1.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life

2.
Anti-de Sitter space
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In mathematics and physics, n-dimensional anti-de Sitter space is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. The anti-de Sitter space and de Sitter space are named after Willem de Sitter, professor of astronomy at Leiden University, Willem de Sitter and Albert Einstein worked together closely in the 1920s in Leiden on the spacetime structure of the universe. Einsteins theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space, the cases of spacetime of constant curvature are de Sitter space, Minkowski space, and anti-de Sitter space. As such, they are solutions of Einsteins field equations for an empty universe with a positive, zero, or negative cosmological constant. Anti-de Sitter space generalises to any number of space dimensions and this non-technical explanation first defines the terms used in the introductory material of this entry. Then, it sets forth the underlying idea of a general relativity-like spacetime. It also explains that Minkowski space, de Sitter space and anti-de Sitter space, as applied to general relativity, finally, it offers some caveats that describe in general terms how this non-technical explanation fails to capture the full detail of the mathematical concept. The space of special relativity is an example, negative curvature means curved hyperbolically, like a saddle surface or the Gabriels Horn surface, similar to that of a trumpet bell. It might be described as being the opposite of the surface of a sphere, general relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from the presence of matter or energy. Energy and matter are equivalent, and space and time can be translated into equivalent units based on the speed of light, of course, in general relativity, both the small and large objects mutually influence the curvature of spacetime. The attractive force of gravity created by matter is due to a curvature of spacetime. As a result, in relativity, the familiar Newtonian equation of gravity F = G m 1 m 2 r 2 is merely an approximation of the gravity-like effects seen in general relativity. However this approximation becomes inaccurate in extreme physical situations, for example, in general relativity, objects in motion have a slightly different gravitation effect than objects at rest. In normal circumstances, gravity bends time so slightly that the differences between Newtonian gravity and general relativity are detectable only with precise instruments, de Sitter space involves a variation of general relativity in which spacetime is slightly curved in the absence of matter or energy. This is analogous to the relationship between Euclidean geometry and non-Euclidean geometry, an intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity. This corresponds to the vacuum having a density and pressure. This spacetime geometry results in initially parallel timelike geodesics diverging, with spacelike sections having positive curvature, an anti-de Sitter space in general relativity is similar to a de Sitter space, except with the sign of the curvature changed. This corresponds to a cosmological constant

3.
Cosmological constant
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In cosmology, the cosmological constant is the value of the energy density of the vacuum of space. It was originally introduced by Albert Einstein in 1917 as an addition to his theory of relativity to hold back gravity and achieve a static universe. Einstein abandoned the concept after Hubbles 1929 discovery that all galaxies outside the Local Group are moving away from each other, from 1929 until the early 1990s, most cosmology researchers assumed the cosmological constant to be zero. When Λ is zero, this reduces to the field equation of general relativity. When T is zero, the equation describes empty space. The cosmological constant has the effect as an intrinsic energy density of the vacuum. In this context, it is moved onto the right-hand side of the equation, and defined with a proportionality factor of 8π, Λ = 8πρvac. It is common to quote values of energy density directly, though using the name cosmological constant. A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, if the energy density is positive, the associated negative pressure will drive an accelerated expansion of the universe, as observed. This ratio is usually denoted ΩΛ, and is estimated to be 0. 6911±0.0062, according to results published by the Planck Collaboration in 2015. In a flat universe ΩΛ is the fraction of the energy of the due to the cosmological constant. Another ratio that is used by scientists is the equation of state, usually denoted w and this ratio is w = −1 for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as quintessence. To counteract this possibility, Einstein added the cosmological constant, likewise, a universe that contracts slightly will continue contracting. However, the cosmological constant remained a subject of theoretical and empirical interest, empirically, the onslaught of cosmological data in the past decades strongly suggests that our universe has a positive cosmological constant. The explanation of this small but positive value is a theoretical challenge. Observations announced in 1998 of distance–redshift relation for Type Ia supernovae indicated that the expansion of the universe is accelerating. When combined with measurements of the microwave background radiation these implied a value of ΩΛ ≈0.7. There are other causes of an accelerating universe, such as quintessence

