Nonperturbative Quantum Gravity J. Ambjørn a, A. G¨orlich b, J. Jurkiewicz b and R. Loll c,d a The Niels Bohr Institute, Copenhagen University Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark. email: [email protected] b Institute of Physics, Jagellonian University, Reymonta 4, PL 30-059 Krakow, Poland. email: [email protected], [email protected] c Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands. email: [email protected] d Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, Canada N2L 2Y5. email: [email protected] Abstract Asymptotic safety describes a scenario in which general relativity can be quan- tized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultra- violet fixed point, for which evidence has been found using renormalization group equations in the continuum. “Causal Dynamical Triangulations” (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the “quantum geometries” which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Hoˇrava-Lifshitz gravitational models. 1 1 Introduction An unsolved problem in theoretical physics is how to reconcile the classical theory of general relativity with quantum mechanics. Consider the gravitational theory defined by the Einstein-Hilbert action, plus possible matter terms. Trying to quan- tize the fluctuations around a given solution to the classical equations of motion one discovers that the corresponding quantum field theory is perturbatively non- renormalizable. Part of the problem is that in spacetime dimension four the mass dimension of the gravitational coupling constant G is 2 in units where ~ = 1 and c = 1. As a result, conventional perturbative quantum− field theory is expected to be applicable only for energies E2 1/G. (1) ≪ Despite being a perfectly good assumption in all experimental situations we can imagine in the laboratory, this relation can be taken as an indication that something “new” has to happen at sufficiently large energies or, equivalently, short distances. If one believes in a fundamental quantum theory of gravity, one would usually read the breakdown of perturbation theory when (1) is no longer satisfied as signaling the appearance of new degrees of freedom as part of a different theory, which is valid at higher energies. A well-known example of this is the electroweak theory, which was described originally by a four-fermion interaction. The latter is not renormalizable and perturbation theory breaks down at sufficiently high energy, namely, when the energy E no longer satisfies (1), with the gravitational coupling constant G replaced by the coupling GF of the four-Fermi interaction, which also has mass dimension 2. The breakdown coincides with the appearance of new degrees of freedom, the W− - and Z-particles. At the same time, the four-Fermi interaction becomes just an approximation to the process where a fermion interacts via W and Z particles with other fermions. The corresponding electroweak theory is renormalizable. Similarly, in the 1960s a model for the scattering of low-energy pions was pro- posed, the so-called non-linear sigma model. It is again nonrenormalizable, with a coupling constant of mass dimension 2 when the model is formulated in four (one time and three space) dimensions. Also− in this case the model did not describe adequately the scattering data at high energy. Nowadays we understand that this happened because the pions cannot be viewed as elementary particles, but are made of quarks and anti-quarks. Again, the correct underlying theory of these quarks, anti-quarks and gluons is a renormalizable quantum field theory. 1.1 What to do about gravity? For the case of gravity there seems to be no simple way of extending it to a renor- malizable quantum field theory by either adding new fields, like in the electroweak theory, or by introducing new fields in terms of which the theory becomes renormal- izable, as in the case of the nonlinear sigma model. It may be possible that this can be done for gravity too, but so far we have not discovered how. 2 There have been alternative proposals which share some of the flavour of the above “resolutions” of nonrenormalizable theories. String theory is an example of a framework which tries to get around the problem of gravity as a nonrenormalizable quantum field theory by adding new degrees of freedom, albeit infinitely many. The ambition of string theory in the 1980s was that of a “theory of everything”, unifying gravity and all matter fields in a single theoretical framework. One problem with this is that one got much more than was asked for, including many unobserved particles, symmetries and spatial dimensions. Another problem is that it has never been entirely clear what kind of theory one is dealing with. Best understood is the perturbative expansion around flat ten-dimensional spacetime, but a genuinely nonperturbative definition of string theory is still missing. This renders the role of (a possibly emergent notion of) space, not to mention time, in string theory somewhat unclear. The world we observe today is not a simple, obvious consequence of the dynamics of string theory. With our present understanding of string theory one has to work hard to extract from it something which even vaguely resembles the world we can observe. The incompleteness of this understanding prevents us moreover from making any predictions for our universe. String theory clearly is a versatile and fascinating theoretical framework, but it is hard to tell whether it is the right one for describing the real world (including quantum gravity), or merely a modern incarnation of epicycles, with sufficiently many free (moduli) parameters to describe anything purely kinematically, but providing no insights into the dynamics which governs nature. Loop quantum gravity represents another bold attempt to circumvent the non- renormalizability of quantum gravity. It does so by adopting a nonstandard pro- cedure of quantization where the Hilbert space of states is nonseparable and the holonomies of connections, viewed as quantum objects, are finite. It is perhaps too early to tell whether this program will be successful in its attempts to quantize four-dimensional gravity, solve all UV problems and provide us with a semiclassical limit which coincides with Einstein gravity in the limit ~ 0. → 1.2 Searching for fixed points A much more mundane approach to quantum gravity, going back to S. Weinberg and known as “asymptotic safety” [1] is inspired by the Wilsonian renormalization group. The key idea is that while a perturbative expansion around a fixed background geometry leads to a nonrenormalizable theory, this merely reflects the infrared end of a renormalization group flow, which originates from a genuinely nonperturbative UV fixed point governing the short-distance physics of quantum gravity. Asymptotic safety refers to the assumption that such an ultraviolet fixed point exists and in its neighbourhood the co-dimension of the critical surface associated with it is finite. As a consequence one only has to adjust a finite number of coupling constants to reach the critical surface, where the physics is identical to that of the UV fixed point. In the abstract coupling-constant space where all possible interactions are allowed, 3 the couplings which need to be adjusted to reach the critical surface are called relevant couplings. In this sense the concept of asymptotic safety is a generalization of the concept of a renormalizable field theory. For renormalizable four-dimensional field theories the fixed point is Gaussian. This implies that the scaling dimension of the fields when approaching the fixed point is just the canonical dimension of the field as it appears in the classical Lagrangian. The above-mentioned finite co- dimension of the critical surface associated with such a Gaussian fixed point is equal to the number of independent polynomials one can form in terms of the fields and their derivatives, such that the coupling constants of the corresponding terms in an action have negative (canonical) mass dimension (in units where c = ~ = 1). For marginal couplings, where the canonical dimension is zero, one needs to calculate the corrections to the canonical dimensions near the fixed point to decide whether they are relevant or irrelevant (or stay marginal). The difficulty one faces in the asymptotic safety scenario is that the fixed point is not Gaussian and standard perturbation theory may tell us little or nothing about its existence, let alone the physics associated with the fixed point if it exists. Do we have reliable methods to check for the existence of a non-Gaussian fixed point? To start with, not too many non-Gaussian fixed points are known outside of two spacetime dimensions (which is very special) and involving only bosonic fields1, with one exception, the Wilson-Fisher fixed point of three-dimensional Euclidean scalar field theory. It plays an important role in nature since it governs the critical behaviour of many materials. It is nonperturbative, and its existence and associ- ated properties have been analyzed in the 4-ε expansion, the 2+ε expansion, by an “exact renormalization group” analysis and by other methods, leading to a general agreement on the values of its critical exponents.