In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Clifford algebra is an
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
generated by a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
, and is a
unital associative algebra with the additional structure of a distinguished subspace. As
-algebras, they generalize the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s,
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s,
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s and several other
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...
systems. The theory of Clifford algebras is intimately connected with the theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s and
orthogonal transformations. Clifford algebras have important applications in a variety of fields including
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
and
digital image processing. They are named after the English mathematician
William Kingdon Clifford (1845–1879).
The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.
Introduction and basic properties
A Clifford algebra is a
unital associative algebra that contains and is generated by a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a
field , where is equipped with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
. The Clifford algebra is the "freest" unital associative algebra generated by subject to the condition
where the product on the left is that of the algebra, and the on the right is the algebra's
multiplicative identity (not to be confused with the multiplicative identity of ). The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a
universal property, as done
below.
When is a finite-dimensional real vector space and is
nondegenerate, may be identified by the label , indicating that has an orthogonal basis with elements with , with , and where indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by
orthogonal diagonalization.
The
free algebra generated by may be written as the
tensor algebra , that is, the
direct sum of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of copies of over all . Therefore one obtains a Clifford algebra as the
quotient of this tensor algebra by the two-sided
ideal generated by elements of the form for all elements . The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. ). Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguished
subspace , being the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of the
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
map. Such a subspace cannot in general be uniquely determined given only a -algebra that is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the Clifford algebra.
If is
invertible in the ground field , then one can rewrite the fundamental identity above in the form
where
is the
symmetric bilinear form associated with , via the
polarization identity.
Quadratic forms and Clifford algebras in characteristic form an exceptional case in this respect. In particular, if it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies , Many of the statements in this article include the condition that the characteristic is not , and are false if this condition is removed.
As a quantization of the exterior algebra
Clifford algebras are closely related to
exterior algebras. Indeed, if then the Clifford algebra is just the exterior algebra . Whenever is invertible in the ground field , there exists a canonical ''linear'' isomorphism between and . That is, they are
naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
since it makes use of the extra information provided by .
The Clifford algebra is a
filtered algebra; the
associated graded algebra is the exterior algebra.
More precisely, Clifford algebras may be thought of as ''quantizations'' (cf.
quantum group) of the exterior algebra, in the same way that the
Weyl algebra is a quantization of the
symmetric algebra.
Weyl algebras and Clifford algebras admit a further structure of a
*-algebra, and can be unified as even and odd terms of a
superalgebra, as discussed in
CCR and CAR algebras.
Universal property and construction
Let be a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a
field , and let be a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
on . In most cases of interest the field is either the field of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s , or the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s , or a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
.
A Clifford algebra is a pair , where is a
unital associative algebra over and is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
that satisfies for all in , defined by the following
universal property: given any unital associative algebra over and any linear map such that
(where denotes the multiplicative identity of ), there is a unique
algebra homomorphism such that the following diagram
commutes (i.e. such that ):
The quadratic form may be replaced by a (not necessarily symmetric)
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
that has the property , in which case an equivalent requirement on is
When the characteristic of the field is not , this may be replaced by what is then an equivalent requirement,
where the bilinear form may additionally be restricted to being symmetric without loss of generality.
A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains , namely the
tensor algebra , and then enforce the fundamental identity by taking a suitable
quotient. In our case we want to take the two-sided
ideal in generated by all elements of the form
for all
and define as the quotient algebra
The
ring product inherited by this quotient is sometimes referred to as the Clifford product to distinguish it from the exterior product and the scalar product.
It is then straightforward to show that contains and satisfies the above universal property, so that is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that is
injective. One usually drops the and considers as a
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
of .
The universal characterization of the Clifford algebra shows that the construction of is in nature. Namely, can be considered as a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the
category of vector spaces with quadratic forms (whose
morphisms are linear maps that preserve the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.
Basis and dimension
Since comes equipped with a quadratic form , in characteristic not equal to there exist
bases for that are
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
. An
orthogonal basis is one such that for a symmetric bilinear form
for
, and
The fundamental Clifford identity implies that for an orthogonal basis
for
, and
This makes manipulation of orthogonal basis vectors quite simple. Given a product
of ''distinct'' orthogonal basis vectors of , one can put them into a standard order while including an overall sign determined by the number of
pairwise swaps needed to do so (i.e. the
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
of the ordering
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
).
