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Non-invertible Symmetry
In physics, a non-invertible symmetry is a symmetry of a quantum field theory that is not described by a group, and which in particular does not have an inverse. Non-invertible symmetries were first studied in 2-dimensional conformal field theory, where fusion categories govern the fusion rules, rather than a group. Four-dimensional examples of non-invertible symmetries can be obtained from Maxwell theory with topological theta term, via a combination of its SL(2,Z) duality and a discrete subgroup of its electric or magnetic 1-form symmetry. References External links "A New Kind of Symmetry Shakes Up Physics"by Kevin Hartnett, ''Quanta Magazine'' "Non-Invertible Symmetries and their Representations" video lecture by Sahand Seifnashri at Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry (physics)
The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some Transformation (function), transformation. A family of particular transformations may be ''continuous'' (such as rotation of a circle) or ''discrete space, discrete'' (e.g., Reflection (physics), reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see ''Symmetry group''). These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as representation of a Lie group, group representations and can, in addition, be exploited to simplify many problems. Arguably the most important examp ... [...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] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an ''invertible element'', also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fusion Category
In mathematics, a fusion category is a category that is abelian, k-linear, semisimple, monoidal, and rigid, and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k is algebraically closed, then the latter is equivalent to \mathrm(1,1)\cong k by Schur's lemma. Examples The Representation Category of a finite group G of cardinality n over a field \mathbb K is a fusion category if and only if n and the characteristic of \mathbb K are coprime. This is because of the condition of semisimplicity which needs to be checked by the Maschke's theorem. Reconstruction Under Tannaka–Krein duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologi ..., every fusion category arises as the representations of a weak Hopf algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fusion Rules
In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of two-dimensional conformal field theory where the relevant group is generated by the Virasoro algebra, the relevant representations are the conformal families associated with a primary field and the tensor product is realized by operator product expansion In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex ...s. The fusion rules contain the information about the kind of families that appear on the right-hand side of these OPEs, including the multiplicities. More generally, integrable models in 2 dimensions which aren't conformal field theories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Formulation Of Classical Electromagnetism
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems. Covariant objects Preliminary four-vectors Lorentz tensors of the following kinds may be used in this article to describe bodies or particles: * four-displacement: x^\alpha = (ct, \mathbf) = (ct, x, y, z) \,. * Four-velocity: u^\alpha = \gamma(c,\mathbf) , where ''γ''(u) is the Lorentz factor at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Vacuum
In quantum field theory, the theta vacuum is the semi-classical vacuum state of non- abelian Yang–Mills theories specified by the vacuum angle ''θ'' that arises when the state is written as a superposition of an infinite set of topologically distinct vacuum states. The dynamical effects of the vacuum are captured in the Lagrangian formalism through the presence of a ''θ''-term which in quantum chromodynamics leads to the fine tuning problem known as the strong CP problem. It was discovered in 1976 by Curtis Callan, Roger Dashen, and David Gross, and independently by Roman Jackiw and Claudio Rebbi. Yang–Mills vacuum Topological vacua The semi-classical vacuum structure of non-abelian Yang–Mills theories is often investigated in Euclidean spacetime in some fixed gauge such as the temporal gauge A_0 = 0. Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations A_i = i\Omega \nabla_i \Omega^, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montonen–Olive Duality
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite" (i.e. they are solitons or topological defects), can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality. It was later proven to hold true when dealing with a ''N'' = 4 supersymmetric Yang–Mills theory. It is named after Finnish physicist Claus Montonen and British physicist David Olive after they proposed the idea in their academic paper '' Magnetic monopoles as gauge particles?'' where they state: S-duality is now a basic ingredient in topological quantum field theories and string theori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
