User:Jheald/sandbox/Geometric algebra

In mathematical physics, a geometric algebra is a multilinear algebra described technically as a Clifford algebra over a real vector space equipped with a non-degenerate quadratic form. Informally, a geometric algebra is a Clifford algebra that includes a geometric product. This allows the theory and properties of the algebra to be built up in a particularly intuitive, geometrically meaningful way. The term is also used in a more general sense to describe the study and application of these algebras: so Geometric algebra is the study of geometric algebras.

Geometric algebra is useful in physics problems that involve rotations, phases or imaginary numbers. Proponents of geometric algebra argue it provides a more compact and intuitive description of classical and quantum mechanics, electromagnetic theory and relativity. Current applications of geometric algebra include computer vision, biomechanics and robotics, and spaceflight dynamics.

Geometric product and geometric algebra

A geometric algebra ${\mathcal {G}}_{n}({\mathcal {V}}_{n})$ is a multigraded algebra similar to Grassmann's exterior algebra, constructed over a vector space ${\mathcal {V}}_{n}$, except that the exterior product is replaced by a more fundamental operator known as the geometric product. In general the result of a geometric product is a multi-graded object called a multivector. A multivector is a linear combination of multivector basis elements, sometimes called basis blades.

More generally, Clifford algebras are associative algebras generated over an underlying vector space ${\mathcal {V}}_{n}$ equipped with a quadratic form $Q({\mathcal {V}}_{n})$ (a metric, if it is positive definite; or often, speaking less precisely, even if it is not). Except for vector spaces defined with certain rather pathological forms of addition, existence of the quadratic form is equivalent to being able to define a symmetric (not necessarily positive) scalar product u.v over the vectors, that can be used to orthogonalise the quadratic form, to give a set of basis vectors {e₁...e_n} such that:

$\mathbf {e} _{i}\cdot \mathbf {e} _{j}={\Bigg \{}{\begin{matrix}-1,0,+1&i=j,\\0&i\not =j\end{matrix}}$

Clifford algebras come about if one assumes the dot product is only the symmetric part of the multiplicative product of two vectors, so that there is a more general (Clifford) vector multiplication uv such that u.v = ½ (uv + vu). The orthogonality relations then imply that for basis vectors e_i and e_j orthogonal,

$\mathbf {e} _{i}\mathbf {e} _{j}=-\mathbf {e} _{j}\mathbf {e} _{i}\qquad i\not =j$

where e_ie_j is neither a scalar, nor a vector, but a new sort of quantity, a bivector.

Imposing closure under multiplication, together with the assumptions of associativity and distributivity, now generates a linear space spanned by 2ⁿ multivector bases, {1, e₁, e₂, e₃, ... , e₁e₂, ... , e₁e₂e₃, ...}. Unlike the original simple bases, these compound bases may or may not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e₁e₂ = - e₂e₁; but e₁(e₂e₃) = + (e₂e₃)e₁.

In the most general case, Clifford algebras are identified as Cℓ(V,Q), where V is the underlying vector space, and Q is the defining quadratic form. Here we shall only consider Clifford algebras defined over real-valued vector spaces, so Clifford algebras generated where the coefficient of each basis element is a real-valued scalar. Such Clifford algebras are often labelled Cℓ_p,q,r(R), where R indicates that the reals are the ground field, and p,q and r indicate that the orthogonalised underlying n dimensional vector space is spanned by p basis elements with (e_i)² = +1, q with (e_i)²= -1, and r with (e_i)²= 0. We shall only usually only consider algebras without underlying bases (e_i)²= 0, so a geometric algebra ${\mathcal {G}}_{p,q}$ will be equivalent to the Clifford algebra Cℓ_p,q,r(R).

Summary of properties of the Geometric product

The properties of the Geometric product in such an environment can then be summarised (indeed characterised) as follows, for all multivectors $A,B,D$:

1. Closure: the product of two multivectors is itself a multivector
   • $AB=C\,$
2. Distributivity over the addition of multivectors:
   • $A(B+D)=AB+AD\,$
   • $(A+B)D=AB+BD\,$
3. Associativity
   • $A(BD)=(AB)D=ABD\,$
4. Unit (scalar) element
   • $1\,A=A$
5. Tensor contraction: for any "vector" (a grade-one element) a, a² is a scalar (real number), not necessarily positive.
   • $\mathbf {a} ^{2}=s$
6. Commutativity of the product by a scalar:
   • $\lambda \,A=A\,\lambda \,$

Properties (1) and (2) are among those needed for an algebra over a field. (3) and (4) mean that a geometric algebra is an associative, unital algebra.

