geometric algebra

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

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.

The

geometric algebra

𝒢

n

=

𝒢

⁢

(

𝒱

n

)

is a multi-graded algebra

similar

to Grassmann’s

exterior algebra

, except that the exterior product is replaced by a more fundamental

multiplication

operation

known as the

geometric product

. In general, the result of the geometric product is a multi-graded object called a

multivector

. A multivector is a linear combination of basis blades.

For vectors

𝐚

,

𝐛

,

𝐜

∈

𝒱

n

and real scalars

α

,

β

∈

𝐑

, the geometric product

satisfies

the following axioms:

associativity:

𝐚

⁢

(

𝐛𝐜

)

=

(

𝐚𝐛

)

⁢

𝐜

𝐚

+

(

𝐛

+

𝐜

)

=

(

𝐚

+

𝐛

)

+

𝐜

commutativity:

α

⁢

β

=

β

⁢

α

α

+

β

=

β

+

α

α

⁢

𝐛

=

𝐛

⁢

α

α

+

𝐛

=

𝐛

+

α

𝐚𝐛

=

1

2

⁢

(

𝐚𝐛

+

𝐛𝐚

)

+

1

2

⁢

(

𝐚𝐛

-

𝐛𝐚

)

𝐚

+

𝐛

=

𝐛

+

𝐚

distributivity:

𝐚

⁢

(

𝐛

+

𝐜

)

=

𝐚𝐛

+

𝐚𝐜

(

𝐛

+

𝐜

)

⁢

𝐚

=

𝐛𝐚

+

𝐜𝐚

𝐥𝐢𝐧𝐞𝐚𝐫𝐢𝐭𝐲

α

⁢

(

𝐛

+

𝐜

)

=

α

⁢

𝐛

+

α

⁢

𝐜

=

(

𝐛

+

𝐜

)

⁢

α

contraction:

𝐚

2

=

𝐚𝐚

=

∑

i

=

1

n

ϵ

i

⁢

|

𝐚

i

|

2

=

α

where

⁢

ϵ

i

∈

{

-

1

,

0

,

1

}

Commutativity

of scalar–

scalar multiplication

and vector–scalar multiplication is

symmetric

; however, in general, vector–

vector multiplication

is not

commutative

. The order of multiplication of vectors is significant. In particular, for

parallel vectors

:

𝐚𝐛

=

𝐛𝐚

and for

orthogonal vectors

:

𝐚𝐛

=

-

𝐛𝐚

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

𝒢

n

generated from

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

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

𝐞

1

,

𝐞

2

∈

𝒱

n

.

ϵ

0

𝐞

1

𝐞

2

𝐞

12

𝐞

1

ϵ

1

𝐞

12

ϵ

1

⁢

𝐞

2

𝐞

2

-

𝐞

12

ϵ

2

-

ϵ

2

⁢

𝐞

1

𝐞

12

-

ϵ

1

⁢

𝐞

2

ϵ

2

⁢

𝐞

1

-

ϵ

1

⁢

ϵ

2

Here,

ϵ

1

and

ϵ

2

represent the contraction rule for

𝐞

1

and

𝐞

2

respectively. Note that the basis vectors of

𝒱

n

become blades themselves in

addition

to the

multiplicative identity

,

ϵ

0

≡

1

and the new

bivector

𝐞

12

≡

𝐞

1

⁢

𝐞

2

. As the table demonstrates, this set of basis blades is closed under the geometric product.

The

geometric product

𝐚𝐛

is related to the

inner product

𝐚

⋅

𝐛

and the exterior product

𝐚

∧

𝐛

by

𝐚𝐛

=

𝐚

⋅

𝐛

+

𝐚

∧

𝐛

=

𝐛

⋅

𝐚

-

𝐛

∧

𝐚

=

2

⁢

𝐚

⋅

𝐛

-

𝐛𝐚

.

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.

David Hestenes,

New

Foundations

for Classical Mechanics

, Kluwer, Dordrecht, 1999

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

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