Mathematical Products: Geometric vs. Tensor

[mikeeey]

hello every body.
may i know what is the difference between geometric product and tensor product ?

 

[Terandol]

I'm not sure if you're looking for a more elementary explanation but the way I understand it, a geometric product is just the algebra multiplication in a (real) Clifford algebra while the tensor product is obtained via the multiplication in the tensor algebra.

There is a relation between these two objects since a Clifford algebra can be obtained as a quotient of the tensor algebra. Concretely, let $V$ be a (real in the case of geometric products) vector space. Then the tensor algebra is defined by
$$\mathcal{T}(V) =\sum_{i=0}^\infty \bigotimes {}^i V$$
and to construct the Clifford algebra associated to some quadratic form $q$ on $V$, you take the quotient $\mathcal{Cl}(V,q)=\mathcal{T} /\mathcal{I}_q(V)$
where $\mathcal{I}_q(V)$ is the ideal generated by the elements $v\otimes v+q(v)1$ where $v\in V$.

So, to get a kind of intuitive picture, the geometric product is obtained in two steps. First you take the tensor multiplication in the tensor algebra, then using the fact that $v\otimes v=-q(v)1$ in the Clifford algebra (the sign is just a convention here, sometimes the positive sign is taken in the definition,) you can get rid of all the squares in the resulting expression using the quadratic form.

 

[mikeeey]

thank you very much Terandol for the useful explanation .

I found hours ago that : the geometric product works only in euclidean space while the tensor product is more generalized in non-euclidean space according to Einstein work .
where the tensor has symmetric and skew-symmetric parts ,where the symmetric part represents the inner product and the skew-symmetric part represents the wedge product.