Quantum Gravity

The research on Quantum Gravity touches on the following disciplines:

• General Relativity
• Quantum Field Theory
• Gauge Field Theory
• Cosmology
• High Energy and Astroparticle Physics
• Mathematical Physics

We will briefly sketch below what these disciplines are, how they connect to each other and why they are relevant for Quantum Gravity. See also the introductory text for more details. After that outline, we explain elements of the research currently conducted at the institute. That section just mentions the relevant physical and mathematical concepts. It is not meant as an introduction to these concepts but rather invites to further studies about them for the interested reader.

General Relativity

General Relativity (GR) is Einstein’s geometric theory of gravity. It is different than the other interactions because the gravitational field, the metric, is not a field that propagates on some space-time, it DEFINES the space-time. This makes the gravitational field self-interacting and interacting with matter in a much more complicated way than matter by itself does. For instance, the Lagrangian of the Standard Model is a fourth order polynomial in all matter fields. The Einstein-Hilbert Lagrangian is not even a polynomial. The interaction of geometry and matter is the content of Einstein’s field equations which relates the curvature of geometry to energy density of matter. Classical GR is our best theory of the gravitational interactions but it has its limitations: The celebrated Penrose-Hawking singularity theorems predict its own failure since the field equations become meaningless inside black holes and close to the big bang. Both curvature and energy density diverge, they become singular. This indicates that the theory has been pushed beyond the limits of validity and must be replaced by a more fundamental one.

Quantum Field Theory

Quantum Field Theory (QFT) is the mathematical framework that has been developed to describe the quantum theory of matter fields in interaction on a given space-time manifold together with a prescribed metric which can be curved. When applying the principles of QFT to GR one runs into a problem: QFT necessarily needs a classical metric in order to define a quantum field. However, if the metric itself is to be quantized this definition becomes inapplicable. Quantum Gravity is the attempt to resolve this problem. QFT on a given curved space-time should be an excellent approximation to Quantum Gravity when the quantum metric fluctuations are small and backreaction of matter on geometry can be neglected, that is, when the matter energy density is small. These conditions are violated close to the singularities and hence QFT must take GR into account. Generically, combining classical GR with QFT is problematic. An illustrative example is the Hawking effect. It states that a black hole radiates free particles (e.g. photons) with a black body frequency spectrum whose temperature is inversely proportional to the Schwarzschild radius of the black hole (which in turn is proportional to its mass). The Schwarzschild radius is the radius at which gravity becomes so strong that not even light can escape, it defines a sphere called the event horizon. The problem is now gravitational redshift: Whatever tiny frequency of a photon one measures far away from the black hole, it was huge when created close to the horizon, it could be comparable to the Planck scale where Quantum Gravity effects should be taken into account which however was not done in Hawking’s calculation.

Also QFT by itself on a given background metric, say Minkowski space-time, has its problems because to date in 4D only a perturbative description of interacting quantum matter is available, however, the individual terms in the perturbation series diverge and can be made finite only by subtracting the divergences in a procedure called renormalisation. The subtracted infinities in principle contribute to the cosmological constant and thus are problematic for Quantum Gravity. It is conceivable that they would disappear if one could define QFT non perturbatively as suggested by Haag’s theorem. This could also render the perturbation series finite which in the present form most probably diverges which means that one cannot trust the perturbative expansions to all orders. For some interactions like QCD a non perturbative formulation is actually not available due to the effect of confinement which means that QCD is strongly coupled at low energies and prevents the existence of free quarks and gluons.

Gauge Theory

All known interactions in nature are gauge theories. This just means that the theoretical description employs fields that are not observable. The best known example is Maxwell theory which uses four fields in the Lagrangian but of which two are redundant: In vacuum an electromagnetic wave has only two independent polarization degrees of freedom. Mathematically this is encoded into the Lagrangian through a symmetry which depends on arbitrary functions which can be given the structure of a group acting everywhere in space-time. The relevant group for QCD is SU(3), for the electroweak interaction it is SU(2)xU(1) and for GR on a space-time manifold M it is its group of diffeomorphisms. Gauge theory is therefore a unifying mathematical framework that encompasses all known interactions and presents a beautiful interface between geometry and physics, in particular differential geometry and the theory of fibre bundles.

