Attempting to quantize geometry

Jan Ambjørn is professor of theoretical high energy physics at the Niels Bohr Institute, University of Copenhagen and at IMAPP, Radboud University.

Jan Ambjørn is professor of theoretical high energy physics at the Niels Bohr Institute, University of Copenhagen and at IMAPP, Radboud University.

The Standard Model of particle physics is a quantum theory. It is born quantum. The observations of the weak and the strong interactions were from the beginning linked to quantum phenomena. For gravity the situation is different. Because the gravitational coupling constant is so small compared to coupling constants in the Standard Model, any observations of quantum aspects of gravity have been ruled out so far. Here we will assume that gravity is a quantum theory. However, quantizing gravity has so far turned out to be difficult. That one might encounter difficulties is maybe not surprising if one recalls that classical gravity is the theory of the geometry of our spacetime. It defines distances between spacetime points. Ordinary quantum field theory tells us about correlations between fields separated by a well defined spacetime distance. Now we are asked to quantize the geometry defining these distances, so the meaning of spacetime distance etc. is not obvious in a quantum theory.

The use of so-called causal dynamical triangulations (CDT) to define a theory of quantum gravity draws upon the tradition of lattice field theory, which has been so successful in the study of the theory of the strong interactions. The lattice spacing a cuts off the high energy modes of the field and makes the quantum theory well defined. The “real” continuum theory is then obtained by letting the lattice spacing a → 0. We attempt to do something similar in the case of quantum gravity, except that the lattice now is our spacetime. The lattice thus has to be dynamic in order to accommodate the quantum fluctuations of spacetime. Questions one can ask are:

(1) is there an average geometry around which spacetime fluctuates?

(2) does this average geometry (which then plays the role of the classical geometry) resemble anything like our present universe?

(3) how large are the quantum fluctuations when the size of the universe becomes small?

Using the formalism of CDT this can be studied by computer simulations. We are thus able to study the quantum universe right after the big bang. The major question we have tried to answer in the present article is if this quantum lattice theory also exists as a genuine continuum quantum theory when we take the lattice spacing a → 0. Such a question is non-trivial even in the Standard Model. The lattice formulation is well suited to answer such a “non-perturbative” question. Our article provides the general setup needed for such an analysis in the context of quantum gravity.

Read the full article in Classical and Quantum Gravity:
Renormalization group flow in CDT
J Ambjørn, A Görlich J Jurkiewicz, A Kreienbuehl and R Loll
Class. Quantum Grav. 31 165003

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