Grades of the algebra, outer product and inner products

Grades

The multivector basis elements naturally fall into different grades, according to the number of simple bases that each compound base is the product of. The multivector basis thus has:

• of grade 0: one basis element, the scalar {1}
• of grade 1: the n basis elements {e₁, ..., e_n} of the underlying original vector space ${\mathcal {V}}_{n}$
• of grade 2: ½n(n-1) bases, {e₁e₂, ... etc.}
• of grade k: n choose k bases, {e₁e₂...e_k, ... etc.}
• up to, at grade n: the final single base {e₁e₂...e_n}.

Wedge product and inner products

In general the product of a multivector of grade k and a multivector of grade l will contain terms of grades | k-l |, 2+ | k-l |, ... k+l, going up in twos. It turns out to be instructive to look at some of these terms individually, by defining the following restricted products, which each consider only the contribution at particular grades:

• the wedge product $\wedge$,

$\langle A\rangle _{k}\wedge \langle B\rangle _{l}=\langle AB\rangle _{k+l}$   (zero if k+l > n),

• the contractive product (the "computer scientist's inner product"),

$\langle A\rangle _{k}\,\,\lrcorner \,\,\langle B\rangle _{l}=\langle AB\rangle _{k-l}$   (zero if l < k),

• the dot product (the "physicist's inner product"),

$\langle A\rangle _{k}\cdot \langle B\rangle _{l}=\langle AB\rangle _{|k-l|\,\,\,\,l,k\neq 0}$

where <A>_k denotes the k-grade part of A, also sometimes written A_<k>.

All of these products are distributive. The wedge product is associative, and matches the definition of the exterior product in Grassmann's exterior algebra. It is anticommutative if kl is odd, but commutative if kl is even; so

$\mathbf {u} \wedge \mathbf {v} =-\mathbf {v} \wedge \mathbf {u}$

but

$\mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w} =\mathbf {w} \wedge \mathbf {u} \wedge \mathbf {v} =\mathbf {v} \wedge \mathbf {w} \wedge \mathbf {u} \,\,\,\,(=-\mathbf {v} \wedge \mathbf {w} \wedge \mathbf {u} )$

if u, v and w are all vectors.

The dot product and contractive product are not associative (for example (e_i . e_j) . e_ie_j = 0; but e_i . (e_j . e_ie_j) = -e_i²e_j² ). Both reduce to the original scalar product when applied to a pair of vectors. The dot product (also known as the "semi-symmetric inner product" is commutative if k(l-k) is even, and anticommutative if k(l-k) is odd; so for the dot product of two vectors

$\mathbf {u} \cdot \mathbf {v} =\mathbf {v} \cdot \mathbf {u}$

but for the dot product of a vector and a bivector

$\mathbf {u} \cdot (\mathbf {v} \wedge \mathbf {w} )=-(\mathbf {v} \wedge \mathbf {w} )\cdot \mathbf {u} .$

Just as a vector $\mathbf {u} \,\;$ can be thought of as a signed quantity (eg length) with an associated orientation, parallel to a particular line in space, the bivector $\mathbf {u} \wedge \mathbf {v}$created by the wedge product can be thought of as a signed quantity associated with the orientation of a particular plane in space. This makes sense because it is easy to show that wedge products of linear combinations of u and v produce scalar multiples of the same bivector:

$(p\mathbf {u} +q\mathbf {v} )\wedge (r\mathbf {u} +s\mathbf {v} )=(ps-qr)\mathbf {u} \wedge \mathbf {v}$

corresponding to linear combinations of two vectors attached to the same point identifying the same plane.

From the useful identity

$A\,\lrcorner \,(C\,\lrcorner \,D)=(A\wedge C)\,\lrcorner \,D$

it follows that

$\mathbf {u} \,\lrcorner \,(\mathbf {u} \,\lrcorner \,B)=(\mathbf {u} \wedge \mathbf {u} )\,\lrcorner \,B=0$

where u is a vector and B is a bivector. Calculating $\mathbf {u} \wedge (\mathbf {u} \,\lrcorner \,B)$ is a little more involved, but if $B=\mathbf {u} \wedge \mathbf {v}$, one can find that

$\mathbf {u} \wedge (\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} ))=(\mathbf {u} \,\lrcorner \,\mathbf {u} )(\mathbf {u} \wedge \mathbf {v} )$

