100- (2 1906)

70- (6 1937)


 

John Stachel -- , . .

 

 

 

John Stachel. History of Relativity in Twentieth century physics. Eds. Laurie M. Brown, Abraham Pais, Sir Brian Pippard. Philadelphia, PA. : Institute of Physics Pub., 1995. Vol. 1, p. 317-319.

 

 

In the paper which developed special-relativistic quantum field theory (see above), Heisenberg and Pauli remarked optimistically 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 wholly analogous to that applied here' [330].  Rosenfeld first applied this formalism to the linearized gravitational field, introducing the term 'gravitational quanta' for the particles associated with this field [331]. A few years later, Bronsteyn undertook a more critical analysis of linearized quantum gravity [332]. He raised the problem of the measurability of the quantized gravitational field, arguing that, in addition to limits imposed by the uncertainty principle, in general relativity there is an 'absolute limit' to the accuracy with which the components of the linearized affine connection within a given volume can be measured. He suggested that application of the formalism of quantum field theory might not yield the desired fusion of quantum theory and gravitation, calling for 'a radical reconstruction of the theory, and in particular, the rejection of a Riemannian geometry... and perhaps also the rejection of our ordinary concepts of space and time, replacing them by some much deeper and nonevident concepts' [333].

 

 

[330] Heisenberg W and Pauli W 1929 Zur Quantenelcktrodynamik der Wellenfelder. Zeit. Physik 56 1-61 (Engl. Transl. from Gorelik G 1992 The first steps of quantum gravity and the Planck values Einstein Studies vol 3.  Studies in the History of General Relativity ed J Eisensaedt and A J Kox (Boston: Birkhauser, p 370)

[331] Rosenfeld L 1930 Zur Quantelung der Wellenfelder Ann. Phys., 5 113-152; 1932 La theorie quantique des champs Institut Henri Poincare Ann. 225-91

[332] Bronsteyn M P 1936 Quantentheories schwacher Gravitationsfelder Phys. Z. Sowjetunion 9 140-157; 1936 Kvantovanie gravitatsionnykh voln Zh. Eksp. Teor. Fiz. 6 140-57

    For an account of Bronsteyn's work, see:

    Gorelik G and Frenkel V J 1985 M P Bronsteyn i kvantovaya teoriya gravitatsii. in Eynshteynovskiy Sbornik 1980-1981 (Moscow: Nauka) p 293

    Gorelik G 1992 The first steps of quantum gravity and the Planck values

[333] Bronsteyn M P 1936 Kvantovanie gravitatsionnykh voln Zh. Eksp. Teor. Fiz. 6 140-57 (Engl. Transl. from Gorelik G 1992 The first steps of quantum gravity and the Planck values, p 377)

 

 

 

 

 John Stachel. The early history of quantum gravity" in Black Holes, Gravitational radiation and the Universe, B. R. Iyer and B. Bhawai (eds.) (Kluwer Academic Publisher, Netherlands 1999). P 528-32

 

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In the late thirties another Pauli coworker, Markus Fierz, developed a formalism to handle free quantum fields of arbitrary spin and mass [12], soon generalized by Pauli and Fierz to include electromagnetic interactions [13]. They noted that the massless spin-two field obeys equations formally identical with the linearized Einstein equations. Pauli prepared a "Report on the general properties of elementary particles" for the projected 1939 Solvay Congress, which was never held due to the outbreak of World War II. At the urging of Heisenberg [14], Pauli's report included a section entitled "Remarks on Gravitational Waves and Gravitational Quanta (Spin 2)." Although the complete report was only published recently [15], most of its contents had already been published elsewhere. The section on gravitation was not, but Heisenberg - and no doubt many another leading physicist - was aware of its contents around 1939 [16].

In this section, Pauli develops a Lagrangian for the massless spin-two field formally identical with that for the linearized Einstein equations. He notes that the invariance of the theory under a group of what (by analogy with the gauge transformations of Maxwell's theory) he calls "gauge transformations of the second kind" corresponds to the invariance under infinitesimal coordinate transformations of the linearized Einstein theory. He quantizes the theory by a method analogous to Fermi's method for electrodynamics, i.e., by requiring the operators corresponding to the linearised harmonic coordinate conditions to annihilate the wave function. This method allows imposition of commutation relations on all components of the linearized metric using the invariant D-function, with which the harmonic constraints are compatible. Pauli concludes:

The gravitational quanta or "gravitons" so defined have the...spin 2 ...It is certainly a limitation of the quantum-mechanical side of this treatment, that one leaves it at that approximation, in which the general-relativistic field equations are linear. This limitation is most closely connected with the well-known divergence difficulties of field theory (p.901).

