Open access peer-reviewed chapter

Some Applications of Clifford Algebra in Geometry

Written By

Ying-Qiu Gu

Submitted: 01 November 2019 Reviewed: 21 July 2020 Published: 03 September 2020

DOI: 10.5772/intechopen.93444

From the Edited Volume

Structure Topology and Symplectic Geometry

Edited by Kamal Shah and Min Lei

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Abstract

In this chapter, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation, and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area, and volume and unifies the calculus of scalar, spinor, vector, and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion, and vector algebra and converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So, Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

Keywords

  • Clifford algebra
  • geometric algebra
  • gamma matrix
  • multi-inner product
  • connection operator
  • Keller connection
  • spin group
  • cross ratio
  • conformal geometric algebra

1. A brief historical review

It is well known that a rotational transformation in the complex plane is equivalent to multiplying the complex number by a factor eθi. How to generalize this simple and elegant operation to three-dimensional space is a difficult problem for many outstanding mathematicians in the early nineteenth century. William Rowan Hamilton (1805–1865) spent much of his later years studying the issue and eventually invented quaternion [1]. This generalization requires four elements 1ijk, and the spatial basis should satisfy the multiplying rules i2=j2=k2=1 and jk=kj=i, ki=ik=j, and ij=ji=k. Although a quaternion is still a vector, it constitutes an associative algebra according to the above rules. However, the commutativity of multiplication is violated. Quaternion can solve the rotational transformation in three-dimensional space very well and simplify the representation of Maxwell equation system of electromagnetic field.

When Hamilton introduced his quaternion algebra, German high school teacher Hermann Gunther Grassmann (1809–1877) was constructing his exterior algebra [2]. He defined the exterior product or outer product ab of two vectors a and b, which satisfies anti-commutative law ab=ba and associativity abc=abc. The exterior product is a generalization of cross product in three-dimensional Euclidian space. Its geometrical meaning is the oriented volume of a parallel polyhedron. Exterior product is now the basic tool of modern differential geometry, but Grassmann’s work was largely neglected in his lifetime.

British mathematician William Kingdon Clifford (1845–1879) was one of the few mathematicians who read and understood Grassmann’s work. In 1878, he combined the algebraic rules of Hamilton and Grassmann to define a new algebraic system, which he himself called geometric algebra [3]. In this algebra, both the inner and exterior products of vectors can be uniquely represented by a linear combination of geometric product. In addition, geometric algebra is always isomorphic to some special matrix algebra.

Clifford algebra combines all the advantages of quaternion with the advantages of vector algebra and uniformly and succinctly describes the contents of geometry and physics. However, the vector calculus introduced by Gibbs had also successfully described the mathematical physics problem in three-dimensional space [4]. Clifford died prematurely at the age of 34, so that the theory of geometric algebra was not deeply researched and fully developed, and people still could not see the superiority of this algebra at that time. Thus, the important insights of Grassmann and Clifford were lost in the late nineteenth century papers. Mathematicians abstracted Clifford algebra from its geometric origins, and, for the most part of a century, it languished as a minor subdiscipline of mathematics and became one more algebra among so many others.

With the establishment of relativity, especially the introduction of Pauli and Dirac’s matrix algebra for spin and the successful application in quantum theory [5], it was felt that there is an urgent need for a mathematical system to deal with problems in high-dimensional space-time. In the 1920s, Clifford algebra re-entered the field of vision and was paid attention and researched by some of the famous mathematicians and physicists such as R. Lipschitz, T. Vahlen, E. Cartan, E. Witt, C. Chevalley, and M. Riesz [6, 7, 8]. When only formal algebra is involved, we usually use the term “Clifford algebra,” but more often use the “geometric algebra” named by Clifford himself if applied to geometric problems.

The first person who realized that Clifford algebra is a unified language in geometry and physics should be David Hestenes. By the 1960s, Hestenes began to restore the geometric meaning behind Pauli and Dirac algebra. Although his initial motivation was to gain insight into the nature of quantum mechanics, he quickly realized that Clifford algebra was a unified language and tool for mathematics, physics, and engineering. He published “space-time algebra” in 1966 and has been working on the promotion of Clifford algebra in teaching and research [9, 10, 11, 12]. Because representation and algorithm in geometric algebra are seemingly as ordinary as arithmetic, his work has been neglected by the scientific community for more than 20 years. Only with the joint impetus of computer-aided design, computer vision and robotics, protein folding, neural networks, modern differential geometry, mathematical physics [13, 14, 15, 16, 17], and especially the Journal “Advances in Applied Clifford algebras” founded by Professor Jaime Keller, geometric algebra began to move towards popularity and prosperity.

As a unified and universal language of natural science, Clifford algebra is developed by many mathematicians, physicists, and engineers according to their different requirements and knowledge background. Such situation leads to “There are a thousand Hamlets in a thousand people’s eyes.” In this chapter, by introducing typical application of Clifford algebra in geometry, we show some special feature and elegance of the algebra.

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2. Application of Clifford algebra in differential geometry

In Euclidean space, we have several important concepts such as vector, length, angle, area, volume, and tensor. The study of relationship between these concepts constitutes the whole content of Euclidean geometry. The mathematical tools previously used to discuss these contents are vector algebra and geometrical method, which are complex and require much fundamental knowledge. Clifford algebra exactly and faithfully describes the intrinsic properties of vector space by introducing concepts such as inner, exterior, and geometric products of vectors and thus becomes a unified language and standard tool for dealing with geometric and physical problems. Clifford algebra has the characteristics of simple concept, standard operation, completeness in conclusion, and easy understanding.

