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Borchers Algebra
In mathematics, a Borchers algebra, Borchers–Uhlmann algebra, or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by , who showed that the Wightman distributions of a quantum field could be interpreted as a state 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 ..., called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra. The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ''ab''−''ba'' for ''a'' and ''b'' having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see #Adjunction and universal property, below). The tensor algebra is important because many other algebras arise as quotient associative algebra, quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Test Function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain \Reals^n forms a vector space, denoted \mathrm^\infty_0(\Reals^n) or \mathrm^\infty_\mathrm(\Reals^n). The dual space of this space endowed with a suitable topology is the space of distributions. Examples The function \Psi : \mathbb \to \mathbb given by \Psi(x) = \begin \exp\left( \frac\right), & \text , x, . In fact, by definition of support, we have that \operatorname(\Psi):=\overline =\overline, where the closure is taken with respect the Euclidean topology of the real line. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function \exp\left(-y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wightman Distribution
In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scattering theory affirmed their significance. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields. Rationale One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectrum o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines 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 inability to describe the wea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State (functional Analysis)
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For ''M'' an operator system in a C*-algebra ''A'' with identity, the set of all states of'' ''M, sometimes denoted by S(''M''), is convex, weak-* closed in the Banach dual space ''M''*. Thus the set of all states of ''M'' with the weak-* topology forms a compact Hausdorff space, known as the state space of ''M'' . In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number). Jordan decomposition States can be viewed as noncommutative generalizations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O*-algebra
In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described by and , who studied some examples of O*-algebras, called Borchers algebras, arising from the Wightman axioms of 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 phy .... and began the systematic study of algebras of unbounded operators. References * * * * * * Operator algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Algebras
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic.''Theory of Operator Algebras I'' By Masamichi Takesaki, Springer 2012, p vi Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory. Overview Operator algebras can be used to study arbitrary sets of operators with little algebraic relation ''simultaneously''. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general, operator algebras are non-commutative rings. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
