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Euclidean Quantum Gravity
In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics. Introduction in layperson's terms The Wick rotation In physics, a Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to dynamics problems in n dimensions, by transposing their descriptions in n + 1 dimensions, by trading one dimension of space for one dimension of time. More precisely, it substitutes a mathematical problem in Minkowski space into a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. It is called a ''rotation'' because when complex numbers are represented as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of \pi/2 radians about the origin. For example, a Wick rotation coul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Quantum Gravity
In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity. Canonical quantization In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADM Formalism
The Arnowitt–Deser–Misner (ADM) formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959. The comprehensive review of the formalism that the authors published in 1962 has been reprinted in the journal '' General Relativity and Gravitation'', while the original papers can be found in the archives of '' Physical Review''. Overview The formalism supposes that spacetime is foliated into a family of spacelike surfaces \Sigma_t, labeled by their time coordinate t, and with coordinates on each slice given by x^i. The dynamic variables of this theory are taken to be the metric tensor of three-dimensional spatial slices \gamma_(t,x^k) and their conjugate momenta \pi^(t,x^k). Using these variables it is possible to define a Hamiltonian, and thereby write the equations of motion for g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Hilbert Action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the action is given as :S = \int R \sqrt \, \mathrm^4x, where g=\det(g_) is the determinant of the metric tensor matrix, R is the Ricci scalar, and \kappa = 8\pi Gc^ is the Einstein gravitational constant (G is the gravitational constant and c is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. If it does not converge, S is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler–Lagrange equation of the Einstein–Hilbert action. The action was proposed by David Hilbert in 1915 as part of his application of the variational principle to a combination of gravity and electromagnetism. Discussion Deriving equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric field on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Integral (QFT)
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a ''perfect'' vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is considerably lower than atmospheric pressure. The Latin term ''in vacuo'' is used to describe an object that is surrounded by a vacuum. The ''quality'' of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum. For example, a typical vacuum cleaner produces enough suction to reduce air pressur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Singularity
A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when”. Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete. General relativity predicts that any object collapsing beyond its Schwarzschild radius would form a black hole, inside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the Closure (topology), closure of not belonging to the Interior (topology), interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a Manifold#Manifold with boundary, different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
