scholarly journals Flat $\mathfrak{so}(\,p,q)$ -connections for manifolds of non-Euclidean signature

2019 ◽  
Vol 36 (16) ◽  
pp. 167002
Author(s):  
Arash Ranjbar ◽  
Jorge Zanelli
Keyword(s):  
Euclidean Signature

scholarly journals Distributions in CFT. Part II. Minkowski space

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Petr Kravchuk ◽  
Jiaxin Qiao ◽  
Slava Rychkov
Keyword(s):  
Minkowski Space ◽  
Statistical Physics ◽  
Minimal Set ◽  
Local Commutativity ◽  
Classic Result ◽  
Lorentzian Signature ◽  
Open Question ◽  
Wightman Axioms ◽  
Euclidean Signature ◽  
3D Ising Model

Abstract CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ, $$ \overline{\rho} $$ ρ ¯ . We prove a key fact that |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ < 1 inside the forward tube, and set bounds on how fast |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ may tend to 1 when approaching the Minkowski space.We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

Euclidean Signature Curves

2011 ◽  
Vol 43 (3) ◽  
pp. 206-213 ◽  
Author(s):  
Mark S. Hickman
Keyword(s):  
Euclidean Signature

scholarly journals Covariant origin of the U(1)3 model for Euclidean quantum gravity

2021 ◽  
Author(s):  
Sepideh Bakhoda ◽  
Thomas Thiemann
Keyword(s):  
Gauge Theory ◽  
Quantum Gravity ◽  
Degrees Of Freedom ◽  
Hamiltonian Formulation ◽  
The Other ◽  
Topological Theory ◽  
Yang Mills ◽  
Consistent Model ◽  
Starting Point ◽  
Euclidean Signature

Abstract If one replaces the constraints of the Ashtekar-Barbero $SU(2)$ gauge theory formulation of Euclidean gravity by their $U(1)^3$ version, one arrives at a consistent model which captures significant structures of its $SU(2)$ version. In particular, it displays a non-trivial realisation of the hypersurface deformation algebra which makes it an interesting testing ground for (Euclidean) quantum gravity as has been emphasised in a recent series of papers due to Varadarajan et al. The simplification from SU(2) to U(1)$^3$ can be performed simply by hand within the Hamiltonian formulation by dropping all non-Abelian terms from the Gauss, spatial diffeomorphism, and Hamiltonian constraints respectively. However, one may ask from which Lagrangian formulation this theory descends. For the SU(2) theory it is known that one can choose the Palatini action, Holst action, or (anti-)selfdual action (Euclidean signature) as starting point all leading to equivalent Hamiltonian formulations. In this paper, we systematically analyse this question directly for the U(1)$^3$ theory. Surprisingly, it turns out that the Abelian analog of the Palatini or Holst formulation is a consistent but topological theory without propagating degrees of freedom. On the other hand, a twisted Abelian analog of the (anti-)selfdual formulation does lead to the desired Hamiltonian formulation. A new aspect of our derivation is that we work with 1. half-density valued tetrads which simplifies the analysis, 2. without the simplicity constraint (which admits one undesired solution that is usually neglected by hand) and 3. without imposing the time gauge from the beginning. As a byproduct, we show that also the non-Abelian theory admits a twisted (anti-)selfdual formulation. Finally, we also derive a pure connection formulation of Euclidean GR including a cosmological constant by extending previous work due to Capovilla, Dell, Jacobson, and Peldan which may be an interesting starting point for path integral investigations and displays (Euclidean) GR as a Yang-Mills theory with non-polynomial Lagrangian.

scholarly journals A note on boundary conditions in Euclidean gravity

2021 ◽  
pp. 2140004
Author(s):  
Edward Witten
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Perturbation Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Conformal Geometry ◽  
Extrinsic Curvature ◽  
Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

THE CONFORMAL FACTOR AND THE COSMOLOGICAL CONSTANT

1990 ◽  
Vol 05 (19) ◽  
pp. 3811-3829 ◽  
Author(s):  
STEVEN B. GIDDINGS
Keyword(s):  
Quantum Gravity ◽  
Cosmological Constant ◽  
Path Integral ◽  
Conformal Factor ◽  
Positive Cosmological Constant ◽  
Lorentzian Signature ◽  
Wrong Sign ◽  
Euclidean Signature

The issue of the conformal factor in quantum gravity is examined for Lorentzian signature spacetimes. In Euclidean signature, the “wrong” sign of the conformal action makes the path integral undefined, but in Lorentzian signature this sign is tied to the instability of gravity and once this is accounted for the path integral should be well-defined. In this approach it is not obvious that the Baum-Hawking-Coleman mechanism for suppression of the cosmological constant functions. It is conceivable that since the multiuniverse system exhibits an instability for positive cosmological constant, the dynamics should force the system to zero cosmological constant.

RIGID QED AND NONLINEAR SIGMA MODELS

1993 ◽  
Vol 08 (27) ◽  
pp. 2585-2592 ◽  
Author(s):  
M. AWADA ◽  
M. MA ◽  
DAVID ZOLLER
Keyword(s):  
Sigma Model ◽  
Beta Function ◽  
Nonlinear Sigma Model ◽  
Coulomb Interactions ◽  
Vacuum Fluctuation ◽  
Strongly Coupled ◽  
Weak Phase ◽  
Running Coupling ◽  
Minkowski Signature ◽  
Euclidean Signature

We show that in the absence of Coulomb interactions the kinetic theory of a recently proposed new model of strongly coupled QED behaves like an enhanced two-dimensional nonlinear sigma model with O(D+1) symmetry in Euclidean signature (which becomes an on-compact O(D, 1) symmetry in Minkowski signature). The beta-function is nontrivial in the absence of virtual fermions due to the non-perturbative vacuum fluctuation of the gauge field. In the presence of Coulomb the running coupling approaches an uv stable fixed point αc of order one. In the weak phase the curvature coupling increases at short distances and decreases at large distances.

scholarly journals A NEWMAN–PENROSE CALCULATOR FOR INSTANTON METRICS

2008 ◽  
Vol 19 (08) ◽  
pp. 1277-1290 ◽  
Author(s):  
TOLGA BIRKANDAN
Keyword(s):  
Exact Solutions ◽  
Vacuum Field ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Gravitational Instantons ◽  
Einstein's Field Equations ◽  
Symbolic Calculator ◽  
Euclidean Signature

We present a Maple11+GRTensorII based symbolic calculator for instanton metrics using Newman–Penrose formalism. Gravitational instantons are exact solutions of Einstein's vacuum field equations with Euclidean signature. The Newman–Penrose formalism, which supplies a toolbox for studying the exact solutions of Einstein's field equations, was adopted to the instanton case and our code translates it for the computational use.

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