Definition 1 For Minkowski space Mn over number field F, if the multiplication rule of vectors satisfies

1.Antisymmetry,xy=yx;E1
2.Associativity,xyz=xyz;E2
3.Distributivity,xay+bz=axy+bxz,a,bF,E3

the algebra is called Grassmann algebra and xyexterior product.

The Grassmann is also called exterior algebra. The geometrical meaning of xy is oriented area of a parallelogram constructed by x and y, and the geometrical meaning of xyz is the oriented volume of the parallelohedron constructed by the vectors (see Figure 1). We call xy two-vector, xyz three-vector, and so on. For k-vector xΛk and l-vector yΛl, we have

Figure 1.

Geometric meaning of exterior products of vectors.

xy=1klyxΛk+l.

By the definition, we can easily check:

Theorem 1 For exterior algebra defined in V=Mn, we have

Wn=FVΛ2VΛnV=r=0nΛrV.

The dimension of the algebra is

dimWn=k=0nCnk=2n.

Under the orthonormal basis e1e2en, the exterior algebra takes the following form:

w=w0+wkek+k<lwklekl+j<k<lwjklejkl++w12ne12n,E4

in which wjklF, ejkl=ejekel, and ejkl=1.

The exterior product of vectors contains alternating combinations of basis, for example:

Vn=x1x2xn=x1jx2kxnlejkl=ϵjklx1jx2kxnle12n=detxjke12n.E5

Definition 2 For any vectors x,y,zMn, Clifford product of vectors is denoted by

xy=xy+xy,E6
xyz=yzxxzy+xyz=yxz,E7
zxy=xzyyzx+xyz=zyx,E8
xyz=yzxxzy+xyz+xyz=xyz.E9

Clifford product is also calledgeometric product.

Similarly, we can define Clifford algebra for many vectors as xyz. In (6), xy=ηabxayb is the scalar product or inner product in Mn. By xy=yx, we find Clifford product is not commutative. By (6), we have

xy=12xy+yx,xy=12xyyx,xx=xx=x2.E10

Definition 3 For Minkowski space Mp,q with metric ηab=diagIpIq, if the Clifford product of vectors satisfies

ekel+elek=2ηkl,orx2=ηklxkxl,

then the algebra

c=c0+ckek+k<lcklekel+j<k<lcjklejekel++c12ne1e2en,E11

is called as Clifford algebra or geometric algebra, which is denoted as Cp,q.

There are several definitions for Clifford algebra [18, 19]. The above definition is the original definition of Clifford. Clifford algebra has also 2n dimensions. Comparing (11) with (4), we find the two algebras are isomorphic in sense of linear algebra, but their definitions of multiplication rules are different. The Grassmann products have clear geometrical meaning, but the Clifford product is isomorphic to matrix algebra and the multiplication of physical variables is Clifford product. Therefore, representing geometrical and physical variables in the form of (4) will bring great convenience [20, 21]. In this case, the relations among three products such as (6)(9) are important.

In physics, we often use curvilinear coordinate system or consider problems in curved space-time. In this case, we must discuss problems in n dimensional pseudo Riemann manifold. At each point x in the manifold, the tangent space TMx is a n dimensional Minkowski space-time. The Clifford algebra can be also defined on the tangent space and then smoothly generalized on the whole manifold as follows.

Definition 4 In n=p+q dimensional manifold TMp,q over R, the element is defined by

dx=γμdxμ=γμdxμ=γaδXa=γaδXa,E12

where γa is the local orthogonal frame and γa the coframe. The distanceds=dx and oriented volumesdVk is defined by

dx2=12γμγν+γνγμdxμdxν=gμνdxμdxν=ηabδXaδXb,E13
dVk=dx1dx2dxk=γμνωdx1μdx2νdxkω,1kn,E14

in which ηab=diagIpIq is Minkowski metric and gμν is Riemann metric.

γμνω=γμγνγωΛkTMp,q

is Grassmann basis. The following Clifford-Grassmann number with basis

c=c0I+cμγμ+cμνγμν++c12nγ12n,ckxRE15

defines real universal Clifford algebraCp,q on the manifold.

The definitions and treatments in this chapter make the corresponding subtle and fallible concepts in differential geometry much simpler. For example, in spherical coordinate system of R3, we have element dx and the area element ds in sphere dr=0 as

dx=σ1dr+σ2rdθ+σ3rsinθdφ,
ds=σ2rdθσ3rsinθ=iσ1r2sinθdθdφ.

We have the total area of the sphere

A=ds=iσ1r2sinθdθdφ=iσ14πr2.

The above definition involves a number of concepts, some more explanations are given in the following:

  1. 1. The geometrical meanings of elements dx,dy,dxdy are shown in Figure 2. The relation between metric and vector basis is given by:

Figure 2.

Geometric meaning of vectors dx,dy and dxdy.

gμν=12γμγν+γνγμ=γμγν,E16
ηab=12γaγb+γbγa=γaγb,E17

which is the most important relation in Clifford algebra. Since Clifford algebra is isomorphic to some matrix algebra, by (17)γa is equivalent to some special matrices [20]. In practical calculation, we need not distinguish the vector basis from its representation matrix. The relation between the local frame coefficientfaμfμa and metric is given by:

γμ=faμγa,γμ=fμaγa,fμafbμ=δba,fμafaν=δμν.
faμfbνηab=gμν,fμafνbηab=gμν.

  1. 2. Assume γaa=12n to be the basis of the space-time, then their exterior product is defined by [22]:

γa1γa2γak1k!σσa1a2akb1b2bkγb1γb2γbk,1kn.

In which σa1a2akb1b2bk is permutation function, if b1b2bk is the even permutation of a1a2ak, it equals 1. Otherwise, it equals −1. The above formula is a summation for all permutations, that is, it is antisymmetrization with respect to all indices. The geometric meaning of the exterior product is oriented volume of a higher dimensional parallel polyhedron. Exterior algebra is also called Grassmann algebra, which is associative.