4.
Topological quantum field theory
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A topological quantum field theory is a quantum field theory which computes topological invariants. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to field theory. In a topological field theory, the correlation functions do not depend on the metric of spacetime and this means that the theory is not sensitive to changes in the shape of spacetime, if the spacetime warps or contracts, the correlation functions do not change. Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics, Minkowski space can be contracted to a point, so a TQFT on Minkowski space computes only trivial topological invariants. Consequently, TQFTs are usually studied on curved spacetimes, such as, for example, most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few theories exist, but they are not very well understood. Quantum gravity is believed to be background-independent, and TQFTs provide examples of independent quantum field theories. This has prompted ongoing theoretical investigation of this class of models, the known topological field theories fall into two general classes, Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also referred to as cohomological field theories. In Schwarz-type TQFTs, the functions or partition functions of the system are computed by the path integral of metric independent action functionals. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action is S = ∫ M B F The spacetime metric does not appear anywhere in the theory, the first example appeared in 1977 and is due to A. Schwarz, its action functional is, ∫ M A ∧ d A. Another more famous example is Chern–Simons theory, which can be used to compute knot invariants, in general partition functions depend on a metric but the above examples are shown to be metric-independent. The first example of the field theories of Witten-type appeared in Wittens paper in 1988. Though its action functional contains the spacetime metric gαβ, after a topological twist it turns out to be metric independent, the independence of the stress-energy tensor Tαβ of the system from the metric depends on whether BRST-operator is closed. Following Wittens example a lot of examples are found in string theory,4. The stress-energy-tensor is of the form T α β = δ G α β for an arbitrary tensor G α β. As an example given a 2-form field B with the differential operator δ which satisfies δ2 =0, the expression δ δ B α β S is proportional to δ G with another 2-form G. In the third equality it was used the fact that δ O i = δ S =0, since ∫ d μ O i G e i S is only a number, the Lie derivative applied on it vanishes

5.
Edward Witten
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Edward Witten is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, in addition to his contributions to physics, Wittens work has significantly impacted pure mathematics. In 1990 he became the first and so far the only physicist to be awarded a Fields Medal by the International Mathematical Union, in 2004, Time magazine stated that Witten is widely thought to be the worlds smartest living theoretical physicist. Witten was born in Baltimore, Maryland, to a Jewish family and he is the son of Lorraine Witten and Louis Witten, a theoretical physicist specializing in gravitation and general relativity. Witten attended the Park School of Baltimore, and received his Bachelor of Arts with a major in history and he published articles in The New Republic and The Nation. In 1968, Witten published an article in The Nation arguing that the New Left had no strategy and he worked briefly for George McGoverns presidential campaign. Witten attended the University of Wisconsin–Madison for one semester as a graduate student before dropping out. He held a fellowship at Harvard University, visited Oxford University, was a fellow in the Harvard Society of Fellows. Witten was awarded the Fields Medal by the International Mathematical Union in 1990, Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems. E has made an impact on contemporary mathematics. In his hands physics is once again providing a source of inspiration. As an example of Wittens work in mathematics, Atiyah cites his application of techniques from quantum field theory to the mathematical subject of low-dimensional topology. In particular, Witten realized that a theory now called Chern–Simons theory could provide a framework for understanding the mathematical theory of knots. Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the energy theorem in general relativity. This theorem asserts that the energy of a gravitating system is always positive. It establishes Minkowski space as a ground state of the gravitational field. While the original proof of this due to Richard Schoen and Shing-Tung Yau used variational methods. Wittens work gave a proof of a classical result, the Morse inequalities

6.
Gauge theory
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In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is a model that holds no matter the mathematical procedure applied to it. This is the concept behind gauge invariance, the idea of fields as described by Michael Faraday in his study of electromagnetism led to the postulate that fields could be described mathematically as scalars and vectors. When a field is transformed, but the result is not, applying gauge theory creates a unification which describes mathematical formulas or models that hold good for all fields of the same class. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian, the transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators, for each group generator there necessarily arises a corresponding field called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations, when such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the theory is referred to as non-abelian. Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups, when they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stricter constraint, in fact, a global symmetry is just a local symmetry whose groups parameters are fixed in spacetime. Gauge theories are important as the field theories explaining the dynamics of elementary particles. Quantum electrodynamics is a gauge theory with the symmetry group U and has one gauge field. The Standard Model is a gauge theory with the symmetry group U×SU×SU and has a total of twelve gauge bosons. Gauge theories are important in explaining gravitation in the theory of general relativity. Its case is unusual in that the gauge field is a tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton, both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of covariance with a true gauge principle with new gauge fields. Historically, these ideas were first stated in the context of classical electromagnetism, however, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below

7.
Solvable group
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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a group is a group whose derived series terminates in the trivial subgroup. Historically, the word solvable arose from Galois theory and the proof of the unsolvability of quintic equation. Specifically, an equation is solvable by radicals if and only if the corresponding Galois group is solvable. Or equivalently, if its derived series, the normal series G ▹ G ▹ G ▹ ⋯. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the least n such that G = is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a group has finite composition length. The Jordan–Hölder theorem guarantees that if one composition series has this property, for the Galois group of a polynomial, these cyclic groups correspond to nth roots over some field. All abelian groups are trivially solvable – a subnormal series being given by just the group itself, but non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable, in particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3, in fact, as the smallest simple non-abelian group is A5, it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a series, giving factor groups isomorphic to A5 and C2. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n >4, we see that Sn is not solvable for n >4. This is a key step in the proof that for every n >4 there are polynomials of n which are not solvable by radicals. This property is used in complexity theory in the proof of Barringtons theorem. The celebrated Feit–Thompson theorem states that every group of odd order is solvable. In particular this implies that if a group is simple

