|
Soler Model
The soler model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 3 spatial and 1 time dimension. It was introduced in 1938 by Dmitri Ivanenko and re-introduced and investigated in 1970 by Mario Soler as a toy model of self-interacting electron. This model is described by the Lagrangian density :\mathcal=\overline \left(i\partial\!\!\!/-m \right) \psi + \frac\left(\overline \psi\right)^2 where g is the coupling constant, \partial\!\!\!/=\sum_^3\gamma^\mu\frac in the Feynman slash notations, \overline=\psi^*\gamma^0. Here \gamma^\mu, 0\le\mu\le 3, are Dirac gamma matrices. The corresponding equation can be written as :i\frac\psi=-i\sum_^\alpha^j\frac\psi+m\beta\psi-g(\overline \psi)\beta\psi, where \alpha^j, 1\le j\le 3, and \beta are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model. Generalizations A commonly considered generalization is :\mathcal=\overline \left(i\partial\!\! ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and 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. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gross–Neveu Model
The Gross–Neveu (GN) model is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum chromodynamics (QCD), the theory of strong interactions. It shares several features of the QCD: GN theory is asymptotically free thus at strong coupling the strength of the interaction gets weaker and the corresponding \beta function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with \mathbb_2 chiral symmetry breaking, and in the large number of flavor (N \to \infty) limit, GN theory behaves as t'Hooft's large N_c limit in QCD. It consists of N Dirac fermions \psi_1, \psi_2, \cdots, \psi_N. The Lagrangian density is :\mathcal=\bar \psi_a \left(i\partial\!\!\!/-m \right) \psi^a + \frac\left bar \psi_a \psi^a\right2. Einstein summation notation is used, \psi^a is a two component spinor objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Dirac Equation
:''See Ricci calculus and Van der Waerden notation for the notation.'' In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-interacting electrons. The nonlinear Dirac equation appears in the Einstein–Cartan–Sciama–Kibble theory of gravity, which extends general relativity to matter with intrinsic angular momentum ( spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action. In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an axial-axial, spin–spin interaction in fermionic matter, which becomes significant only at extremely high densities. Consequently, the Dirac equation becomes nonlinear (cubic) in the spi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, ''antimatter'', previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a ''theoretical'' justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton Wave
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the " Wave of Translation". Definition A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalizability
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Symmetry
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuous'' (such as rotation of a circle) or ''discrete'' (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see '' Symmetry group''). These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems. Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Neveu
André Neveu (; born 28 August 1946) is a French physicist working on string theory and quantum field theory who coinvented the Neveu–Schwarz algebra and the Gross–Neveu model. Biography Neveu studied in Paris at the École Normale Supérieure (ENS). In 1969 he received his diploma (Thèse de troisième cycle) at University of Paris XI in Orsay with and Claude Bouchiat and in 1971 he completed his doctorate (Doctorat d'État) there. In 1969 he and his classmate from ENS and Orsay, Joël Scherk, together with John H. Schwarz and David Gross at Princeton University, examined divergences in one-loop diagrams of the bosonic string theory (and discovered the cause of tachyon divergences). From 1971 to 1974 Neveu was at the Laboratory for High Energy Physics of the University of Paris XI where he and Scherk showed that spin-1 excitations of strings could describe Yang–Mills theories. In 1971, Neveu with John Schwarz in Princeton developed, at the same time as Pierre Ramo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. Gross is the Chancellor's Chair Professor of Theoretical Physics at the Kavli Institute for Theoretical Physics (KITP) of the University of California, Santa Barbara (UCSB), and was formerly the KITP director and holder of their Frederick W. Gluck Chair in Theoretical Physics. He is also a faculty member in the UCSB Physics Department and is currently affiliated with the Institute for Quantum Studies at Chapman University in California. He is a foreign member of the Chinese Academy of Sciences. Early life and education Gross was born to a Jewish family in Washington, D.C., in February 1941. His parents were Nora (Faine) and Bertram Myron Gross (1912–1997). Gross received his bachelor's degree from the Hebrew University of Jerusalem, Isr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mathbb). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin- particles. In Dirac representation, the four contravariant gamma matrices are :\begin \gamma^0 &= \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Fermion
In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category. Description In particle physics, all fermions in the standard model have distinct antiparticles (''perhaps'' excepting neutrinos) and hence are Dirac fermions. They are named after Paul Dirac, and can be modeled with the Dirac equation. A Dirac fermion is equivalent to two Weyl fermions. The counterpart to a Dirac fermion is a Majorana fermion, a particle that must be its own antiparticle. Dirac quasi-particles In condensed matter physics, low-energy excitations in graphene and topological insulators, among others, are fermionic quasiparticles described by a pseudo-relativistic Dirac equation. See also * Dirac spinor, a wavefunction-like description of a Dirac fermion * Dirac–Kähler fermion, a geometric formulation of Dirac fermions * Majorana fermion, an alternate category of fermion, possibly de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Internal Symmetry")