  1. 3. By (12) and (13) we find that, using Clifford algebra to deal with the problems on a manifold or in the tangent space, the method is the same. Unless especially mentioned, we always use the Greek alphabet to stand for the index in curved space-time, and the Latin alphabet for the index in tangent space. We use Einstein summation convention.

  2. 4. In Eq. (15), each grade-k term is a tensor. For example, c0IΛ0 is a scalar, cμγμΛ1 is a true vector, and cμνγμνΛ2 is an antisymmetric tensor of rank-2, which is also called a bivector, and so on. In practical calculation, coefficient and basis should be written together, because they are one entity, such as (12) and (15). In this form, the variables become coordinate free. The coefficient is the value of tensor, which is just a number table, but the geometric meaning and transformation law of the tensor is carried by basis.

The real difficulty in learning modern mathematics is that in order to get a little result, we need a long list of subtle concepts. Mathematicians are used to defining concepts over concepts, but if the chain of concepts breaks down, the subsequent contents will not be understandable. Except for the professionals, the common readers impossibly have so much time to check and understand all concepts carefully. Fortunately, the Clifford algebra can avoid this problem, because Clifford algebra depends only on a few simple concepts, such as numbers, vectors, derivatives, and so on. The only somewhat new concept is the Clifford product of the vector bases, which is isomorphic to some special matrix algebra; and the rules of Clifford algebra are also standardized and suitable for brainless operations, which can be well mastered by high school students.

Definition 5 For vector x=γμxμΛ1 and multivector m=γθ1θ2θkmθ1θ2θkΛk, their inner product is defined as

xm=γμγθ1θ2θkxμmθ1θ2θk,mx=γθ1θ2θkγμxμmθ1θ2θk,E18

in which

γμγθ1θ2θkgμθ1γθ2θkgμθ2γθ1θ3θk++1k+1gμθkγθ1θk1,E19
γθ1θ2θkγμ1k+1gμθ1γθ2θk+1kgμθ2γθ1θ3θk++gμθkγθ1θk1.E20

Theorem 2 For basis of Clifford algebra, we have the following relations

γμγθ1θ2θk=γμγθ1θ2θk+γμθ1θk,E21
γθ1θ2θkγμ=γθ1θ2θkγμ+γθ1θkμ.E22
γa1a2an1=ϵa1a2anγ12nγan,E23
γa1a2an2=12!ϵa1a2anγ12nγan1an,E24
γa1a2ank=1k!ϵa1a2anγ12nγank+1an.E25

Proof. Clearly γμγθ1θ2θkΛk1Λk+1, so we have

γμγθ1θ2θk=a1gμθ1γθ2θk+a2gμθ2γθ1θ3θk++akgμθkγθ1θk1+Aγμθ1θk.E26

Permuting the indices θ1 and θ2, we find a2=a1. Let μ=θ1, we get a1=1. Check the monomial in exterior product, we get A=1. Thus, we prove (21). In like manner, we prove (22). For orthonormal basis γa, by (22) we have:

γa1a2an1γan=ϵa1a2anγ12n.E27

Again by γanγan=1 (not summation), we prove (23). Other equations can be proved by antisymmetrization of indices. The proof is finished.

Likewise, we can define multi-inner productAkB between multivectors as follows:

γμνγαβ=gμβγναgμαγνβ+gναγμβgνβγμα,E28
γμν2γαβ=gμβgναgμαgνβ,γμνkγαβ=0,k>2.E29

We use AkB rather AkB, because the symbol “” is too small to express exponential power. Then for the case γμ1μ2μfγθ1θ2θk, we have similar results. For example, we have

γμνγαβ=γμν2γαβ+γμνγαβ+γμναβ.E30

In C1,3, denote the Pauli matrices by

σa100101100ii01001,E31
σ0=σ˜0=I,σ˜k=σk,k=1,2,3.E32

We use k,f,j standing for spatial indices. Define Dirac γ matrix by:

γa=0σ˜aσa0,γ5=diagII.E33

γa forms the grade-1 basis of Clifford algebra C1,3. In equivalent sense, the representation (33) is unique. By γ-matrix (33), we have the complete bases of C1,3 as follows [21]:

I,γa,γab=i2ϵabcdγcdγ5,γabc=iϵabcdγdγ5,γ0123=iγ5.E34

Based on the above preliminaries, we can display some enlightening examples of application, which show how geometric algebra works efficiently. For a skew-symmetrical torsion TμνωgμβTνωβ in M1,3, by Clifford calculus, we have:

T=Tμνωγμνω=Tabcγabc=Tabcϵabcdγdiγ5iγdγ5Td=iγαγ5Tα,E35

and then

Tα=fdαTabcϵabcd=Tμνωfaμfbνfcωfdαϵabcd=1gϵμνωαTμνω,E36

where g=detgμν. So we get:

Tμνω=gϵμνωαTα,TμνωTω=0,TμναTν=0.E37

So, the skew-symmetrical torsion is equivalent to a pseudo vector in M1,3. This example shows the advantages to combine variable with basis together.