8.
Quantum gravity
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Quantum gravity is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored. The current understanding of gravity is based on Albert Einsteins general theory of relativity, the necessity of a quantum mechanical description of gravity is sometimes said to follow from the fact that one cannot consistently couple a classical system to a quantum one. This is false as is shown, for example, by Walds explicit construction of a consistent semiclassical theory, the problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, a theory of quantum gravity that is also a grand unification of all known interactions is sometimes referred to as The Theory of Everything. As a result, quantum gravity is a mainly theoretical enterprise, much of the difficulty in meshing these theories at all energy scales comes from the different assumptions that these theories make on how the universe works. Quantum field theory, if conceived of as a theory of particles, General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves. Historically, the most obvious way of combining the two ran quickly into what is known as the renormalization problem, another possibility is to focus on fields rather than on particles, which are just one way of characterizing certain fields in very special spacetimes. This solves worries about consistency, but does not appear to lead to a version of full general theory of relativity. Quantum gravity can be treated as a field theory. Effective quantum field theories come with some high-energy cutoff, beyond which we do not expect that the theory provides a description of nature. The infinities then become large but finite quantities depending on this finite cutoff scale and this same logic works just as well for the highly successful theory of low-energy pions as for quantum gravity. Indeed, the first quantum-mechanical corrections to graviton-scattering and Newtons law of gravitation have been explicitly computed. In fact, gravity is in ways a much better quantum field theory than the Standard Model. Specifically, the problem of combining quantum mechanics and gravity becomes an issue only at high energies. This problem must be put in the context, however. While there is no proof of the existence of gravitons. The predicted find would result in the classification of the graviton as a force similar to the photon of the electromagnetic field. Many of the notions of a unified theory of physics since the 1970s assume, and to some degree depend upon

9.
Killing form
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In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. The Killing form was introduced into Lie algebra theory by Élie Cartan in his thesis. The name Killing form first appeared in a paper of Armand Borel in 1951, Borel admits that the name seems to be a misnomer, and that it would be more correct to call it the Cartan form. A basic result Cartan made use of was Cartans criterion, which states that the Killing form is non-degenerate if, consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad of g with the help of the Lie bracket, as a d =. Now, supposing g is of dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form B = t r a c e, with values in K. The Killing form B is bilinear and symmetric, the Killing form is an invariant form, in the sense that it has the associativity property B = B, where is the Lie bracket. If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a multiple of the Killing form. The Killing form is also invariant under automorphisms s of the algebra g, the Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate. The Killing form of a nilpotent Lie algebra is identically zero, if I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form. The orthogonal complement with respect to B of an ideal is again an ideal, if a given Lie algebra g is a direct sum of its ideals I1. In, then the Killing form of g is the direct sum of the Killing forms of the individual summands. Given a basis ei of the Lie algebra g, the elements of the Killing form are given by B i j = t r / I a d where Iad is the Dynkin index of the adjoint representation of g. Here = = = c i m n c j k m e n in Einstein summation notation, the index k functions as column index and the index n as row index in the matrix adad. In the above indexed definition, we are careful to distinguish upper and lower indices. This is because, in cases, the Killing form can be used as a metric tensor on a manifold. When the Lie algebra is semisimple over a field, its Killing form is nondegenerate. In this case, it is possible to choose a basis for g such that the structure constants with all upper indices are completely antisymmetric. The Killing form for some Lie algebras g are, Suppose that g is a semisimple Lie algebra over the field of real numbers R, by Cartans criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries ±1

