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Twisted Sector
In theoretical physics, a twisted sector is a subspace of the full Hilbert space of closed string states in a particular theory over a (good) orbifold. In the first quantized formalism of string theory (or in two-dimensional conformal field theory) the target space is an orbifold M/G if the observables of the string are only defined modulo G. Consequently, the value of the field after one cycle around the closed string need only be the same as its original value modulo some G transformation. i.e. there exists some g\in G such that :X(\sigma+2\pi,\tau)=g (\sigma,\tau)/math> For each conjugacy class of G, we have a different superselection sector In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originall ... (wrt the worldsheet). The conjugacy class consisting of the identity gives rise to th ... [...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 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 Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed String
In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often have an interest in string theories because theories in which the fundamental entities are strings rather than point particles automatically have many properties that some physicists expect to hold in a fundamental theory of physics. Most notably, a theory of strings that evolve and interact according to the rules of quantum mechanics will automatically describe quantum gravity. Overview In string theory, the strings may be open (forming a segment with two endpoints) or closed (forming a loop like a circle) and may have other special properties. Prior to 1995, there were five known versions of string theory incorporating the idea of supersymmetry, which differed in the type of strings and in other aspects. Today these different string theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Theory
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. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. 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. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and conde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Conformal Field Theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models. Basic structures Geometry Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces. Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Target Space
Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, fixed at a specific target * Color chart (or reference card), the reference target used in digital imaging for accurate color reproduction Places * Target, Allier, France * Target Lake, a lake in Minnesota Terms * Target market, marketing strategy ** Target audience, intended audience or readership of a publication, advertisement, or type of message * In mathematics, the target of a function is also called the codomain * Target (cricket), the total number of runs a team needs to win People * Target (rapper), stage name of Croatian hip-hop artist Nenad Šimun * DJ Target, stage name of English grime DJ Darren Joseph, member of Roll Deep * Gui-Jean-Baptiste Target (1733–1807), French lawyer Art and media * The Target, a comic book charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugacy Class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^. for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superselection Sector
In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states \psi_1 and \psi_2 are separated by a selection rule if \langle \psi_1 , H , \psi_2 \rangle = 0 for the given Hamiltonian H , while they are separated by a superselection rule if \langle \psi_1 , A , \psi_2 \rangle = 0 for ''all ''physical observables A . Because no observable connects \langle \psi_1 , and , \psi_2 \rangle they cannot be put into a quantum superposition \alpha , \psi_1 \rangle + \beta , \psi_2 \rangle , and/or a quantum superposition cannot be distinguished from a classical mixture of the two states. It also implies that there is a class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate (group Theory)
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^. for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold Projection
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
