Clifford Algebra

The Clifford algebra is a mathematical object from algebra named after William Kingdon Clifford , which extends the complex and hyper-complex number systems. It is used in differential geometry as well as in quantum physics . You used to define the spin group and its representations , the construction of spinor fields / -bündeln, in turn, to the description of electrons and other elementary particles are important, and the determination of invariants of manifolds .

The question of complex units

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In mathematics there are number systems ( division algebras with one element) with complex units, more precisely the complex numbers , the quaternions and octaves . In each of these, 1, 3 or 7 elements can be fixed which, together with the 1, span the number space as a real vector space and which (not only) satisfy. Sometimes that's not enough. For any number , structures are searched that contain the real numbers and elements and in which a product is defined which meets the conditions ${\displaystyle \mathbf {i} _{k}}$ ${\displaystyle (\mathbf {i} _{k})^{2}=-1}$ ${\displaystyle n}$ ${\displaystyle \mathbf {i} _{1},\dots ,\mathbf {i} _{n}}$ ${\displaystyle \circ }$

${\displaystyle \mathbf {i} _{k}\circ \mathbf {i} _{l}+\mathbf {i} _{l}\circ \mathbf {i} _{k}=2\sigma _{k}\delta _{kl}}$

fulfilled, where the Kroneckersymbol is and . The link symbol is often left out. ${\displaystyle \delta _{kl}}$ ${\displaystyle \sigma _{k}=\pm 1}$

The elements are called the generators of Clifford algebra. The product of all generators is designated . The square of can be +1 or −1. ${\displaystyle \mathbf {i} _{k}}$ ${\displaystyle \omega }$ ${\displaystyle \omega =\mathbf {i} _{1}\mathbf {i} _{2}\cdots \mathbf {i} _{n}}$ ${\displaystyle \omega }$

With the exception of the examples mentioned, this structure is not a number system in the above sense, but can only be realized as an algebra in which the generators are. One such algebra is called the Clifford algebra after William Kingdon Clifford who discovered it in 1878. It is denoted by or , if ${\displaystyle \mathbf {i} _{k}}$ ${\displaystyle Cl(p,q)}$ ${\displaystyle Cl(p,q,\mathbb {R} )}$

${\displaystyle \sigma _{1}=\dots =\sigma _{p}=-1}$ and ${\displaystyle \sigma _{p+1}=\dots =\sigma _{p+q}=1}$

and otherwise no algebraic relation of the generators holds.

So far we have established formal rules of calculation, but we do not yet know anything about the existence, uniqueness and structure of such an algebra. This problem is solved immediately if one can represent the Clifford algebra as part of a real matrix algebra.

More general consideration

In the mathematical part, the calculation rules are supplemented by a universal property and the Clifford algebra is constructed from a tensor algebra . It should only be noted for now that the generators span a real (sub) vector space of dimension n = p + q within the algebra. If the defining property is added up over the coordinate representation of a vector of this vector space, a coordinate-free (in physical language: covariant ) representation of the defining algebraic relation results . ${\displaystyle \mathbf {i} _{1},\dots ,\mathbf {i} _{n}}$ ${\displaystyle V}$ ${\displaystyle {\vec {v}}=x^{1}i_{1}+\dots +x^{n}i_{n}}$

${\displaystyle {\vec {v}}\circ {\vec {v}}=-Q({\vec {v}})\cdot 1_{\mathbb {R} }}$, in which

${\displaystyle Q({\vec {v}}):=(x^{1})^{2}+\dots +(x^{p})^{2}-(x^{p+1})^{2}-\dots -(x^{n})^{2}}$ is a quadratic function on which defines a (pseudo-) scalar product : ${\displaystyle V}$

${\displaystyle Q(t{\vec {v}})=t^{2}Q({\vec {v}})}$ and

${\displaystyle \langle {\vec {v}},{\vec {w}}\rangle :={\frac {1}{4}}Q({\vec {v}}+{\vec {w}})-{\frac {1}{4}}Q({\vec {v}}-{\vec {w}})}$.

