Quantum Gravity

Ashtekar/Bianchi, in their 2021 paper A Short Review of Loop Quantum Gravity, clearly articulate recent progress within this leading theory of quantum gravity. Its thesis statement:

An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes – near the big bang and inside black holes– where Einstein’s smooth continuum breaks down.

It goes on to explain the difficulties, the need for introducing

a new syntax to describe all of classical physics, that of Riemannian geometry. Thus, spacetime is represented as a 4-dimensional manifold M equipped with a (pseudo-) Riemannian metric g_ab, and matter is represented by tensor fields.

another newer syntax –that of a quantum Riemannian geometry– where one only has a probability amplitude for various spacetime geometries in place of a single metric. Creation of this syntax was truly challenging because all of twentieth physics presupposes a classical spacetime with a metric, its sharp light cones, precise geodesics and proper time assigned to clocks. How do we do physics if we do not have a specific spacetime continuum in the background to anchor the habitual notions we use?

Rovelli published an open article on LQG two decades ago.

Also available is Rovelli’s 2000 talk at the Grossman Conference in Rome (from A history of LQG developments, published in 2008, further documenting the headwinds and achievements of 20th century LQG development.)

An evident peculiarity of the research in quantum gravity is that all along its development it can be separated into three main lines of research. The relative weight of these lines has changed, there have been important intersections and connections between the three, and there has been research that does not fit into any of the three lines. Nevertheless, the three lines have maintained a distinct individuality across 70 years of research. The three main lines are often denoted “covariant”, “canonical”, and “sum over histories”, even if these names can be misleading and are often confused. They cannot be characterized by a precise definition, but within each line there is a remarkable methodological unity, and a remarkable consistency in the logic of the development of the research.

The covariant line of research is the attempt to build the theory as a quantum field theory of the fluctuations of the metric over a flat Minkowski space, or some other background metric space. The program was started by Rosenfeld, Fierz and Pauli in the thirties. The Feynman rules of general relativity (GR, from now on) were laboriously found by DeWitt and Feynman in the sixties. t’Hooft and Veltman, Deser and Van Nieuwen-huizen, and others, found firm evidence of non-renormalizability at the beginning of the seventies. Then, a search for an extension of GR giving a renormalizable or finite perturbation expansion started. Through high derivative theory and supergravity, the search converged successfully to string theory in the late eighties.

The canonical line of research is the attempt to construct a quantum theory in which the Hilbert space carries a representation of the operators corresponding to the full metric, or some functions of the metric, without background metric to be fixed. The program was set by Bergmann and Dirac in the fifties. Unraveling the canonical structure of GR turned out to be laborious. Bergmann and his group, Dirac, Peres, Arnowitz,  Deser, and Misner completed the task in the late fifties and early sixties. The formal equations of the quantum theory were then written down by Wheeler and DeWitt in the middle sixties, but turned out to be too ill-defined. A well defined version of the same equations was successfully found only in the late eighties, with loop quantum gravity.

The sum over histories line of research is the attempt to use some version of Feynman’s functional integral quantization to define the theory. Hawking’s Euclidean quantum gravity, introduced in the seventies, most of the the discrete (lattice-like, posets . . . ) approaches, and the spin foam models, recently introduced, belong to this line.

There are of course other ideas that have been explored. So far, however, none of these alternatives has been developed into a largescale research program:

• Twistor theory has been more fruitful on the mathematical side than on the strictly physical side, but it is still actively developing.
• Noncommutative geometry has been proposed as a key mathematical tool for describing Planck scale geometry, and has recently obtained very surprising results, particularly with the work of Connes and collaborators.
• Finkelstein, Sorkin, and others, pursue courageous and intriguing independent paths.
• Penrose idea of a gravity induced quantum state reduction have recently found new life with the perspective of a possible experimental test.
• …

He follows with a valuable history of ideas over the prior half century and summarizes:

The lines of research that I have summarized have found many points of contact in the course of their development and have often intersected each other. For instance, there is a formal way of deriving a sum over over histories formulation from a canonical theory and viceversa; the perturbative expansion can also be obtained expanding the sum over histories; string theory today faces the problem of a finding its nonperturbative formulation, and thus the typical problems of a canonical theory, while loop quantum gravity has mutated into the spin foam models, a sum over history formulation, using techniques that can be traced to a development of string theory of the early nineties. Recently, Lee Smolin has been developing an attempt to connect non- perturbative string theory and loop quantum gravity. However, in spite of this continuous cross fertilization, the three main lines of development have kept their essential separation.

The paradigm shift when transitioning from classic to atomic to Planck scale dimensions presents  obstacles to arriving at a satisfactory quantum gravity theory. The first step in LQG was to systematically construct a specific theory of quantum Riemannian geometry from basic principles, a task completed in 1990s. The new syntax arose from two principal ideas:

• a reformulation of GR (with matter) in the language of gauge theory that successfully describes the other three basis forces of nature, but now without reference to any background field, not even a spacetime metric
• subsequent passage to quantum theory using non-perturbative techniques from gauge theories, again without reference to a background.

If a theory of spacetime has no background field, it has access only to an underlying manifold and therefore is covariant with respect to its diffeomorphisms, those transformations that preserve the manifold structure. Theorists then explain how diffeomorphism covariance, together with non-perturbative methods, naturally lead to a notional, in-built discreteness in geometry that can enable ultraviolet finiteness.

Einstein’s familiar spacetime continuum becomes emergent in two senses:

• it is built out of certain fields that feature naturally in gauge theories, without any reference to a spacetime metric
• it emerges only on coarse graining of the fundamental discrete structures, the ‘atoms of geometry’, of the quantum Riemannian framework.

The discretized elements of quantum spacetime are connected by a spin network in the sense of Penrose. Spinfoam is the name given the network of loop connections between ‘atoms’ of discrete spacetime. An effective theory of spin foam is being formulated which is computable on current computers; simulations (to date, tiny models) show hints that semi-classical scaling  may be demonstrable as models scale up. Spinfoam is where the canonical LQG  meets the sums-over-history lines of reasoning, a major unification of approaches within LQG.

Summarizing LQG ‘niceness’:

• One can recover Einstein GR from a natural, background independent gauge theory, which has the further advantage that it enormously simplifies the constraints as well as evolution equations.
• Since the Hamiltonian constraint is purely quadratic in momenta with no potential term, solutions to evolution equations have a natural geometrical interpretation as geodesics of the ‘supermetric’ on the (infinite dimensional) space of connections.
• One can extend the LQG theory to the fields that feature in the standard model (scalar, Dirac, Yang-Mills).
• From a gauge theory perspective, the Riemannian geometry that underlies GR becomes a secondary, emergent structure.

LQG is well enough established to attract a new generation of physicists to give it the attention it deserves. See String Theory Meets LQG.

The story of any discipline within physics is not complete without acknowledging the men and women who advanced the cause.  The following stars in the recent physics firmament approximate straight lines of development from relativistic to quantum nature of reality:

• General Relativity (GR – 1916 – Nobelists Lorentz/EINSTEIN/Penrose/Thorne; also Minkowski, Bergmann, Misner, Bondi, Newman et al.)
• Quantum Mechanics (QM – 1927 – Nobelists Planck/EINSTEIN/Bohr/Heisenberg/de Broglie/Pauli/Dirac/Schrödinger/Yang/Feynman/t’Hooft; also Jordan, Rosenfeld, Bohm, Bell, Wheeler, DeWitt et al.)
• Relational Quantum Lorentzian Spacetime (LQG – 1980 – Bergmann, Misner, Regge, Smolin, Rovelli, Ashtekar, Thiemann, Dittrich et al.)