From Quantum Field Theory to Quantum Gravity

Home , Klaus Fredenhagen, Quantum gravity

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 =⇒ Ansätzeare 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 d4xϕ(x)f (x)

ﬁeld ϕ 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: Poincarésymmetry Classification of irreducible representations with positive energy (Wigner): Parametrized by 1 mass m > 0, spin s ∈ 2 N0, and 1 m = 0, helicity h ∈ 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)ϕ = 0

2 m and ξ parameter, d’Alembert operator 1 √ = √ ∂ 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 with 2-point function u ω(ϕ(x)ϕ(y)) = + v ln(σ) + w σ with smooth functions u, v, w and the Synge function σ (the signed square of the geodesic distance between x and y). u andv are uniquely defined. A direct use of this characterization of singularities is cumbersome. A break through was obtained by the observation of Radzikowski that the singularity can be completely characterized by its wave front set. Therefore the techniques from microlocal analysis became available and allowed a full analysis of the ultraviolet divergences in renormalized perturbation theory (Brunetti-F-Köhler 1996, Brunetti-F 2000, Hollands- Wald 2001).

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Covariance is generalized to local covariance (Hollands-Wald 2001, Brunetti-F-Verch 2003): Quantum ﬁeld theory is a (covariant) functor from the category of globally hyperbolic time oriented spacetimes (with isometric, orientation and causality preserving embeddings as morphisms) to the category of unital operator algebras. Restriction to subregions of a given spacetime provides a covariant Haag-Kastler net. The microlocal spectrum condition characterizes a contravariant functor to the category of state spaces.

Klaus Fredenhagen From quantum field theory to quantum gravity Second step: Consider the gravitational field as a fluctuation around classical metrics (not necessarily solutions of Einstein’s equation). Consistency requirement: Invariance under infinitesimal changes of the background. Complication: Invariance under diffeomorphisms has to be treated in analogy to gauge theories, therefore introduction of auxiliary fields (ghosts etc.) (Brunetti-F-Rejzner 2014). Major conceptual problem: No local observables in the usual sense.

Klaus Fredenhagen From quantum field theory to quantum gravity Third step: Choose covariant fields as coordinates and express other covariant fields in terms of these coordinates (relational observables)(Rovelli): Implementation in quantum field theory (Brunetti-F-Hack-Pinamonti-Rejzner (2016)): a Choose scalar fields XΓ , a = 1,... 4 which are local functionals of the configuration Γ = (g, ϕ, . . .) and equivariant, i.e. for a diffeomorphism χ acting on Γ

a a Xχ∗Γ = XΓ ◦ χ .

Assume that for a given background conﬁguration Γ0 = (g0, ϕ0,...) the map

X : x 7→ (X 1 (x),..., X 4 (x)) ∈ 4 Γ0 Γ0 Γ0 R is injective.

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Then let for Γ near to Γ0 −1 αΓ = XΓ ◦ XΓ0 .

We then set for any other equivariant scalar ﬁeld AΓ

AΓ = AΓ ◦ αΓ . We obtain gauge invariant ﬁelds

AΓ(x) := AΓ(αΓ(x)) . Here gauge invariance is obtained by evaluating the field at a point xΓ = αΓ(x) which is shifted in a Γ-dependent way such that the fields XΓ chosen as coordinates assume at this point the same values as XΓ0 at the point x. In perturbation theory the observables enter only by their Taylor expansion around the background Γ0. These expressions involve the classical fields at the background Γ0 and polynomials of quantum fields which can be interpreted in terms of normal products.

Klaus Fredenhagen From quantum field theory to quantum gravity E.g., up to first order δA ∂A δαµ A = A + h Γ (Γ ), δΓi + Γ0 h Γ (Γ ), δΓi . Γ0+δΓ Γ0 δΓ 0 ∂xµ δΓ 0 The last term on the right hand side is necessary in order to get gauge invariant fields (up to 1st order). We find !µ δαµ ∂X −1 δX a Γ (Γ ) = − Γ0 Γ (Γ ) . δΓ 0 ∂x δΓ 0 a Observations:

If AΓ vanishes on the background, then it is gauge invariant at ﬁrst order.