The generators then form an orthonormal basis . ${\displaystyle (V,\langle \cdot ,\cdot \rangle )}$

Such a pair of real vector space and a quadratic function defined on it is the starting point for the mathematical theory of Clifford algebras . ${\displaystyle (V,Q)}$

definition

Let be a body and a finite dimensional square space . ${\displaystyle k}$ ${\displaystyle (V,Q)}$

Then the Clifford algebra of quadratic space is defined as the greatest non-commutative algebra over that is generated by and the unit element and the multiplication of which is the relation ${\displaystyle Cl(V,Q)}$ ${\displaystyle (V,Q)}$ ${\displaystyle k}$ ${\displaystyle V}$ ${\displaystyle 1_{Cl}}$

${\displaystyle v\cdot v=-Q(v)1_{Cl}}$

Fulfills.

This is well-defined , since it can be shown that a linear embedding (i.e. a vector space homomorphism ) in an associative -algebra with one, so that the relation ${\displaystyle j\colon (V,Q)\to A}$ ${\displaystyle k}$

${\displaystyle j(v)\cdot j(v)=-Q(v)1_{Cl}}$

holds, can be continued to an -algebra homomorphism . Therefore the Clifford algebra is unique except for isomorphism. ${\displaystyle k}$ ${\displaystyle {\tilde {f}}\colon Cl(V,Q)\to A}$

Examples

Complex numbers

The complex numbers can be understood as the simplest Clifford algebra with a single generator. The vector space is one-dimensional and generated so and the square shape is . As a real vector space, algebra is two-dimensional with and as basic elements; it can be identified with the algebra of the 2x2 matrices of the form ${\displaystyle \mathbb {C} }$ ${\displaystyle V}$ ${\displaystyle i}$ ${\displaystyle V=i\mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle Q(x)=x^{2}}$ ${\displaystyle 1}$ ${\displaystyle i}$

${\displaystyle {\begin{pmatrix}a&-b\\b&a\end{pmatrix}}}$.

Such matrices thus satisfy the equation

${\displaystyle x\cdot x=-x^{2}\cdot {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$.

This Clifford algebra is also noted by means, since it is an example of a real Clifford algebra . This is defined later in this article. ${\displaystyle Cl(\mathbb {R} ,x^{2})}$ ${\displaystyle Cl(1,0)}$

Quaternions

The quaternions result from the Clifford algebra . The generators have a nontrivial product , from the defining properties of the product it follows that it corresponds to the product of the quaternions . The vector space is real two-dimensional, the algebra real four-dimensional. A matrix representation is the partial algebra of the complex 2x2 matrices ${\displaystyle Cl(2,0)}$ ${\displaystyle (i,j)}$ ${\displaystyle k=i\cdot j}$ ${\displaystyle V}$

${\displaystyle {\begin{pmatrix}a&b\\-{\bar {b}}&{\bar {a}}\end{pmatrix}}}$,

by inserting the real 2x2 matrices of the complex numbers and a partial algebra of the real 4x4 matrices results. ${\displaystyle a}$ ${\displaystyle b}$

Abnormally complex numbers

The algebra of abnormally complex numbers has a generator with square 1. Therefore elements of real 2-dimensional algebra can be split into two summands , of which the first keeps its sign when multiplied by and the second changes its sign. In the multiplication of two elements, these summands are multiplied separately, as in the multiplication of two diagonal matrices. The algebra is isomorphic to the direct sum of two copies of , . ${\displaystyle Cl(0,1)}$ ${\displaystyle i}$ ${\displaystyle a+b\,\mathrm {i} }$ ${\displaystyle {\tfrac {1}{2}}\,(a+b)\,(1+\mathrm {i} )+{\tfrac {1}{2}}\,(a-b)\,(1-\mathrm {i} )}$ ${\displaystyle i}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle Cl(0,1)\cong \mathbb {R} \oplus \mathbb {R} }$

Graßmann algebra

The Graßmann algebra of a real vector space is the Clifford algebra with the trivial quadratic form . The Graßmann algebra can be constructed within any Clifford algebra by defining the wedge product as - and analogously as an alternating sum for more than two factors. ${\displaystyle \Lambda V}$ ${\displaystyle V}$ ${\displaystyle Cl(V,0)}$ ${\displaystyle Q=0}$ ${\displaystyle u\wedge v={\tfrac {1}{2}}(uv-vu)}$

Conversely, any Clifford algebra can be constructed within Graßmann algebra by defining a new product in it as ${\displaystyle Cl(V,Q)}$ ${\displaystyle \Lambda V}$ ${\displaystyle \circ }$

${\displaystyle v\circ w:=v\wedge w-q(v,w)}$.

