# geometric algebra

Geometric algebra is a Clifford algebra   (http://planetmath.org/CliffordAlgebra2) which has been used with great success in the modeling of a wide variety   of physical phenomena. Clifford algebra is considered a more general algebraic  framework than geometric algebra. The primary  distinction is that geometric algebra utilizes only real numbers as scalars and to represent magnitudes. The underlying philosophical justification for this is the interpretation   that the unit imaginary  has geometric significance which naturally arises from the properties of the algebra and the interaction of its various subspaces  .

Let ${\cal{V}}^{n}$ be an $n$–dimensional vector space  over the real numbers. As with traditional vector algebra, the vector space is spanned by a set of $n$ linearly independent  basis vectors. Any vector in this space may be represented by a linear combination  of the basis vectors. In the geometric algebra literature, such basis entities are also called blades.

Since vectors are one-dimensional directed quantities, they are assigned a grade of 1. Scalars are considered to be grade-0 entities. In geometric algebra, there exist higher dimensional analogues to vectors. Two-dimensional directed quantites are termed bivectors and they are grade-2 entites. In general a $k$-dimensional entity is known as a $k$-vector.

For vectors $\mathbf{a},\mathbf{b},\mathbf{c}\in{\cal{V}}^{n}$ and real scalars $\alpha,\beta\in\mathbf{R}$, the geometric product satisfies the following axioms:

 $\begin{array}[]{lll}\mbox{{associativity:}}&\mathbf{a}(\mathbf{b}\mathbf{c})=(% \mathbf{a}\mathbf{b})\mathbf{c}&\mathbf{a}+(\mathbf{b}+\mathbf{c})=(\mathbf{a}% +\mathbf{b})+\mathbf{c}\\ \mbox{{commutativity:}}&\alpha\beta=\beta\alpha&\alpha+\beta=\beta+\alpha\\ &\alpha\mathbf{b}=\mathbf{b}\alpha&\alpha+\mathbf{b}=\mathbf{b}+\alpha\\ &\mathbf{ab}=\frac{1}{2}(\mathbf{ab}+\mathbf{ba})+\frac{1}{2}(\mathbf{ab}-% \mathbf{ba})&\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}\\ \mbox{{distributivity:}}&\mathbf{a}(\mathbf{b}+\mathbf{c})=\mathbf{a}\mathbf{b% }+\mathbf{a}\mathbf{c}&(\mathbf{b}+\mathbf{c})\mathbf{a}=\mathbf{b}\mathbf{a}+% \mathbf{c}\mathbf{a}\\ \mbox{{linearity}}&\alpha(\mathbf{b}+\mathbf{c})=\alpha\mathbf{b}+\alpha% \mathbf{c}=(\mathbf{b}+\mathbf{c})\alpha\\ \mbox{{contraction:}}&\mathbf{a}^{2}=\mathbf{a}\mathbf{a}=\sum_{i=1}^{n}% \epsilon_{i}|\mathbf{a}_{i}|^{2}=\alpha&\mbox{where }\epsilon_{i}\in\{-1,0,1\}% \end{array}$
 $\mathbf{a}\mathbf{b}=\mathbf{b}\mathbf{a}$
 $\mathbf{a}\mathbf{b}=-\mathbf{b}\mathbf{a}$

The parallelism of vectors is encoded as a symmetric property, while orthogonality of vectors is encoded as an antisymmetric property.

The contraction rule specifies that the square of any vector is a scalar equal to the sum of the square of the magnitudes of its components   in each basis direction. Depending on the contraction rule for each of the basis directions, the magnitude of the vector may be positive, negative, or zero. A vector with a magnitude of zero is called a null vector.

The graded algebra  ${\cal{G}}_{n}$ generated from ${\cal{V}}^{n}$ is defined over a $2^{n}$-dimensional linear space. This basis entities for this space can be generated by successive application of the geometric product to the basis vectors of ${\cal{V}}^{n}$ until a closed set of basis entities is obtained. The basis entites for the space are known as blades. The following multiplication table illustrates the generation of basis blades from the basis vectors $\mathbf{e}_{1},\mathbf{e}_{2}\in{\cal{V}}^{n}$.

 $\begin{array}[]{cccc}\epsilon_{0}&\mathbf{e}_{1}&\mathbf{e}_{2}&\mathbf{e}_{12% }\\ \mathbf{e}_{1}&\epsilon_{1}&\mathbf{e}_{12}&\epsilon_{1}\mathbf{e}_{2}\\ \mathbf{e}_{2}&-\mathbf{e}_{12}&\epsilon_{2}&-\epsilon_{2}\mathbf{e}_{1}\\ \mathbf{e}_{12}&-\epsilon_{1}\mathbf{e}_{2}&\epsilon_{2}\mathbf{e}_{1}&-% \epsilon_{1}\epsilon_{2}\end{array}$

Here, $\epsilon_{1}$ and $\epsilon_{2}$ represent the contraction rule for $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively. Note that the basis vectors of ${\cal{V}}^{n}$ become blades themselves in addition to the multiplicative identity  , $\epsilon_{0}\equiv 1$ and the new bivector $\mathbf{e}_{12}\equiv\mathbf{e}_{1}\mathbf{e}_{2}$. As the table demonstrates, this set of basis blades is closed under the geometric product.

The geometric product $\mathbf{a}\mathbf{b}$ is related to the inner product $\mathbf{a}\cdot\mathbf{b}$ and the exterior product $\mathbf{a}\wedge\mathbf{b}$ by

 $\mathbf{a}\mathbf{b}=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\wedge\mathbf{b}=% \mathbf{b}\cdot\mathbf{a}-\mathbf{b}\wedge\mathbf{a}=2\mathbf{a}\cdot\mathbf{b% }-\mathbf{b}\mathbf{a}.$

In the above example, the result of the inner (dot) product is a scalar (grade-0), while the result of the exterior (wedge) product is a bivector (grade-2).

## Bibliography

1. 1.

David Hestenes, New Foundations for Classical Mechanics, Kluwer, Dordrecht, 1999

2. 2.

David Hestenes, Garret Sobczyk, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1984

 Title geometric algebra Canonical name GeometricAlgebra Date of creation 2013-03-22 13:17:03 Last modified on 2013-03-22 13:17:03 Owner PhysBrain (974) Last modified by PhysBrain (974) Numerical id 15 Author PhysBrain (974) Entry type Definition Classification msc 15A66 Classification msc 15A75 Synonym Clifford algebra Related topic ExteriorAlgebra Related topic CliffordAlgebra2 Related topic CCliffordAlgebra Related topic SpinGroup Defines geometric product