Euclidean Sector Research Articles

Spacetime metric from quantum-gravity corrected Feynman propagators

Differentiation of the scalar Feynman propagator with respect to the spacetime coordinates yields the metric on the background spacetime that the scalar particle propagates in. Now Feynman propagators can be modified in order to include quantum-gravity corrections as induced by a zero-point length [Formula: see text]. These corrections cause the length element [Formula: see text] to be replaced with [Formula: see text] within the Feynman propagator. In this paper, we compute the metrics derived from both the quantum-gravity free propagators and from their quantum-gravity corrected counterparts. We verify that the latter propagators yield the same spacetime metrics as the former, provided one measures distances greater than the quantum of length [Formula: see text]. We perform this analysis in the case of the background spacetime [Formula: see text] in the Euclidean sector.

Schwinger effect in compact space

We consider a theory of scalar QED on a spatially compact $1+1$-dimensional spacetime. By considering a constant electric field pointing down the compact dimension, we compute the quantum effective action by integrating out the scalar degrees of freedom in the Euclidean sector. Working in the saddle-point approximation, we uncover two novel branches/physical regimes upon analytically continuing back to real time and discover a new result, hitherto unreported in previous literature. Implications of our results are discussed.

Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes

We discuss the quantization of vacuum Bianchi I spacetimes in the modified formalism of loop quantum cosmology recently proposed by Dapor and Liegener. This modification is based on a regularization procedure where both the Euclidean and Lorentzian terms of the Hamiltonian are treated independently. Whereas the Euclidean part has already been dealt with in the literature for Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum mechanically. After a brief review of the quantum kinematics and the quantization of the Euclidean sector, we represent the Lorentzian part of the Hamiltonian constraint by an operator according to the factor ordering rules of the Martin-Benito--Mena Marugan--Olmedo prescription. We study the general properties of this quantum operator and the superselection rules derived therefrom, resulting in an action similar to that of the Euclidean operator except that the orientation of the densitized triad is not preserved, a fact which leads to a generic enlargement of the superselection sectors. We conclude with an explanation of the mechanism that prevents this enlargement in the isotropic case and a comment on the effect of alternative prescriptions for the implementation of the improved dynamics.

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc

Let Δ ⊊ C \Delta \subsetneq \mathbb {C} be a simply connected domain, let f : D → Δ f:\mathbb {D} \to \Delta be a Riemann map, and let { z k } ⊂ Δ \{z_k\}\subset \Delta be a compactly divergent sequence. Using Gromov’s hyperbolicity theory, we show that { f − 1 ( z k ) } \{f^{-1}(z_k)\} converges non-tangentially to a point of ∂ D \partial \mathbb {D} if and only if there exists a simply connected domain U ⊊ C U\subsetneq \mathbb {C} such that Δ ⊂ U \Delta \subset U and Δ \Delta contains a tubular hyperbolic neighborhood of a geodesic of U U and { z k } \{z_k\} is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if ( ϕ t ) (\phi _t) is a non-elliptic semigroup of holomorphic self-maps of D \mathbb {D} with Koenigs function h h and h ( D ) h(\mathbb {D}) contains a vertical Euclidean sector, then ϕ t ( z ) \phi _t(z) converges to the Denjoy-Wolff point non-tangentially for every z ∈ D z\in \mathbb {D} as t → + ∞ t\to +\infty . Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but is oscillating, in the sense that the slope of the trajectories is not a single point.

Thermodynamics of Lorentzian Taub-NUT spacetimes

The thermodynamics of the Taub-NUT solution has been predominantly studied in the Euclidean sector, upon imposing the condition for the absence of Misner strings. Such thermodynamics is quite exceptional: the periodicity of the Euclidean time is restricted and thence the NUT charge cannot be independently varied, the entropy is not equal to a quarter of the area, and the thermodynamic volume can be negative. In this paper we revisit this paradigm and study the thermodynamics of the Lorentzian Taub-NUT solution, maintaining (as recently shown relatively harmless) Misner strings. We argue that in order to formulate a full cohomogeneity first law where the NUT parameter can be independently varied, it is natural to introduce a new charge together with its conjugate quantity. We consider two scenarios: one in which the entropy is given by the Lorentzian version of the Noether charge, the other in which the entropy is given by the standard Bekenstein--Hawking area law. In both cases consistent thermodynamics with positive thermodynamic volume can be formulated.

Predictions of the quantum landscape multiverse

The 2015 Planck data release has placed tight constraints on the class of inflationary models allowed. The current best fit region favors concave downwards inflationary potentials, since they produce a suppressed tensor to scalar index ratio r. Concave downward potentials have a negative curvature , therefore a tachyonic mass square that drives fluctuations. Furthermore, their use can become problematic if the field rolls in a part of the potential away from the extrema, since the semiclassical approximation of quantum cosmology, used for deriving the most probable wavefunction of the universe from the landscape and for addressing the quantum to classical transition, breaks down away from the steepest descent region. We here propose a way of dealing with such potentials by inverting the metric signature and solving for the wavefunction of the universe in the Euclidean sector. This method allows us to extend our theory of the origin of the universe from a quantum multiverse, to a more general class of concave inflationary potentials where a straightforward application of the semiclassical approximation fails. The work here completes the derivation of modifications to the Newtonian potential and to the inflationary potential, which originate from the quantum entanglement of our universe with all others in the quantum landscape multiverse, leading to predictions of observational signatures for both types of inflationary models, concave and convex potentials.

Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Schwarzschild Black Hole

In this article, we will discuss a Lorentzian sector calculation of the entropy of a minimally coupled scalar field in the Schwarzschild black hole background using the brick wall model of 't Hooft. In the original article, the WKB approximation was used for the modes that are globally stationary. In a previous article, we found that the WKB quantization rule together with a proper counting of the states, leads to a new expression of the scalar field entropy which is not proportional to the area of the horizon. The expression of the entropy is logarithmically divergent in the brick wall cut-off parameter in contrast to an inverse power divergence obtained earlier. In this article, we will consider the entropy for a thin shell of matter field of a given thickness surrounding the black hole horizon. The thickness is chosen to be large compared with the Planck length and is of the order of the atomic scale. When expressed in terms of a covariant cut-off parameter, the entropy of a thin shell of matter field of a given thickness and surrounding the horizon in the Schwarzschild black hole background is given by an expression proportional to the area of the black hole horizon. This leading order divergent term in the cut-off parameter remains to be logarithmically divergent. The logarithmic divergence is expected from the nature of the solution in the near-horizon region. We will find that these discussions are significant in the context of the continuation to the Euclidean sector and the corresponding regularization schemes used to evaluate the thermodynamical properties of matter fields in curved spaces. These are related with the geometric aspects of curved spaces. The above discussions are also important in presence of cosmological event horizon.

Wave function of the Universe, preferred reference frame effects and metric signature transition

Gravitational model of non-minimally coupled Brans Dicke (BD) scalar field 0 with dynamical unit time-like four vector field is used to study flat Robertson Walker (RW) cosmology in the presence of variable cosmological parameter V (ϕ) = Λϕ. Aim of the paper is to seek cosmological models which exhibit metric signature transition. The problem is studied in both classical and quantum cosmological approach with large values of BD parameter ω >> 1. Scale factor of RW metric is obtained as which describes nonsingular inflationary universe in Lorentzian signature sector. Euclidean signature sector of our solution describes a re-collapsing universe and is obtained from analytic continuation of the Lorentzian sector by exchanging . Dynamical vector field together with the BD scalar field are treated as fluid with time dependent barotropic index. They have regular (dark) matter dominance in the Euclidean (Lorentzian) sector. We solved Wheeler De Witt (WD) quantum wave equation of the cosmological system. Assuming a discrete non-zero ADM mass we obtained solutions of the WD equation as simple harmonic quantum Oscillator eigen functionals described by Hermite polynomials. Absolute values of these eigen functionals have nonzero values on the hypersurface in which metric field has signature degeneracy. Our eigen functionals describe nonzero probability of the space time with Lorentzian (Euclidean) signature for . Maximal probability corresponds to the ground state j = 0.

Free energy of a Lovelock holographic superconductor

We study thermodynamics of black hole solutions in Lanczos-Lovelock AdS gravity in d+1 dimensions coupled to nonlinear electrodynamics and a Stueckelberg scalar field. This class of theories is used in the context of gauge/gravity duality to describe a high-temperature superconductor in d dimensions. Larger number of coupling constants in the gravitational side is necessary to widen a domain of validity of physical quantities in a dual QFT. We regularize the gravitational action and find the finite conserved quantities for a planar black hole with scalar hair. Then we derive the quantum statistical relation in the Euclidean sector of the theory, and obtain the exact formula for the free energy of the superconductor in the holographic quantum field theory. Our result is analytic and it includes the effects of backreaction of the gravitational field. We further discuss on how this formula could be used to analyze second order phase transitions through the discontinuities of the free energy, in order to classify holographic superconductors in terms of the parameters in the theory.

Quantum field theory in de Sitter and quasi–de Sitter spacetimes revisited

It is possible to associate temperatures with the nonextremal horizons of a large class of spherically symmetric spacetimes using periodicity in the Euclidean sector, and this procedure works for the de Sitter spacetime as well. But unlike, e.g., the black hole spacetimes, the de Sitter spacetime also allows a description in Friedmann coordinates. This raises the question of whether the thermality of the de Sitter horizon can be obtained working entirely in the Friedmann coordinates, without reference to the static coordinates or using the symmetries of de Sitter spacetime. We discuss several aspects of this issue for de Sitter and approximately de Sitter spacetimes in the Friedmann coordinates (with a time-dependent background and the associated ambiguities in defining the vacuum states). The different choices for the vacuum states, the behavior of the mode functions and the detector response are studied in both ($1+1$) and ($1+3$) dimensions. We compare and contrast the differences brought about by the different choices. In the last part of the paper, we also describe a general procedure for studying quantum field theory in spacetimes which are approximately de Sitter and, as an example, derive the corrections to the thermal spectrum due to the presence of pressure-free matter.