The dimension of algebra is retained, it is , with . ${\displaystyle 2^{n}}$ ${\displaystyle n=\dim(V)}$

This relationship is important for the quantization of supersymmetric field theories , among other things .

Alternative definitions

From a mathematical point of view, the Clifford algebra is a natural construct for a vector space with a square shape defined on it, because it can be characterized as the initial object of a category .

As an initial object

Consider the category of all associative algebras in which is embedded, i.e. all pairs with linear, which also have the property ${\displaystyle \mathbb {K} }$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle (A,j)}$ ${\displaystyle j\colon V\to A}$

${\displaystyle j(v)\cdot j(v)=-Q(v)\cdot 1_{A}}$for all of ${\displaystyle v}$ ${\displaystyle V}$

or the equivalent statement

${\displaystyle j(v)\cdot j(w)+j(w)\cdot j(v)=-2q(v,w)\cdot 1_{A}}$

for everyone to meet from . The morphisms of this category are algebra morphisms that convert the embedded copies of V into one another, that is , not only fulfills , but also . ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle V}$ ${\displaystyle \phi \colon (A,j)\to (B,k)}$ ${\displaystyle \phi (ab)=\phi (a)\phi (b)}$ ${\displaystyle \phi (j(v))=k(v)}$

An initial object of a category is characterized by the fact that there is exactly one morphism for every other object in the category. If there are several initial objects, then these are isomorphic. Every initial object of the category considered here, if one exists at all, is called Clifford algebra . For each additional pair that is the category there are a uniquely determined Algebrenmorphismus with . ${\displaystyle (A,j)}$ ${\displaystyle Cl(V,Q)=A}$ ${\displaystyle (B,k)}$ ${\displaystyle \varphi \colon Cl(V,Q)\to B}$ ${\displaystyle k=\varphi \circ j}$

It will be identified below with its embedding , that is, the mapping is no longer explicitly mentioned. ${\displaystyle V}$ ${\displaystyle j(V)\subset Cl(V,Q)}$ ${\displaystyle j}$

Construction in tensor algebra

Let the ideal be defined in the tensor algebra . Then the quotient is a realization of the Clifford algebra . ${\displaystyle T(V)}$ ${\displaystyle {\mathcal {I}}:={\mbox{span}}_{T(V)}\{v\otimes w+w\otimes v+q(v,w):\;v\,,w\in V\}}$ ${\displaystyle T(V)/{\mathcal {I}}}$ ${\displaystyle Cl(V,Q)}$

Special Clifford algebras

Real Clifford algebras

In the following we assume an n-dimensional vector space. ${\displaystyle V\cong \mathbb {R} ^{n}}$

• If equipped with the standard scalar product, the Clifford algebra generated by it is also referred to as. The generators are then the canonical basis vectors , the quadratic form that is induced from the standard scalar product is the square sum of the coordinates. ${\displaystyle V}$ ${\displaystyle Cl(n,0)}$ ${\displaystyle \mathbf {i} _{k}:=\mathbf {e} _{k}}$

• Is the room equipped with the Minkowski shape with the signature , so that applies. Then the square shape is through ${\displaystyle V}$ ${\displaystyle (p,q)}$ ${\displaystyle n:=p+q}$

${\displaystyle Q({\vec {x}})=x_{1}^{2}+\dots +x_{p}^{2}-x_{p+1}^{2}-\dots -x_{n}^{2}}$

given. So the real Clifford algebra is also noted. ${\displaystyle Cl(p,q)=Cl(p,q,\mathbb {R} )}$

Complex Clifford Algebras

For every real Clifford algebra, the complexified algebra

${\displaystyle \mathbb {\mathbb {C} } l(p+q):=Cl(p,q,\mathbb {R} )\otimes \mathbb {C} }$