Apart from signaling these divergence difficulties, common to all quantum field theories at that time, Pauli, like Rosenfeld before him, seemed satisfied with linearized quantum gravity. At any rate, he did not offer any indication that quantizing the gravitational field might confront unique physical problems.

But in 1935, well before Pauli's work and apparently unknown to him, a young physicist working at the Leningrad Physico-Technical Institute, Matvei Petrovich Bronstein, had already applied Fermi's quantization technique to the linearized Einstein equations. If his technical approach was similar to Pauli's, Bronstein drew much more skeptical conclusions about the physical significance of the resulting formalism.

Bronstein was no newcomer to the problem of quantum gravity in 1935. He had been pondering the issue since at least 1930, when he opined: "It is a task of the nearest future to identify the ties between quantum mechanics and the theory of gravitation" (cited from Gorelik and Frenkel 1994, p. 88). In 1931, he summed up an article on unified field theories by stating: "It seems that Einstein's program of unified field theory will prove impossible... what will be needed is a sort of marriage between relativity and quantum theory" (ibid.). By 1933 he was stressing the crucial significance of the three dimensional constants, c, G, and h, in defining the past and future development of physics (Bronstein 1933):

After relativistic quantum theory is formulated, the next task would be to realize the third step .... namely, to merge quantum theory (h constant) with special relativity (c constant) and the theory of gravitation (G constant) into a single whole (cited from Gorelik and Frenkel 1994, p.90) [17].

In 1935 Bronstein presented his Doctoral Thesis on "Quantizing Gravitational Waves" to a committee that included Vladimir Fock and Igor Tamm [18]. In 1936, the work was published in Russian [19], as well as in a condensed German version [20]. After carrying out the quantization of the linear theory by the Fermi method, developing the quantum analogue of Einstein's quadrupole radiation formula, and deducing the Newtonian law of attraction from the interchange of longitudinal gravitational quanta, Bronstein proceeds to some critical reflections on the physical significance of his results.

He carries out an analysis of the measurability of the (linearized) Christoffel symbols, which he takes to be the components of the gravitational field. By analogy with the then-recent Bohr-Rosenfeld analysis of the measurability of the electromagnetic field components, he shows that there are limitations on the measurability of the gravitational field components implied by the uncertainly relations between position and momentum of a test body, the acceleration of which is used to measure the gravitational field. But he notes that there is an additional gravitational complication, which has no electromagnetic analogue: To measure the components of the electromagnetic field, it is permissible to introduce electrically neutral test bodies, which have no effect on the field being measured. But in the gravitational case, due to the universality of gravitational interactions, the effect of the energy-momentum of the test bodies on the gravitational field cannot be neglected - even in linear approximation. Bronstein derives an expression for the minimum uncertainty in a measurement of a component of the Christoffel symbols that depends inversely on the mass density of the test body, just as Bohr-Rosenfeld's corresponding result does on the charge density r of the test body. He then states what he sees as the crucial difference between the two cases:

Here we should take into account a circumstance that reveals the fundamental distinction between quantum electrodynamics and the quantum theory of the gravitational field. Formal quantum electrodynamics that ignores the structure of the elementary charge does not, in principle, limit the density of r. When it is large enough we can measure the electric field's components with arbitrary precision. In nature, there are probably limits to the density of electric charge ... but formal quantum electrodynamics does not take these into account... The quantum theory of gravitation represents a quite different case: it has to take into account the fact that the gravitational radius of the test body (krV) must be less than its linear dimensions krV < V1/3 (Bronstein 1936b, p.217, transl. from Gorelik and Frenkel 1994, p.105 and Gorelik 1992, pp.376-377).

He acknowledges that "this was a rough approximation, because with the measuring device's relatively large mass departures from the superposition principle [i.e., the linearized approximation] will probably be noticeable;" but thinks "a similar result will survive in a more exact theory since it does not follow from the superposition principle. It merely corresponds to the fact that in General Relativity there cannot be bodies of arbitrarily large mass with the given volume" (Gorelik and Frenkel, 1994, pp. 105-106). He concludes:

The elimination of the logical inconsistencies connected with this result 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 rejection of our ordinary concepts of space and time, replacing them by some much deeper and nonevident concepts. Wer's nicht glaubt, bezahlt einen Taler (Bronstein 1936b, transl. from Gorelik 1992, p.377) [21].

