: Pergamon Press, 1955) p. 70
29. Dirac P The Principles of Quantum Mechanics [Translated into Russian (Moscow: Gostehteorizdat, 1932)]
30. Bronstein M P "Dirac P. Printsipy kvantovoy mehaniki (Oxford, 1930)" (Review) ("The Principles of Quantum Mechanics (Oxford, 1930)" Review) Usp. Fiz. Nauk 11 355-358 (1931)
31. Schweber S S Phys. Today 54 (11) 56 (2001) <http://www.physicstoday.org/pt/vol-54/iss-11/p56.html>
32. Bohr N, Rosenfeld L "Zur Frage der Messbarkeit der elektromag-netischen Feldgrossen" Kgl. Danske Vidensk. Selskab., Math.-Fys. Medd. 12 (8) 3-65 (1933) ["On the question of the measurability of electromagnetic field quantities", in Quantum Theory and Measurement (Eds J A Wheeler, W H Zurek) (Princeton, NJ: Princeton Univ. Press, 1983) pp. 470-522]
33. Bohr N Izbrannye nautchnye trudy T. 2 (Selected Scientific Works Vol. 2) (Moscow: Nauka, 1971) p. 121
34. Bronstein M P "K voprosu o relyativistskom obobshchenii printsi-pa neopredelennosti" ("On the Question of Relativistic Generalization of Indeterminacy Principle") DAN SSSR1 388-390 (1934)
35. Landau L D "K teorii zvezd (1932)" ("On the Theory of Stars) (1932)"), in Landau L D Sobranie trudov T. 1 (Moscow: Nauka, 1969) p. 89
36. Bronstein M P "On the expanding universe" Phys. Z. Sowjetunion 3 73-82(1933)
37. Bohr N Collected Works Vol. 9 Nuclear Physics, 1929-1952 (Amsterdam: North-Holland, 1986) p. 571
38. Bohr N "On the Method of Correspondence in the Theory of Electrone (1934)" [Translated into Russian in: Bohr N Izbrannye nautchnye trudy T. 2 (Moscow: Nauka, 1971)] p. 172
39. Pol' Dirak i fizika XX veka (Paul Dirac and Physics of the XXth Century) (Ed. Medvedeva, Sost. A B Kozhevnikov) (Moscow: Nauka, 1990) p. 167
40. Pauli W "Laws of Conservation in the Theory of Relativity and Atomic Physics (1937)" [Translated into Russian in: Fizika: Probl-emy, istoriya, lyudi. Sb. nautch. tr. (Ed. V M Tutchkevitch) (Leningrad: Nauka, 1986)] p. 224,228
41. Peierls R Bird of Passage: Recollections of a Physicist (Princeton, NJ: Princeton Univ. Press, 1985) p. 66
42. Bridgman P Analiz razmernostey (Ed. SI Vavilova) (Leningrad-Moscow: ONTI, 1934)
43. Einshteinovskiisbornik, 1980-1981 (Ed. I Yu Kobzarev, Sost. G E Gorelik) (Moscow: Nauka, 1985)
44. Born M "Tainstvennoe chislo 137" ("Secret Number 137") Usp. Fiz. Nauk 16 717 (1936)
45. Hawking S W, Ellis G F R The Large Scale Structure of Space-Time. (Cambridge Univ. Press, 1975) p. 365
46. Bronstein M P "O magnitnom rasseyanii neytronov" Zh. Eksp. Tear. Fiz. 7 357-362(1937)
47. Bronstein M P "O vozmojnosti spontannogo rasschepleniya foto-nov" ("On the Possibility of Spontaneous Splitting of Photons") Zh. Eksp. Tear. Fiz. 1 335-358 (1937); Kratkii variant (Abridged Version): Bronstein M Phys. Z. Sowjetunion 10 686-688 (1936))
48. Zeldovich Ya B, Novikov I D Stroenie i evolyutsiya Vselennoy (Structure and Evolution of the Universe) (Moscow: Nauka, 1975) p. 124
49. Bronstein M P "Sovremennoe sostoyanie relyativistskoy kosmolo-gii" ("Modern State of Relativistic Cosmology") Usp. Fiz. Nauk 11 124-184(1931)
50. Halpern O "Scattering processes produced by electrons in negative energy states" Phys. Rev. 44 855-856 (1933); Heitler W The Quantum Theory of Radiation (Oxford: The Clarendon Press, 1936) p. 193