To be defined. This definition is independent of the complexized scalar product, because there is exactly one uniquely determined, non-degenerate square form. ${\displaystyle \mathbb {C} ^{n}}$

properties

graduation

The image

${\displaystyle {\begin{aligned}j_{-}\colon V&\to Cl(V,Q)\\v&\mapsto j_{-}(v):=-v\end{aligned}}}$

also fulfills the defining identity , so because of the universal property there is an algebra isomorphism with for all and . The Clifford algebra thus breaks down into an even part ${\displaystyle j_{-}(v)^{2}=-Q(v)}$ ${\displaystyle \kappa \colon Cl(V,Q)\to Cl(V,Q)}$ ${\displaystyle \kappa (v)=-v}$ ${\displaystyle v\in V}$ ${\displaystyle \kappa ^{2}=\mathrm {id} }$

${\displaystyle Cl^{0}(V,Q):=\mathrm {Kern} (\mathrm {id} -\kappa )=\mathrm {Bild} (\mathrm {id} +\kappa )}$

and an odd part ${\displaystyle Cl^{1}(V,Q):=\mathrm {Kern} (\mathrm {id} +\kappa )=\mathrm {Bild} (\mathrm {id} -\kappa )\,.}$

This decomposition produces a - graduation of the algebra, products even-even and odd-odd produce even elements, products even-odd produce odd elements. Products with an even number of factors from V are even, products with an odd number of factors from V are odd. ${\displaystyle \mathbb {Z} _{2}}$

${\displaystyle Cl^{0}(V,Q)}$is a sub-algebra of the Clifford algebra and is also referred to as the second Clifford algebra, is only a module relating to . ${\displaystyle Cl^{1}(V,Q)}$ ${\displaystyle Cl^{0}(V,Q)}$

Filtered algebra

Since the Clifford algebra can be understood as a quotient from the tensor algebra and the tensor algebra has a natural filtration , a filtration can also be explained for the Clifford algebra. The figure is the natural projection of the tensor algebra into the quotient space and the filtration of the tensor algebra. If one sets this, the Clifford algebra also becomes a filtered algebra. ${\displaystyle \pi _{Q}\colon T(V)\to Cl(V,Q)}$ ${\displaystyle Cl(V,Q)}$ ${\displaystyle T_{1}(V)\subset T_{2}(V)\subset \cdots \subset T(V)}$ ${\displaystyle Cl_{i}(V,Q)=\pi _{Q}(T_{i}(V))}$

Relationship to the orthogonal group

Let be a vector space with non-degenerate symmetric bilinear form and . In the Clifford algebra , reflections can then be represented in. To do this, an elementary conclusion from the structure of the product is used: ${\displaystyle V}$ ${\displaystyle q}$ ${\displaystyle Q(v)=q(v,v)}$ ${\displaystyle Cl(V,Q)}$ ${\displaystyle V}$

${\displaystyle {\frac {vxv}{\langle v,v\rangle }}=-{\frac {(2\langle v,x\rangle +xv)v}{\langle v,v\rangle }}=-2{\frac {\langle v,x\rangle v}{\langle v,v\rangle }}+x.}$

Is a unit vector , then the Figure , the reflection in the to vertical hyperplane . Each reflection is an orthogonal map , so the group created by the reflections is a subset of the orthogonal group . ${\displaystyle v}$ ${\displaystyle |\langle v,v\rangle |=1}$ ${\displaystyle v\mapsto S(v)}$ ${\displaystyle S(v)x:={\tfrac {vxv}{\langle v,v\rangle }}=\pm vxv}$ ${\displaystyle v}$

The pin group

Conversely, every orthogonal mapping can be broken down into a product of reflections, see Household transformation or QR decomposition . The decomposition is not unique, but the Clifford products of the unit vectors of the mirror matrices differ at most in sign.