At least one physicist outside the Soviet Union acknowledged, and indeed extended, Bronstein's views. In 1938 the French physicist Jacques Solomon after summarizing Bronstein's argument concluded [22]:

In the case when the gravitational field is not weak, the very method of quantization based on the superposition principle fails, so that it is no longer possible to apply a relation such as [the equation setting a lower limit on the measurability of the linearized field strength] in an unambiguous way ... Such considerations are of a sort to put seriously in doubt the possibility of reconciling the present formalism of field quantization with the non-linear theory of gravitation (p.484).

In one of the many tragic ironies of history, both of these pre-war advocates of the need for a radically different approach to quantum gravity perished prematurely. On August 6,1937, Bronstein was arrested by the Soviet State Security agency (NKVD); only twenty years later did his widow, the writer Lydia Chukovskaya, learn the exact date of his death: February 18, 1938 [23]. References to his name disappeared from the annals of Soviet physics for several decades. Jacques Solomon, a Communist militant active in the underground resistance to the German occupation of France, was arrested together with his wife, Helene Langevin, in March 1942. He was killed by the Germans on May 23, 1942; she was sent to Auschwitz, but survived the war [24]. Between them, Stalin and Hitler saw to it that the post-World-War-II discussion of quantum gravity took place without what could have been two significant voices.

Bronstein was a student of Yakov Frenkel, who was quite skeptical of the whole project of quantizing the gravitational field equations. In an article prepared for the Schilpp volume (see footnote 4), he argued against the analogy between the gravitational and electromagnetic fields that is the basis of the graviton approach. Since this article is unavailable to me, I shall quote a summary that includes citations from Gorelik and Frenkel 1994:

[Frenkel] argued that "the electromagnetic field [was] matter while the gravitational field merely [determined] the metrical properties of the spacetime continuum," He insisted that "strictly speaking there [were] no such things as gravitational energy or gravitational momentum since the corresponding values [did] not form a true tensor and [were] nothing more than a pseudotensor." It was his conviction that the attempts to quantize gravitation were senseless since "the gravitational field [had] macroscopic, rather than microscopic meaning; it merely [supplied] a framework for physical events occurring in space and time while quantizing [could] be applied only to microscopic processes in material fields." These considerations were little affected by the developments in physics that occurred after November 1935 - this and his remark during Bronstein's defense ... allows us to surmise that his position was the same in 1935 (p.85).

Another physicist who expressed doubts during the pre-war period about the need to quantize general relativity was van Dantzig. He considered general relativity to be a sort of thermodynamic limit of a deeper, underlying theory of interactions between particles [25].

Thus, by the mid-1930's the three positions that were to characterize post-World War II discussions of quantum gravity (among physicists not still wedded to Einstein's unified field theory program) had already been enunciated:

1) Quantum gravity should be formulated by analogy with quantum electrodynamics. In particular, one should start from quantization of the linearized gravitational field equations. Technical problems that arise will be similar to those arising in quantum electrodynamics and will presumably be solved pari passu with the problems of the latter (Rosenfeld, Pauli, Fierz) [26].

2) The unique features of gravitation will require special treatment. In particular, the full theory, with its non-linear field equations must be quantized. This implies that, while the general techniques of quantum field theory may be relevant, they must be generalized in such a way as to be applicable in the absence of a background metric (Bronstein, Solomon).

3) General relativity is essentially a macroscopic theory, to which the techniques of quantum field theory should not be (mis)applied (Frenkel, van Dantzig). It is sobering to realize how little real progress has been made on the problem, of quantum gravity since these alternatives were posed sixty years ago, particularly when one recalls that sixty years is the time span that separates Maxwell's treatise from the mid-thirties!

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References

1.    Albert Einstein, "Näherungsweise Integration der Feldgleichungen der Gravitation," Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1916): 688-696, translated as "Approximative Integration of the Field Equations of Gravitation," in The Collected Papers of Albert Einstein, vol. 6, The Berlin Years: Writings 1914-1917, English Translation of Selected Texts (Princeton University Press, 1997), Alfred Engel, transl., pp,201-210.