The following example discusses the absolute differential of tensors. The definition of vector, tensor, and spinor in differential geometry involving a number of refined concepts such as vector bundle and dual bundle, which are too complicated for readers in other specialty. Here, we inherit the traditional definitions based on the bases γa and γμ. In physics, basis of tensors is defined by direct products of grade-1 bases γμ. For metric, we have [23]:

g=gμνγμγν=gμνγμγν=δμνγμγν=ηabγaγb=ηabγaγb=δabγaγb.E38

For simplicity, we denote tensor basis by:

γμ1μ2μn=γμ1γμ2γμn,γμ1μ2μ3μn=γμ1γμ2γμn,E39

In general, a tensor of rank n is given by:

T=Tμ1μ2μnγμ1μ2μn=Tμ2μnμ1γμ1μ2μ3μn=E40

The geometrical information of the tensor such as transformation law and differential connection are all recorded by basis γμ, and all representations of rank rs tensor denote the same one practical entity Tx. Tμν is just a quantity table similar to cμν in (15), but the physical and geometrical meanings of the tensor T are represented by basis γμ. Clifford algebra is a special kind of tensor with exterior product. Its algebraic calculus exactly reflects the intrinsic property of space-time and makes physical calculation simple and clear.

For the absolute differential of vector field A=γμAμ, we have

dAlimΔx0Ax+ΔxAx=αAμγμ+Aμdαγμdxα=αAμγμ+Aμdαγμdxα.E41

We call dαconnection operator [23]. According to its geometrical meanings, connection operator should satisfy the following conditions:

  1. It is a real linear transformation of basis γμ,

  2. It satisfies metric consistent condition dg=0.

Thus, the differential connection can be generally expressed as:

dαγμ=Παβμ+Tαβμγβ,Παβμ=Πβαμ,Tαβμ=Tβαμ.E42

For metric g=gμνγμγν, by metric consistent condition we have:

0=dg=dgμνγμγν=αgμνγμγν+gμνdαγμγν+gμνγμdαγνdxα=αgμνgνβΠαμβgμβΠανβdxαgνβTαμβ+gμβTανβdxαγμγν.E43

By (43), we have:

αgμνgνβΠαμβgμβΠανβdxαgνβTαμβ+gμβTανβdxα=0.E44

Since dxαδXa is an arbitrary vector in tangent space, (44) is equivalent to:

αgμνgνβΠαμβgμβΠανβ=gνβTαμβ+gμβTανβ.E45

(45) is a linear nonhomogeneous algebraic equation of ΠαβμTαβμ.

Solving (45), we get the symmetrical particular solution “Christoffel symbols” as follows;

Πμνα=12gαβμgβν+νgμββgμν+πμνα=Γμνα+πμνα,E46

in which Γμνα is called Levi-Civita connection determined by metric, πμνα=πνμα is a symmetrical post-metric part of connection. In this chapter, the “post-metric connection” means the parts of connection cannot be determined by metric, i.e., the components πμνα and Tμνα different from Levi-Civita connection Γμνα. Denote

Tμνα=gμβTναβ,πμνα=gμβπναβ,Kμνα=πμνα+Tμνα,E47

where Kμνα is called contortion with total n3 components [24]. Substituting (46) and (47) into metric compatible condition (45), we get 12n+1n2 constraints for Kμνα,

Kμνα+Kνμα=0=πμνα+πνμα+Tμνα+Tνμα.E48

By (48), Kμνα has only 12n1n2 independent components. Noticing torsion Tμνα has just 12n1n2 independent components, so Kμνα or πμνα can be represented by Tμνα.

Theorem 3 For post-metric connections we have the following relations

πμνα=Tναμ+Tανμ,E49
Kμνα=Tναμ+Tανμ+Tμνα,E50
Tμνα=13παμνπνμα+T˜μνα,E51

and consistent condition

πμνα+παμν+πναμ=0.E52

T˜=T˜μνωγμνωΛ3 is an arbitrary skew-symmetrical tensor.

Proof If we represent πμνα by Tμνα, by (48) and symmetry we have solution as (49). By (49), we get consistent condition (52). By (49) and (47), we get (50).

If we represent Tμνα by πμνα, we generally have linear relation

Tμνα=kπνμαπαμν+T˜μνα,E53

in which k is a constant to be determined, T˜μνα is particular solution as πμνα0. T˜μνα satisfies

T˜μνα=T˜αμν=T˜ναμ=T˜μαν=T˜νμα=T˜ανμ.E54

So this part of torsion is a skew-symmetrical tensor T˜=T˜μνωγμνωΛ3, which has Cn3=16n2n1n independent components. Substituting (53) into (48), we get

k1πμνα+πνμα=2kπαμν.E55

Calculating the summation of (55) for circulation of μνα, we also get consistent condition (52). Substituting (52) into (55) we get k=13. Again by (53), we get solution (51). It is easy to check, (49) and (51) are the inverse representation under condition (52). The proof is finished.

Substituting (42) into

0=dg=δνμdαγμγν+γμdαγνdxα,E56

we get

dαγμ=Γαμν+παμν+Tαμνγν.E57

To understand the meaning of πμνα and Tμνα, we examine the influence on geodesic.

dvdsdvαdsγα+vαdμγαvμ=dvαds+Γμνα+πμνα+Tμναvμvνγα,=ddsvα+Γμναvμvνγα+πμναvμvνγα.E58

The term Tμναvμvν=0 due to Tμνα=Tνμα. So the symmetrical part πμνα influences the geodesic, but the antisymmetrical part Tμνα only influences spin of a particle. This means πμνα0 violates Einstein’s equivalent principle. In what follows, we take πμνα=0.

By (42) and (57), we get:

Theorem 4 In the case πμνα0, the absolute differential of vector A is given by

dA=αAμγμdxα=αAμγμdxα,E59

in which α denotes the absolute derivatives of vector defined as follows:

αAμ=A;αμ+TαβμAβ,A;αμ=αAμ+ΓανμAν,E60
αAμ=Aμ;αTαμβAβ,Aμ;α=αAμΓαμνAν,E61

where A;αμ and Aμ;α are usual covariant derivatives of vector without torsion. Torsion TμνωΛ3 is an antisymmetrical tensor of Cn3 independent components.