Confirmation

Using the result for vectors that

$\mathbf {u} A=(\mathbf {u} \,\lrcorner \,A)+(\mathbf {u} \wedge A)$

it follows that

$\mathbf {u} ^{2}\mathbf {v} =\mathbf {u} \,\lrcorner \,(\mathbf {u} \,\lrcorner \,\mathbf {v} )\,+\,\mathbf {u} \wedge (\mathbf {u} \,\lrcorner \,\mathbf {v} )\,+\,\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} )\,+\,\mathbf {u} \wedge (\mathbf {u} \wedge \mathbf {v} )$

But

$\mathbf {u} \wedge \mathbf {u} \wedge \mathbf {v} =(\mathbf {u} \wedge \mathbf {u} )\wedge \mathbf {v} =0$

and

$\mathbf {u} \,\lrcorner \,(\mathbf {u} \,\lrcorner \,\mathbf {v} )=(\mathbf {u} \wedge \mathbf {u} )\,\lrcorner \,\mathbf {v} =0$

and, since u is a vector,

$\mathbf {u} ^{2}=\mathbf {(} \mathbf {u} \,\lrcorner \,\mathbf {u} ),$

a scalar, so

$\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} )=(\mathbf {u} \,\lrcorner \,\mathbf {u} )\mathbf {v} -(\mathbf {u} \,\lrcorner \,\mathbf {v} )\mathbf {u}$

Thus

${\begin{aligned}\mathbf {u} \wedge (\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} ))&=(\mathbf {u} \,\lrcorner \,\mathbf {u} )(\mathbf {u} \wedge \mathbf {v} )-(\mathbf {u} \,\lrcorner \,\mathbf {v} )(\mathbf {u} \wedge \mathbf {u} )\\&=(\mathbf {u} \,\lrcorner \,\mathbf {u} )(\mathbf {u} \wedge \mathbf {v} )\\\end{aligned}}$

as advertised.

The vector $\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} )$ thus represents a vector still in the plane represented by $\mathbf {u} \wedge \mathbf {v}$ but orthogonal to u — an orthogonal projection of the bivector $\mathbf {u} \wedge \mathbf {v}$, orthogonal to u. Dividing through by $\mathbf {u} \,\lrcorner \,\mathbf {u}$ gives

$\mathbf {v} _{\perp }={\frac {1}{(\mathbf {u} \,\lrcorner \,\mathbf {u} )}}\,(\mathbf {u} \,\lrcorner \,(\mathbf {u} \wedge \mathbf {v} ))$

as the projection of v perpendicular to u. Any vector v' which has the same $\mathbf {v} _{\perp }$ perpendicular to u. $\mathbf {u} \wedge \mathbf {v}$ can therefore often be associated with the oriented area of a parallelogram with the vectors u and v for sides. orientation.

All of this carries over straightforwardly to higher dimensions, with the word blade being used to denote a multivector that can be expressed as a single string of wedge products of independent vectors,

$A=\mathbf {a} _{1}\wedge \mathbf {a} _{2}\wedge \dots \wedge \mathbf {a} _{k}$

Such a blade identifies a k-dimensional oriented and scaled subspace of Rⁿ, one in which any vector a satisfies

$\mathbf {a} \wedge A=A\wedge \mathbf {a} =0$

Otherwise each new vector a_k+1 can be orthogonalised

$(\mathbf {a} _{k+1})_{\perp }={\frac {1}{(A\,\lrcorner \,A)}}\,(A\,\lrcorner \,(A\wedge \mathbf {a} _{k+1}))$

with

$A\wedge \mathbf {a} _{k+1}=A\wedge (\mathbf {a} _{k+1})_{\perp }$

establishing that the magnitude of A can be identified with the (hyper-)volume of the (hyper-)parallelepiped that a₁, a₂ ... a_k mark out.

It should be noted that while all blades are "pure" multivectors (homogeneous multivectors),

$\langle A\rangle _{k}=A$

not all pure multivectors are blades. Thus for example, a general bivector can always be written as a blade $\mathbf {u} \wedge \mathbf {v}$ in R³; but in higher dimensions it may not necessarily be possible to combine $\mathbf {u} \wedge \mathbf {v} +\dots +\mathbf {w} \wedge \mathbf {z}$ into a simpler form.

---

is apparent that they are made up of one real scalar, n choose k compound bases of grade k. These multivector bases can be described by introducing the Defining a pure

A geometric algebra ${\mathcal {G}}_{n}({\mathcal {V}}_{n})$ is an algebra constructed over a vector space ${\mathcal {V}}_{n}$ in which a geometric product is defined. The elements of geometric algebra are multivectors.

The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as

$\mathbf {a} \,\mathbf {b} =\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \wedge \mathbf {b}$

The original vector space ${\mathcal {V}}$ is constructed over the real numbers as scalars. From now on, a vector is something in ${\mathcal {V}}$ itself. Vectors will be represented by boldface, small case letters.

