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

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.

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

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 vector equation by direct quantization of the 4-momentum vector with modulus m: Pψ=mψ (no slash!), the spinor ψ taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DEproperties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors Xµ appear instead of Dirac’s Yµ-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=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 4×4 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 generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.

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

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 matrices1,2 is superfluous. One can write down a coordinate-free manifestly covariant vector equation by direct quantization of the 4-momentum vector with modulus m: Pψ=mψ (no slash!), the spinor ψ 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 Xµ appear instead of Dirac’s Yµ-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=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 4×4 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 generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.

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

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.

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

A new representation for spin 1/2 in the even 3D subalgebra of the spacetime algebra (STA) combines in a single geometric object the roles of the standard Pauli spin vector and spin state. It is a vector quantity comprising a gauge phase. In the one-particle case the representation (1) is Hermitian; (2) chiral; (3) reproduces all standard expectation values, including the total one-particle spin modulus ; (4) constrains a spinor basis representing opposite spins to preserve handiness (chirality); (5) the gauge phase allows to explicitly formalize irreversibility in spin measurement. In the two-particle case it (1) identifies entangled spin pairs as having opposite handiness and precise gauge phase relations; (2) doubles the dimensionality of the spin space due to variation of handiness; (3) the four maximally entangled states are naturally derived by pairing spins that are reflections (triplets) and inversions (singlet) of each-other. The cross-product terms in the expression for the squared total spin of two particles can be affected by experiment and they yield the standard expectation values after measurement. Here I directly define and transform spin in 3D orientation space, without invoking concepts like abstract Hilbert space and tensor product as in the standard formulation. The STA formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.

Spacetime algebra for the reformulation of fluid field equations

In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.