Black hole motion in Euclidean space as a diffusion process. II

A diffusion equation approach to black hole thermodynamics in Euclidean sector is proposed. A diffusion equation for a generic Kerr-Newman black hole in Euclidean sector is derived from the Bloch equation. Black hole thermodynamics is also derived and it is found, in particular, that the entropy of a Kerr-Newman black hole is the same, apart from the logarithmic corrections, as the Bekenstein-Hawking entropy of the black hole.

New cases of the universality theorem for gravitational theories

The ‘universality theorem’ for gravity shows that f(R) theories (in their metric-affine formulation) in vacuum are dynamically equivalent to vacuum Einstein equations with suitable cosmological constants. This holds true for a generic (i.e. except sporadic degenerate cases) analytic function f(R), and standard gravity without a cosmological constant is reproduced if f is the identity function (i.e. f(R) = R). The theorem is here extended introducing in dimension 4 a one-parameter family of invariants βR inspired by the Barbero–Immirzi formulation of general relativity (GR) (which in the Euclidean sector includes also self-dual formulation). It will be proven that f(βR) theories so defined are dynamically equivalent to the corresponding metric-affine f(R) theory. In particular for the function f(R) = R the standard equivalence between GR and Holst Lagrangian is obtained.

Partition functions of three-dimensional quantum gravity and the black hole entropy

We analyze aspects of the holographic principle relevant to the quantum gravity partition functions in Euclidean sector of AdS3. The sum of known contributions to the partition functions can be presented exactly (including corrections) in the form where the Patterson-Selberg zeta function involves.

Semiclassical universe from first principles

Causal dynamical triangulations in four dimensions provide a background-independent definition of the sum over space–time geometries in non-perturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor.

The Tits boundary of a $\text {CAT}(0)$ 2-complex

We investigate the Tits boundary of CAT(0) 2-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

TOPOLOGICAL INTERPRETATION OF THE HORIZON TEMPERATURE

A class of metrics gab(xi) describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics gab(xi;λ)without horizons when λ→0. We construct specific examples in which the curvature corresponding gab(xi;λ) becomes a Dirac delta function and gets concentrated on the horizon when the limit λ→0 is taken, but the action remains finite. When the horizon is interpreted in this manner, one needs to remove the corresponding surface from the Euclidean sector, leading to winding numbers and thermal behavior. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [g00=ε2+a2x2; gμν=-δμν] or [g00=-gxx=(ε2+4a2x2)1/2, gyy=gzz=-1] when ε→0. In the Euclidean sector, the curvature gets concentrated on the origin of tE-x plane in a manner analogous to Aharanov–Bohm effect (in which the vector potential is a pure gauge everywhere except at the origin) and the curvature at the origin leads to nontrivial topological features and winding number.

Chern-Simons formulation of noncommutative gravity in three dimensions

We formulate noncommutative three-dimensional (3d) gravity by making use of its connection with 3d Chern-Simons theory. In the Euclidean sector, we consider the particular example of topology $T^2 \times R$ and show that the 3d black hole solves the noncommutative equations. We then consider the black hole on a constant U(1) background and show that the black hole charges (mass and angular momentum) are modified by the presence of this background.

Nonperturbative lorentzian path integral for gravity

We construct a well-defined regularized path integral for Lorentzian quantum gravity in terms of dynamically triangulated causal space-times. Each Lorentzian geometry and its action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time and, for finite lattice volume, the associated transfer matrix is self-adjoint, bounded, and strictly positive. The degenerate geometric phases found in dynamically triangulated Euclidean gravity are not present.

A problem with Euclidean preferences in spatial models of politics

Euclidean public sector preferences can not be induced from a strictly quasiconcave primitive utility function and a linear constraint.

Relativistic invariance and charge conjugation in quantum field theory

We prove that superselection sectors with finite statistics in the sense of Doplicher, Haag, and Roberts are automatically Poincare covariant under natural conditions (e.g. split property for space-like cones and duality for contractible causally complete regions). The same holds for topological charges, namely sectors localized in space-like cones, providing a converse to a theorem of Buchholz and Fredenhagen. We introduce the notion of weak conjugate sector that turns out to be equivalent to the DHR conjugate in finite statistics. The weak conjugate sector is given by an explicit formula that relates it to the PCT symmetry in a Wightman theory. Every Euclidean convariant sector (possibly with infinite statistics) has a weak conjugate sector and the converse is true under the above natural conditions. On the same basis, translation covariance is equivalent to the property that sectors are sheaf maps modulo inner automorphisms, for a certain sheaf structure given by the local algebras. The construction of the weak conjugate, sector also applies to the case of local algebras onS1 in conformal theories. Our main tools are the Bisognano-Wichmann description of the modular structure of the von Neumann algebras associated with wedge regions in the vacuum sector and the relation between Jones index theory for subfactors and the statistics of superselection sectors.