Conservation laws in the Dirac theory

Boudet, Roger

doi:10.1063/1.526613

Cited by 15 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and engineering [8,9,10,11,12,13,14,15]. Theoretically we have some equivalent definitions for Clifford algebras [16,17].…”

mentioning

confidence: 99%

A Note on the Representation of Clifford Algebra

Gu¹

2020

Preprint

View full text Add to dashboard Cite

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

show abstract

“…and engineering [8,9,10,11,12,13,14,15]. Theoretically we have some equivalent definitions for Clifford algebras [16,17].…”

mentioning

confidence: 99%

A Note on the Representation of Clifford Algebra

Gu¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…All our wave equations have a ( ) are consequence of the Lagrangian formalism which is a consequence of the 16 other equations. This was first seen by Boudet [50] in the frame of the linear Dirac theory of the electron. Our study proves that it is general: the numeric equations equivalent to the wave equations of the "matter" (spinor waves) may be split into two parts: a dynamical part containing rotational-like terms, and a conservative part containing divergence-like terms, and the conservative part is a consequence of the dynamical equations.…”

Section: Lessons Of This Calculationmentioning

confidence: 75%

Towards the Unification of All Interactions (The First Part: The Spinor Wave)

Daviau¹,

Bertrand²,

Girardot³

2016

JMP

View full text Add to dashboard Cite

For the unification of gravitation with electromagnetism, weak and strong interactions, we use a unique and very simple framework, the Clifford algebra of space () Cl M 3 2 = ». We enlarge our previous wave equation to the general case, including all leptons, quarks and antiparticles of the first generation. The wave equation is a generalization of the Dirac equation with a compulsory non-linear mass term. This equation is form invariant under the Cl 3 * group of the invertible elements in the space algebra. The form invariance is fully compatible with the () () () U SU SU 1 2 3 × × gauge invariance of the standard model. The wave equations of the different particles come by Lagrange equations from a Lagrangian density and this Lagrangian density is the sum of the real parts of the wave equations. Both form invariance and gauge invariance are exact symmetries, not only partial or broken symmetries. Inertia is already present in the () U 1 part of the gauge group and the inertial chiral potential vector simplifies weak interactions. Relativistic quantum physics is then a naturally yet unified theory, including all interactions.

show abstract

“…The paradigmatic example of the convenience of interpreting geometrically the imaginary unit i is Hestenes' reformulation of the Dirac equation [8,9,10,11,12]. However, complex numbers appear in many fields and in many equations of physics and they cannot be translated by the same geometric meaning in every case.…”

Section: Advances In Applied Cliffordmentioning

confidence: 98%

Complex Lorentz rotations on vectors⊕pseudovectors

Pozo

Parra

2001

AACA

View full text Add to dashboard Cite

Abstract. We study the complex structure of the space of vectors and pseudovectors A_ ~ AI@A3 arising from the properties of the Hodge duality in 4-dimensionM spacetimes. This structure appears naturally in the framework of its real Clifford geometric algebra Ceu3 or C~3,1, in which the Hodge duality is the simple multiplication by the volume unit ~. Interpreting the linear combination of a scalar anda pseudoscalar c~ + ~~ asa complex number (A0@A4 --~ C), the odd subspace A_ E C~ is a left/right complex linear space, previously studied by Sobczyk. From the real metric defining C_~, A_ inherits a natural complex bilinear norm N(a). Corresponding to this norm there is the group of complex Lorentz rotations Spinl,3(C), for which we find a formulation using exclusively the real algebra ~. We discuss its behaviour in some examples and find expressions for different decompositions into real and selfdual, real and anti-selfdual, and real and pure imaginary angle rotations. We finally apply our results to the implementation of the rotations corresponding to Euclidean and null signatures inside the real Lorentzian Clifford algebra C 91

show abstract

Conservation laws in the Dirac theory

Cited by 15 publications

References 10 publications

A Note on the Representation of Clifford Algebra

A Note on the Representation of Clifford Algebra

Towards the Unification of All Interactions (The First Part: The Spinor Wave)

Complex Lorentz rotations on vectors⊕pseudovectors

Contact Info

Product

Resources

About