## Periodic Reporting for period 1 - QLO-QG (Quasi-local observables in quantum gravity)

Reporting period: 2016-07-01 to 2018-06-30

Although the theory of quantum gravity is unknown at the fundamental level, valid quantum gravitational predictions can be made at low energies and large distances compared to the Planck scale, which is about 10^(-33) cm where the precise theory of quantum gravity becomes relevant. At low energies, quantum gravity can be treated as an effective field theory by quantising metric fluctuations around a classical background (this approach is called “perturbative quantum gravity”, or pQG). Experimentally, quantum gravity effects manifest themselves in the cosmological microwave background, where one of the lowest-order predictions of pQG (the scalar power spectrum) has already been experimentally confirmed, and higher-order corrections are likely to be tested by next-generation experiments.

An open issue in this context was the identification of suitable observables in pQG corresponding to experimentally observed quantities. In contrast to other well-known gauge theories where the gauge symmetry only concerns internal degrees of freedom, the gauge symmetry of pQG – diffeomorphisms – moves points on the underlying manifold, and local observables (defined at a fixed point of the background metric) are not gauge-invariant and hence unphysical. The primary objective of the project was the study of non-local observables in pQG, which contributes both to our understanding of the fundamental forces of nature, and to a correct interpretation of future experimental results.

Besides advances in the mathematical description of these observables and the proof that a certain class of such observables are unsuitable from a physical point of view, the main result of the project is the construction of an observable that quantifies in a mathematically sound and gauge-invariant way the local expansion rate of the universe (the Hubble rate). Loop corrections to this observable were then calculated, which confirm a physical picture that was conjectured 25 years ago: the exponential expansion of spacetime during the inflationary period of the early universe produces large amounts of gravitons, whose mutual attraction then slows down the expansion.

An open issue in this context was the identification of suitable observables in pQG corresponding to experimentally observed quantities. In contrast to other well-known gauge theories where the gauge symmetry only concerns internal degrees of freedom, the gauge symmetry of pQG – diffeomorphisms – moves points on the underlying manifold, and local observables (defined at a fixed point of the background metric) are not gauge-invariant and hence unphysical. The primary objective of the project was the study of non-local observables in pQG, which contributes both to our understanding of the fundamental forces of nature, and to a correct interpretation of future experimental results.

Besides advances in the mathematical description of these observables and the proof that a certain class of such observables are unsuitable from a physical point of view, the main result of the project is the construction of an observable that quantifies in a mathematically sound and gauge-invariant way the local expansion rate of the universe (the Hubble rate). Loop corrections to this observable were then calculated, which confirm a physical picture that was conjectured 25 years ago: the exponential expansion of spacetime during the inflationary period of the early universe produces large amounts of gravitons, whose mutual attraction then slows down the expansion.

1. It was shown that correlation functions defined at fixed geodesic distance can be renormalised by treating the geodesic embedding coordinates as fields, and performing a wave function renormalisation for them. A scalar field correlation function at fixed geodesic distance was computed in flat space, including one-loop graviton corrections, with the final result being gauge-invariant but dependent on the finite parts of counterterms even for separated points.

2. It was further shown that correlation functions using a field-dependent coordinate system are renormalisable using the usual local counterterms in flat space. A scalar field correlation function was computed including one-loop graviton corrections, and the final result was gauge-invariant and uniquely defined for non-coincident points.

3. The field-dependent coordinate system was generalised to cosmological (Friedmann-Lemaître-Robertson-Walker) spacetimes, and the graviton propagator in a gauge adapted to this coordinate system was determined.

4. A non-local observable corresponding to the local expansion rate was constructed, and its one-loop expectation value was computed for spacetimes of physical interest: matter- and radiation-dominated eras, and slow-roll inflation. For slow-roll inflation, the one-loop quantum corrections lead to a diminution of the slow-roll parameters; in the matter-dominated era a secular decrease of the expansion rate was found, and for radiation domination the result vanishes at one-loop order.

5. Ward identities for general gauge theories (including gauge theories with open gauge algebras) were studied in detail in algebraic quantum field theory, and a gauge-invariant quantum state was constructed to all orders in perturbation theory. These were used to show the renormalisability of the observable mentioned in Point 4 to all orders using the usual local counterterms.

6. Quantum corrections to the Newtonian potential for spinning particles have been computed in flat space and de Sitter space, and it was found that there exists a term (proportional to the Hubble constant squared) that grows logarithmically with the distance to the particle. When resummed, this term leads to a slower fall-off of the gravitational force with distance in de Sitter space (with respect to flat space).