10.
AdS/CFT correspondence
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On one side are anti-de Sitter spaces which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories which are quantum field theories, the duality represents a major advance in our understanding of string theory and quantum gravity. It also provides a toolkit for studying strongly coupled quantum field theories. This fact has been used to study aspects of nuclear. The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997, important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov, and by Edward Witten. By 2015, Maldacenas article had over 10,000 citations and our current understanding of gravity is based on Albert Einsteins general theory of relativity. Formulated in 1915, general relativity explains gravity in terms of the geometry of space and time and it is formulated in the language of classical physics developed by physicists such as Isaac Newton and James Clerk Maxwell. The other nongravitational forces are explained in the framework of quantum mechanics, developed in the first half of the twentieth century by a number of different physicists, quantum mechanics provides a radically different way of describing physical phenomena based on probability. Quantum gravity is the branch of physics that seeks to describe gravity using the principles of quantum mechanics, currently, the most popular approach to quantum gravity is string theory, which models elementary particles not as zero-dimensional points but as one-dimensional objects called strings. In the AdS/CFT correspondence, one typically considers theories of quantum gravity derived from string theory or its modern extension, in everyday life, there are three familiar dimensions of space, and there is one dimension of time. Thus, in the language of physics, one says that spacetime is four-dimensional. The quantum gravity theories appearing in the AdS/CFT correspondence are typically obtained from string and this produces a theory in which spacetime has effectively a lower number of dimensions and the extra dimensions are curled up into circles. A standard analogy for compactification is to consider an object such as a garden hose. Thus, an ant crawling inside it would move in two dimensions, the application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles. In the AdS/CFT correspondence, one considers, in addition to a theory of quantum gravity and this is a particularly symmetric and mathematically well behaved type of quantum field theory. In the AdS/CFT correspondence, one considers string theory or M-theory on an anti-de Sitter background and this means that the geometry of spacetime is described in terms of a certain vacuum solution of Einsteins equation called anti-de Sitter space. It is closely related to space, which can be viewed as a disk as illustrated on the right

11.
Monster group
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The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, the Monster group contains all but six of the other sporadic groups as subquotients. Robert Griess has called these 6 exceptions pariahs, and refers to the other 20 as the happy family and it is difficult to make a good constructive definition of the Monster because of its complexity. Martin Gardner wrote an account of the monster group in his June 1980 Mathematical Games column in Scientific American. The Monster was predicted by Bernd Fischer and Robert Griess as a group containing a double cover of Fischers Baby Monster group as a centralizer of an involution. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and it was not clear in the 1970s that the Monster actually existed. Griess constructed M as the group of the Griess algebra. In his 1982 paper he referred to the Monster as the Friendly Giant, John Conway and Jacques Tits subsequently simplified this construction. Griesss construction showed that the Monster existed, Thompson showed that its uniqueness would follow from the existence of a 196, 883-dimensional faithful representation. A proof of the existence of such a representation was announced by Norton, Griess, Meierfrankenfeld & Segev gave the first complete published proof of the uniqueness of the Monster. The Monster was a culmination of a development of simple groups and can be built from any 2 of 3 subquotients, the Fischer group Fi24, the Baby Monster. The Schur multiplier and the automorphism group of the Monster are both trivial. The minimal degree of a complex representation is 196,883. The smallest faithful linear representation over any field has dimension 196,882 over the field with 2 elements, the smallest faithful permutation representation of the Monster is on 24 ·37 ·53 ·74 ·11 ·132 ·29 ·41 ·59 ·71 points. The Monster can be realized as a Galois group over the rational numbers, the Monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of small representations, for example, the simple groups A100 and SL20 are far larger, but easy to calculate with as they have small permutation or linear representations. All sporadic groups other than the Monster also have linear representations small enough that they are easy to work with on a computer. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes

12.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized

13.
Gravitational anomaly
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The adjective gravitational is derived from the symmetry of a gravitational theory, namely from general covariance. A gravitational anomaly is generally synonmous with diffeomorphism anomaly, since general covariance is symmetry under coordinate reparametrization, general covariance is the basis of general relativity, the current theory of gravitation. Therefore, all gravitational anomalies must cancel out, the anomaly usually appears as a Feynman diagram with a chiral fermion running in the loop with n external gravitons attached to the loop where n =1 + D /2 where D is the spacetime dimension. Field-theoretic pure gravitational anomalies occur only in even spacetime dimensions, however, diffeomorphism anomalies can occur in the case of an odd-dimensional spacetime manifold with boundary. Consider a classical gravitational field represented by the vielbein e μ a and a quantized Fermi field ψ. Einstein anomaly δ ξ W = − ∫ d 4 x e ξ ν, weyl anomaly δ σ W = ∫ d 4 x e σ ⟨ T μ μ ⟩, which indicates that the trace is non-zero. Mixed anomaly Green–Schwarz mechanism Alvarez-Gaumé, Luis, Edward Witten