Similarly, we can calculate the absolute differential for any tensor. The example also shows the advantages to combine variable with basis.

Now we take spinor connection as example to show the power of Clifford algebra. For Dirac equation in curved space-time without torsion, we have [23, 25, 26]:

γμiμ+Γμϕ=,Γμ=14γνμγν+Γμανγα.E62

Γμ is called spinor connection. Representing γμΓμΛ1Λ3 in the form of (15), we get:

αμp̂μϕsμΩμϕ=mγ0ϕ,E63

where αμ is current operator, p̂μ is momentum operator, and sμ spin operator. They are defined respectively by:

αμ=diagσμσ˜μ,p̂μ=iμ+ϒμeAμ,sμ=12diagσμσ˜μ,E64

where σμ=faμσa and σ˜μ=faμσ˜a are the Pauli matrices in curved space-time. ϒμΛ1 is called Keller connection, and ΩμΛ3 is called Gu-Nester potential, which is a pseudo vector [23, 26, 27]. They are calculated by:

ϒμ=12faνμfνaνfμa,Ωα=12ϵabcdfdαfaμfbνμfνeηce=14ϵdabcfdαfaβSbcμνβgμν,E65

where Sabμνfa{μfbν}signab for LU decomposition of metric. In the Hamiltonian of a spinor, we get a spin-gravity coupling potential sμΩμ. If the metric of the space-time can be orthogonalized, we have Ωμ0.

If the gravitational field is generated by a rotating ball, the corresponding metric, like the Kerr one, cannot be diagonalized. In this case, the spin-gravity coupling term has nonzero coupling effect. In asymptotically flat space-time, we have the line element in quasi-spherical coordinate system [28]:

dx=γ0Udt+Wdφ+Vγ1dr+γ2rdθ+γ3U1rsinθdφ,E66
dx2=Udt+Wdφ2Vdr2+r2dθ2U1r2sin2θdφ2,E67

in which UVW is just functions of rθ. As r we have:

U12mr,W4Lrsin2θ,V1+2mr,E68

where mL are mass and angular momentum of the star, respectively. For common stars and planets, we always have rmL. For example, we have m=̇3 km for the sun. The nonzero tetrad coefficients of metric (66) are given by:

ft0=U,fr1=V,fθ2=rV,fφ3=rsinθU,fφ0=UW,f0t=1U,f1r=1V,f2θ=1rV,f3φ=Ursinθ,f3t=UWrsinθ.E69

Substituting it into (65) we get

Ωα=f0tf1rf2θf3φ0θgrg0=Vr2sinθ10θUWrUW04Lr402rcosθsinθ0.E70

By (70), we find that the intensity of Ωα is proportional to the angular momentum of the star, and its force line is given by:

dxμds=Ωμdr=2rcosθsinθr=Rsin2θ.E71

(71) shows that the force lines of Ωα is just the magnetic lines of a magnetic dipole. According to the above results, we know that the spin-gravity coupling potential of charged particles will certainly induce a macroscopic dipolar magnetic field for a star, and it should be approximately in accordance with the Schuster-Wilson-Blackett relation [29, 30, 31].

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3. Representation of Clifford algebra

The matrix representation of Clifford algebra is an old problem with a long history. As early as in 1908, Cartan got the following periodicity of 8 [18, 19].

Theorem 5 For real universal Clifford algebra Cp,q, we have the following isomorphism

Cp,qMat2n2R,ifmodpq8=0,2Mat2n12RMat2n12R,ifmodpq8=1Mat2n12,ifmodpq8=3,7Mat2n22H,ifmodpq8=4,6Mat2n32HMat2n32H,ifmodpq8=5.E72

For C0,2, we have C=tI+xγ1+yγ2+zγ12 with

γ12=γ22=γ122=1,γ1γ2=γ2γ1=γ12,γ2γ12=γ12γ2=γ1,γ12γ1=γ1γ12=γ2.E73

By (73), we find C is equivalent to a quaternion, that is, we have isomorphic relation C0,2H.

Similarly, for C2,0, we have C=tI+xγ1+yγ2+zγ12 with

γ12=γ22=γ122=1,γ1γ2=γ2γ1=γ12,γ2γ12=γ12γ2=γ1,γ12γ1=γ1γ12=γ2.E74

By (74), the basis is equivalent to

γ1=0110,γ2=1001,γ12=0110.E75

Thus, (75) means C2,0Mat2R.

In geometry and physics, the matrix representation of generators of Clifford algebra is more important and fundamental than the representation of whole algebra. Define γμ by

γμ=0ϑ˜μϑμ0Γμm,ϑμ=diagσμ,σμ,,σμm,ϑ˜μ=diagσ˜μ,σ˜μ,,σ˜μm.E76

which forms the generator or grade-1 basis of Clifford algebra C1,3. To denote γμ by Γμm is for the convenience of representation of high dimensional Clifford algebra. For any matrices Cμ satisfying C1,3 Clifford algebra, we have [20, 32]:

Theorem 6 Assuming the matrices Cμ satisfy anti-commutative relation of C1,3

CμCν+CνCμ=2ημν,E77

then there is a natural number m and an invertible matrix K, such that K1CμK=Γμm.

This means in equivalent sense, we have unique representation (76) for generator of C1,3. In [20], we derived complex representation of generators of Cp,q based on Theorem 6 and real representations according to the complex representations as follows.

Theorem 7 Let

γ5=idiagEE,EdiagI2kI2l,k+l=n.E78

Other γμ,μ3 are given by(76). Then the generators of Clifford algebra C1,4 are equivalent to γμ,μ=0,1,2,3,5.