51. Solomon J "Gravitation et quanta" /. Phys. Radium Vn Ser. 9(11) 479-485 (1938); Bustamante M C "Jacques Solomon (1908-1942): profil d'un physicien theoricien dans la France des annees trentes" ["Jacques Solomon (1908 -1942): Profile of a theoretical physicist in France in the 1930s"] Rev. d'Histoire Sci. 50 (1&2) 49-87 (1997)
52. Gorelik G E "V.A. Fock: filosofiya tyagoteniya i tyajest' filosofii" ("V A Fock: Philosophy of Gravity and Weight of Philosophy") Priroda(10)81(1993)
53. Blohintsev D I, Gal'perin F M "Gipoteza neytrino i zakon sohraneniya energii" ("Neutrino Hypothesis and Conservation of Energy") Podznamenem marksizma (6) 147-157 (1934)
54. Landau L D, Abrikosov A A, Khalatnikov IM "Asimptoticheskoe vyrazhenie dlya fotonnoy funktsii Grina v kvantovoy elektrodina-mike" ("Asymptotic Expression for the Green's Function of a Photon in Quantum Electrodynamics") Doklady Akademii Nauk SSSR 95 1177-1180 (1954)
55. Klein O "Aktuella Problem Kring Fysikens Sma och Stora Tal" Kosmos (Sweden) 32 33 (1954); "Generalization of Einstein's Theory of Gravitation Considered from the Point of View of Quantum Field Theory" Fuenfzig Jahre Relativitotstheorie. Bern, 11-16 Juli 1955 (Helvetica Physica Acta, Suppl. 4, Eds A Mercier, M Kervaire) (Basel: Birkhauser, 1956) pp. 58-68
56. Wheeler J A "Geons" Phys. Rev. 97 511 - 536 (1955)
57. Misner C W, Wheeler J A "Classical physics as geometry: gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space" Ann. Phys. (New York) 2 525 - 603 (1957)
58. Rosenfeld L "On quantization of fields" Nucl. Phys. 40 353-356 (1963)
59. Sakharov A D "Vakuumnye kvantovye fluktuatsii v iskrivlennom prostranstve i teoriya gravitatsii "("Vacuum Quantum Fluctuations in the Curved Space and Gravity Theory") DAN SSSR 177 70-71 (1967); Usp. Fiz. Nauk 161 (5) 64-66 (1991)
60. Misner C W, Thorne K S, Wheeler J A Gravitation (San Francisco: W. H. Freeman, 1973) p. 426-428; Wheeler J A "Beyond the End of Time",in The World of Physics: a Small Library of the Literature of Physics from Antiquity to the Present Vol. 3 The Evolutionary Cosmos and the Limits of Science (Ed. J H Weaver) (New York: Simon and Schuster, 1987) p. 675-694
61. Tamm I E "Teoretitcheskaya fizika" ("Theoretical Physics"), in Oktyabr' i nautchny progress T. 1 (Ed. M V Keldysha et al.; Sost. V A Boyarskii, G D Komkov) (Moscow: Izd-vo APN, 1967) p. 170
62. Isham C J, Penrose R, Sciama D W (Eds) Quantum Gravity: an Oxford Symp. (Oxford: Clarendon Press, 1975) p. 605; written versions of talks at a symposium held at the Rutherford Laboratory on Feb. 15-16,1974
63. Gorelik G E "Pervye shagi kvantovoy gravitatsii i plankovskie velitchiny" ("First Steps of Quantum Gravity and Planck Values"), in Einshteinovskii sbornik, 1978-1979 (Ed. U I Frankfurta) (Moscow: Nauka, 1983)pp. 334-364
64. Stachel J "History of relativity", in Twentieth Century Physics (Eds L M Brown, A Pais, Sir B Pippard) Vol. 1 (Philadelphia, PA.: Institute of Physics Publ., 1995) p. 317-319
65. Kiefer C Quantum Gravity (New York: Oxford Univ. Press, 2004) ix, 308 p.; Rovelli C Quantum Gravity (Cambridge: Cambridge Univ. Press, 2004) xxiii, 455 p.
66. Einstein A "Autobiographical notes", in Albert Einstein — Philosopher-Scientist (The Library of Living Philosophers, Vol. 7, Ed. P A Schilpp) (Evanston, 111.: Library of Living Philosophers, 1949)
67. Dyson F "The world on a string" New York Rev. Books 51 (8) (2004); http://www.nybooks.com/articles/17094