Dirac spinorIn quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group. Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks.
Gamma matricesIn mathematical physics, the gamma matrices, 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 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.
SpinorIn geometry and physics, spinors spɪnɚ are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state.
Pauli matricesIn mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
Dirac equationIn 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.
Higher-dimensional gamma matricesIn mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors.
Pure spinorIn the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated under the Clifford action by a maximal isotropic subspace of the space of vectors with respect to the scalar product determining the Clifford algebra. They were introduced by Élie Cartan in the 1930s to classify complex structures. Pure spinors were a key ingredient in the study of spin geometry and twistor theory, introduced by Roger Penrose in the 1960s.
Spin groupIn mathematics the spin group Spin(n) is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2) The group multiplication law on the double cover is given by lifting the multiplication on . As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
ChiralityChirality kaɪˈrælɪtiː is a property of asymmetry important in several branches of science. The word chirality is derived from the Greek χειρ (kheir), "hand", a familiar chiral object. An object or a system is chiral if it is distinguishable from its ; that is, it cannot be superimposed onto it. Conversely, a mirror image of an achiral object, such as a sphere, cannot be distinguished from the object. A chiral object and its mirror image are called enantiomorphs (Greek, "opposite forms") or, when referring to molecules, enantiomers.
Dirac seaThe Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons (electrons traveling near the speed of light). The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, before its experimental discovery in 1932.
Chirality (chemistry)In chemistry, a molecule or ion is called chiral (ˈkaɪrəl) if it cannot be superposed on its by any combination of rotations, translations, and some conformational changes. This geometric property is called chirality (kaɪˈrælɪti). The terms are derived from Ancient Greek χείρ (cheir) 'hand'; which is the canonical example of an object with this property. A chiral molecule or ion exists in two stereoisomers that are mirror images of each other, called enantiomers; they are often distinguished as either "right-handed" or "left-handed" by their absolute configuration or some other criterion.
Relativistic quantum mechanicsIn physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics.
Paul DiracPaul Adrien Maurice Dirac (dɪˈræk; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is considered to be one of the founders of quantum mechanics and quantum electrodynamics. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University and the University of Miami, and a 1933 Nobel Prize in Physics recipient. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics.
Relativistic mechanicsIn physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light c. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles.
Quantum mechanicsQuantum 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.
Spin (physics)Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin should not be understood as in the "rotating internal mass" sense: spin is a quantized wave property. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.
Spin structureIn differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.
Interpretations of quantum mechanicsAn interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.
Spinor bundleIn differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors . A section of the spinor bundle is called a spinor field. Let be a spin structure on a Riemannian manifold that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering of the special orthogonal group by the spin group.
Spin representationIn mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. Elements of a spin representation are called spinors.
