Search

We argue that a consistent coupling of a quantum theory to gravity requires an extension of ordinary `first order' Riemannian geometry to second order Riemannian geometry, which incorporates both a line element and an area element. This extension results in a misalignment between the dimension of the manifold and the dimension of the tangent spaces. In particular, we find that for a 4-dimensional spacetime, tangent spaces become 18-dimensional. We then discuss the construction of physical theories within this framework, which involves the introduction of terms that are quadratic in derivatives in the action. On a flat spacetime, the quadratic sector is perpendicular to the first order sector and only affects the normalization of the path integral, whereas in a curved spacetime the quadratic sector couples to the first order sector. Moreover, we show that, despite the introduction of higher order derivatives, the Ostragradski instability can be avoided, due to an order mixing of the two sectors. Finally, we comment on extensions to higher order geometry and on relations with non-commutative and generalized geometry., Comment: 32+12 pages; v2: section 6 improved

We propose a stochastic interpretation of spacetime non-commutativity starting from the path integral formulation of quantum mechanical commutation relations. We discuss how the (non-)commutativity of spacetime is inherently related to the continuity or discontinuity of paths in the path integral formulation. Utilizing Wiener processes, we demonstrate that continuous paths lead to commutative spacetime, whereas discontinuous paths correspond to non-commutative spacetime structures. As an example we introduce discontinuous paths from which the $\kappa$-Minkowski spacetime commutators can be obtained. Moreover we focus on modifications of the Leibniz rule for differentials acting on discontinuous trajectories. We show how these can be related to the deformed action of translation generators focusing, as a working example, on the $\kappa$-Poincar\'e algebra. Our findings suggest that spacetime non-commutativity can be understood as a result of fundamental discreteness of spacetime., Comment: 22+9 pages

In this work, we discuss the matching of the Vilkovisky-DeWitt effective action of quantum gravity to an example of an ultra-violet complete theory of quantum gravity. We show how this matching enables a calculation of the local Wilson coefficients of the effective action. We provide several examples in string theory. Moreover, we discuss the matching conditions within the context of the swampland program., Comment: 11 pages

We consider quantum gravitational corrections to the Oppenheimer-Snyder metric describing time-dependent dust ball collapse. The interior metric also describes Friedmann-Lemaitre-Robertson-Walker cosmology and our results are interpreted in that context. The exterior corrections are an example of quantum hair, and are shown to persist throughout the collapse. Our results show the quantum hair survives throughout the horizon formation and that the internal state of the resulting black hole is accessible to outside observers., Comment: 13 pages

We construct an explicit one-to-one correspondence between non-relativistic stochastic processes and solutions of the Schrodinger equation and between relativistic stochastic processes and solutions of the Klein-Gordon equation. The existence of this equivalence suggests that the Lorentzian path integral can be defined as an Ito integral, similar to the definition of the Euclidean path integral in terms of the Wiener integral. Moreover, the result implies a stochastic interpretation of quantum theories., Comment: 9 pages

We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection., Comment: 113 pages; preprint of a book published by Springer Nature

We discuss the Schwartz-Meyer second order geometry framework and its relevance to theories of quantum gravity that incorporate a notion of spacetime stochasticity or quantum foam. We illustrate the framework in the context of Nelson's stochastic quantization., Comment: 6 pages; contribution to the proceedings of the 14th international workshop on Lie Theory and its Applications in Physics (LT-14)

We review some recent results obtained for black holes using effective field theory methods applied to quantum gravity, in particular the unique effective action. Black holes are complex thermodynamical objects that not only have a temperature but also have a pressure. Furthermore, they have quantum hair which provides a solution to the black hole information paradox., Comment: to appear in the proceedings of QGRAV-2021

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. These observations establish the existence of ubiquitous quantum hair due to gravitational effects., Comment: 12 pages

We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson's stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman-Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a quantum particle on a curved spacetime., Comment: 15+13 pages. v.2: accepted in J. Math. Phys

We calculate quantum gravitational corrections to the entropy of black holes using the Wald entropy formula within an effective field theory approach to quantum gravity. The corrections to the entropy are calculated to second order in curvature and we calculate a subset of those at third order. We show that, at third order in curvature, interesting issues appear that had not been considered previously in the literature. The fact that the Schwarzschild metric receives corrections at this order in the curvature expansion has important implications for the entropy calculation. Indeed, the horizon radius and the temperature receive corrections. These corrections need to be carefully considered when calculating the Wald entropy.