2.    Albert Einstein, "Über Gravitationswellen," Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1918):154-167.

3.    Albert Einstein, "Spielen Gravitationsfelder im Aufbau der materiellen Elementarteichen eine wesentliche Rolle?," Preussische Akademie der Wissenschaften (Berlin). Sitzungberichte (1919): 349-356, translated as "Do Gravitational Fields Play an Essential Part in the Structure of the Elementary Particles of Matter?" in The Principle of Relativity, Otto Blumenthal, ed., W. Ferret and J. B. Jeffery, transit (Methuen, London, 1923), reprint 1952 (Dover, New York), pp. 191-198.

4.    There is an intriguing comment by Y. I. Frenkel, in a paper written for the Schilpp volume Albert Einstein: Philosopher- Scientist, but not submitted: "Einstein was probably the first to assimilate gravitational waves and the corresponding particles in a conversation with the author back in 1925" (quotation from Gennady E. Gorelik and Victor Y. Frenkel, "Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties", Birkhauser, Cambridge, MA, 1994, p,85, cited hereafter).

5.    Oskar Klein, "Zur fünfdimensionalen Darstellung der Relativittstheorie," Zeitschrift fur Physik 46 (1927): 188. Klein was working with Bohr in Copenhagen at this time, and his comments may well reflect Bohr's views.

6.    Werner Heisenberg and Wolfgang Pauli, "Zur Quantenelcktrodynamik der Wellenfelder," Zeitschrift fur Physik 56 (1929): 1-61.

7.    Leon Rosenfeld, "Zur Quantelung der Wellenfelder," Annalen der Physik 5 (1930): 1113-152; "Uber die Gravitationswirkungen des Lichtes," Zeitschrift fur Physik 65 (1930): 589-599.

8.    In response to a query from Pauli (see Pauli to Rosenfeld, 12 April 1931, in Wolfgang Pauli, "Scientific correspondence with Bohr, Einstein, Heisenberg and others", vol.3, 1940-1949, Karl von Meyenn, ed., Springer Verlag, New York, 1993, p.746) Rosenfeld added a supplement to his paper showing that the gravitational self-energy of any one-photon state is infinite. Solomon soon showed that this divergence was not due to the zero-point energy of the field. See Jacques Solomon, "Nullpunktsenergie der Strahlung und Quantentheorie der Gravitation," Zeitschrift für Physik 71 (1931): 162-170.

9.    Rosenfeld himself later came to question the necessity of quantizing the gravitational field, and did not include his work on this topic in a collection of papers that he selected; instead he reprinted critical comments on field quantization, including arguments against the need for a quantum gravity. See Leon Rosenfeld, "On Quantization of Fields," in Selected Papers of Leon Rosenfeld, Robert S. Cohen and John Stachel, eds. (Reidel, Dordrecht/Boston/London, 1979) (hereafter Rosenfeld 1979), pp.442-445, and "Quantum Theory and Gravitation," ibid., pp.598-608.

10.    See Wolfgang Pauli, Wissenschaftlicher Briefwechsel, vol. 2, 1930-1939, Karl von Meyenn, ed. (Springer Verlag, Berlin/Heidelberg/New York/Tokyo, 1985), Bohr to Pauli, 15 March 1934, p.308: "The idea was that the neutrino, for which one assumes a zero rest mass, could hardly be anything else than a gravitational wave with appropriate quantization" (transl. from Niels Bohr, Collected Works, vol. 7, Foundations of Quantum Physics II (1933-1958), J. Kalcar, ed. (Elsevier, Amsterdam/Lausanne/New York/Oxford/Shannon/Tokyo, 1996), p.479). Fermi had evidently had a similar idea, but was aware of the problem of the different spins. See ibid., Pauli to Heisenberg 6 February 1934, p.277: "Fermi would prefer to make a connection between neutrinos and ha!f gravitational quanta." As late as November 1934, Pauli cautiously stated: "White up to now it has been held almost certain that gravitational phenomena play practically no role in nuclear physics, it now seems that the possibility cannot be immediately rejected, that the phenomena of bet eradiation could be connected with the square root of kappa [the gravitational constant] ("Raum, Zeit, und Kausalität in der modernen Physik," Scientia 59 (1936): 65-76, p.76). This suggests that Pauli may have had in mind construction of a graviton from two neutrinos, along the lines of DeBroglie's neutrino theory of light.