If the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of over is and is an orthogonal basis of , then is
free over with a basis
The empty product () is defined as being the multiplicative
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
. For each value of there are
basis elements, so the total dimension of the Clifford algebra is
Examples: real and complex Clifford algebras
The most important Clifford algebras are those over
real and
complex vector spaces equipped with
nondegenerate quadratic forms.
Each of the algebras and is isomorphic to or , where is a
full matrix ring with entries from , , or . For a complete classification of these algebras see ''
Classification of Clifford algebras''.
Real numbers
Clifford algebras are also sometimes referred to as
geometric algebras, most often over the real numbers.
Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
where is the dimension of the vector space. The pair of integers is called the
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
of the quadratic form. The real vector space with this quadratic form is often denoted The Clifford algebra on is denoted The symbol means either or , depending on whether the author prefers positive-definite or negative-definite spaces.
A standard
basis for consists of mutually orthogonal vectors, of which square to and of which square to . Of such a basis, the algebra will therefore have vectors that square to and vectors that square to .
A few low-dimensional cases are:
* is naturally isomorphic to since there are no nonzero vectors.
* is a two-dimensional algebra generated by that squares to , and is algebra-isomorphic to , the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s.
* is a two-dimensional algebra generated by that squares to , and is algebra-isomorphic to the
split-complex number
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...
s.
* is a four-dimensional algebra spanned by . The latter three elements all square to and anticommute, and so the algebra is isomorphic to the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s .
* is isomorphic to the algebra of
split-quaternions.
* is an 8-dimensional algebra isomorphic to the
direct sum , the
split-biquaternions.
* , also called the
Pauli algebra, is isomorphic to the algebra of
biquaternions.
Complex numbers
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension is equivalent to the standard diagonal form
Thus, for each dimension , up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on with the standard quadratic form by .
For the first few cases one finds that
* , the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
* , the
bicomplex numbers
* , the
biquaternions
where denotes the algebra of matrices over .
Examples: constructing quaternions and dual quaternions
Quaternions
In this section, Hamilton's
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s are constructed as the even subalgebra of the Clifford algebra .
Let the vector space be real three-dimensional space , and the quadratic form be the usual quadratic form. Then, for in we have the bilinear form (or scalar product)
Now introduce the Clifford product of vectors and given by
Denote a set of orthogonal unit vectors of as , then the Clifford product yields the relations
and
The general element of the Clifford algebra is given by
The linear combination of the even degree elements of defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements as
which shows that the even subalgebra is Hamilton's real
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
algebra.
To see this, compute
and
Finally,
Dual quaternions
In this section,
dual quaternions are constructed as the even subalgebra of a Clifford algebra of real four-dimensional space with a degenerate quadratic form.
Let the vector space be real four-dimensional space and let the quadratic form be a degenerate form derived from the Euclidean metric on For in introduce the degenerate bilinear form
This degenerate scalar product projects distance measurements in onto the hyperplane.
The Clifford product of vectors and is given by
Note the negative sign is introduced to simplify the correspondence with quaternions.
Denote a set of mutually orthogonal unit vectors of as , then the Clifford product yields the relations
and
The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements and the dual unit as
This provides the correspondence of with
dual quaternion algebra.
To see this, compute
and
The exchanges of and alternate signs an even number of times, and show the dual unit commutes with the quaternion basis elements .
Examples: in small dimension
Let be any field of characteristic not .
Dimension 1
For , if has diagonalization , that is there is a non-zero vector such that , then is algebra-isomorphic to a -algebra generated by an element that satisfies , the quadratic algebra .
In particular, if (that is, is the zero quadratic form) then is algebra-isomorphic to the
dual numbers algebra over .
If is a non-zero square in , then .
Otherwise, is isomorphic to the quadratic field extension of .
Dimension 2
For , if has diagonalization with non-zero and (which always exists if is non-degenerate), then is isomorphic to a -algebra generated by elements and that satisfies , and .
