scholarly journals Some space-time integrability estimates of the solution for heat equations in two dimensions

2011 ◽  
Keyword(s):  
Space Time ◽  
Two Dimensions ◽  
Heat Equations

QUANTIZATION OF THE LIOUVILLE MODE AND STRING THEORY

1989 ◽  
Vol 04 (11) ◽  
pp. 1033-1041 ◽  
Author(s):  
SUMIT R. DAS ◽  
SATCHIDANANDA NAIK ◽  
SPENTA R. WADIA
Keyword(s):  
String Theory ◽  
Dimensional Space ◽  
Scalar Fields ◽  
Space Time ◽  
Two Dimensions ◽  
General Covariance ◽  
World Sheet ◽  
Lorentzian Signature ◽  
Dimensional Space Time ◽  
Background Fields

We discuss the space-time interpretation of bosonic string theories, which involve d scalar fields coupled to gravity in two dimensions, with a proper quantization of the world-sheet metric. We show that for d>25, the theory cannot describe string modes consistently coupled to each other. For d=25 this is possible; however, in this case the Liouville mode acts as an extra timelike variable and one really has a string moving in 26-dimensional space-time with a Lorentzian signature. By analyzing such a string theory in background fields, we show that the d=25 theory possesses the full 26-dimensional general covariance.

Bosonization in two dimensions: A space-time tunneling approach

1994 ◽  
Vol 50 (17) ◽  
pp. 12733-12743 ◽  
Author(s):  
D. Schmeltzer ◽  
A. R. Bishop
Keyword(s):  
Space Time ◽  
Two Dimensions

scholarly journals Enabling unification of the general theory of relativity and quantum electro-dynamics by visualizing a particle energy configuration

2016 ◽  
Vol 2 (11) ◽  
pp. 288
Author(s):  
William S. Oakley
Keyword(s):  
General Theory ◽  
Radial Strain ◽  
Space Time ◽  
Particle Energy ◽  
Two Dimensions ◽  
Emergent Properties ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Curved Space ◽  
Strong Force

<p class="abstract">The long standing major issue in physics has been the inability to unify the two main theories of quantum electro-dynamics (QED) and the general theory of relativity (GTR), both of which are well proven and cannot accommodate significant change. The problem is resolved by combining the precepts of GTR and QED in a conceptual model describing the electron as electromagnetic (EM) energy localized in relativistic quantum loops near an event horizon. EM energy is localized by propagating in highly curved space-time of closed geometry, the local metric index increases, and the energy is thus relativistic to the observer at velocity v &lt; c, with the curved space-time thereby evidencing gravity. The presence of gravity leads to the observer notion of mass. Particle energy is in dynamic equilibrium with relativistic loop circumferential metric strain at the strong force scale opposed by radial metric strain. The resulting particle is a quantum black hole with the circumferential strong force in the curved metric orthogonal in two dimensions to all particle radials. The presence of energy E is thus evident in observer space reduced by c<sup>2</sup> to E/c<sup>2</sup> = mass. The circumferential strain diminishes as it extends into the surrounding metric as the particle’s gravitational field. The radial strain projects outward into observer space and is therein evident as electric field. Gravity, unit charge, and their associated fields are emergent properties and Strong and electric forces are equal within the particle, quantizing gravity and satisfying the Planck scale criteria of force equality. A derived scaling factor produces the gravity effect experienced by the observer and the GRT-QED unification issue is thereby largely resolved.</p>

scholarly journals STRING HAMILTONIAN FROM GENERALIZED YANG–MILLS GAUGE THEORY IN TWO DIMENSIONS

2004 ◽  
Vol 19 (02) ◽  
pp. 205-225 ◽  
Author(s):  
FLORIAN DUBATH ◽  
SIMONE LELLI ◽  
ANNA RISSONE
Keyword(s):  
Gauge Theory ◽  
String Theory ◽  
Space Time ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Bosonic Field ◽  
Standard Case ◽  
Free Fermions ◽  
Yang Mills ◽  
Large N

Two-dimensional SU (N) Yang–Mills theory is known to be equivalent to a string theory, as found by Gross in the large N limit, using the 1/N expansion. Later it was found that even a generalized YM theory leads to a string theory of the Gross type. In the standard YM theory case, Douglas and others found the string Hamiltonian describing the propagation and the interactions of states made of strings winding on a cylindrical space–time. We address the problem of finding a similar Hamiltonian for the generalized YM theory. As in the standard case we start by writing the theory as a theory of free fermions. Performing a bosonization, we express the Hamiltonian in terms of the modes of a bosonic field, that are interpreted as in the standard case as creation and destruction operators for states of strings winding around the cylindrical space–time. The result is similar to the standard Hamiltonian, but with new kinds of interaction vertices.

