From quantum field theory to quantum gravity Klaus Fredenhagen II. Institut f¨urTheoretische Physik, Hamburg Klaus Fredenhagen From quantum field theory to quantum gravity Introduction Since about a century, the relation between quantum physics and gravity is not fully understood. Quantum physics: very successful in nonrelativistic physics where precise mathematical results can be compared with experiments, somewhat less successful in elementary particle physics where theory delivers only the first terms of a formal power series, but up to now also excellent agreement with experiments. Gravity (general relativity): spectacular confirmation e.g. by gravitational waves. Deviations can be explained by plausible assumptions (dark matter, dark energy). Essentially open: Consistent theory which combines general relativity and quantum physics. Klaus Fredenhagen From quantum field theory to quantum gravity Approaches to quantum gravity are difficult because: Field equations of general relativity are complicated nonlinear partial differential equations. Quantization of field theory leads to divergences which are more dangerous in gravity than in particle physics. Meaning of general covariance after quantization is unclear. Observed phenomena seem to give no hint on the form of the theory. Needed: Input from experiments, mathematics and natural philosophy. Klaus Fredenhagen From quantum field theory to quantum gravity Attempts for quantum gravity: String theory: surprising cancellation of divergences, rich and unexpected mathematical structures, but unclear concepts. Loop Quantum Gravity: clear concept of quantizing a discretized version of general relativity, but problems with the relation to continuum theory. Perturbative Quantum Field Theory: explicit and well established methods, good connection to known nongravitational phenomena, but severe problems with divergences and with the physical interpretation. Other approaches (causal sets, noncommutative spaces, . ) are less developed. Klaus Fredenhagen From quantum field theory to quantum gravity Big problem for Quantum Gravity: Lack of visible effects =) Ans¨atzeare tested by consistency, but not by observations. Consistency requires Internal consistency In the limit where quantum effects become small it should reproduce Classical General Relativity. If the back reaction of matter fields on the gravitational fields are neglegible it should agree with Quantum Field Theory on Lorentzian manifolds. Klaus Fredenhagen From quantum field theory to quantum gravity Claim: Perturbative Quantum Gravity is consistent as an effective quantum field theory. It reproduces General Relativity and Quantum Field Theory on curved spacetime in appropriate limits. In addition, it has already been tested via cosmological perturbation theory in Cosmic Microwave Background. Problems of perturbative Quantum Gravity: Nonrenormalizability Existence of local observables? What is the meaning of spacetime after quantization? Klaus Fredenhagen From quantum field theory to quantum gravity Tentative answers: Renormalization at every order is well defined, hence perturbative Quantum Gravity is an effective field theory whose validity for small energies depends on the size of the new coupling constants occuring in higher orders. In addition there are indications that Quantum Gravity might be asymptotically safe (Reuter et al.). Local observables in the sense of relative observables (Rovelli) can be defined (see later). Spacetime after quantization is defined in terms of quantum fields which are interpreted as coordinates. Klaus Fredenhagen From quantum field theory to quantum gravity Quantum Field Theory on curved spacetime First step towards quantum gravity: Quantum field theory on Lorentzian spacetimes (external gravitational field) Meaningful since the back reaction on the gravitational field is usually very small. Surprise: Contrary to classical fields there are problems for quantum fields. Klaus Fredenhagen From quantum field theory to quantum gravity Particles Origin: Role of particles. Connection quantum fields and particles (e.g. electromagnetical field and photons): Quantum field theory is a quantum theory, i.e. observables correspond to Hilbert space operators. Additional structure: Localizability of observables Example: Z d4x'(x)f (x) field ' at point x, averaged with the function f . Einstein causality (locality): observables in spacelike separarted regions commute. Klaus Fredenhagen From quantum field theory to quantum gravity Spacetime symmetries: act by unitary or antiunitary operators on the Hilbert space. Product of symmetry transformations corresponds to product of operators (up to a factor) (Wigner) Minkowski space: Poincar´esymmetry Classification of irreducible representations with positive energy (Wigner): Parametrized