The definition and the associativity of geometric product entails the concept of the inverse of a vector (or division by vector). Thus, one can easily set and solve vector algebra equations that otherwise would be cumbersome to handle. In addition, one gains a geometric meaning that would be difficult to retrieve, for instance, by using matrices. Although not all the elements of the algebra are invertible, the inversion concept can be extended to multivectors. Geometric algebra allows one to deal with subspaces directly, and manipulate them too. Furthermore, geometric algebra is a coordinate-free formalism.

Geometric objects like $\mathbf {a} \wedge \mathbf {b}$ are called bivectors. A bivector can be pictured as a plane segment (a parallelogram, a circle etc.) endowed with orientation. One bivector represents all planar segments with the same magnitude and direction, no matter where they are in the space that contains them. However, once either the vector $\mathbf {a}$ or $\mathbf {b}$ is meant to depart from some preferred point (e.g. in problems of Physics), the oriented plane $B=\mathbf {a} \wedge \mathbf {b}$ is determined unambiguously.

The outer product (the exterior product, or the wedge product) $\wedge$ is defined such that the graded algebra (exterior algebra of Hermann Grassmann) $\wedge ^{n}{\mathcal {V}}_{n}$ of multivectors is generated. Multivectors are thus the direct sum of grade k elements (k-vectors), where k ranges from 0 (scalars) to n, the dimension of the original vector space ${\mathcal {V}}$. Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively).

Inverting a vector

As a meaningful result one can consider a fixed non-zero vector $\mathbf {v}$, from a point chosen as the origin, in the usual Euclidean space, $\mathbb {R} ^{3}$. The set of all vectors $\mathbf {x}$ such that $\mathbf {x} \wedge \mathbf {v} =B$ , $B$ denoting a given bivector containing $\mathbf {v}$, determines a line $l$ parallel to $\mathbf {v}$. Since $B$ is a directed area, $l$ is uniquely determined with respect to the chosen origin. The set of all vectors $\mathbf {x}$ such that $\mathbf {x} \cdot \mathbf {v} =s$, $s$ denoting a given (real) scalar, determines a plane P orthogonal to $\mathbf {v}$. Again, P is uniquely determined with respect to the chosen origin. The two information pieces, $B$ and $s$, can be set independently of one another. Now, what is (if any) the vector $\mathbf {y}$ that satisfies the system {$\mathbf {y} \wedge \mathbf {v} =B$ , $\mathbf {y} \cdot \mathbf {v} =s$} ? Geometrically, the answer is plain: it is the vector that departs from the origin and arrives at the intersection of $l$ and P. By geometric algebra, even the algebraic answer is simple: $\mathbf {y} \mathbf {v} =s+B=>\mathbf {y} =(s+B)/\mathbf {v} =(s+B)\mathbf {v}$^-1, where the inverse of a non-zero vector is expressed by $\mathbf {z}$^-1 $=\mathbf {z} /(\mathbf {z} \cdot \mathbf {z} )$. Note that the division by a vector transforms the multivector $s+B$ into the sum of two vectors. Note also that the structure of the solution does not depend on the chosen origin.

The contraction rule

The connection between Clifford algebras and quadratic forms come from the contraction property. This rule also gives the space a metric defined by the naturally derived inner product. It is to be noted that in geometric algebra in all its generality there is no restriction whatsoever on the value of the scalar, it can very well be negative, even zero (in that case, the possibility of an inner product is ruled out if you require $\langle x,x\rangle \geq 0$).

The contraction rule can be put in the form:

$Q(\mathbf {a} )=\mathbf {a} ^{2}=\epsilon _{a}{\Vert \mathbf {a} \Vert }^{2}$

where $\Vert \mathbf {a} \Vert$ is the modulus of vector a, and $\epsilon _{a}=0,\,\pm 1$ is called the signature of vector a. This is especially useful in the construction of a Minkowski space (the spacetime of special relativity) through $\mathbb {R} _{1,3}$. In that context, null-vectors are called "lightlike vectors", vectors with negative signature are called "spacelike vectors" and vectors with positive signature are called "timelike vectors" (these last two denominations are exchanged when using $\mathbb {R} _{3,1}$ instead).