An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations

2019 ◽  
Vol 29 (9) ◽  
pp. 3437-3453 ◽  
Author(s):  
Yunxia Sun ◽  
Xufeng Xiao ◽  
Zhiming Gao ◽  
Xinlong Feng
Keyword(s):  
Upper Bound ◽  
Operator Splitting ◽  
Splitting Method ◽  
Mass Conservation ◽  
Space Time ◽  
Time Operator ◽  
High Dimensional ◽  
Heat Equations ◽  
Content Type ◽  
Operator Splitting Method

Purpose The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations. Design/methodology/approach Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement. Findings The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method. Practical implications Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method. Originality/value The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

Manual interception of moving targets in two dimensions: Performance and space-time accuracy

2009 ◽  
Vol 1250 ◽  
pp. 202-217 ◽  
Author(s):  
J.R. Tresilian ◽  
A.M. Plooy ◽  
W. Marinovic
Keyword(s):  
Space Time ◽  
Two Dimensions ◽  
Moving Targets ◽  
Time Accuracy

scholarly journals A renormalisation group equation for transport coefficients in (2 + 1)-dimensions derived from the AdS/CMT correspondence

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Brian P. Dolan
Keyword(s):  
Magnetic Field ◽  
Charge Density ◽  
Transport Coefficients ◽  
Space Time ◽  
Two Dimensions ◽  
Renormalisation Group ◽  
Ac Conductivities ◽  
Group Equation ◽  
De Sitter ◽  
Renormalisation Group Equation

Abstract Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS4 space-time. The $$ \mathcal{Q} $$ Q -factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.

scholarly journals Fault-tolerant renormalization group decoder for abelian topological codes

2014 ◽  
Vol 14 (9&10) ◽  
pp. 721-740
Author(s):  
Guillaume Duclos-Cianci ◽  
David Poulin
Keyword(s):  
Renormalization Group ◽  
Failure Probability ◽  
Perfect Matching ◽  
Fault Tolerant ◽  
Three Dimensional ◽  
Space Time ◽  
Two Dimensions ◽  
Decoding Algorithm ◽  
Bit Flip ◽  
Toric Code

We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in \cite{DP09a,DP10a}. We also provide a complete detailed description of the structure of the algorithm, which should be sufficient for anyone interested in implementing it. This 3D implementation extends our previous 2D algorithm by incorporating a failure probability of the syndrome measurements, i.e., it enables fault-tolerant decoding. We report a fault-tolerant storage threshold of $\sim1.9(4)\%$ for Kitaev's toric code subject to a 3D bit-flip channel (i.e. including imperfect syndrome measurements). This number is to be compared with the $2.9\%$ value obtained via perfect matching \cite{H04a}. The 3D generalization inherits many properties of the 2D algorithm, including a complexity linear in the space-time volume of the memory, which can be parallelized to logarithmic time.

Continuous-Valued Cellular Automata in Two Dimensions

2003 ◽  
Author(s):  
Rudy Rucker
Keyword(s):  
Cellular Automata ◽  
Wave Equations ◽  
Reaction Diffusion ◽  
Space Time ◽  
Two Dimensions ◽  
Nonlinear Wave Equations ◽  
Flow Of Time ◽  
Action At A Distance ◽  
Necessary Evil ◽  
Update Process

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of nonlinear wave equations. In addition we cast some of Hans Meinhardt’s activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW, a. CA simulator, are presented. A cellular automaton, or CA, is a computation made up of finite elements called cells. Each cell contains the same type of state. The cells are updated in parallel, using a rule which is homogeneous, and local. In slightly different words, a CA is a computation based upon a grid of cells, with each cell containing an object called a state. The states are updated in discrete steps, with all the cells being effectively updated at the same time. Each cell uses the same algorithm for its update rule. The update algorithm computes a cell’s new state by using information about the states of the cell’s nearby space-time neighbors, that is, using the state of the cell itself, using the states of the cell’s nearby neighbors, and using the recent prior states of the cell and its neighbors. The states do not necessarily need to be single numbers, they can also be data structures built up from numbers. A CA is said to be discrete valued if its states are built from integers, and a CA is continuous valued if its states are built from real numbers. As Norman Margolus and Tommaso Toffoli have pointed out, CAs are well suited for modeling nature [7]. The parallelism of the CA update process mirrors the uniform flow of time. The homogeneity of the CA update rule across all the cells corresponds to the universality of natural law. And the locality of CAs reflect the fact that nature seems to forbid action at a distance. The use of finite space-time elements for CAs are a necessary evil so that we can compute at all. But one might argue that the use of discrete states is an unnecessary evil.

scholarly journals A Portable Demonstration of Orbits in Curved Space-Time

1990 ◽  
Vol 105 ◽  
pp. 223-224
Author(s):  
David B. Friend ◽  
Kevin Forkey
Keyword(s):  
Video Camera ◽  
Space Time ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Small Ball ◽  
Rubber Sheet ◽  
Curved Space ◽  
Circular Symmetry ◽  
Good Grasp ◽  
Metal Frame

Many introductory astronomy students, when confronted with the idea of curved space-time in discussions of relativity and cosmology, don’t have a very good grasp of what this really means. They are told that the presence of mass “curves” space-time, but it is often not clear to them what is meant by that statement. We have developed a simple portable demonstration of what is meant by curved space-time, by using a two-dimensional analog. We have stretched a thin rubber sheet over a circular metal frame which is supported a few inches above a table top. By placing a heavy weight (about a kilogram) in the center of the sheet, we can mimic the effect of the curvature of space-time around a massive object on an orbiting body (such as a planet orbiting a star). We roll a small ball around the sheet, and the ball describes approximately elliptical orbits around the weight (see the photograph). The orbits aren’t exactly elliptical for two reasons: energy is dissipated by friction between the ball and the sheet, and the sheet doesn’t quite have the correct shape (it is shaped roughly like In r instead of 1/r: see this by solving Laplace’s equation in two dimensions with circular symmetry). We like to project this demonstration with a video camera placed directly above the sheet, so that the students can see the “orbit” without actually seeing the curvature of the sheet.

Sign in / Sign up

Export Citation Format

Share Document