First, the pin group is defined as the set of all products of unit vectors:

${\displaystyle \operatorname {Pin} (V):=\{v_{1}\dots v_{k}:k\in \mathbb {N} ,v_{i}\in V,\langle v_{i},v_{i}\rangle =\pm 1\}.}$

This amount is a submonoid the multiplicative monoid of Clifford algebra and becomes the group by the existence of an inverse: . There are products whose factors are different, but which designate the same element of the pin group, for example applies to orthogonal unit vectors and with and every pair ${\displaystyle v_{1}\dots v_{k}v_{k}\dots v_{1}=\pm 1}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle Q(v)=Q(w)}$ ${\displaystyle (c,s)=(\cos \,\alpha ,\sin \,\alpha )}$

${\displaystyle (cv-sw)(sv+cw)=vw}$.

However, it applies that each element consists of exactly one orthogonal mapping ${\displaystyle \operatorname {Pin} (V)}$

${\displaystyle {\tilde {S}}(v_{1}\dots v_{k})(\cdot ):=v_{1}\dots v_{k}(\cdot )(v_{1}\dots v_{k})^{-1}=S(v_{1})S(v_{2})\dots S(v_{k})(\cdot )}$

whose independence from the chosen factorization follows from the uniqueness of the inverse. It is also known that is surjective of order 2, i.e. H. a double overlay . The prototypes of the same orthogonal mapping differ only in terms of the sign. ${\displaystyle {\tilde {S}}\colon \operatorname {Pin} (V)\to O(V)}$

The spin group

→ Main article : Spin group

However, a subgroup of the pin group, the spin group, is physically and geometrically significant

${\displaystyle {\mbox{Spin}}(V):=\{v_{1}\dots v_{2k}\in {\mbox{Pin}}(V):k\in \mathbb {N} \}={\mbox{Pin}}(V)\cap Cl^{0}(V)}$

the products with an even number of factors (the playful reinterpretation of the spin group as a “special pin group” resulted in the term “pin” group). It is known from this that it is a two-fold superposition of the special orthogonal group , and that, provided the dimension of the underlying vector space is greater than 2, it is simply connected , i.e. universal superposition . Since the matrix group is a representation of weight 2 of, it is also said in physics that representations of the spin group of weight 1 are spin representations of the orthogonal group. ${\displaystyle SO(V)}$ ${\displaystyle SO(n)}$ ${\displaystyle {\mbox{Spin}}(n)}$ ${\displaystyle {\tfrac {1}{2}}}$

Representations

A representation of an algebra is an embedding of this in the algebra of the endomorphisms of a vector space, i.e. (according to the basic choice) in a matrix algebra. The matrices can have real, complex or quaternionic entries.

It can be shown that every Clifford algebra is isomorphic to a matrix algebra or the direct sum of two matrix algebras over the real numbers , the complex numbers or the quaternions . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {H} }$

Real Clifford Algebra

The assignment and dimension of the real Clifford algebras are tabulated as follows:

( p - q ) mod 8	ω 2	Cl ( p , q , ℝ) ( p + q = 2 m )	( p - q ) mod 8	ω 2	Cl ( p , q , ℝ) ( p + q = 2 m + 1)
0	+	M (2 m , ℝ)	1	-	M (2 m , ℂ)
2	-	M (2 m −1 , ℍ)	3	+	M (2 m −1 , ℍ) ⊕ M (2 m −1 , ℍ)
4th	+	M (2 m −1 , ℍ)	5	-	M (2 m , ℂ)
6th	-	M (2 m , ℝ)	7th	+	M (2 m , ℝ) ⊕ M (2 m , ℝ)

The following general isomorphies apply:

• ${\displaystyle Cl(d,0)\otimes Cl(0,2)\cong Cl(0,d+2)}$
• ${\displaystyle Cl(0,d)\otimes Cl(2,0)\cong Cl(d+2,0)}$
• ${\displaystyle Cl(p,q)\otimes Cl(1,1)\cong Cl(p+1,q+1)}$

Complex Clifford Algebra

The representation of the complex Clifford algebra is simpler than that of the real one. It is true

${\displaystyle \mathbb {C} l(n)\cong {\begin{cases}M\left(2^{\frac {n}{2}},\mathbb {C} \right)&n\ {\mbox{gerade}}\\M\left(2^{\frac {n-1}{2}},\mathbb {C} \right)\oplus M\left(2^{\frac {n-1}{2}},\mathbb {C} \right)&n\ {\mbox{ungerade}}\,.\end{cases}}}$

