# Quantum Field Theory

## 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.