14.
Holographic principle
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First proposed by Gerard t Hooft, it was given a precise string-theory interpretation by Leonard Susskind who combined his ideas with previous ones of t Hooft and Charles Thorn. As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. Cosmological holography has not been made mathematically precise, partly because the horizon has a non-zero area. The holographic principle was inspired by black hole thermodynamics, which conjectures that the entropy in any region scales with the radius squared. In the case of a hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory, however, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called Wheelers bags of gold, the existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood. An object with relatively high entropy is microscopically random, like a hot gas, a known configuration of classical fields has zero entropy, there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einsteins equations, they were not to have any entropy either. But Jacob Bekenstein noted that this leads to a violation of the law of thermodynamics. If one throws a hot gas with entropy into a hole, once it crosses the event horizon. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. One way of salvaging the second law is if black holes are in random objects with an entropy that increases by an amount greater than the entropy of the consumed gas. Bekenstein assumed that black holes are maximum entropy objects—that they have more entropy than anything else in the same volume, in a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The only known limit is gravitational, when there is too much energy the gas collapses into a black hole, Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is proportional to the area of the event horizon. Stephen Hawking had shown earlier that the horizon area of a collection of black holes always increases with time. The horizon is a boundary defined by light-like geodesics, it is those light rays that are just barely unable to escape, if neighboring geodesics start moving toward each other they eventually collide, at which point their extension is inside the black hole

15.
Quantum field theory in curved spacetime
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In particle physics, quantum field theory in curved spacetime is an extension of standard, Minkowski space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields, for non-zero cosmological constants, on curved spacetimes quantum fields lose their interpretation as asymptotic particles. Only in certain situations, such as in asymptotically flat spacetimes, can the notion of incoming and outgoing particle be recovered, even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer. Another observation is that unless the metric tensor has a global timelike Killing vector. The concept of a vacuum is not invariant under diffeomorphisms and this is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If t′ is a diffeomorphism, in general, the Fourier transform of exp will contain negative frequencies even if k >0, creation operators correspond to positive frequencies, while annihilation operators correspond to negative frequencies. This is why a state which looks like a vacuum to one observer cannot look like a state to another observer. Indeed, the viewpoint of local quantum physics is suitable to generalize the procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained, the most striking application of the theory is Hawkings prediction that Schwarzschild black holes radiate with a thermal spectrum. A related prediction is the Unruh effect, accelerated observers in the measure an thermal bath of particles. This formalism is used to predict the primordial density perturbation spectrum arising from cosmic inflation. Since this spectrum is measured by a variety of cosmological measurements—such as the CMB - if inflation is correct this particular prediction of the theory has already been verified, the Dirac equation can be formulated in curved spacetime, see Dirac equation in curved spacetime for details. The theory of field theory in curved spacetime can be considered as a first approximation to quantum gravity. A second step towards that theory would be semiclassical gravity, which would include the influence of particles created by a gravitational field on the spacetime. However gravity is not renormalizable in QFT, so merely formulating QFT in curved spacetime is not a theory of quantum gravity. Field Statistical field theory Topological quantum field theory Local quantum field theory General relativity Quantum geometry Quantum spacetime Quantum field theory N. D. Birrell & P. C. W, aspects of quantum field theory in curved space-time. Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quantum field theory in curved space-time and black hole thermodynamics. Quantum Field Theory in Curved Spacetime, Local Wick polynomials and time ordered products of quantum fields in curved space-time

16.
Hawking radiation
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Hawking radiation is blackbody radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. Hawking radiation reduces the mass and energy of black holes and is also known as black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink, micro black holes are predicted to be larger net emitters of radiation than larger black holes and should shrink and dissipate faster. In June 2008, NASA launched the Fermi space telescope, which is searching for the terminal gamma-ray flashes expected from evaporating primordial black holes. In the event that speculative large extra dimension theories are correct, CERNs Large Hadron Collider may be able to create black holes. In September 2010, a signal that is related to black hole Hawking radiation was claimed to have been observed in a laboratory experiment involving optical light pulses. However, the results remain unverified and debatable, other projects have been launched to look for this radiation within the framework of analog gravity. Black holes are sites of immense gravitational attraction, classically, the gravitation is so powerful that nothing, not even electromagnetic radiation, can escape from the black hole. It is yet unknown how gravity can be incorporated into quantum mechanics, nevertheless, far from the black hole the gravitational effects can be weak enough for calculations to be reliably performed in the framework of quantum field theory in curved spacetime. Hawking showed that quantum effects allow black holes to emit exact black body radiation, the electromagnetic radiation is produced as if emitted by a black body with a temperature inversely proportional to the mass of the black hole. Physical insight into the process may be gained by imagining that particle–antiparticle radiation is emitted from just beyond the event horizon. This radiation does not come directly from the hole itself. As the particle–antiparticle pair was produced by the holes gravitational energy. An alternative view of the process is that vacuum fluctuations cause a particle–antiparticle pair to appear close to the event horizon of a black hole, one of the pair falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the hole must have had a negative energy. This causes the black hole to lose mass, and, to an outside observer, in another model, the process is a quantum tunnelling effect, whereby particle–antiparticle pairs will form from the vacuum, and one will tunnel outside the event horizon. This leads to the black hole information paradox, however, according to the conjectured gauge-gravity duality, black holes in certain cases are equivalent to solutions of quantum field theory at a non-zero temperature. This means that no information loss is expected in black holes, if this is correct, then Hawkings original calculation should be corrected, though it is not known how