In this context isomorphism applies

${\displaystyle \mathbb {C} l(n)\otimes M(2,\mathbb {C} )\cong \mathbb {C} l(n+2)\,,}$

which is also essential for the proof of the representation. Is straight, then with the natural graduation in this context one calls the spinor module . ${\displaystyle n}$ ${\displaystyle \mathbb {C} ^{m}}$ ${\displaystyle m=2^{\frac {n}{2}}}$ ${\displaystyle \mathbb {R} ^{m}\oplus \mathbb {R} ^{m}}$

Low dimensional examples

The dimension of as a real vector space is 2 ^{p + q} . The Clifford algebra can thus be represented by real matrices of this dimension, which describe the multiplication in the algebra. This representation is not minimal; H. there are matrices of smaller dimensions which do the same thing, see [1] and the examples below. ${\displaystyle Cl(p,q)}$

• ${\displaystyle Cl(1,0)\cong \mathbb {C} }$

has the generator with . So there is a complex one-dimensional representation that maps onto the imaginary unit i , and the corresponding real two-dimensional representation. ${\displaystyle e_{1}}$ ${\displaystyle e_{1}^{2}=-1}$ ${\displaystyle e_{1}}$

• ${\displaystyle Cl(0,1)\cong \mathbb {R} \oplus \mathbb {R} }$

The generator is with . Each element of the algebra can be used in two terms and are split. Since the following applies, this split is maintained under product formation. The Clifford algebra is isomorphic to the with component-wise product, where the element corresponds and the unity element corresponds to the element . This direct sum of two algebras can also be implemented as an algebra of the 2x2 diagonal matrices. ${\displaystyle e:=e_{1}}$ ${\displaystyle e^{2}=1}$ ${\displaystyle a+be}$ ${\displaystyle {\tfrac {1}{2}}(a+b)(1+e)}$ ${\displaystyle {\tfrac {1}{2}}(a-b)(1-e)}$ ${\displaystyle (1+e)(1-e)=0}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle e}$ ${\displaystyle (1,-1)}$ ${\displaystyle (1,1)}$

• ${\displaystyle Cl(2,0)\cong \mathbb {H} }$

has the generators and and their product k = ij with the relations ${\displaystyle i:=e_{1}}$ ${\displaystyle j:=e_{2}}$

${\displaystyle i^{2}=j^{2}=-1,\;k=ij=-ji,\;ijk=k^{2}=-ijji=-1,\;}$.

One calculates that this is isomorphic to the algebra of the quaternions .

• ${\displaystyle Cl(1,1)\cong M_{2}(\mathbb {R} )}$

has the generators and , , and . Make sure that the generators correspond to the following real 2x2 matrices: ${\displaystyle i}$ ${\displaystyle e}$ ${\displaystyle i^{2}=-1}$ ${\displaystyle e^{2}=1}$ ${\displaystyle ie=-ei}$

${\displaystyle 1={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\;e={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\;i={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\;ie={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$

thus all real matrices can be reached.

• ${\displaystyle Cl(0,2)\cong M_{2}(\mathbb {R} )}$

has the generators and with square 1, the product of which has the square , so this algebra is isomorphic to the previous one. ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle i:=e_{1}e_{2}}$ ${\displaystyle -1}$

Quantum physically significant examples

• ${\displaystyle Cl(3,0)\cong Cl(2,0)\otimes Cl(0,1)\cong \mathbb {H} \oplus \mathbb {H} }$( Biquaternions )

has the producers , and with the relations ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$

${\displaystyle (\mathbf {e} _{i})^{2}=-1}$, , , . ${\displaystyle \mathbf {e} _{i}\mathbf {e} _{k}=-\mathbf {e} _{k}\mathbf {e} _{i}}$ ${\displaystyle (\mathbf {e} _{i}\mathbf {e} _{k})^{2}=-1}$ ${\displaystyle (\mathbf {e} _{1}\mathbf {e} _{2}\mathbf {e} _{3})^{2}=1}$

