Wer's nicht glaubt (in der quantengravitation), zahlt einen Taler / Whosoever does not believe (in quantum gravity), must pay a euro

A tentative historical introduction for a book that it is currently possible to continue now:

Here, as [Matveï] 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... 

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. 

... 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.
[...] 

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. [...]

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 ρ. 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 must be less than its linear dimensions ... (Bronstein 1936b, p. 217)

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 non evident 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).

[...]

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. 

First Steps of Quantum Gravity and the Planck Values

by Gennady Gorelik (1992)

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

* blogger's comment : in the Bronstein's diagram at the beginning of this post it is hard to miss the lack of a block labeled "Thermodynamics / Statistical physics" on the first line but this is another story for another post with the following question : is there any universal constant one should associate to the missing block, like a large integer N ...? 

(**addendum, 1st March 2015, the tale is titled "Vom klugen Schneiderlein" or The cunning little tailor, vielen Dank an meine Kollege Elizabeth M. ;-)
 
A possible first chapter of the book

Our knowledge of spacetime is described by two existing theories:

• General Relativity
• The Standard Model of particle physics

General relativity describes spacetime as far as large scales are concerned and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein, and Minkowski, by a curved spacetime metric whose components form the gravitational potential. The basic equations are Einstein equations... which have a clear geometric meaning and are derived from a simple action principle. Many processes in physics can be understood in terms of an action principle, which says, roughly speaking, that the actual observed process minimizes some functional, the action, over the space of possible processes...

This Einstein-Hilbert action S_EH, from which the Einstein equations are derived in empty space, is replaced in the presence of matter by the combination (1) S = S_EH + S_SM where the second term S_SM is the standard model action which encapsulates our knowledge of all the different kinds of elementary particles to be found in nature. While the Einstein-Hilbert action S_EH has a clear geometric meaning, the additional term S_SM ... is quite complicated (it takes about four hours to typeset the formula) and is begging for a better understanding... 

Our goal in this expository text is to explain that a conceptual understanding of the full action functional is now available (joint work with A. Chamseddine and M. Marcolli [6], [8]) and shows that the additional term S_SM exhibits the fine texture of the geometry of spacetime. This fine texture appears as the product of the ordinary 4-dimensional continuum by a very specific finite discrete space F. Just to get a mental picture one may, in first approximation,think of F as a space consisting of two points. The product space then appears as a 4-dimensional continuum with “two sides”. As we shall see, after a judicious choice of F, one obtains the full action functional (1) as describing pure gravity on the product space M × F. It is crucial, of course, to understand the “raison d’être” of the space F and to explain why crossing the ordinary continuum with such a space is necessary from first principles. As we shall see below such an explanation is now available...

ON THE FINE STRUCTURE OF SPACETIME

Alain Connes

A summary of the second chapter under way

... we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of the two-sided relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y,Y' from the four-dimensional manifold M⁴to S⁴. The intuitive picture using the two maps from M⁴ to S⁴ is that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively the Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).

Geometry and the Quantum: Basics

Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov

(Submitted on 4 Nov 2014)