17.
Black hole
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A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. The theory of relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon, although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways a black hole acts like a black body. Moreover, quantum theory in curved spacetime predicts that event horizons emit Hawking radiation. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. Black holes were considered a mathematical curiosity, it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality, black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings, by absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus that supermassive black holes exist in the centers of most galaxies, despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter that falls onto a black hole can form an accretion disk heated by friction. If there are other stars orbiting a black hole, their orbits can be used to determine the black holes mass, such observations can be used to exclude possible alternatives such as neutron stars.3 million solar masses. On 15 June 2016, a detection of a gravitational wave event from colliding black holes was announced. The idea of a body so massive that light could not escape was briefly proposed by astronomical pioneer John Michell in a letter published in 1783-4. Michell correctly noted that such supermassive but non-radiating bodies might be detectable through their effects on nearby visible bodies. In 1915, Albert Einstein developed his theory of general relativity, only a few months later, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the solution for the point mass. This solution had a peculiar behaviour at what is now called the Schwarzschild radius, the nature of this surface was not quite understood at the time

18.
Black hole complementarity
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Black hole complementarity is a conjectured solution to the black hole information paradox, proposed by Leonard Susskind and Larus Thorlacius, and Gerard t Hooft. But how can this be possible if information cannot escape the event horizon without traveling faster than light and this seems to rule out Hawking radiation as the carrier of the missing information. It also appears as if information cannot be reflected at the event horizon as there is nothing special about it locally. According to an observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon. He also postulated a stretched horizon, which is a membrane hovering about a Planck length outside the event horizon, according to the external observer, infalling information heats up the stretched horizon, which then reradiates it as Hawking radiation, with the entire evolution being unitary. However, according to an observer, nothing special happens at the event horizon itself. This isnt to say there are two copies of the lying about — one at or just outside the horizon. Instead, an observer can only detect the information at the horizon itself, or inside, complementarity is a feature of the quantum mechanics of noncommuting observables, and Susskind proposed that both stories are complementary in the quantum sense. To an infalling observer, information and entropy pass through the horizon with nothing strange happening, to an external observer, the information and entropy is absorbed into the stretched horizon which acts like a dissipative fluid with entropy, viscosity and electrical conductivity. See the membrane paradigm for more details, the stretched horizon is conducting with surface charges which rapidly spread out over the horizon. Global symmetries dont exist in quantum gravity, baryon number is violated, but only at very small scales, and the proton has a very long lifetime. But with a short enough time resolution, the proton oscillates between different baryon numbers and the time warping near the horizon magnifies that, alternatively, the hot temperatures of the stretched horizon cause the proton to decay. But an infalling observer never has time to see the proton decay, recently, it appears that black hole complementarity combined with the monogamy of entanglement suggests the existence of a firewall

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Black hole information paradox
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The black hole information paradox is a puzzle resulting from the combination of quantum mechanics and general relativity. Calculations suggest that information could permanently disappear in a black hole. A fundamental postulate of the Copenhagen interpretation of quantum mechanics is that information about a system is encoded in its wave function up to when the wave function collapses. The evolution of the function is determined by a unitary operator. There are two principles in play, Quantum determinism means that given a present wave function, its future changes are uniquely determined by the evolution operator. Reversibility refers to the fact that the operator has an inverse. The combination of the two means that information must always be preserved, specifically, Hawkings calculations indicated that black hole evaporation via Hawking radiation does not preserve information. Today, many believe that the holographic principle demonstrates that Hawkings conclusion was incorrect. In 2004 Hawking himself conceded a bet he had made, agreeing that black hole evaporation does in fact preserve information, in 1975, Stephen Hawking and Jacob Bekenstein showed that black holes should slowly radiate away energy, which poses a problem. From the no-hair theorem, one would expect the Hawking radiation to be independent of the material entering the black hole. This violates Liouvilles theorem and presents a physical paradox, but since everything within the interior of the black hole will hit the singularity within a finite time, the part which is traced over partially might disappear completely from the physical system. Hawking remained convinced that the equations of black-hole thermodynamics together with the no-hair theorem led to the conclusion that quantum information may be destroyed and this annoyed many physicists, notably John Preskill, who bet Hawking and Kip Thorne in 1997 that information was not lost in black holes. The solution to the problem that concluded the battle is the holographic principle, with this, Susskind quashes Hawking in quarrel over quantum quandary. There are various ideas about how the paradox is solved and his argument assumes the unitarity of the AdS/CFT correspondence which implies that an AdS black hole that is dual to a thermal conformal field theory. When announcing his result, Hawking also conceded the 1997 bet, according to Roger Penrose, loss of unitarity in quantum systems is not a problem, quantum measurements are by themselves already non-unitary. Penrose claims that quantum systems will in no longer evolve unitarily as soon as gravitation comes into play. The Conformal Cyclic Cosmology advocated by Penrose critically depends on the condition that information is in fact lost in black holes, the significance of the findings was subsequently debated by others. Information is irretrievably lostAdvantage, Seems to be a consequence of relatively non-controversial calculation based on semiclassical gravity