In order to express the general representation of generators, we introduce some simple notations. Im stands for m×m unit matrix. For any matrix A=Aab, denote block matrix

AIm=AabIm,ABC=diagABC.E79

in which the direct product of matrix is Kronecker product. Obviously, we have I2I2=I4, I2I2I2=I8, and so on. In what follows, we use Γμm defined in (76). For μ0,1,2,3, Γμm is 4m×4m matrix, which constitute the generator of C1,3. Similar to the above proofs, we can check the following theorem by method of induction.

Theorem 8

  1. 1. In equivalent sense, for C4m, the matrix representation of generators is uniquely given by

ΓμnΓμn22Γμn22I2Γμn24Γμn24Γμn24Γμn24I22,Γμn26Γμn26Γμn26Γμn26Γμn26Γμn26Γμn26Γμn26I23.E80

in which n=2m1N, where N is any given positive integer. All matrices are 2m+1N×2m+1N type.

  1. 2. For C4m+1, besides(80)we have another real generator

γ4m+1=EEEEEEEE,E=I2kI2l.E81

If and only if k=l, this representation can be uniquely expanded as generators of C4m+4.

  1. 3. For any Cp,q, pqp+q4mmodp+q41, the combination of p+q linear independent generators γμiγν taking from(80)constitutes the complete set of generators. In the case pqp+q4mmodp+q4=1, besides the combination of γμiγν, we have another normal representation of generator taking the form(81)with kl.

  2. 4. For Cm,m<4, we have another 2×2 Pauli matrix representation for its generators σ1σ2σ3.

Then, we get all complex matrix representations for generators of real Cp,q explicitly.

The real representation of Cp,q can be easily constructed from the above complex representation. In order to get the real representation, we should classify the generators derived above. Let Gcn stand for any one set of all complex generators of Cn given in Theorem 8, and set the coefficients before all σμ and σ˜μ as 1 or i. Denote Gc+ stands for the set of complex generators of Cn,0 and Gc for the set of complex generators of C0,n. Then, we have:

Gc=Gc+Gc,GciGc+.E82

By the construction of generators, we have only two kinds of γμ matrices. One is the matrix with real nonzero elements and the other is that with imaginary nonzero elements. This is because all nonzero elements of σ2 are imaginary but all other σμμ2 are real. Again assume

Gc+=GrGi,Gr=γrμγrμis real,Gi=γiμγiμis imaginary.E83

Denote J2=iσ2, we have J22=I2. J2 becomes the real matrix representation for imaginary unit i. Using the direct products of complex generators with I2J2, we can easily construct the real representation of all generators for Cp,q from Gc+ as follows.

Theorem 9

  1. For Cn,0, we have real matrix representation of generators as

    Gr+=γμI2ifγμGriγνJ2ifγνGi.E84

  2. For C0,n, we have real matrix representation of generators as

    Gr=γμJ2γμGr+.E85

  3. For Cp,q, we have real matrix representation of generators as

    Gr=Γ+μaΓνbΓ+μa=γμaGr+,a=12pΓνb=γνbGr,b=12q.E86

Obviously we have CnpCnq=Cnp2 choices for the real generators of Cp,q from each complex representation.

Proof. By calculating rules of block matrix, it is easy to check the following relations:

γμI2γνJ2+γνJ2γμI2=γμγν+γνγμJ2,E87
γμJ2γνJ2+γνJ2γμJ2=γμγν+γνγμI2.E88

By these relations, Theorem 9 becomes a direct result of Theorem 8.

For example, we have 4×4 real matrix representation for generators of C0,3 as follows:

iσ1σ2σ3σ1J2iσ2I2σ3J2Σ1Σ2Σ3=000100100100100000100001100001000100100000010010.E89

It is easy to check

ΣkΣl+ΣlΣk=2δkl,ΣkΣlΣlΣk=2ϵklmΣm.E90
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4. Transformation of Clifford algebra

Assume V is the base vector space of Cp,q, then Clifford algebra has the following global properties [22, 33, 34]:

Cp,q=k=0nΛkV=Cp,q+Cp,q,E91
Cp,q+k=evenΛkV,Cp,qk=oddΛkV,E92
Cp,qCp,q+1+.E93

Cp,q is a 2-graded superalgebra, and Cp,q+ is a subalgebra of Cp,q. We have:

C+C+=CC=C+,C+C=CC+=C.E94

Definition 6 The conjugation of element in Cp,q is defined by

γk1k2km=1mγkmk2k1=112mm+1γk1k2km,0mn.E95

The main involution of element is defined by

αγk1k2km=1mγk1k2km,0mn.E96

The norm and inverse of element are defined by

NXXX,X1=X/NXifNX0.E97

By the definition, it is easy to check

γk=γk,γab=γab,γabc=γabc,E98
αx=αx,αγk=γk,αγab=γab,E99
g1=g,g=g1g2gmgkΛ1Ngk=1.E100

Definition 7 The Pin group and Spin group of Cp,q are defined by

Pinp,q=gCp,qNg=±1αgxgVxV,E101
Spinp,q=gCp,q+Ng=±1gxgVxV=PinC+.E102

The transformation xαgxg is called sandwich operator. Pin or Spin group consists of two connected components with Ng=1 or Ng=1,

Spinp,q+=gCp,q+Ng=+1gxgVxV,E103
Spinp,q=gCp,q+Ng=1gxgVxV.E104

For gPinp,q,xV, the sandwich operator is a linear transformation for vector in V,

x=αgxgX=KX,X=x1x2xnT.E105

In all transformations of vector, the reflection and rotation transformations are important in geometry. Here, we discuss the transformation in detail. Let mΛ1 be a unit vector in V, then the reflection transformation of vector XΛ1 with respect to n1 dimensional mirror perpendicular to m is defined by [35]:

X=mXm=mXm.E106

Let m=γama,X=γaXa, substituting it into (106) and using (21), we have:

X=mX+maXbγabm=mXmmaXbmcγabγc=mXmmaXbmcgbcγagacγb+γabc=2mXm+X=XX.E107

Eq. (107) clearly shows the geometrical meaning of reflection. By (106), we learn reflection transformation belongs to Pinp,q group (Figure 3).

