Quantum Mechanics in the Geometry of Space-Time

Boudet, Roger

doi:10.1007/978-3-642-19199-2

Cited by 13 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Part II in my thesis studied models of fermions represented as ideals in D 4 . Just as columns are left ideals in matrix algebra, minimal left ideals in D 4 can be defined by idempotents (serving as projection operators); thus…”

Section: Genesis Of Spacetime Algebramentioning

confidence: 99%

The Genesis of Geometric Algebra: A Personal Retrospective

Hestenes

2016

Adv. Appl. Clifford Algebras

View full text Add to dashboard Cite

Abstract. Even today mathematicians typically typecast Clifford Algebra as the "algebra of a quadratic form," with no awareness of its grander role in unifying geometry and algebra as envisaged by Clifford himself when he named it Geometric Algebra. It has been my privilege to pick up where Clifford left off-to serve, so to speak, as principal architect of Geometric Algebra and Calculus as a comprehensive mathematical language for physics, engineering and computer science. This is an account of my personal journey in discovering, revitalizing and extending Geometric Algebra, with emphasis on the origin and influence of my book Space-Time Algebra. I discuss guiding ideas, significant results and where they came from-with recollection of important events and people along the way. Lastly, I offer some lessons learned about life and science.

show abstract

Section: Genesis Of Spacetime Algebramentioning

confidence: 99%

The Genesis of Geometric Algebra: A Personal Retrospective

Hestenes

2016

Adv. Appl. Clifford Algebras

View full text Add to dashboard Cite

show abstract

“…The geometrical interpretation of Equation (347) has been extensively discussed by Hestens in several of his articles [17,[35][36][37] and also by Boudet [63]. The Dirac theory is regarded as a fundamental theory and also a general theory.…”

Section: The Dirac Equation In Complex Vector Spacetimementioning

confidence: 99%

Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

Muralidhar

2015

Mathematics

View full text Add to dashboard Cite

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n − 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.

show abstract

“…; η e = η 1e η 2e To get the relativistic invariance the Dirac theory uses x = x μ σ μ and x satisfying x = x 0 + x 3 x 1 − ix 2 x 1 + ix 2 x 0 − x 3 ; x = x μ σ μ = x 0 + x 3 x 1 − ix 2 x 1 + ix 2 x 0 − x 3 (1.6) This is equivalent to identify the space algebra Cl 3 to the matrix algebra M 2 (C) 1 . Noting a * the conjugate of a and with…”

Section: Form Invariance Of the Dirac Equationmentioning

confidence: 99%

“…The transformation (2.23) explains by itself why only the SL(2, C) part of Cl * 3 was previously seen. The P rotor which plays a central role in the Hestenes' work [11] and in the Boudet's work [1], actually an element of SL(2, C), is defined such as…”

Section: Dirac Equation and Electromagnetism In CLmentioning

confidence: 99%

Gauge Group of the Standard Model in Cl 1,5

Daviau¹

2015

Adv. Appl. Clifford Algebras

View full text Add to dashboard Cite

Abstract. Describing a wave with spin 1/2, the Dirac equation is form invariant under SL(2, C), subgroup of Cl * 3 = GL(2, C) which is the true group of form invariance of the Dirac equation. Firstly we use the Cl3 algebra to read all features of the Dirac equation for a wave with spin 1/2. We extend this to electromagnetic laws. Next we get the gauge group of electro-weak interactions, first in the leptonic case, electron+neutrino, next in the quark case. The complete wave for all objects of the first generation uses the Clifford algebra Cl1,5. The gauge group is then enlarged into a U (1) × SU (2) × SU (3) Lie group. We consolidate both the standard model and the use of Clifford algebras, true mathematical frame of quantum physics.Mathematics Subject Classification. 15A66; 35Q41; 81T13; 83E15.

show abstract

Quantum Mechanics in the Geometry of Space-Time

Abstract: The Clifford Algebra Associated with an Euclidean Space. .. 2.2 The Clifford Algebras and the ''Imaginary Number'' ffiffiffiffiffiffi ffi À1 p. .. . 2.3 The Field of the Hamilton Quaternions and the Ring of the Biquaternion as Cl þ ð3; 0Þ and Clð3; 0Þ ' Cl þ ð1;

Cited by 13 publications

References 0 publications

The Genesis of Geometric Algebra: A Personal Retrospective

The Genesis of Geometric Algebra: A Personal Retrospective

Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

Gauge Group of the Standard Model in Cl 1,5

Contact Info

Product

Resources

About