11,    Dmitri Ivanovich Blokhmtsev and F, M. Gal'per in, "Gipoteza neitrino i zakon sokhraneniya energii," Pod znamenem marxisma (1934), no. 6, pp,147-157. As cited by Gorelik and Frenkel, they wrote: "The comparison displayed above indicates that the graviton and the neutrino have much in common, This probably testifies that in general the highly improbable process of graviton radiation becomes practically observable in beta-decay. If the neutrino turns out to be the graviton this would mean that contemporary physics had approached the limits beyond which there would be no present insurmountable barrier between gravitation and electromagnetism. Due to theoretical considerations it is hard to identify gravitons with the neutrino since it is hard to admit that they have the same spin 1/2 as the neutrino. In this respect gravitons have much more in common with light quanta. It is impossible, however, to totally rule out a theoretical possibility of their identification. So far it is much more correct to regard the neutrino as an independent type of particle" (Gorelik and Frenkel 1994, p.97).

12.    Marcus Fierz, "Über die relativistische Theorie kraftefreier Teilchen mit beliebigem Spin," Helvetica Physica Acta 12 (1939): 3-37.

13.    Wolfgang Pauli and Marcus Fierz, "Uber relativistische Wellengleichungen von Teilchen mit beliebigem Spin im elektromagnetischen Peld," Helvetica Physica Acta 12 (1939): 207-300; ibid., "On relativistic wave equations for particles of arbitrary spin in an electromagnetic field," Royal Society (London). Proceedings A173 (1939): 211-232.

14.    See Pauli to Heisenberg, 10 June 1939, in Wolfgang Pauli, Wissenschaftliche Briefwechsel, vol 2, 1930-1939, Karl von Meyenn, ed. (Springer, Berlin/Heidelberg/New York/Tokyo, 1985), p.662; and Heisenberg to Pauli, 12 June 1939, ibid., p.665.

15.    See Wolfgang Pauli, Wissenschaftliche Briefwechsel, vol 2, 1930-1939, Karl von Meyenn, ed. (Springer, Berlin/Heidelberg/New York/Tokyo, 1985), pp.833-901; the section on gravitation is on pp.897-901.

16.    See, for example, Pauli to Schrodinger, 5 November 1939, ibid., p.823-825.

17.    In 1934 Pauli also discussed "the three fundamental natural constants," but added: "for the sake of simplicity we ignore gravitational phenomena for the present" (the article was not published until 1936 in Scientia; see the reference in note in).

18.    For Bronstein's life and work, see Gorelik and Frenkel 1994.

19.    Matvei Petrovich Bronstein, "Kvantovanie gravitatsionnykh voln [Quantization of gravitational waves]," Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki 6 (1936): 195-236.

20.    Matvei Petrovich Bronstein, "Quantentheorie schwacher Gravitationsfelder," Physikalische Zeitschrift der Sowjetunion 9 (1936); 140-157 (hereafter Bronstein, 1936b).

21.    The German phrase - "Let him who doubts it pay a Thaler"- comes from the Grimm brother's tale, "Der tapfere Schneider."

22.    Jacques Solomon, "Gravitation et Quanta," Journal de Physique et de Radium 9 (1938): 479-485.

23.    See Gorelik and Frenkel 1994, pp. 144-147; and Lydia Chukovskaya, The Akhmatova Journals/ Volume I 1938-41 (Farrar, Strauss and Giroux, New York 1994).

24.    See Leon Rosenfeld, "Jacques Solomon," in Rosenfeld 1979, pp.297-301; Martha Cecilia Bustamente, "Jacques Solomon (1908-1942): Profil d'un physicien théoricien dans la France des années trente," Revue d'histoire des sciences 50, 49-87, 1997.

25.    See D. van Dantzig, "Some possibilities of the future development of the notions of space and time," Etkenntnis 7 (1938): 142-146; "On the Relation Between Geometry and Physics and the Concept of Space-time," in Fünfzig Jahre Relativitätstheorie, P. KervaJrc, ed,, Helvetica Physica Acta Supplementum IV, (Birkhauser, Basel 1956), pp.48-53.

26.    Indeed, in 1949 Bryce DeWitt, using Schwinger's covariant technique, recalculated the gravitational self-energy of the photon and showed that it vanished identically. See Carl Bryce Seligman [DeWitt], "I. The Theory of Gravitational Interactions. II. The Interactions of Gravity With Light," Ph.D. Thesis, Harvard University, December 1949. Note that DeWitt emphasized the need to quantize the full, nonlinear theory, and never regarded quantization of the linearized equations as more than a preliminary exercise.

 

 

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