%0 Journal Article %1 boudet:les-algebres-de-clifford-et-les-transformations-des-multivecteurs-lalgebre-de-clifford-de-r13-et-la-constante-de-planck-clifford-algebras-and-multivectorial-transformations-the-clifford-algebra-r13-and-the-planck-constant %A Boudet, R. %D 1992 %J Fundam. Theor. Phys. %K Clifford Dirac Planck algebra; constant; inner momentum-energy motion; multivectorial particle} products; reduced tensor; {Darboux %P 343-352 %T Les algebres de Clifford et les transformations des multivecteurs. L'algebre de Clifford de $R(1,3)$ et la constante de Planck. (Clifford algebras and multivectorial transformations. The Clifford algebra $R(1,3)$ and the Planck constant). %V 47 %X For the entire collection see Zbl. 745.00055.Using a convenient form of the inner products $$x.X,\ X.x,\ xE,\ XınE,\ E=Rq,n-q$$ and considering the relations $$xX=x.X+xX,\ Xy=X.y+Xy$$ as definitions one proves, as a theorem, the relation $x(Xy)=(xX)y$ which allows one to construct the Clifford algebra $C(E)$ as a multivectorial algebra. The infinitesimal operators associated with some transformations of multivectors, using specific properties of $C(E)$, are defined and studied.As an application, a Euclidean interpretation of the reduced Planck constant $\hbar$ is drawn up in the following way. $\hbar$ appears in the kinetic part $(\hbar c/2)L$ of the momentum-energy tensor of the Dirac particle. The tensor $L$ is constructed a priori in a purely geometrical way, expressing a generalized ``Darboux motion'' of a plane in spacetime $R1,3$.

@article{boudet:les-algebres-de-clifford-et-les-transformations-des-multivecteurs-lalgebre-de-clifford-de-r13-et-la-constante-de-planck-clifford-algebras-and-multivectorial-transformations-the-clifford-algebra-r13-and-the-planck-constant, abstract = {{[For the entire collection see Zbl. 745.00055.]\par Using a convenient form of the inner products $$x.X,\ X.x,\ x\in E,\ X\in\wedge E,\ E=R\sp{q,n-q}$$ and considering the relations $$xX=x.X+x\wedge X,\ Xy=X.y+X\wedge y$$ as definitions one proves, as a theorem, the relation $x(Xy)=(xX)y$ which allows one to construct the Clifford algebra $C(E)$ as a multivectorial algebra. The infinitesimal operators associated with some transformations of multivectors, using specific properties of $C(E)$, are defined and studied.\par As an application, a Euclidean interpretation of the reduced Planck constant $\hbar$ is drawn up in the following way. $\hbar$ appears in the kinetic part $(\hbar c/2)L$ of the momentum-energy tensor of the Dirac particle. The tensor $L$ is constructed a priori in a purely geometrical way, expressing a generalized ``Darboux motion'' of a plane in spacetime $R\sp{1,3}$.}}, added-at = {2008-03-02T02:12:02.000+0100}, author = {Boudet, R.}, biburl = {https://www.bibsonomy.org/bibtex/216d8d774235092f6b05a88e4a6b05633/dmartins}, classmath = {{*15A66 Clifford algebras 15A90 Appl. of matrix theory to physics}}, description = {robotica-bib}, interhash = {bbd44a4e2ac358c6c42c3b599f861d45}, intrahash = {16d8d774235092f6b05a88e4a6b05633}, journal = {Fundam. Theor. Phys.}, keywords = {Clifford Dirac Planck algebra; constant; inner momentum-energy motion; multivectorial particle} products; reduced tensor; {Darboux}, language = {French. English summary}, note = {Clifford algebras and their applications in mathematical physics, Proc. 2nd Workshop }, pages = {343-352}, timestamp = {2008-03-02T02:12:22.000+0100}, title = {{Les algebres de Clifford et les transformations des multivecteurs. L'algebre de Clifford de $R(1,3)$ et la constante de Planck. (Clifford algebras and multivectorial transformations. The Clifford algebra $R(1,3)$ and the Planck constant).} }, volume = 47, year = 1992 }