Setting of relativistic physics in geometric algebra
In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics."[1]: ix
Spacetime algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or 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.[2]: 40, 43, 97, 113 It is also the natural parent algebra of spinors in special relativity.[2]: 333 These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.[1]: v
In comparison to related methods, STA and Dirac algebra are both Clifford Cl1,3 algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars.
The STA spacetime split is similar to the algebra of physical space (APS, Pauli algebra) approach. APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.[3]: 225–266
spacetimealgebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetimealgebra provides...
geometric algebras applied in physics include the spacetimealgebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric...
quantum spacetime, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example. Snyder's Lie algebra was made...
(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)...
The action is the relativistic arclength of the path of the particle in spacetime, minus the potential energy contribution, plus an extra contribution which...
\mathbf {B} \right)=0} We can identify APS as a subalgebra of the spacetimealgebra (STA) C ℓ 1 , 3 ( R ) {\displaystyle C\ell _{1,3}(\mathbb {R} )} ...
real and one complex coordinate, or using tensors. Spacetimealgebra is a type of geometric algebra that is closely related to Minkowski space, and is...
relativity, a theory that explains the force of gravity and the structure of spacetime at the macro-level. The other is quantum mechanics, a completely different...
representations of a Lie supergroup. The general supersymmetry algebra for spacetime dimension d, and with the fermionic piece consisting of a sum of...
form is not a Lorentz transformation of the underlying spacetime. The Clifford algebra of spacetime used in physics thus has more structure than Cl4(C)....
depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if spacetime warps or contracts, the...
= Cl2(C) = Cl1,2(R), the Pauli algebra Cl3,0(R), and the even part Cl[0] 1,3(R) = Cl[0] 3,1(R) of the spacetimealgebra. Let {1, i, j, k} be the basis...
} is the 4x4 unit matrix. The Dirac algebra can be regarded as a complexification of the real spacetimealgebra Cl1,3( R {\displaystyle \mathbb {R} }...
mathematical suites are computer algebra systems that use symbolic mathematics. They are designed to solve classical algebra equations and problems in human...
function fields. Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: Boolean algebra used in logic gates...
In physics, spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum...
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important...
algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics...
representation Pure spinor Spin-1/2 Spinor bundle Supercharge Twistor theory Spacetimealgebra Spinors in three dimensions are points of a line bundle over a conic...
of the Pauli matrices); and the spacetimealgebra Cl1,3( R {\displaystyle \mathbb {R} } ). The elements of the algebra Clp,q( R {\displaystyle \mathbb...
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also...
as a collection of mathematical methods such as real analysis, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics...
are lost in the original Arabic at this time. However, his other book on algebra remains. In the early 12th century, Latin translations of said al-Khwarizmi...
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which...
geometric property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of...
linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A2 maps to a nonnegative real...
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