Spacetime algebraIn mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted.
Dirac fermionIn physics, a Dirac fermion is a spin-1⁄2 particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. 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.
Green's theoremIn vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem. Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is anticlockwise.
Aix-en-ProvenceAix-en-Provence (UKˌɛks_ɒ̃_prɒˈvɒ̃s, USˌeɪks_ɒ̃_proʊˈvɒ̃s,ˌɛks-), or simply Aix (medieval Occitan: Aics), is a city and commune in southern France, about north of Marseille. A former capital of Provence, it is the subprefecture of the arrondissement of Aix-en-Provence, in the department of Bouches-du-Rhône, in the region of Provence-Alpes-Côte d'Azur. The population of Aix-en-Provence is approximately 145,000. Its inhabitants are called Aixois or, less commonly, Aquisextains.
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.
Lie algebra extensionIn the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Lie superalgebraIn mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
Special forcesSpecial forces or special operations forces (SOF) are military units trained to conduct special operations. NATO has defined special operations as "military activities conducted by specially designated, organized, selected, trained and equipped forces using unconventional techniques and modes of employment". Special forces emerged in the early 20th century, with a significant growth in the field during World War II, when "every major army involved in the fighting" created formations devoted to special operations behind enemy lines.
Algebra of physical spaceIn physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl3,1(R).
