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Yang–Mills Existence And Mass Gap
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution. The problem is phrased as follows:Arthur Jaffe and Edward WittenQuantum Yang-Mills theory. Official problem description. :''Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on \mathbb^4 and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in , and . In this statement, a quantum Yang–Mills theory is a non-abelian quantum field theory similar to that underlying the Standard Model of particle physics; \mathbb^4 is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must prove that: * Yang–Mills th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glueball
In particle physics, a glueball (also gluonium, gluon-ball) is a hypothetical composite particle. It consists solely of gluon particles, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong interaction between themselves. Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states. In pure gauge theory, glueballs are the only states of the spectrum and some of them are stable. Theoretical calculations show that glueballs should exist at energy ranges accessible with current collider technology. However, due to the aforementioned difficulty (among others), they have so far not been observed and identified with certainty, although phenomenological calculations have suggested that an experimentally identified glueball candidate, denoted f_(1710), has properties consistent with those expected of a Standard Model glueball. The prediction that glueballs exist is one of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representations Of The Poincaré Group
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It covers topics including literary, historical, and cultural studies. The founding editorial board was chaired by Stephen Greenblatt and Svetlana Alpers. ''Representations'' frequently publishes thematic special issues, for example, the 2007 issue on the legacies of American Orientalism, the 2006 issue on cross-cultural mimesis Mimesis (; grc, μίμησις, ''mīmēsis'') is a term used in literary criticism and philosophy that carries a wide range of meanings, including ''imitatio'', imitation, nonsensuous similarity, receptivity, representation, mimicry, the act ..., and the 2005 issue on political and intellectual redress. Topics covered * The Body, Gender, and Sexuality * Culture and Law * Empire, Imperialism, and The New Worl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In general, the unitary equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. Overview A Minkowski spacetime isometry has the property that the interval between events is left invariant. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. A time or space reversal (a reflection) is also an isometry of this group. In Minkowski space (i.e. ignoring the effec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure State
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen ato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences. Von Neumann made major contributions to many fields, including mathematics (foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, ballistics, nuclear physics and quantum statistical mechanics), economics ( game theory and general equilibrium theory), computing ( Von Neumann architecture, linear programming, numerical meteo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wightman Axioms
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 Prize Problems, Millennium Problems is to realize the Yang–Mills existence and mass gap, 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 unitary representation, unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-time
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize Special relativity, relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based wikisource:Translation:On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