20.
Black-hole thermodynamics
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In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. The second law of thermodynamics requires that black holes have entropy, if black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the hole more than compensates for the decrease of the entropy carried by the object that was swallowed. Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. In 1973 Bekenstein suggested /4 π as the constant of proportionality, asserting that if the constant was not exactly this, the next year in 1974, Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature. This is often referred to as the Bekenstein–Hawking formula, the subscript BH either stands for black hole or Bekenstein-Hawking. The black hole entropy is proportional to the area of its event horizon A, the fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound was the main observation that led to the holographic principle. In fact, so called no hair theorems appeared to suggest that black holes could have only a single microstate, various studies are in progress, but this has not yet been elucidated. In Loop quantum gravity it is possible to associate a geometrical interpretation to the microstates, LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. It is possible to derive, from the covariant formulation of quantum theory the correct relation between energy and area, the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes, there seems to be also discussed the calculation of Bekenstein-Hawking entropy from the point of view of LQG. The four laws of black hole mechanics are physical properties that black holes are believed to satisfy, the laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking, and James Bardeen. The laws of black hole mechanics are expressed in geometrized units, the horizon has constant surface gravity for a stationary black hole. The horizon area is, assuming the weak condition, a non-decreasing function of time. This law was superseded by Hawkings discovery that black holes radiate and it is not possible to form a black hole with vanishing surface gravity. κ =0 is not possible to achieve, the zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the gravity is analogous to temperature. T constant for thermal equilibrium for a system is analogous to κ constant over the horizon of a stationary black hole

21.
ER=EPR
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ER=EPR is a conjecture in physics stating that entangled particles are connected by a wormhole. The conjecture was proposed by Leonard Susskind and Juan Maldacena in 2013 and they proposed that a nontraversable wormhole is equivalent to a pair of maximally entangled black holes. The symbol is derived from the first letters of the surnames of authors who wrote the first paper on wormholes, the two papers were published in 1935, but the authors did not claim any connection between the concepts. This is a resolution to the AMPS firewall paradox. Whether or not there is a firewall depends upon what is thrown into the other distant black hole, however, as the firewall lies inside the event horizon, no external superluminal signalling would be possible. They backed up their conjecture by showing that the production of charged black holes in a background magnetic field leads to entangled black holes. Susskind and Maldacena envisioned gathering up all the Hawking particles and smushing them together until they collapse into a black hole and that black hole would be entangled, and thus connected via wormhole, with the original black hole. That trick transformed a confusing mess of Hawking particles — paradoxically entangled with both a hole and each other — into two black holes connected by a wormhole. Entanglement overload is averted, and the problem goes away. This conjecture sits uncomfortably with the linearity of quantum mechanics, an entangled state is a linear superposition of separable states. Presumably, separable states are not connected by any wormholes, the conjecture leads to a grander conjecture that the geometry of space, time and gravity is determined by entanglement. ER = EPR or Whats Behind the Horizons of Black Holes

22.
Gravitational singularity
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The quantities used to measure gravitational field strength are the scalar invariant curvatures of space-time, which includes a measure of the density of matter. Since such quantities become infinite within the singularity, the laws of normal space-time could not exist, the Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity, according to modern general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity. Many theories in physics have mathematical singularities of one kind or another, equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a piece in the theory, as in the Ultraviolet Catastrophe, re-normalization. Some theories, such as the theory of quantum gravity suggest that singularities may not exist. A conical singularity occurs when there is a point where the limit of every diffeomorphism invariant quantity is finite, thus, space-time looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable system is used. An example of such a singularity is a cosmic string. Solutions to the equations of general relativity or another theory of gravity often result in encountering points where the metric blows up to infinity, however, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, such quantities are the same in every coordinate system, so these infinities will not go away by a change of coordinates. An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole, in coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, space-time at the event horizon is regular, the regularity becomes evident when changing to another coordinate system, where the metric is perfectly smooth. On the other hand, in the center of the hole, where the metric becomes infinite as well. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i. e. R μ ν ρ σ R μ ν ρ σ, such a singularity may also theoretically become a wormhole. For example, any observer inside the event horizon of a black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time, extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite space-time curvature. Until the early 1990s, it was believed that general relativity hides every singularity behind an event horizon