Thus is isomorphic to the (generalized)
quaternion algebra . We retrieve Hamilton's quaternions when , since .
As a special case, if some in satisfies , then .
Properties
Relation to the exterior algebra
Given a vector space , one can construct the
exterior algebra , whose definition is independent of any quadratic form on . It turns out that if does not have characteristic then there is a
natural isomorphism between and considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if . One can thus consider the Clifford algebra as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on with a multiplication that depends on (one can still define the exterior product independently of ).
The easiest way to establish the isomorphism is to choose an ''orthogonal'' basis for and extend it to a basis for as described
above. The map is determined by
Note that this works only if the basis is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.
If the
characteristic of is , one can also establish the isomorphism by antisymmetrizing. Define functions by
where the sum is taken over the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
on elements, . Since is
alternating, it induces a unique linear map . The
direct sum of these maps gives a linear map between and . This map can be shown to be a linear isomorphism, and it is natural.
A more sophisticated way to view the relationship is to construct a
filtration on . Recall that the
tensor algebra has a natural filtration: , where contains sums of tensors with
order . Projecting this down to the Clifford algebra gives a filtration on . The
associated graded algebra
is naturally isomorphic to the exterior algebra . Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of in for all ), this provides an isomorphism (although not a natural one) in any characteristic, even two.
Grading
In the following, assume that the characteristic is not .
Clifford algebras are -
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
s (also known as
superalgebras). Indeed, the linear map on defined by (
reflection through the origin) preserves the quadratic form and so by the universal property of Clifford algebras extends to an algebra
automorphism
Since is an
involution (i.e. it squares to the
identity) one can decompose into positive and negative eigenspaces of
where
Since is an automorphism it follows that:
where the bracketed superscripts are read modulo 2. This gives the structure of a -
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
. The subspace forms a
subalgebra of , called the ''even subalgebra''. The subspace is called the ''odd part'' of (it is not a subalgebra). -grading plays an important role in the analysis and application of Clifford algebras. The automorphism is called the ''main
involution'' or ''grade involution''. Elements that are pure in this -grading are simply said to be even or odd.
''Remark''. The Clifford algebra is not a -graded algebra, but is -
filtered, where is the subspace spanned by all products of at most elements of .
The ''degree'' of a Clifford number usually refers to the degree in the -grading.
The even subalgebra of a Clifford algebra is itself isomorphic to a Clifford algebra. If is the
orthogonal direct sum of a vector of nonzero norm and a subspace , then is isomorphic to , where is the form restricted to . In particular over the reals this implies that:
In the negative-definite case this gives an inclusion , which extends the sequence
Likewise, in the complex case, one can show that the even subalgebra of is isomorphic to .
Antiautomorphisms
In addition to the automorphism , there are two
antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the
tensor algebra comes with an antiautomorphism that reverses the order in all products of vectors:
Since the ideal is invariant under this reversal, this operation descends to an antiautomorphism of called the ''transpose'' or ''reversal'' operation, denoted by . The transpose is an antiautomorphism: . The transpose operation makes no use of the -grading so we define a second antiautomorphism by composing and the transpose. We call this operation ''Clifford conjugation'' denoted
Of the two antiautomorphisms, the transpose is the more fundamental.
Note that all of these operations are
involutions. One can show that they act as on elements that are pure in the -grading. In fact, all three operations depend on only the degree modulo . That is, if is pure with degree then
where the signs are given by the following table:
:
Clifford scalar product
When the characteristic is not , the quadratic form on can be extended to a quadratic form on all of (which we also denoted by ). A basis-independent definition of one such extension is
where denotes the scalar part of (the degree- part in the -grading). One can show that
where the are elements of – this identity is ''not'' true for arbitrary elements of .
The associated symmetric bilinear form on is given by
One can check that this reduces to the original bilinear form when restricted to . The bilinear form on all of is
nondegenerate if and only if it is nondegenerate on .
