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Reeh–Schlieder Theorem
The Reeh–Schlieder theorem is a result in relativistic local quantum field theory published by Helmut Reeh and Siegfried Schlieder in 1961. The theorem states that the vacuum state \vert \Omega \rangle is a Cyclic subspace#Definition, cyclic vector for the field algebra \mathcal(\mathcal) corresponding to any open set \mathcal in Minkowski space. That is, any state \vert \psi \rangle can be approximated to arbitrary precision by acting on the vacuum with an operator selected from the local algebra, even for \vert \psi \rangle that contain excitations arbitrarily far away in space. In this sense, states created by applying elements of the local algebra to the vacuum state are not localized to the region \mathcal. For practical purposes, however, local operators still generate quasi-local states. More precisely, the long range effects of the operators of the local algebra will diminish rapidly with distance, as seen by the cluster properties of the Wightman functions. And with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Quantum Field Theory
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . 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 set \_ 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 condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Reeh
Helmut is a German name. Variants include Hellmut, Helmuth, and Hellmuth. From old German, the first element deriving from either ''heil'' ("healthy") or ''hiltja'' ("battle"), and the second from ''muot'' ("spirit, mind, mood"). Helmut may refer to: People A–L *Helmut Angula (born 1945), Namibian politician * Helmut Ashley (1919–2021), Austrian director and cinematographer *Helmut Bakaitis (born 1944), Australian director and actor *Helmut Berger (1944–2023), Austrian actor *Helmut Dantine (1917–1982), Austrian actor * Helmut Deutsch (born 1945), Austrian classical pianist *Helmut Ditsch (born 1962), Argentine painter * Hellmut Diwald (1924–1993), German historian * Helmut Donner (born 1941), Austrian high jumper *Helmut Duckadam (1959–2024), Romanian footballer * Helmut Fischer (1926–1997), German actor * Hellmut von Gerlach (1866–1935), German journalist * Helmut Goebbels (1935–1945), only son of Joseph Goebbels * Helmut Graeb, German electrical engineer *Helm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegfried Schlieder
Siegfried is a German-language male given name, composed from the Germanic elements ''sig'' "victory" and ''frithu'' "protection, peace". The German name has the Old Norse cognate ''Sigfriðr, Sigfrøðr'', which gives rise to Swedish ''Sigfrid'' (hypocorisms ''Sigge, Siffer''), Danish/Norwegian ''Sigfred''. In Norway, ''Sigfrid'' is given as a feminine name. official statistics at Statistisk Sentralbyrå, National statistics office of Norway, https://www.ssb.no; Statistiska Centralbyrån, National statistics office of Sweden, https://www.scb.se/ The name is medieval and was borne by the legendary dragon-slayer also known as . It did survive in marginal use into the modern period, but after 1876 it enjoyed renewed popular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum State
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. However, the quantum vacuum is not a simple empty space, but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field. The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s, it was reformulated by Feynman, Tomonaga, and Schwinger, who jointly received the Nobel prize for this work in 1965. For a historical discussion, see for example For the Nobel prize details and the Nobel lectures by these authors, see Today, the electromagnetic interactions and the weak interactions are unified (at very high energies only) in the theory of the electroweak interaction. The Standard Model is a generalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Subspace
In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector ''v'' in a vector space ''V'' and a linear transformation ''T'' of ''V'' is called the ''T''-cyclic subspace generated by ''v''. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra. Definition Let T:V\rightarrow V be a linear transformation of a vector space V and let v be a vector in V. The T-cyclic subspace of V generated by v, denoted Z(v;T), is the subspace of V generated by the set of vectors \. In the case when V is a topological vector space, v is called a cyclic vector for T if Z(v;T) is dense in V. For the particular case of finite-dimensional spaces, this is equivalent to saying that Z(v;T) is the whole space V. There is another equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others said it "was grown on experimental physical grounds". Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events.This makes spacetime distance an inva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wightman Functions
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm \, T\, of a linear map T : X \to Y is the maximum factor by which it "lengthens" vectors. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \text v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Entanglement
Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurement#Quantum mechanics, Measurements of physical properties such as position (vector), position, momentum, Spin (physics), spin, and polarization (waves), polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior ... [...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] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipartite Entanglement
In the case of systems composed of m > 2 subsystems, the classification of quantum-entangled states is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states. Full and partial separability The definitions of fully separable and fully entangled multipartite states naturally generalizes that of separable and entangled states in the bipartite case, as follows. Full ''m''-partite separability (''m''-separability) of ''m'' systems The state \; \varrho_ of \; m subsystems \; A_1, \ldots, A_m with Hilbert space \; \mathcal_=\mathcal_\otimes\ldots\otimes \mathcal_ is fully separable if and only if it can be written in the form :\; \varrho_ = \sum_^k p_i \varrho_^i \otimes \ldots \otimes \varrho_^i. Correspondingly, the state \; \varrho_ is fully entangled if it cannot be written in the above form. As in the bipartite case, the set of \; m-sep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
