by Seth K. Asante, Bianca Dittrich, and Hal M. Haggard
Fifty years ago this December the astronauts of the Apollo 8 mission were the first humans to ever see the far side of the moon. As they passed behind the moon they lost radio contact with mission control in Houston. They were completely isolated. Only recently have cockpit recordings of their reactions become public [1]. At first they couldn’t see the moon at all, but then the command module pilot James A. Lovell Jr. exclaims “Hey, I got the moon!”. William A. Anders, the lunar module pilot, asks excitedly “Is it below us?” and Lovell begins “Yes, and it’s—” when Anders interrupts him having spotted it. Deeply enthused the astronauts have dropped their technical patter and systems checks, which make up the main fabric of the recordings. Anders marvels “I have trouble telling the bumps from the holes.” In his excitement Anders completely loses his technical jargon. He can’t even recall the word ‘crater’. He is reacting to the moon. It is easy to feel his enthusiasm at this hidden wonder.
Hal Haggard, Seth Asante, and Bianca Dittrich form a triangle area, the main variable in their new study of discrete gravity [5]. If you squint the image is even a bit like The Dark Side of the Moon’s cover art. The picture is taken in front of artwork by Elizabeth McIntosh hanging in the main atrium of the Perimeter Institute.
The craters of the moon are named for famous scientists, philosophers, mathematicians and explorers. On the near side there are the craters Galileo, Grimaldi, Aristarchus, Plato, Copernicus, Tycho, Kepler and many others. But, hidden on the far side are two craters named for Amalie Emmy Noether and Giulio Racah. The work of these two great scientists has surprising connections to quantum gravity.
Indeed, this year marks both the golden jubilee (50 years) of the first model of quantum gravity [2], for which Racah unwittingly laid the foundations, and the centennial of the extraordinary paper “Invariant Variation Problems” by Noether [3]. Noether’s remarkable achievement was to connect the notion of symmetry to the evolution of physical systems, both when their dynamics is completely determined by the variables of the system and when there is a gauge symmetry that constrains the dynamics.
Although the advent of general relativity was what drew Noether into this circle of ideas, it is still in the context of gravity where our understanding is murkiest; for it seems that the problem of quantizing gravity is centrally one of recognising what are the appropriate variables in which to cast the theory and its symmetries and to invent a quantization procedure that interfaces smoothly with curved spacetime and its invariance under changes of coordinates, so-called diffeomorphism symmetry. Indeed, investigation of an alternative set of area variables for discrete gravity is the main theme of our paper [5]. Racah also played a surprising role in one formulation of this problem.
Fifty years ago, in 1968 G. Ponzano and Tullio Regge studied the small ℏ limit of the Racah coefficients (also known as 6j-symbols, these are more symmetric versions of Clebsch-Gordan coefficients). They found that these coefficients, usually associated with atomic physics and the recoupling of angular momenta, perfectly coded the geometry of a tetrahedron. Not only did the edge lengths and angles between the faces of the tetrahedron emerge in this limit, but the prefactor in their formula is expressed in terms of the tetrahedron’s volume. This would have been quite remarkable enough, but Ponzano and Regge recognised in their result the potential for a quantum theory of gravity.
Seven years earlier Regge had constructed a classical, discrete version of Einstein’s theory of gravity [4]. The idea was to cut spacetime up into small building blocks; in three-dimensions he used tetrahedra because they were the polyhedra with the smallest number of faces. Each tetrahedral piece had a flat interior, but by gluing them together appropriately he built extended regions that carried curvature. In essence, this was the first example of a lattice gauge theory, the usual symmetries of flat spacetime applied to each tetrahedral grain, but non-trivial effects, in this case curvature, could be encoded in paths that visited multiple tetrahedra.
The semiclassical phase that emerged in Ponzano and Regge’s analysis of the Racah symbol was precisely the exponential of the action of a single tetrahedron in Regge’s treatment of discretized general relativity. They immediately recognised that if a large number of Racah symbols were multiplied and summed that this would give a quantum mechanical description of a discretized spacetime path integral. The volume factor that they found was precisely what was needed to make the theory diffeomorphism invariant. This was the first proposal for a concrete quantum theory of gravity and it has survived to this day. Tullio Regge was so amazed by the richness of the Racah symbols and so appreciative of Giulio Racah’s work that he put in the application for a lunar crater to be named for Racah.
Ponzano and Regge’s work and its developments provide a full theory of three-dimensional quantum spacetimes, but generalisation to the four dimensions of space and time that we live in is difficult, with several aspects that are still open. Two principal challenges in extending these ideas to higher dimensions are in identifying the appropriate variables and in implementing the symmetries of general relativity. The variables of classical Regge gravity are the edge lengths of the polyhedra, or more generally, higher-dimensional simplicial building blocks. There is no clear principle that identifies these as the key variables and our recent work [5], which you can check out here, takes seriously triangle areas as the fundamental degrees of freedom of the discrete theory and investigates the interplay of these variables with diffeomorphism symmetry. We look forward to many more celebrations of the rich project that is quantum gravity!
Seth K. Asante is a graduate student at the Perimeter Institute for Theoretical Physics and the University of Waterloo.
Bianca Dittrich is Faculty at the Perimeter Institute for Theoretical Physics.
Hal M. Haggard is a Professor in the Physics Program at Bard College.
[1] J. Kluger, Apollo 8: The Thrilling Story of the First Mission to the Moon. New York: Henry Holt & Co. (2017)
[2] G. Ponzano and T. Regge, “Semiclassical limit of Racah coefficients”. In Bloch, F. Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publ. Co. (1968) 1.
[3] A.E. Noether, “Invariante Variationsprobleme,”Nachr. Akad. Wiss. Gottinqen, Math.-Phys. Kl. l (1918) 235.
For an English translation of the original article see:
A. Tavel , “Invariant Variation Problems,” Transport Theory of Statis. Physics I (1971) 186.
[4] T. Regge, “General Relativity Without Coordinates,” Nuovo Cim. 19 (1961) 558.
[5] S. K. Asante, B. Dittrich, and H. M. Haggard, “The degrees of freedom of area Regge calculus: dynamics, non-metricity, and broken diffeomorphisms,” Class. and Quant. Gravit. 35 (2018) 135009
This work is licensed under a Creative Commons Attribution 3.0 Unported License.
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