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Conformal Field Theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Killing Equation
In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flow defines conformal transformations, that is, preserve g up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g. \mathcal_g = \lambda g for some function \lambda on the manifold. For n \ne 2 there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields. Densitized metric tensor and Conformal Killing vectors A vector field X is a Killing vector field if and only if its flow preserves the metric tensor g (strictly speaking for each compact subsets of ... [...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] |
Minimal Model (physics)
In 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 p ..., a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified, giving rise to an ADE classification. Most minimal models have been solved, i.e. their 3-point structure constants have been computed analytically. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra. Relevant representations of the Virasoro algebra Representations In minimal models, the central charge of the Virasoro algebra takes values of the type : c_ = 1 - 6 \ . where p, q are coprime integers s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (physics)
State most commonly refers to: * State (polity), a centralized political organization that regulates law and society within a territory **Sovereign state, a sovereign polity in international law, commonly referred to as a country **Nation state, a state where the majority identify with a single nation (with shared culture or ethnic group) ** Constituent state, a political subdivision of a state ** Federated state, constituent states part of a federation *** U.S. state * State of nature, a concept within philosophy that describes the way humans acted before forming societies or civilizations State may also refer to: Arts, entertainment, and media Literature * '' State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * '' Our State'', a monthly magazine published in North Carolina and formerly called ''The State'' * The State (Larry Niven), a fictional future governme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group Flow
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying physical laws (codified in a quantum field theory) as the energy (or mass) scale at which physical processes occur varies. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales (self-similarity), where under the fixed point of the renormalization group flow the field theory is conformally invariant. As the scale varies, it is as if one is decreasing (as RG is a semi-group and doesn't have a well-defined inverse operation) the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally consist of self-sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C-theorem
In quantum field theory the ''C''-theorem states that there exists a positive real function, C(g^_i,\mu), depending on the coupling constants of the quantum field theory considered, g^_i, and on the energy scale, \mu^_, which has the following properties: *C(g^_i,\mu) decreases monotonically under the renormalization group (RG) flow. *At fixed points of the RG flow, which are specified by a set of fixed-point couplings g^*_i, the function C(g^*_i,\mu)=C_* is a constant, independent of energy scale. The theorem formalizes the notion that theories at high energies have more degrees of freedom than theories at low energies and that information is lost as we flow from the former to the latter. Two-dimensional case Alexander Zamolodchikov proved in 1986 that two-dimensional quantum field theory always has such a ''C''-function. Moreover, at fixed points of the RG flow, which correspond to conformal field theories, Zamolodchikov's ''C''-function is equal to the central charge of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Zamolodchikov
Alexander Borisovich Zamolodchikov (; born September 18, 1952) is a Russian-American theoretical physicist, known for his contributions to conformal field theory, statistical mechanics, string theory and condensed matter physics. He is widely regarded as one of the most accomplished theoretical physicists for his profound contributions to fundamental physics and especially to Quantum Field Theories, for which he was awarded the Breakthrough Prize in Fundamental Physics in 2024. He is currently the C.N. Yang – Wei Deng Endowed Chair of Physics at Stony Brook University. Biography Born in Novo-Ivankovo, now part of Dubna, Zamolodchikov earned a M.Sc. in nuclear engineering (1975) from Moscow Institute of Physics and Technology, a Ph.D. in physics from the Institute for Theoretical and Experimental Physics (1978). He joined the research staff of Landau Institute for Theoretical Physics (1978) where he got a Doctor of Sciences degree (1983). He co-authored the famous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Anomaly
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. In quantum field theory when we set Planck constant \hbar to zero we have only Feynman tree diagrams, which is a "classical" theory (equivalent to the Fredholm theory of a classical field theory). One-loop (''N''-loop) Feynman diagrams are proportional to \hbar (\hbar^N). If a current is conserved classically (\hbar=0) but develops a divergence at loop level in quantum field theory (\propto \hbar), we say there is an anomaly. A famous example is the axial current anomaly where massless fermions will have a classically conserved axial current, but which develops a nonzero divergence in the presence of gauge fields. A scale invariant theory, one in which there are no mass scales, will have a conserved Noether current called the "scale current." This is derived by performing scale transformations on the coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virasoro Algebra
In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro. Structure The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy ,L_n0 and The factor of \frac is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra or Schottenloher, Thm. 5.1, pp. 79. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. The generators L_ are called annihilation modes, while L_ are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set : \mathcal = \Big\_ For L\in \mathcal, let , L, = \sum_^k n_i, then _0,L= , L, L. Representation theory In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Extension
In the theory of Lie groups, Lie algebras and Lie algebra representation, their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from Projective representation, projective group representations. Such a Lie algebra will contain central charges. Starting with a Loop algebra, polynomial loop algebra over finite-dimensional Simple Lie group, simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the ''Killing vector'' will not distort distances on the object. Definition Specifically, a vector field X is a Killing vector field if the Lie derivative with respect to X of the metric tensor g vanishes: : \mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is : g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 for all vectors Y and . In local coordinates, this amounts to the Killing equation : \nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,. This condition is expressed in covarian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Algebra
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C 'z'',''z''−1 There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Élie Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s. Basis A basis for the Witt algebra is given by the vector fields L_n=-z^ \frac, for ''n'' in ''\mathbb Z''. The Lie bracket of two basis vector fields is given by : _m,L_n(m-n)L_. This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal field theory and string theory. Note that by restricting ''n'' to 1,0,-1, one gets a subalgebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