If AΓ0 depends only on 1 variable, the correction involves only the ﬁeld !µ ∂X −1 δX a xµ = − Γ0 h Γ (Γ ), δΓi . 1 ∂x δΓ 0 a

Klaus Fredenhagen From quantum field theory to quantum gravity If AΓ0 = 0, the second order correction is δA 2∂ h Γ (Γ ), δΓi · xµ µ δΓ 0 1 and involves in general all coordinates. Conclusion: A perturbative calculation of observables in quantum gravity is possible. The quantized spacetime may be described by quantum fields which are chosen as coordinates. It will in general depend on this choice. Question: Can the theory be experimentally checked? Problem: Quantum effects of gravity are expected to be of the size of Planck units: −35 Planck length lP = 1.62 · 10 m −44 Planck time tP = 5.39 · 10 s Planck energy 1.22 · 1019GeV

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Application to cosmology

But: Fluctuations of the cosmological microwave background (CMB) can successfully be explained by quantum effects. Model: Einstein’s gravity coupled to a scalar field (inflaton). Background: Metric g = dt2 − a(t)2(dx2 + dy 2 + dz2) (flat Friedmann-Lemaitre-Robertson-Walker spacetime) . Inflaton field φ spatially constant solution of Klein-Gordon equation with some potential. Linearized fluctuations can be related to temperature fluctuations in the CMB by an ad hoc quantization.

Klaus Fredenhagen From quantum field theory to quantum gravity We observe that the expansion of physical observables contains contributions of the physical coordinates expanded around the background. Difficulty: background not generic, therefore not sufficiently many physical coordinates Solution: use ϕ as time coordinate and add auxiliary fields mimicking fields of the standard model Toy model: 3 minimally coupled scalar fields X a, a = 1, 2, 3. Background

2 2 2 a a g0 = a (τ)(dτ − dx ) , ϕ0 = φ , X0 = x

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Comparison with cosmological perturbation theory:

−2A (∂ B + V )t δg = a2 i i −∂i B + Vi 2(∂i ∂j E + δij D + ∂(i Wj) + Tij )

Interesting observables: Spatial curvature, deﬁned as curvature of the metric tensor dϕ ⊗ dϕ h = g − g −1(dϕ, dϕ)

On surfaces of constant ϕ, h is nondegenerate and Riemannian.

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Scalar spatial curvature in linear order 4H R(ϕ) = ∆µ 1 φ0

0 dφ H = aH (conformal Hubble parameter), φ = dτ , φ0 µ = δϕ − H D Mukhanov-Sasaki variable Here no 1st order correction, since the 0th order vanishes.

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Lapse function (up to 1st order)

−1 − 1 a a 0 0 N = |g (dϕ, dϕ)| 2 = − + (δϕ − Aφ ) φ0 φ02 Correction term a φ00 N = N + − H δϕ φ02 φ0

On shell one obtains 2a N = − ∆Ψ φ03 Ψ Bardeen potential (analogue of the Newtonian potential)

0 Ψ = A − (∂τ + H)(B + E )

Klaus Fredenhagen From quantum field theory to quantum gravity Fluctuations in the microwave background are explained by the Sachs-Wolfe effect: δT 1 = Ψ T 3 where Ψ in 1st order is considered as a quantum field. It involves besides the inflaton field also gravitational degrees of freedom.

Klaus Fredenhagen From quantum field theory to quantum gravity Brunetti-F-Hack-Pinamonti-Rejzner (2016): This is the first order of full perturbative quantum gravity (Brunetti-F-Rejzner 2014). Problem: Background non generic, thus not sufficiently many covariant local coordinates. Way out: Nonlocal covariant coordinates can be introduced which reproduce the linear theory, but problems with renormalization. Fröb(2017): Renormalization up to 2nd order is possible. Future work: Compute second order effects and compare with observations.

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity Conclusion

Perturbative quantum gravity is a consistent theory which in principle makes testable predictions.

Provided the precision of cosmological observations increases as in the past decades there is a good chance that many of you will see the ﬁrst experimental conﬁrmation of the theory.

Klaus Fredenhagen From quantum ﬁeld theory to quantum gravity