We quantize the Helmholtz equation (plus perturbative interactions) in two dimensions to illustrate a manifestly local description of quantum field theory. Using the general boundary formulation we describe the quantum dynamics both in a traditional time evolution setting as well as in a setting referring to finite disk (or annulus) shaped regions of spacetime. We demonstrate that both descriptions are equivalent when they should be.
We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.
We show that the Bethe-Salpeter equation for the scattering amplitude in the limit of zero incident energy can be transformed into a purely Euclidean form, as it is the case for the bound states. The decoupling between Euclidean and Minkowski amplitudes is only possible for zero energy scattering observables and allows determining the scattering length from the Euclidean Bethe-Salpeter amplitude. Such a possibility strongly simplifies the numerical solution of the Bethe-Salpeter equation and suggests an alternative way to compute the scattering length in Lattice Euclidean calculations without …
Studying transition amplitudes in (2+1)-dimensional causal dynamical triangulations, Cooperman and Miller discovered speculative evidence for Lorentzian quantum geometries emerging from its Euclidean path integral. On the basis of this evidence, Cooperman and Miller conjectured that Lorentzian de Sitter spacetime, not Euclidean de Sitter space, dominates the ground state of the quantum geometry of causal dynamical triangulations on large scales, a scenario akin to that of the Hartle-Hawking no-boundary proposal in which Lorentzian spacetimes dominate a Euclidean path integral. We argue against…
We prove the unitarity of the Euclidean nonlocal scalar field theory to all perturbative orders in the loop expansion. The amplitudes in the Euclidean space are calculated assuming that all the particles have purely imaginary energies, and afterwards they are analytically continued to real energies. We show that such amplitudes satisfy the Cutkowsky rules and that only the cut diagrams corresponding to normal thresholds contribute to their imaginary part. This implies that the theory is unitary. This analysis is then exported to nonlocal gauge and gravity theories by means of Becchi-Rouet-Stor…
We shortly review different methods to obtain the scattering solutions of the Bethe-Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude.
We consider the backreaction of the winding condensate on the cigar background. We focus on the case of the $SL(2,\mathbb{R})_k/U(1)$ cigar associated with, e.g., the near-horizon limit of $k$ NS5 black-branes. We solve the equations of motion numerically in the large $k$ limit as a function of the amplitude, $A$, of the winding mode at infinity. We find that there is a critical amplitude, $A_c=\exp(-γ/2)$, that admits a critical solution. In string theory, the exact $SL(2,\mathbb{R})_k/U(1)$ cigar CFT fixes completely the winding amplitude, $A_s$, at infinity. We find that in the large $k$ li…
A spacelike surface with cylinder topology can be described by various sets of canonical variables within pure AdS3 gravity. Each is made of one real coordinate and one real momentum. The Hamiltonian can be either $H=0$ or it can be nonzero and we display the canonical transformations that map one into the other, in two relevant cases. In a choice of canonical coordinates, one of them is the cylinder aspect $q$, which evolves nontrivially in time. The time dependence of the aspect is an analytic function of time $t$ and an "angular momentum" $J$. By analytic continuation in both $t$ and $J$ we…
We propose a definition for the Lorentzian Jackiw-Teitelboim (JT) gravity path integral that includes Lorentzian topology changing configurations. The construction is inspired by the bosonic string genus expansion on singular Lorentzian worldsheets, with geometries known as lightcone diagrams playing a prominent role. The Lorentzian path integral is defined through a suitable analytic continuation of the Euclidean path integral, and includes metrics that become degenerate at isolated points allowing for Lorentzian topology changing transitions. We discuss the relation between Euclidean JT ampl…
It has long been known that the coarse-grained approximation to the black hole density of states can be computed using classical Euclidean gravity. In this work we argue for another entry in the dictionary between Euclidean gravity and black hole physics, namely that Euclidean wormholes describe a coarse-grained approximation to the energy level statistics of black hole microstates. To do so we use the method of constrained instantons to obtain an integral representation of wormhole amplitudes in Einstein gravity and in full-fledged AdS/CFT. These amplitudes are non-perturbative corrections to…
In Euclidean path integrals, quantum mechanical tunneling amplitudes are associated with instanton configurations. We explain how tunneling amplitudes are encoded in real-time Feynman path integrals. The essential steps are borrowed from Picard-Lefschetz theory and resurgence theory.
We consider states of holographic conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the extent to which arbitrary bulk coherent states can be represented by such Euclidean path-integrals in the CFT. We construct the associated dual Lorentzian spacetimes perturbatively in the sources. Extending earlier work, we provide explicit formulae for the Lorentzian fields to first order in the sources for general scalar field and metric perturbations in arbitrary dimensions. We check the results by holographically comp…
I give a local description of the Euclidean regime $(M, g_{ab}, u^a)$ of Lorentzian spacetimes $(M, g_{ab})$ based on timelike geodesics $u^a$ passing through an arbitrary event $p_0 \in M$. I show that, to leading order, the Euclidean Einstein-Hilbert action $I_E$ is proportional to the Einstein tensor $G_{ab}u^a u^b$ of $g_{ab}$. The positivity of $I_E$ follows if $G_{ab}u^a u^b>0$ holds. I suggest an interpretation of this result in terms of the amplitude $\mathcal{A}[Σ_0]=\exp[{-I_E}]$ for a single space-like hypersurface $Σ_0 \in I^{+}(p_0)$ to emerge at a constant geodesic distance $λ_0$…
In this work, we consider tunneling between non-metastable states in gravitational theories. Such processes arise in various contexts, e.g., in inflationary scenarios where the inflaton potential involves multiple fields or multiple branches. They are also relevant for bubble wall nucleation in some cosmological settings. However, we show that the transition amplitudes computed using the Euclidean method generally do not approach the corresponding field theory limit as $M_{p}\rightarrow \infty$. This implies that in the Euclidean framework, there is no systematic expansion in powers of $G_{N}$…
The one-loop effective action of quantum electrodynamics in four dimensions is shown to be controlled by the Euclidean Dirac propagator $G$ in a background potential. After separating the photon self-energy and photon-photon scattering graphs from the effective action the remainder is known to be the logarithm of an entire function of the electric charge of order 4 under mild regularity assumptions on the potential. This input together with QED's lack of an ultrastable vacuum constrain the strong field behavior of $G$. It is shown that $G$ vanishes in the strong field limit. The relevance of t…
We discuss new possible tunneling processes in the presence of gravity. We formulate quantum tunneling using the Wheeler-deWitt canonical quantization and the WKB approximation. The distinctive feature of our formulation is that it accommodates the coexistence of Euclidean and Lorentzian evolution. It opens up a new possibility of quantum tunneling; e.g. a bubble wall itself tunnels the potential barrier pulling the field nearby, where the wall region experiences the Euclidean evolution while the other regions experience the Lorentzian evolution simultaneously. We execute numerical analysis an…
We investigate a nonperturbative formulation of quantum gravity defined via Euclidean dynamical triangulations (EDT) with a non-trivial measure term in the path integral. We are motivated to revisit this older formulation of dynamical triangulations by hints from renormalization group approaches that gravity may be asymptotically safe and by the emergence of a semiclassical phase in causal dynamical triangulations (CDT). We study the phase diagram of this model and identify the two phases that are well known from previous work: the branched polymer phase and the collapsed phase. We verify that…
We employ a scalar model to exemplify the use of contour deformations when solving Lorentz-invariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation…