Cosmology

Cosmology is the physics of the universe. As far as the geometric aspects are concerned it is a chapter in GR focussing on solutions to Einstein’s equations which are spatially homogeneous. The best known solution and the one most relevant for our own universe is the Friedmann-Robertson-Walker (FRW) solution which is not only homogeneous but also isotropic and roughly describes the universe on very large scales beyond the extensions of galaxies, clusters of galaxies and super clusters and correctly describes its expansion. But Cosmology is much more than that: Cosmologists nowadays can precisely measure the inhomogeneities and anisotropies in the universe and describe how they came into existence when using the machinery from classical GR and QFT. This is the theory of structure formation. There are many puzzles that we face today. Very recent measurements, among others by the satellites WMAP and PLANCK, which focus on the Cosmological Background Microwave Radiation (CMBR), show that the universe is filled with only 5% baryonic matter, 25% dark matter that clumps around galaxies and 70% of dark energy. The adjective “dark” just indicates that we do not what it is, we just know that it is there because both leave its trace through their gravitational interaction. Dark matter could be a yet unknown particle in the minimal supersymmetric extension of the standard model while dark energy could be a cosmological constant. Another puzzle is the horizon problem: If we assume that shortly after the big bang the dominant energy component was radiation which would be the case if there was no matter else than that contained in the standard model of elementary particle physics, then it is not easy to explain why the universe is so isotropic. This is because the visible part of universe was causally highly disconnected when it was created and even when the CMBR was created some 300.000 years after the big bang, not sufficient time had elapsed to explain why possible initial anisotropies should have evened out if no signal can travel faster than light. To solve the puzzle one can speculate on the existence of an exotic field that that has have long decayed. It is called the inflaton because it inflated the universe at an exponential rate, thus removing the causal (horizon) problem. Quantum Gravity could remove the horizon problem in a more drastic way: Since it should remove the big bang singularity, there should be a time before the big bang and thus the whole universe was always causally connected. Quantum Gravity also might add insight to the dark energy problem because the cosmological constant term would also be quantized. In general, cosmology is interesting for Quantum Gravity because the big bang singularity can in principle be seen indirectly by gravitational wave detectors of sufficient resolution while black hole singularities are hidden by the event horizon.

High Energy and Astroparticle Physics

The energy scale at which Quantum Gravity becomes non negligible is the Planck scale of 10¹⁹ GeV. This is the energy at which the Compton wavelength of a lump of energy such as an elementary particle equals its Schwarzschild radius which then would force the particle to become a black hole. This is 16 orders of magnitude away from the best man made microscope to date which is the Large Hadron Collider (LHC) at the CERN with an energy resolution of a few TeV. As such one would expect that one could safely ignore QG effects in all experiments in the forseeable future. While that is true for man made experiments it is wrong for processes that are relevant for astrophysics. Apart from the unknown physics that is going on close to the GR singularities as described above, ultra high energetic (UHE) cosmic rays are known to reach energies of the order of 10¹⁶ eV which is only 9 orders of magnitude away from the Planck scale. As we can probe the structure of space and time only with elementary particles, the above line of thought suggests that in fact it is meaningless to try to resolve energies beyond the Planck energy or distances beyond the Planck length of 10^-33 cm or times beyond the Planck time of 10^-43 s. This of course would have a tremendous conceptual impact on the foundations of QFT since then quantum space-time would resemble a discrete lattice rather than a continuum. Accordingly, research in Quantum Gravity is not at all of academic interest only and one can hope to find signatures in astroparticle physics. High energy physics, which uses the mathematical framework of QFT, most importantly scattering matrix theory and Feynman graphs, is of course also interesting in its own right for Quantum Gravity because every matter species couples gravitationally. Thus it makes a big difference whether or not the LHC discovers evidence for the Higgs particle and/or supersymmetry.

Mathematical Physics

Having mentioned the importance of cosmology and high energy astroparticle physics for experimental signatures of Quantum Gravity, another important guideline for the construction of a theory of Quantum Gravity is mathematical consistency. It turns out that it is highly non trivial to combine the principles of GR and QFT into one framework without running into contradictions. This has two aspects: First, mathematical techniques beyond those of QFT and GR need to be developed. Secondly, as there is no input from experiment so far there is no physical intuition about which aspects of the mathematical framework should be treated with care and which aspects can be handled in an intuitive way. All aspects must be analyzed rigorously. Accordingly, research in Quantum Gravity benefits greatly from interactions with several mathematical disciplines such as operator theory, functional analysis and measure theory. Quite in general methods from Mathematical Physics will play an important role in the development of Quantum Gravity.

The main research topics in Erlangen

• Dynamics of the Quantum Einstein Equations:
  How do quantum effects influence the physics at the Planck scale?
• Semiclassical aspects of Quantum Gravity:
  Does the classical limit of Quantum Gravity reproduce QFT (on curved spacetimes) and GR?
• Canonical and covariant approaches of Quantum Gravity:
  How are these two approaches related?
• Quantum cosmology:
  What are the cosmological consequences of a theory of quantum gravity?
• Quantum Gravity and black holes:
  How can Quantum Gravity explain the entropy of black holes and Hawking radiation?
• Representation theory for Quantum Gravity:
  Are there other possible representations for the quantum theory and what are their properties and physical consequences?