SPECIAL SECTION: 100 YEARS OF GENERAL RELATIVITY CURRENT SCIENCE, VOL. 109, NO. 12, 25 DECEMBER 2015 2243 *e-mail: ashtekar@gravity.psu.edu Quantum general relativity Abhay Ashtekar* Institute for Gravitation and the Cosmos and Physics Department, Penn State, University Park, PA 16802, USA After a brief historical introduction on quantum gra- vity as a whole, the current status of loop quantum gravity is discussed. Because of space limitation, I could only illustrate recent advances through one example – cosmology of the very early universe – and provide references for results in other main areas. Keywords: General relativity, loop quantum gravity, quantum gravity. Introduction EINSTEIN’S reservations on foundations of quantum mechanics are well-known. However, being a founding father of that subject, he was well-aware of the limitation of classical theories and emphasized 1 , already in 1916, that quantum theory would have to modify not only Max- wellian electrodynamics but also general relativity. Three decades later he was even more explicit saying, in the context of cosmology 1 ‘One may not assume the validity of field equations at very high density of field and matter and one may not conclude that the beginning of the expansion should be a singularity in the mathematical sense.’ By now, we know that classical physics cannot always be trusted even in the astronomical world because quantum phenomena are not limited just to tiny, microscopic sys- tems. For example, neutron stars owe their very existence to a quintessentially quantum effect: the Fermi degeneracy pressure. At the nuclear density of ~ 10 15 g/cm 3 encoun- tered in neutron stars, this pressure becomes strong enough to counterbalance the mighty gravitational pull and halt the collapse. The Planck density is some eighty orders of magnitude higher! Astonishing as the reach of GR is, it cannot be stretched into the Planck regime; here one needs a grander theory that unifies the principles un- derlying both general relativity and quantum physics. Early developments Serious attempts at constructing such a theory date back to the 1930s with papers on the quantization of the lin- earized gravitational field by Rosenfeld 2 and Bronstein 3 . Bronstein’s papers are particularly prescient in that he gave a formulation in terms of the electric and magnetic parts of the Weyl tensor and his equations have been periodically rediscovered all the way to 2002 (ref. 4)! Analysis of interactions between gravitons began only in the 1960s when Feynman extended his calculational tools from QED to general relativity 5 . Soon after, DeWitt com- pleted this analysis by systematically formulating the Feynman rules for calculating the scattering amplitudes among gravitons and between gravitons and matter quanta. He showed that the theory is unitary order by order in perturbation theory (for summary, see, e.g. ref. 6). In 1974, ‘t Hooft and Veltman 7 used elegant symmetry arguments to show that pure general relativity is renor- malizable to 1 loop but they also found that this feature is destroyed when gravity is coupled to even a single scalar field. For pure gravity, there was a potential divergence at two loops because of a counter term that is cubic in the Riemann tensor. However there was no general argument to say that its coefficient is necessarily non-zero. A heroic calculation by Goroff and Sagnotti 8 settled this issue by showing that the coefficient is (209/2880(4) 2 )! Thus in perturbation theory off Minkowski space, pure gravity fails to be renormalizable at 2 loops, and when coupled to a scalar field, already at 1-loop. The question then arose whether one should modify Einstein gravity at short distances and/or add astutely chosen matter which would improve its ultra-violet be- haviour. The first avenue led to higher derivative theories. Stelle, Tomboulis and others showed that such a theory can be not only renormalizable but asymptotically free 9 . But it soon turned out that the theory fails to be unitary and its Hamiltonian is unbounded below. The discovery of supersymmetry suggested another avenue: with a suit- able combination of fermions and bosons, perturbative in- finities in the bosonic sector could be cancelled by those in the fermionic sector, improving the ultraviolet behav- iour. This hope was shown to be realized to 2 loops by a number of authors 10 . However, by the late 1980s a consensus emerged that all supergravity theories would diverge by 3 loops and are therefore not viable (see, e.g. ref. 11; note 1). A series of parallel developments was sparked in the canonical approach by Dirac’s analysis of constrained Hamiltonian systems. In the 1960s, this framework was applied to general relativity by Dirac, Bergmann, Ar- nowitt, Deser, Misner and others 12–16 . The basic canonical variable was the 3-metric on a spatial slice and general relativity could be interpreted as a dynamical theory of