Figure 3.

Reflection transformation X=XX.

The rotation transformationRSpinp,q,

X=RXR1.E108

The group elements of elementary transformation in Λ2 are given by [22, 36]:

coshυab2+γabsinhυab21=coshυab2γabsinhυab2,υabR,E109
cosθab2+γabsinθab21=cosθab2γabsinθab2,θabππ.E110

The total transformation can be expressed as multiplication of elementary transformations as follows:

R=ηaaηbb=1coshυab2+γabsinhυab2ηaaηbb=1cosθab2+γabsinθab2.E111

(111) has 12n1n generating elements like SOn. In (111), we have commutative relation as follows:

coshυab2+γabsinhυab2cosθcd2+γcdsinθcd2=2sinhυab2sinθcd2γabγcd,E112
cosθab2+γabsinθab2cosθcd2+γcdsinθcd2=2sinθab2sinθcd2γabγcd,E113

in which

γabγcd=ηbcγadηacγbd+ηadγbcηbdγacΛ2.E114

If abcd, the right hand terms vanish, and then two elementary transformations commute with each other.

R forms a Lie Group of 12n1n paraments. In the case Cn,0 or C0,n, R is compact group isomorphic to SOn. Otherwise, R is noncompact one similar to Lorentz transformation. The infinitesimal generators of the corresponding Lie group is γab, and the Lie algebra is given by:

R=εabγab,γabγcd=2γabγcdΛ2,εabR.E115

Thus, Λ2Mp,q is just the Lie algebra of proper Lorentz transformation of the space-time Mp,q.

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5. Application in classical geometry

Suppose the basic space of projective geometry is n-dimensional Euclidean space π (see Figure 4), and the basis is γaa=12n. The coordinate of point x is given by x=γaxa. The projective polar is P, and its height from the basic space π is h. The total projective space is n+1 dimensional, and an auxiliary basis γn+1=γp is introduced. The coordinate of the polar P is p=γμpμ. In this section, we use Greek characters for n+1 indices. Assume the unit directional vector of the projective ray is t=γμtμ, the unit normal vector of the image space π is n=γμnμ, coordinate in π is y=γμyμ, and the intercept of π with the n+1 coordinate axis is a. Then, we have:

Figure 4.

Diagram of parameter setting for projective geometry.

yan=0,oryn=nμyμ=anp.E116

The equation of projective ray is given by:

s=p+λt,E117

where λ is parameter coordinate of the line. In the basic space π, we have sn+1=0 and λ=h/tp, so the coordinate of the line in π reads

x=phtpt.E118

Let s=y and substitute (117) into (116) we get image equation as follows:

y=p+anppntnt,λ=anppntn.E119

In the above equation tn0, which means t cannot be perpendicular to n; otherwise, the projection cannot be realized. Eliminating coordinate t in (118) and (119), we find the projective transformation yx is nonlinear. In (119), only the parameters an are related to image space π; so, all geometric variables independent of two parameters an are projective invariants. In what follows we prove the fundamental theorems of projective geometry by Clifford algebra.

Theorem 10 For 4 different points y1y2y3y4 on a straight line L, the following cross ratio is a projective invariant

1234y1y3y2y3y2y4y1y4.E120

Proof Substituting (119) into (120) we get

1234=t3nt1t1nt3t3nt2t2nt3t4nt2t2nt4t4nt1t1nt4.E121

By (19) and (20), we get

tbntatantb=tatbn=±tatbmn,E122

where m is the unit normal vector of the plane spanned by tatb, which is independent of the image space π. Substituting it into (121), we get

1234=t3t1t3t2t4t2t4t1.E123

(123) is independent of an; so, it is a projective invariant. Likewise, 1324 and 1423 are also projective invariants. The proof is finished.

Now we examine affine transformation. In this case, the polar P at infinity and the directional vector t of rays becomes constant vector. The equation of rays is given by y=x+λt. Substituting it into (116), we get the coordinate transformation from basic space π to image space π,

y=x+anpnxtnt,λ=anpnxtn.E124

Since t and n are constant vectors for all rays, the affine transformation yx is linear. A variable independent of an is an affine invariant.

Theorem 11 Assume x1x2x3 are 3 points on a straight line L in basic space π, and y1y2y3 are respectively their projective images on line L in π. Then the simple ratio

1213y2y1y3y1E125

is an affine invariant.

Proof By equation of transformation (124) we get

yk=xk+anpnxktnt.E126

In (126), only the parameters an are related to image space π. Substituting (126) into (125), we have:

1213=tnx2x1nx2x1ttnx3x1nx3x1t=x2x1tnx3x1tn.E127

Denote the unit directional vector of line L by k, then we have

x2x1=±x2x1k,x3x1=±x3x1k.E128

Substituting them into (127) we get:

1213=y2y1y3y1=x2x1x3x1.E129

This proves the simple ratio 1213 is an affine invariant. Likewise, we can prove 1223 and 1323 are also affine invariants. The proof is finished.

The treatment of image information by computer requires concise and general algebraic representation for geometric modeling as well as fast and robust algebraic algorithm for geometric calculation. Conformal geometry algebra was introduced in this context. By establishing unified covariant algebra representation of classical geometry, the efficient calculation of invariant algebra is realized [13, 14, 15]. It provides a unified and concise homogeneous algebraic framework for classical geometry and algorithms, which can thus be used for complicated symbolic geometric calculations. This technology is currently widely applied in high-tech fields such as computer graphics, vision calculation, geometric design, and robots.