by 1 mass m > 0, spin s 2 2 N0, and 1 m = 0, helicity h 2 2 Z (m = 0 representations with infinite spin incompatible with quantum field theory) Klaus Fredenhagen From quantum field theory to quantum gravity Wigner: irreducible representations correspond to particles. Result of quantum field theory (Haag-Ruelle): To each particle there exist the corresponding multiparticle states, in two versions: outgoing and incoming particles Transition amplitudes = S-matrix-elements Conditions: existence of a vacuum state, locality, translation covariance and properties of the energy-momentum spectrum (positive energy, isolated mass shell). Klaus Fredenhagen From quantum field theory to quantum gravity Limitations of the particle interpretation of quantum field theory: No particle interpretation at finite time in interacting theories. Haag-Ruelle theory at m = 0 replaced by Buchholz theory but: no isolated mass shell for massive particles (infrared divergences) Charged particles do not have a sharp mass (infra particles)(Schroer, Buchholz). Finite temperature: Interacting particles cannot be eigenstates of the mass operator (Narnhofer, Requardt, Thirring). Klaus Fredenhagen From quantum field theory to quantum gravity Globally hyperbolic spacetimes External gravitational fields = curved spacetime A spacetime is globally hyperbolic, if it has a Cauchy surface. Cauchy surface = spacelike hypersurface which is met by each non prolongable causal curve exactly once. Causal: tangent vector timelike or lightlike. Globally hyperbolic: Initial value problem can be formulated. Causality structure similar to Minkowski space. In particular: no closed timelike curves. Klaus Fredenhagen From quantum field theory to quantum gravity Example: Klein-Gordon equation 2 ( + m + Rξ)' = 0 2 m and ξ parameter, d'Alembert operator 1 p = p @ g µν g@ g µ ν µν gµν spacetime metric with inverse g , determinant g and Ricci curvature scalar R. Globally hyperbolic: equation has unique advanced and retarded Green functions. But: Feynman propagator and Wightman-2-point function not uniquely determined. Reason: Decomposition according to positive and negative frequences coordinate dependent. Klaus Fredenhagen From quantum field theory to quantum gravity Consequence: Fock space quantization with creation and annihilation operators not well defined. Vacuum state for one decomposition is multiparticle state for another decompositon, with, in general, infinitely many particles. Thus: Many concepts of quantum field theory loose their meaning: There is no distinguished vacuum state. The concept of a particle is not well defined. Hence the S-matrix becomes meaningless. The Hilbert space of state vectors is not uniquely determined. Klaus Fredenhagen From quantum field theory to quantum gravity In addition technical problems: The momentum space depends on the choice of the coordinate system. A euclidian version (imaginary time) does not exist, in general. The path integral is not uniquely defined. Klaus Fredenhagen From quantum field theory to quantum gravity Algebraic Quantum Field Theory Appropriate framework: Algebraic Quantum Field Theory Basic structure: Algebras A(O) of observables associated to spacetime regions O. Axioms: If the observable A is associated to a region, then also to all larger regions (Isotony). Two observables which are associated to spacelike separated regions commute (Local commutativity). If O contains a Cauchy surface of the region O1, then all observables in O1 are contained in A(O)(Time-Slice-Axiom). Spacetime symmetries g act by algebra isomorphisms αg : A(O) ! A(gO)(Covariance). On static spacetimes the theory has a ground state (Stability) Klaus Fredenhagen From quantum field theory to quantum gravity Axioms work well on Minkowski space. On globally hyperbolic spacetimes the causal relations are similar to Minkowski space, hence the axioms of isotony, local commutativity and time slice remain meaningful. But problems with covariance (trivial symmetry group in the generic case) and stability (no useful concept of energy). Klaus Fredenhagen From quantum field theory to quantum gravity Solution: Stability is replaced by a local concept (microlocal spectrum condition): Wave front sets of correlation functions of quantum fields satisfy a positivity condition (Radzikowski 1995). Example: 2-point function of a free scalar field on Minkowski space: Positive frequency part of the causal propagator ∆ = ∆ret − ∆adv. On a curved globally hyperbolic spacetime: ∆ is uniquely determined, but a split into positive and negative frequencies depends on the choice of coordinates. Klaus Fredenhagen From quantum field theory to quantum gravity It is, however, possible to make a split up to smooth functions so that the singular part is unique. This defines the so-called Hadamard states