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Braid Statistics
In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase gain of \pi (2 \pi) under the exchange of identical particles, a particle with braid statistics leads to a rational fraction of \pi under such exchange or even a non-trivial unitary transformation in the Hilbert space (see non-Abelian anyons). A similar notion exists using a loop braid group. Braid statistics are applicable to theoretical particles such as the two-dimensional anyons and their higher-dimensional analogues known as plektons. See also * Braid symmetry * Parastatistics In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alt ... * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Statistics
Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble (an idealization comprising the state space of possible states of a system, each labeled with a probability) that emphasizes properties of a large system as a whole at the expense of knowledge about parameters of separate particles. When an ensemble describes a system of particles with similar properties, their number is called the particle number and usually denoted by ''N''. Classical statistics In classical mechanics, all particles (elementary particle, fundamental and composite particles, atoms, molecules, electrons, etc.) in the system are considered identity (philosophy), distinguishable. This means that individual particles in a system can be tracked. As a consequence, switching the positions of any pair of particles in the system leads to a different configuration of the system. Furthermore, there is no restriction on p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin–statistics Theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that move in 3 dimensions have either integer spin or half-integer spin. Background Quantum states and indistinguishable particles In a quantum system, a physical state is described by a state vector. A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position. While the physical state does not change under the exchange of the particles' po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spin (,, ...). Every observed subatomic particle is either a boson or a fermion. Bosons are named after physicist Satyendra Nath Bose. Some bosons are elementary particles and occupy a special role in particle physics unlike that of fermions, which are sometimes described as the constituents of "ordinary matter". Some elementary bosons (for example, gluons) act as force carriers, which give rise to forces between other particles, while one (the Higgs boson) gives rise to the phenomenon of mass. Other bosons, such as mesons, are composite particles made up of smaller constituents. Outside the realm of particle physics, superfluidity arises because composite bosons (bose particles), such as low temperature helium-4 atoms, follow Bose–E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anyon
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as ''abelian'' or ''non-abelian''. Abelian anyons (detected by two experiments in 2020) play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected, although this is an active area of research. Introduction The statistical mechanics of large many-body systems obeys laws described by Maxwell–Boltzmann statistics. Quantum statistics is more complicated because of the different behaviors of two different kinds of particles called fermions and bosons. Quoting a recent, simple description:In the three-dimensional world we live in, there are only two types of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Braid Group
The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop-like topologies within three dimensions of space and time. The basic operations which generate a loop braid group for ''n'' loops are exchanges of two adjacent loops, and passing one adjacent loop through another. The topology forces these generators to satisfy some relations, which determine the group. To be precise, the loop braid group on ''n'' loops is defined as the motion group of n disjoint circles embedded in a compact three-dimensional "box" diffeomorphic to the three-dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding ''n'' circles into the 3-disk. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plekton
In particle physics, a plekton is a theoretical kind of particle that obeys a different style of statistics with respect to the interchange of identical particles. That is, it would be neither a boson nor a fermion, but subject to a braid statistics. Such particles have been discussed as a generalization of the braid characteristics of the anyon to dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ... > 2. References Hypothetical particles Parastatistics Braids {{particle-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Symmetry
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands ( warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parastatistics
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions. Herbert S. Green is credited with the creation of parastatistics in 1953. Formalism Consider the operator algebra of a system of ''N'' identical particles. This is a *-algebra. There is an ''SN'' group (symmetric group of order ''N'') acting upon the operator algebra with the intended interpretation of permuting the ''N'' particles. Quantum mechanics requires focus on observables having a physical meaning, and the observables would have to be invariant under all possible permutations of the ''N'' particles. For example, in the case ''N'' = 2, ''R''2 − ''R''1 cannot be an observable because it changes si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