The operator of left (respectively right) Clifford multiplication by the transpose of an element is the
adjoint of left (respectively right) Clifford multiplication by with respect to this inner product. That is,
and
Structure of Clifford algebras
''In this section we assume that characteristic is not , the vector space is finite-dimensional and that the associated symmetric bilinear form of is nondegenerate.''
A
central simple algebra over is a matrix algebra over a (finite-dimensional) division algebra with center . For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
* If has even dimension then is a central simple algebra over .
* If has even dimension then the even subalgebra is a central simple algebra over a quadratic extension of or a sum of two isomorphic central simple algebras over .
* If has odd dimension then is a central simple algebra over a quadratic extension of or a sum of two isomorphic central simple algebras over .
* If has odd dimension then the even subalgebra is a central simple algebra over .
The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that has even dimension and a non-singular bilinear form with
discriminant , and suppose that is another vector space with a quadratic form. The Clifford algebra of is isomorphic to the tensor product of the Clifford algebras of and , which is the space with its quadratic form multiplied by . Over the reals, this implies in particular that
These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the
classification of Clifford algebras.
Notably, the
Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends on only the signature . This is an algebraic form of
Bott periodicity.
Lipschitz group
The class of Lipschitz groups (
Clifford groups or Clifford–Lipschitz groups) was discovered by
Rudolf Lipschitz.
In this section we assume that is finite-dimensional and the quadratic form is
nondegenerate.
An action on the elements of a Clifford algebra by its
group of units
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the ele ...
may be defined in terms of a twisted conjugation: twisted conjugation by maps , where is the ''main involution'' defined
above.
The Lipschitz group is defined to be the set of invertible elements that ''stabilize the set of vectors'' under this action, meaning that for all in we have:
This formula also defines an action of the Lipschitz group on the vector space that preserves the quadratic form , and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements of for which is invertible in , and these act on by the corresponding reflections that take to . (In characteristic these are called orthogonal transvections rather than reflections.)
If is a finite-dimensional real vector space with a
non-degenerate quadratic form then the Lipschitz group maps onto the orthogonal group of with respect to the form (by the
Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field . This leads to exact sequences
Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.
Spinor norm
In arbitrary characteristic, the
spinor norm is defined on the Lipschitz group by
It is a homomorphism from the Lipschitz group to the group of non-zero elements of . It coincides with the quadratic form of when is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of , , or on . The difference is not very important in characteristic other than 2.
The nonzero elements of have spinor norm in the group ( of squares of nonzero elements of the field . So when is finite-dimensional and non-singular we get an induced map from the orthogonal group of to the group , also called the spinor norm. The spinor norm of the reflection about , for any vector , has image in , and this property uniquely defines it on the orthogonal group. This gives exact sequences:
Note that in characteristic the group has just one element.
From the point of view of
Galois cohomology of
algebraic groups, the spinor norm is a
connecting homomorphism on cohomology. Writing for the
algebraic group of square roots of 1 (over a field of characteristic not it is roughly the same as a two-element group with trivial Galois action), the short exact sequence
yields a long exact sequence on cohomology, which begins
The 0th Galois cohomology group of an algebraic group with coefficients in is just the group of -valued points: , and , which recovers the previous sequence
where the spinor norm is the connecting homomorphism .
Spin and pin groups
In this section we assume that is finite-dimensional and its bilinear form is non-singular.
The
pin group is the subgroup of the Lipschitz group of elements of spinor norm , and similarly the
spin group is the subgroup of elements of
Dickson invariant in . When the characteristic is not , these are the elements of determinant . The spin group usually has index in the pin group.
Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define the
special orthogonal group to be the image of . If does not have characteristic this is just the group of elements of the orthogonal group of determinant . If does have characteristic , then all elements of the orthogonal group have determinant , and the special orthogonal group is the set of elements of Dickson invariant .
There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm . The kernel consists of the elements and , and has order unless has characteristic . Similarly there is a homomorphism from the Spin group to the special orthogonal group of .
In the common case when is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when has dimension at least . Further the kernel of this homomorphism consists of and . So in this case the spin group, , is a double cover of . Note, however, that the simple connectedness of the spin group is not true in general: if is for and both at least then the spin group is not simply connected. In this case the algebraic group is simply connected as an algebraic group, even though its group of real valued points is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.
