scholarly journals Spacetime algebra as a powerful tool for electromagnetism

2015 ◽  
Vol 589 ◽  
pp. 1-71 ◽  
Author(s):  
Justin Dressel ◽  
Konstantin Y. Bliokh ◽  
Franco Nori
Keyword(s):  
Spacetime Algebra

scholarly journals Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

2021 ◽  
Author(s):  
Joseph Wilson ◽  
Matt Visser
Keyword(s):  
Efficient Method ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Spin Representation ◽  
Lorentz Transformations ◽  
Rodrigues Formula ◽  
Dimension Formula ◽  
Spacetime Algebra ◽  
Geometric Algebras

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

scholarly journals Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

2021 ◽  
Author(s):  
Sokol Andoni
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Physics ◽  
Present Contribution ◽  
Spacetime Algebra ◽  
Time Space ◽  
Modern Physics ◽  
Present Formalism ◽  
Covariant Equation ◽  
Physical Spacetime ◽  
Internal Degrees Of Freedom

Abstract The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices is superfluous. One can write down a coordinate-free manifestly covariant equation by direct quantization of the energy-momentum 4-vector P with modulus m: P(psi) = m(psi) (no slash!), the spinor (psi) taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DE properties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors x0, x1, x2, x3 formally substitute Dirac’s gamma-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5, which in 3D transforms a parity-odd vector x into a parity-even vector x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4 x 4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for the generators of polar vectors and boosts and the other for the generators of axial vectors and rotors, comprising Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. By proving that Dirac’s matrices are redundant, because all the information from them is contained in spacetime-reflection, it becomes relevant to reexamine those areas of modern physics that take Dirac matrices and their generalizations as fundamental.

Complex Aberration Effect in Moving Dispersive DNG Media: A Spacetime Algebra Approach

PIERS Online ◽  
2008 ◽  
Vol 4 (6) ◽  
pp. 611-614 ◽  
Author(s):  
Sérgio A. Matos ◽  
João R. Canto ◽  
Carlos R. Paiva ◽  
Afonso M. Barbosa
Keyword(s):  
Spacetime Algebra ◽  
Algebra Approach
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