Both real and complex decay as representations , wherein null space of the projector and null space of the projector with is. It is true that both subspaces generate subrepresentations that are independent of one another. ${\displaystyle V=V_{+}\oplus V_{-}}$ ${\displaystyle V_{+}}$ ${\displaystyle (1-\omega )/2}$ ${\displaystyle V_{-}}$ ${\displaystyle (1+\omega )/2}$ ${\displaystyle \omega :=e_{1}e_{2}e_{3}}$ ${\displaystyle e_{k}\omega =\omega e_{k}}$

A purely negative representation, i. H. with , is directly isomorphic to the quaternion algebra, ${\displaystyle V_{+}=0}$

${\displaystyle e_{1}\to i,e_{2}\to j,e_{3}\to k}$,

a purely positive is conjugated isomporphic,

${\displaystyle e_{1}\to -i,e_{2}\to -j,e_{3}\to -k}$.

In both cases, what has been said applies. ${\displaystyle Cl(2,0,\mathbb {R} )}$

• ${\displaystyle Cl(2,1)\cong Cl(1,1)\otimes Cl(1,0)\cong M_{2}(\mathbb {C} )}$

• ${\displaystyle Cl(1,2)\cong Cl(1,1)\otimes Cl(0,1)\cong M_{2}(\mathbb {R} )\oplus M_{2}(\mathbb {R} )}$

• ${\displaystyle Cl(0,3)\cong Cl(0,2)\otimes Cl(1,0)\cong \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} }$

• ${\displaystyle Cl(4,0)\cong Cl(2,0)\otimes Cl(0,2)\cong M_{2}(\mathbb {H} )}$

The even part of this algebra that contains the group is too isomorphic. It is generated by , e.g. B. . ${\displaystyle Spin_{4}}$ ${\displaystyle Cl(3,0)}$ ${\displaystyle \mathbf {f} _{1}:=\mathbf {e} _{1}\mathbf {e} _{4},\;\mathbf {f} _{2}:=\mathbf {e} _{2}\mathbf {e} _{4}\;\mathbf {f} _{3}:=\mathbf {e} _{3}\mathbf {e} _{4}}$ ${\displaystyle \mathbf {e} _{1}\mathbf {e} _{2}=\mathbf {f} _{1}\mathbf {f} _{2}}$

• ${\displaystyle Cl(3,1)\cong Cl(1,1)\otimes Cl(2,0)\cong M_{2}(\mathbb {H} )}$

${\displaystyle Cl^{0}(3,1)\cong Cl(3,0)\cong \mathbb {H} \oplus \mathbb {H} }$ or

${\displaystyle Cl^{0}(3,1)\cong Cl(2,1)\cong M_{2}(\mathbb {C} )}$

• ${\displaystyle Cl(1,3)\cong Cl(1,1)\otimes Cl(0,2)\cong M_{4}(\mathbb {R} )}$

${\displaystyle Cl^{0}(1,3)\cong Cl(0,3)\cong \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} }$ or

${\displaystyle Cl^{0}(1,3)\cong Cl(1,2)\cong M_{2}(\mathbb {R} )\oplus M_{2}(\mathbb {R} )}$

literature

• Bartel L. van der Waerden : Algebra. 9th edition. Volume 1. Springer, Berlin et al. 1993, ISBN 3-540-56799-2 .
• Bartel L. van der Waerden: A history of Algebra. From al-Khwārizmī to Emmy Noether. Springer, Berlin et al. 1985, ISBN 3-540-13610-X .
• H. Blaine Lawson , Marie-Louise Michelsohn: Spin Geometry (= Princeton Mathematical Series. Vol. 38). Princeton University Press, Princeton NJ 1989, ISBN 0-691-08542-0 .

Web links

• José Figueroa-O'Farrill: Majorana spinors and general representation theory. (PDF file; 239 kB).

Individual evidence

1. ^ William Kingdon Clifford . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ^ ^{A b} Nicole Berline , Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= The basic teachings of the mathematical sciences in individual representations. Vol. 298). Springer, Berlin et al. 1992, ISBN 0-387-53340-0 , p. 100.
3. ^ HB Lawson, M. Michelsohn: Spin Geometry . Princeton University Press, 1989, ISBN 978-0-691-08542-5 , pp. 8f.
4. ↑ HB Lawson, M.-L. Michelsohn: Spin Geometry. 1989, pp. 9-10.