Spinors
Clifford algebras , with even, are matrix algebras that have a complex representation of dimension . By restricting to the group we get a complex representation of the Pin group of the same dimension, called the
spin representation. If we restrict this to the spin group then it splits as the sum of two ''half spin representations'' (or ''Weyl representations'') of dimension .
If is odd then the Clifford algebra is a sum of two matrix algebras, each of which has a representation of dimension , and these are also both representations of the pin group . On restriction to the spin group these become isomorphic, so the spin group has a complex spinor representation of dimension .
More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the
structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra.
For examples over the reals see the article on
spinors.
Real spinors
To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The
pin group, is the set of invertible elements in that can be written as a product of unit vectors:
Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group . The
spin group consists of those elements of that are products of an even number of unit vectors. Thus by the
Cartan–Dieudonné theorem Spin is a cover of the group of proper rotations .
Let be the automorphism that is given by the mapping acting on pure vectors. Then in particular, is the subgroup of whose elements are fixed by . Let
(These are precisely the elements of even degree in .) Then the spin group lies within .
The irreducible representations of restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of .
To classify the pin representations, one need only appeal to the
classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)
and realize a spin representation in signature as a pin representation in either signature or .
Applications
Differential geometry
One of the principal applications of the exterior algebra is in
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
where it is used to define the
bundle of
differential forms on a
smooth manifold. In the case of a (
pseudo-)
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, the
tangent spaces come equipped with a natural quadratic form induced by the
metric. Thus, one can define a
Clifford bundle in analogy with the
exterior bundle. This has a number of important applications in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
. Perhaps more important is the link to a
spin manifold, its associated
spinor bundle and manifolds.
Physics
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra that has a basis that is generated by the matrices , called
Dirac matrices, which have the property that
where is the matrix of a quadratic form of signature (or corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra , whose
complexification is , which, by the
classification of Clifford algebras, is isomorphic to the algebra of complex matrices . However, it is best to retain the notation , since any transformation that takes the bilinear form to the canonical form is ''not'' a Lorentz transformation of the underlying spacetime.
The Clifford algebra of spacetime used in physics thus has more structure than . It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by
This is in the convention, hence fits in .
The Dirac matrices were first written down by
Paul Dirac when he was trying to write a relativistic first-order wave equation for the
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the
Dirac equation and introduce the
Dirac operator. The entire Clifford algebra shows up in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
in the form of
Dirac field bilinears.
The use of Clifford algebras to describe quantum theory has been advanced among others by
Mario Schönberg, by David Hestenes in terms of geometric calculus, by David Bohm and Basil Hiley and co-workers in form of a Basil Hiley#Hierarchy of Clifford algebras, hierarchy of Clifford algebras, and by Elio Conte et al.
Computer vision
Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford analysis#The Fourier transform, Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.
Generalizations
* While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to a Module (mathematics), module over any unital, associative, commutative ring.
* Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.
History
See also
* Algebra of physical space
* Cayley–Dickson construction
*
Classification of Clifford algebras
* Clifford analysis
* Clifford module
* Complex spin structure
*
Dirac operator
* Exterior algebra
* Fierz identity
* Gamma matrices
* Generalized Clifford algebra
* Geometric algebra
* Higher-dimensional gamma matrices
* Hypercomplex number
* Octonion
* Paravector
* Quaternion
* Spin group
* Spin structure
*
Spinor
* Spinor bundle
Notes
Citations
References
*
*
* , section IX.9.
*
*
*
*
*
*
*
*
*
*
* . An advanced textbook on Clifford algebras and their applications to differential geometry.
*
*
*
*
*
*
*
*
* ; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in ''The Collected Mathematics Papers of James Joseph Sylvester'' (Cambridge University Press, 1909) v III
onlinean
further
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*
Further reading
*
External links
*
(unverified)
John Baez on Clifford algebrasClifford Algebra: A Visual IntroductionClifford Algebra Explorer : A Pedagogical Tool
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Clifford algebras,
Ring theory
Quadratic forms