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String theory
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In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how strings propagate through space and interact with each other. On distance scales larger than the scale, a string looks just like an ordinary particle, with its mass, charge. In string theory, one of the vibrational states of the string corresponds to the graviton. Thus string theory is a theory of quantum gravity, String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. Despite much work on problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows to choose the details. String theory was first studied in the late 1960s as a theory of the nuclear force. Subsequently, it was realized that the properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, one of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and these issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics, one of these frameworks was Albert Einsteins general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other was quantum mechanics, a different formalism for describing physical phenomena using probability. In spite of successes, there are still many problems that remain to be solved. One of the deepest problems in physics is the problem of quantum gravity. The general theory of relativity is formulated within the framework of classical physics, in addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe. String theory is a framework that attempts to address these questions

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Bosonic string theory
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Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it only contains bosons in the spectrum, in the 1980s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus. Although bosonic string theory has many features, it falls short as a viable physical model in two significant areas. First, it only the existence of bosons whereas many physical particles are fermions. Second, it predicts the existence of a mode of the string with imaginary mass, in addition, bosonic string theory in a general spacetime dimension displays inconsistencies due to the conformal anomaly. But, as was first noticed by Claud Lovelace, in a spacetime of 26 dimensions, the dimension for the theory. This would leave only the four dimensions of spacetime visible to low energy experiments. The existence of a dimension where the anomaly cancels is a general feature of all string theories. There are four possible bosonic string theories, depending on whether open strings are allowed, recall that a theory of open strings also must include closed strings, open strings can be thought as having their endpoints fixed on a D25-brane that fills all of spacetime. A specific orientation of the means that only interaction corresponding to an orientable worldsheet are allowed. A sketch of the spectra of the four theories is as follows, Note that all four theories have a negative energy tachyon. The rest of this article applies to the closed, oriented theory, corresponding to borderless, G is the metric on the target spacetime, which is usually taken to be the Minkowski metric in the perturbative theory. Under a Wick rotation, this is brought to a Euclidean metric G μ ν = δ μ ν, M is the worldsheet as a topological manifold parametrized by the ξ coordinates. T is the tension and related to the Regge slope as T =12 π α ′. I0 has diffeomorphism and Weyl invariance, a normalization factor N is introduced to compensate overcounting from symmetries. While the computation of the partition function correspond to the cosmological constant, the symmetry group of the action actually reduces drastically the integration space to a finite dimensional manifold. One still has to quotient away diffeomorphisms, the fundamental problem of perturbative bosonic strings therefore becomes the parametrization of Moduli space, which is non-trivial for genus h ≥4. At tree-level, corresponding to genus 0, the cosmological constant vanishes, Z0 =0

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M-theory
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M-theory is a theory in physics that unifies all consistent versions of superstring theory. The existence of such a theory was first conjectured by Edward Witten at a string theory conference at the University of Southern California in the spring of 1995, Wittens announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Wittens announcement, string theorists had identified five versions of superstring theory, although these theories appeared, at first, to be very different, work by several physicists showed that the theories were related in intricate and nontrivial ways. In particular, physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality, Wittens conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity. Modern attempts to formulate M-theory are typically based on theory or the AdS/CFT correspondence. Investigations of the structure of M-theory have spawned important theoretical results in physics and mathematics. More speculatively, M-theory may provide a framework for developing a theory of all of the fundamental forces of nature. One of the deepest problems in physics is the problem of quantum gravity. The current understanding of gravity is based on Albert Einsteins general theory of relativity, however, nongravitational forces are described within the framework of quantum mechanics, a radically different formalism for describing physical phenomena based on probability. String theory is a framework that attempts to reconcile gravity. In string theory, the particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how strings propagate through space and interact with each other, in a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the scale, a string will look just like an ordinary particle, with its mass, charge. In this way, all of the different elementary particles may be viewed as vibrating strings, one of the vibrational states of a string gives rise to the graviton, a quantum mechanical particle that carries gravitational force. There are several versions of string theory, type I, type IIA, type IIB, the different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings and closed strings, while types IIA and IIB include only closed strings, each of these five string theories arises as a special limiting case of M-theory. This theory, like its string theory predecessors, is an example of a theory of gravity. It describes a force just like the familiar gravitational force subject to the rules of quantum mechanics, in everyday life, there are three familiar dimensions of space, height, width and depth