# Mathematical products

1. Aug 27, 2014

### mikeeey

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

2. Aug 27, 2014

### 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.

3. Aug 27, 2014

### 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.