The algebraic representation of a geometric object is homogeneous, which means that any two algebraic expressions representing this object differ by only one nonzero factor and any such algebraic expressions with different nonzero multiple represent the same geometric object. The embedding space provided by conformal geometric algebra for n dimensional Euclidean space is n+2 dimensional Minkowski space. Since the orthonormal transformation group of the embedding space is exactly double coverage of the conformal transformation group of the Euclidean space, this model is also called the conformal model. The following is a brief introduction to the basic concepts and representation for geometric objects of conformal geometric algebra. The materials mainly come from literature [13].

In conformal geometry algebra, an additional Minkowski plane M1,1 is attached to n dimensional Euclidean space Rn, M1,1 has an orthonormal basis e+e, which has the following properties:

e+2=1,e2=1,e+e=0.E130

In practical application, e+e is replaced by null basis e0e

e0=12ee+,e=e+e+.E131

They satisfy

e02=e2=0,ee0=1.E132

A unit pseudo-scalar E for M1,1 is defined by:

E=ee0=e+e=e+e.E133

In conformal geometric algebra, we work with Mn+1,1=RnM1,1.

Define the horosphere of Rn by:

Nen=xMn+1,1x2=0xe=1.E134

Nen is a homogeneous model of Rn. The powerful applications of conformal geometry come from this model. By calculation, for xRn we have:

x=x+12x2e+e0,E135

which is a bijective mapping xRnxNen, we have NenRn. x is referred to as the homogeneous point of x. Clearly, 0Rne0Nen and RneNen are in homogeneous coordinate.

Now we examine how conformal geometric algebra represents geometric objects. For a line passing through points a and b, we have

eab=eab+baE.E136

Since ab=aba is the moment for a line through point a with tangent ab, eab characterizes the line completely.

Again by using (135) and (136), we get

eabc=eabc+bacaE.E137

We recognize abc as the moment of a plane with tangent baca. Thus, eabc represents a plane through points {a, b, c}, or, more specifically, the triangle (2-simplex) with these points as vertices.

For a sphere with radius ρ and center pRn, we have xp2=ρ2. By (135), the equation in terms of homogeneous points becomes

xp=12ρ2.E138

Using xe=1, we get:

xs=0,s=p12ρ2e=p+e0+12p2ρ2e,E139

where

s2=ρ2,es=1.E140

From these properties, the form (139) and center p can be recovered. Therefore, every sphere in Rn is completely characterized by a unique vector sMn+1,1. According to (140), s lies outside the null cone. Analysis shows that every such vector determines a sphere.

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6. Discussion and conclusion

The examples given above are only applications of Clifford algebra in geometry, but we have seen the power of Clifford algebra in solving geometrical problems. In fact, Clifford algebra is more widely used in physics. Why does Clifford algebra work so well? As have been seen from the above examples, the power of Clifford algebra comes from the following features:

  1. In the geometry of flat space, the basic concepts are only length, angle, area, and volume, which are already implicitly included in the definition of Clifford algebra. So, Clifford algebra summarizes these contents of classical geometry and algebraize them all. By introducing the concepts of inner, exterior, and direct products of vector, Clifford algebra summarizes the operations of scalars, vectors, and tensors and then can represent all the physical variables in classical physics, because only these variables are included in classical physics.

  2. By localizing the basis or frame of space-time, Clifford algebra is naturally suitable for the tangent space in a manifold. If the differential μ and connection operator dμγν are introduced, Clifford algebra can be used for the whole manifold, so it contains Riemann geometry. Furthermore, Clifford algebra can express all contents of classical physics, including physical variables, differential equations, and algebraic operations. Clifford algebra transforms complicated theories and relations into a unified and standard calculus with no more or less contents, and all representations are neat and elegant [23, 36].

  3. If the above contents seem to be very natural, Clifford algebra still has another unusual advantage, that is, it includes the theory of spinor. So, Clifford algebra also contains quantum theory and spinor connection. These things are far beyond the human intuition and have some surprising properties.

  4. There are many reasons to make Clifford algebra become a unified and efficient language and tool for mathematics, physics, and engineering, such as Clifford algebra generalizes real number, complex number, quaternion, and vector algebra; Clifford algebra is isomorphic to matrix algebra; the derivative operator γμμ contains grad, div, curl, etc. However, the most important feature of Clifford algebra should be taking the physical variable and the basis as one entity, such as g=gμνγμγν and T=Tμνωγμνω. In this representation, the basis is an operator without ambiguity. Clifford algebra calculus is an arithmetic-like operation which can be well understood by everyone.

    “But, if geometric algebra is so good, why is it not more widely used?” As Hestenes replied in [11]: “Its time will come!” The published geometric algebra literature is more than sufficient to support instruction with geometric algebra at intermediate and advanced levels in physics, mathematics, engineering, and computer science. Though few faculty are conversant with geometric algebra now, most could easily learn what they need while teaching. At the introductory level, geometric algebra textbooks and teacher training will be necessary before geometric algebra can be widely taught in the schools. There is steady progress in this direction, but funding is needed to accelerate it. Malcolm Gladwell has discussed social conditions for a “tipping point” when the spread of an idea suddenly goes viral. Place your bets now on a Tipping Point for Geometric Algebra!

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Acknowledgments

I would like to thank Dr. Min Lei for her kind invitation and help. The discussion on torsion is completed under the inspiration and guidance of Prof. James M. Nester. The content of conformal geometric algebra is added according to the suggestion of Dr. Isiah Zaplana. The chapter has been improved according to the comments of a referee.

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Written By

Ying-Qiu Gu

Submitted: 01 November 2019 Reviewed: 21 July 2020 Published: 03 September 2020