7. It was shown that in cosmological spacetimes local measurements of the graviton are equivalent to local measurements of the linearised Weyl tensor and another local infrared-convergent tensor, or more technically, that that graviton field smeared with a compactly supported test function is equal to a smearing of the linearised Weyl tensor and this other tensor with a compactly supported test function determined in a precise way from the first.

8. It was shown that the two tensors mentioned in Point 7, together with the linearised Einstein tensor, form a complete set of local observables at linear order in cosmological spacetimes.

The above results have been published in 10 research articles in high-impact journals, and presented in 22 conferences, workshops and seminars. They have already attracted the attention of eminent researchers in the field.

2. It was further shown that correlation functions using a field-dependent coordinate system are renormalisable using the usual local counterterms in flat space. A scalar field correlation function was computed including one-loop graviton corrections, and the final result was gauge-invariant and uniquely defined for non-coincident points.

3. The field-dependent coordinate system was generalised to cosmological (Friedmann-Lemaître-Robertson-Walker) spacetimes, and the graviton propagator in a gauge adapted to this coordinate system was determined.

4. A non-local observable corresponding to the local expansion rate was constructed, and its one-loop expectation value was computed for spacetimes of physical interest: matter- and radiation-dominated eras, and slow-roll inflation. For slow-roll inflation, the one-loop quantum corrections lead to a diminution of the slow-roll parameters; in the matter-dominated era a secular decrease of the expansion rate was found, and for radiation domination the result vanishes at one-loop order.

5. Ward identities for general gauge theories (including gauge theories with open gauge algebras) were studied in detail in algebraic quantum field theory, and a gauge-invariant quantum state was constructed to all orders in perturbation theory. These were used to show the renormalisability of the observable mentioned in Point 4 to all orders using the usual local counterterms.

6. Quantum corrections to the Newtonian potential for spinning particles have been computed in flat space and de Sitter space, and it was found that there exists a term (proportional to the Hubble constant squared) that grows logarithmically with the distance to the particle. When resummed, this term leads to a slower fall-off of the gravitational force with distance in de Sitter space (with respect to flat space).

7. It was shown that in cosmological spacetimes local measurements of the graviton are equivalent to local measurements of the linearised Weyl tensor and another local infrared-convergent tensor, or more technically, that that graviton field smeared with a compactly supported test function is equal to a smearing of the linearised Weyl tensor and this other tensor with a compactly supported test function determined in a precise way from the first.

8. It was shown that the two tensors mentioned in Point 7, together with the linearised Einstein tensor, form a complete set of local observables at linear order in cosmological spacetimes.

The above results have been published in 10 research articles in high-impact journals, and presented in 22 conferences, workshops and seminars. They have already attracted the attention of eminent researchers in the field.

Several different advances have been made beyond the state of the art. While in general the renormalisation of non-local operators in quantum field theory is an open issue, it has been shown that both the correlation functions at fixed geodesic distance and the observables defined using a field-dependent coordinate system can be renormalised. It has further been shown that the correlation functions at fixed geodesic distance, which had been proposed in cosmology as a solution to the problem of large infrared logarithms, are not physically viable, while the correlation functions in a field-dependent coordinate system can be used to quantify quantum gravitational corrections in a gauge-invariant and physically reasonable way. Based on this, the observable measuring the local expansion rate that we have constructed does not suffer from any of the drawbacks that previous proposals had (such as being not fully gauge-invariant, or their nonlocality being non-causal). Lastly, we have shown that the Ward identities previously proven in the algebraic approach to quantum field theory for gauge theories with closed gauge algebras also hold for theories with open gauge algebras (such as supersymmetric theories), and derived identities also for interacting time-ordered products which is necessary to treat the above observables.

The main impact of the results will be on research in the subfield. Since it has been shown that observables defined using a field-dependent coordinate system can be renormalised and give physically sensible results, other observables corresponding to other quantities of interest can now be constructed and loop corrections to them can be computed. Ultimately, predictions made from these calculations will be compared to future experiments.

The main impact of the results will be on research in the subfield. Since it has been shown that observables defined using a field-dependent coordinate system can be renormalised and give physically sensible results, other observables corresponding to other quantities of interest can now be constructed and loop corrections to them can be computed. Ultimately, predictions made from these calculations will be compared to future experiments.