In this essay we show that quantum gravity and the spin-statistics theorem have very interesting consequences for dark matter candidates. Quantum gravity can lead to fifth force type interactions that lead to a lower bound on the masses of bosonic candidates. In the case of fermions, the spin-statistics theorem leads to a lower bound on fermion masses. For both bosonic and fermionic dark matter candidates, quantum gravity leads to a decay of dark matter particles. A comparison of their lifetime with the age of the universe leads to an upper bound on their masses. For singlet scalar dark matter fields, we find $10^{-3}{\rm eV} \lesssim m\lesssim 10^{7}{\rm eV}$., Comment: Honorable Mention in the 2021 Gravity Research Foundation Essay Competition. arXiv admin note: text overlap with arXiv:2009.11575

It was shown recently that stochastic quantization can be made into a well defined quantization scheme on (pseudo-)Riemannian manifolds using second order differential geometry, which is an extension of the commonly used first order differential geometry. In this letter, we show that restrictions to relativistic theories can be obtained from this theory by imposing a stochastic energy-momentum relation. In the process, we derive non-perturbative quantum corrections to the line element as measured by scalar particles. Furthermore, we extend the framework of stochastic quantization to massless scalar particles., Comment: v1: 12 pages. v2: clarifications added. v3: appendix added, to appear in J.Math.Phys

We embed Nelson's stochastic quantization in the Schwartz-Meyer second order geometry framework. The result is a non-perturbative theory of quantum mechanics on (pseudo)-Riemannian manifolds. Within this approach, we derive stochastic differential equations for massive spin-0 test particles charged under scalar potentials, vector potentials and gravity. Furthermore, we derive the associated Schr\"odinger equation. The resulting equations show that massive scalar particles must be conformally coupled to gravity in a theory of quantum gravity. We conclude with a discussion of some prospects of the stochastic framework., Comment: 48 pages; v2: minor revisions

In this letter, we show that quantum gravity leads to lower and upper bounds on the masses of dark matter candidates. These bounds depend on the spins of the dark matter candidates and the nature of interactions in the dark matter sector. For example, for singlet scalar dark matter, we find a mass range $10^{-3}{\rm eV} \lesssim m_\phi \lesssim 10^{7}{\rm eV}$. The lower bound comes from limits on fifth force type interactions and the upper bound from the lifetime of the dark matter candidate.

In this paper we consider very weakly interacting and ultra light scalar and pseudoscalar dark matter candidates. We show that quantum gravity has important implications for such models and that the masses of the singlet scalar and pseudoscalar fields must be heavier than $3\times 10^{-3}$ eV. However, if they are gauged, their masses could be much lighter and as light as $10^{-22}\, {\rm eV}$. The existence of new gauge forces in the dark matter sector can thus be probed by atomic clocks or quantum sensors experiments.

In this paper we derive the leading quantum gravitational corrections to the geodesics and the equations of motion for a scalar field in the spacetime containing a constant density star. It is shown that these corrections can be calculated in quantum gravity reliably and in a model independent way. Furthermore, we find that quantum gravity gives rise to an additional redshift that results from the gradient instead of the amplitude of the density profile., Comment: 13 pages

In this paper we consider a static and regular fluid generating a locally spherically symmetric and time-independent space-time and calculate the leading quantum corrections to the metric to first order in curvature. Starting from a singularity free classical solution of general relativity, we show that singularities can be introduced in the curvature invariants by quantum gravitational corrections calculated using an effective field theory approach to quantum gravity. We identify non-trivial conditions that ensure that curvature invariants remain singularity free to leading order in the curvature expansion of the effective action., Comment: 10 pages

In this paper we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, this effective field theory contains two new degrees of freedom which have important implications for the derivation of these energy theorems: a massive spin-2 field and a massive spin-0 field. Using an explicit mapping of this theory from the Jordan frame to the Einstein frame, we show that both the cosmological and the black hole singularity theorems may not hold due to the presence of a massive spin-2 field in the particle spectrum of quantum gravity. Furthermore, we show that the massive scalar field can lead to a violation of the assumptions used to derive Hawking's singularity theorem. On the other hand, it does not affect Penrose's singularity theorem.

In this paper we consider the full set of quantum gravitational corrections to a star metric to second order in curvature. As we use an effective field theoretical approach, these corrections apply to any model of quantum gravity that is based on general coordinate invariance. We then discuss the black hole limit and identify an interesting phenomenon which could shed some light on the nature of astrophysical black holes: while star metrics receive corrections at second order in curvature, vacuum solutions such as black hole metrics do not. What happens to these corrections when a star collapses?

Bifurcation analysis is used to study an effective model of QCD$_4$ with four-fermi interactions. Our analysis supports the scenario of a fixed point merger at the lower edge of the conformal window. This indicates square root scaling of the anomalous scaling dimensions of the fermion fields just above the lower edge and exponential scaling just below. We also predict existence of new fixed points in this model whose (dis)appearance may indicate transitions of the flow within the conformal window. Furthermore, we make new predictions for the critical value $(N_{f}/N_{c})_{\textrm{crit}}$ at the lower edge. We also obtain exotic spiraling flows that are generated by complex scaling dimensions of the effective four-fermi interactions. Finally, we extend the model by adding a scalar field that couples with a Yukawa interaction term and study the modifications it causes to RG-flows., Comment: 31+26 pages; references added, typos corrected