|
Local Quantum Field Theory
The Haag–Kastler axiomatic framework for quantum field theory, introduced by , is an application to local quantum physics of C*-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those. Haag–Kastler axioms Let \mathcal be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a net \_ of von Neumann algebras \mathcal(O) on a common Hilbert space \mathcal satisfying the following axioms: * ''Isotony'': O_1 \subset O_2 implies \mathcal(O_1) \subset \mathcal(O_2). * ''Causality'': If O_1 is space-like separated from O_2, then mathcal(O_1),\mathcal(O_2)0. * ''Poincaré covariance'': A strongly continuous unitary representation U(\mathcal) of the Poincaré group \mathcal on \mathcal exists such that \mathcal(gO) = U(g) \mathcal(O) U(g)^*, g \in \mathcal. * ''Spectrum condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic Framework
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system. Properties An axiomatic system is said to be ''consistent'' if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven ( principle of explosion). In an axiomatic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events. The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships. Tangent vectors If \,(M,g) is a Lorentzian manifold (for metric g on manifold M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: * timelike if \,g(X,X) 0 Here we use t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNS Construction
GNS may refer to: Places * Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia * Gainesville station (Georgia), an Amtrak station in Georgia, United States Companies and organizations * Gesellschaft für Nuklear-Service, a German nuclear-waste services company * Ghana Nuclear Society, nuclear energy advocacy organization * Glenlyon Norfolk School in Victoria, British Columbia, Canada * GNS Healthcare, an American data analytics company * GNS Science, a New Zealand earth-science research institute * Gordon-North Sydney Hockey Club, based in Sydney, Australia * Government of National Salvation, in Serbia during the Second World War * Gunns, a defunct Australian timber company Other uses * Gelfand–Naimark–Segal construction, a theorem in functional analysis * General News Service, a BBC-internal news-distribution service * GEOnet Names Server, a database of place names and locations * Global Namespace, computer networking concept * GNS theory, in role-playing game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map f between topological spaces X and Y: #The map f is continuous in the topological sense; #Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f(x) (continuous in the sequential sense). While it is necessarily true that condition 1 implies condition 2 (The truth of the condition 1 ensures the truth of the conditions 2.), the reverse implication is not nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation. Details Suppose V, W are finite-dimensional vector spaces over a field, with dimensions m and n, respectively. For any space A, let L(A) denote the space of linear operators on A. The partial trace over W is then written as \operatorname_W: \operatorname(V \otimes W) \to \operatorname(V). It is defined as follows: For T\in \operatorname(V \otimes W), let e_1, \ldots, e_m , and f_1, \ldots, f_n , be bases for ''V'' and ''W'' respectively; then ''T'' has a matrix representation : \ \quad 1 \leq k, i \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Linear Functional
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive element (ordered group), positive elements v \in V, that is v \geq 0, it holds that f(v) \geq 0. In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V is a Complex numbers, complex vector space, it is assumed that for all v\ge0, f(v) is real. As in the case when V is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W\subseteq V, and the partial order does not extend to all of V, in which case the positive elements of V are the positive elements of W, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x ... [...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 . 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 of probability measures. By ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Completion
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be ''causal factors'' for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space. Causality is an abstraction that indicates how the world progresses. As such a basic concept, it is more apt as an explanation of other concepts of progression than as something to be explained by others more basic. The concept is like those of agency and efficacy. For this reason, a leap of intuition may be needed to grasp it. Accordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Operation
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