Inner and outer product

The usual dot product and cross product of traditional vector algebra (on $\mathbb {R} ^{3}$) find their places in geometric algebra ${\mathcal {G}}_{3}$ as the inner product

$\mathbf {a} \cdot \mathbf {b} ={\frac {1}{2}}(\mathbf {a} \mathbf {b} +\mathbf {b} \mathbf {a} )$

(which is symmetric) and the outer product

$\mathbf {a} \wedge \mathbf {b} ={\frac {1}{2}}(\mathbf {a} \mathbf {b} -\mathbf {b} \mathbf {a} )$

with

$\mathbf {a} \times \mathbf {b} =-i(\mathbf {a} \wedge \mathbf {b} )$

(which is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the mere distinction between vectors and bivectors (elements of grade two). The $i$ here is the unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property $i^{2}=-1$.

While the cross product can only be defined in a three-dimensional space, the inner and outer products can be generalized to any dimensional ${\mathcal {G}}_{p,q,r}$.

Let $\mathbf {a} ,\,\mathbf {A} _{\langle k\rangle }$ be a vector and a homogeneous multivector of grade k, respectively. Their inner product is then

$\mathbf {a} \cdot \mathbf {A} _{\langle k\rangle }={1 \over 2}\,\left(\mathbf {a} \,\mathbf {A} _{\langle k\rangle }+(-1)^{k+1}\,\mathbf {A} _{\langle k\rangle }\,\mathbf {a} \right)=(-1)^{k+1}\mathbf {A} _{\langle k\rangle }\cdot \mathbf {a}$

and the outer product is

$\mathbf {a} \wedge \mathbf {A} _{\langle k\rangle }={1 \over 2}\,\left(\mathbf {a} \,\mathbf {A} _{\langle k\rangle }-(-1)^{k+1}\,\mathbf {A} _{\langle k\rangle }\,\mathbf {a} \right)=(-1)^{k}\mathbf {A} _{\langle k\rangle }\wedge \mathbf {a}$

Applications of geometric algebra

A useful example is $\mathbb {R} _{3,1}$, and to generate ${\mathcal {G}}_{3,1}$, an instance of geometric algebra called spacetime algebra by Hestenes. The electromagnetic field tensor, in this context, becomes just a bivector $\mathbf {E} +i\mathbf {B}$ where the imaginary unit is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks".

Boosts in this Lorenzian metric space have the same expression $e^{\mathbf {\beta } }$ as rotation in Euclidean space, where $\mathbf {\beta }$ is of course the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

History

The geometric algebra of David Hestenes et al. (1984) reinterprets Clifford algebras over the reals, and is claimed to return to the name and interpretation Clifford originally intended. Emil Artin's Geometric Algebra discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry.

See also

• Clifford algebra
• algebra of physical space
• spinor
• quaternion

References

• Baylis, W. E., ed., 1996. Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering. Boston: Birkhäuser.
• Baylis, W. E., 2002. Electrodynamics: A Modern Geometric Approach, 2nd ed. Birkhäuser. ISBN 0-8176-4025-8
• Nicolas Bourbaki, 1980. Eléments de Mathématique. Algèbre. Chpt. 9, "Algèbres de Clifford". Paris: Hermann.
• Chris Doran and Anthony Lasenby, 2003. Geometric Algebra for Physicists. Cambridge Univ. Press. ISBN 0521480221
• David Hestenes and Garret Sobczyk, 1984. Clifford Algebra to Geometric Calculus, Springer Verlag ISBN 90-277-1673-0
• Hestenes, D., 1999. New Foundations for Classical Mechanics, 2nd ed. Springer Verlag ISBN 0-7923-5302-1
• Lasenby, J., Lasenby, A. N., and Doran, C. J. L., 2000, "A Unified Mathematical Language for Physics and Engineering in the 21st Century," Philosophical Transactions of the Royal Society of London A 358: 1-18.
• Pertti Lounesto, 2001, Clifford Algebras and Spinors, 2nd ed. Cambridge Univ. Press. ISBN 0521005515

External links

Research groups

• Geometric Calculus International. Links to Research groups, Software, and Conferences, worldwide.
• Cambridge Geometric Algebra group. Full-text online publications, and other material.
• University of Amsterdam group
• Geometric Calculus research & development (University of Arizona).
• GA-Net blog and newsletter archive. Geometric Algebra/Clifford Algebra development news.

Further reading

• Imaginary Numbers are are not Real - the Geometric Algebra of Spacetime. Introduction (Cambridge GA group).
• Physical Applications of Geometric Algebra. Final-year undergraduate course (Cambridge GA group; see also 1999 version).
• Maths for (Games) Programmers: 5 - Multivector methods. Comprehensive introduction and reference for programmers, from Ian Bell.
• A Geometric Algebra Primer, especially for computer scientists.
• Clifford/Geometric algebra overview by Martin Baker
• "Geometric Algebra". PlanetMath.

Category:Clifford algebras Category:Ring theory