Some Applications of Clifford Algebra in Geometry

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. ,


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 e θi . 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 1, i, j, k f g, and the spatial basis should satisfy the multiplying rules i 2 ¼ j 2 ¼ k 2 ¼ À1 and jk ¼ Àkj ¼ i, ki ¼ Àik ¼ j, and ij ¼ Àji ¼ Àk. 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. by introducing typical application of Clifford algebra in geometry, we show some special feature and elegance of the algebra.

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  n over number field , if the multiplication rule of vectors satisfies 1: Antisymmetry, x ∧ y ¼ Ày ∧ x; (1) 3: Distributivity, x ∧ ay þ bz the algebra is called Grassmann algebra and x ∧ y exterior product. The Grassmann is also called exterior algebra. The geometrical meaning of x ∧ y is oriented area of a parallelogram constructed by x and y, and the geometrical meaning of x ∧ y ∧ ⋯ ∧ z is the oriented volume of the parallelohedron constructed by the vectors (see Figure 1). We call x ∧ y two-vector, x ∧ y ∧ z three-vector, and so on. For k-vector x ∈ Λ k and l-vector y ∈ Λ l , we have By the definition, we can easily check: Theorem 1 For exterior algebra defined in V ¼  n , we have The dimension of the algebra is Under the orthonormal basis e 1 , e 2 , ⋯, e n f g , the exterior algebra takes the following form: in which ∀w jk⋯l ∈ , e jk⋯l ¼ e j ∧ e k ∧ ⋯ ∧ e l , and ∀|e jk⋯l | ¼ 1.
The exterior product of vectors contains alternating combinations of basis, for example: Definition 2 For any vectors x, y, z ∈  n , Clifford product of vectors is denoted by Clifford product is also called geometric product. Similarly, we can define Clifford algebra for many vectors as xy⋯z. In (6), x Á y ¼ η ab x a y b is the scalar product or inner product in  n . By x ∧ y ¼ Ày ∧ x, we find Clifford product is not commutative. By (6), we have then the algebra is called as Clifford algebra or geometric algebra, which is denoted as Cℓ p,q . There are several definitions for Clifford algebra [18,19]. The above definition is the original definition of Clifford. Clifford algebra has also 2 n 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 T x ð Þ 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 ¼ p þ q dimensional manifold T p,q over , the element is defined by where γ a is the local orthogonal frame and γ a the coframe. The distance ds ¼ |dx| and oriented volumes dV k is defined by in which η ab ð Þ ¼ diag I p , ÀI q À Á is Minkowski metric and g μν is Riemann metric.
is Grassmann basis. The following Clifford-Grassmann number with basis defines real universal Clifford algebra Cℓ p,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  3 , we have element dx and the area element ds in sphere dr ¼ 0 as We have the total area of the sphere The above definition involves a number of concepts, some more explanations are given in the following: 1. The geometrical meanings of elements dx, dy, dx ∧ dy are shown in Figure 2.
The relation between metric and vector basis is given by: which is the most important relation in Clifford algebra. Since Clifford algebra is isomorphic to some matrix algebra, by (17) γ a 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 coefficient and metric is given by: 2. Assume γ a ja ¼ 1, 2⋯n f g to be the basis of the space-time, then their exterior product is defined by [22]: In which σ b 1 b 2 ⋯b k a 1 a 2 ⋯a k is permutation function, if b 1 b 2 ⋯b k is the even permutation of a 1 a 2 ⋯a k , it equals 1. Otherwise, it equals À1. 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.
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.
4. In Eq. (15), each grade-k term is a tensor. For example, c 0 I ∈ Λ 0 is a scalar, c μ γ μ ∈ Λ 1 is a true vector, and c μν γ μν ∈ Λ 2 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 ¼ γ μ x μ ∈ Λ 1 and multivector m ¼ γ θ 1 θ 2 ⋯θ k m θ 1 θ 2 ⋯θ k ∈ Λ k , their inner product is defined as (18) in which Theorem 2 For basis of Clifford algebra, we have the following relations Permuting the indices θ 1 and θ 2 , we find a 2 ¼ Àa 1 . Let μ ¼ θ 1 , we get a 1 ¼ 1. Check the monomial in exterior product, we get A ¼ 1. Thus, we prove (21). In like manner, we prove (22). For orthonormal basis γ a , by (22) we have: γ a 1 a 2 ⋯a nÀ1 γ a n ¼ ϵ a 1 a 2 ⋯a n γ 12⋯n : Again by γ a n γ a n ¼ 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 product A ⊙ k B between multivectors as follows: We use A ⊙ k B rather AÁ k B, because the symbol "Á" is too small to express exponential power. Then for the case γ μ 1 μ 2 ⋯μ f γ θ 1 θ 2 ⋯θ k , we have similar results. For example, we have In Cℓ 1,3 , denote the Pauli matrices by We use k, f , j standing for spatial indices. Define Dirac γÀ matrix by: γ a forms the grade-1 basis of Clifford algebra Cℓ 1,3 . In equivalent sense, the representation (33) is unique. By γ-matrix (33), we have the complete bases of Cℓ 1,3 as follows [21]: Based on the above preliminaries, we can display some enlightening examples of application, which show how geometric algebra works efficiently. For a skewsymmetrical torsion T μνω g μβ T β νω in  1,3 , by Clifford calculus, we have: and then where g ¼ |det g μν |. So we get: So, the skew-symmetrical torsion is equivalent to a pseudo vector in  1,3 . 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 γ a and γ μ . In physics, basis of tensors is defined by direct products of grade-1 bases γ μ . For metric, we have [23]: For simplicity, we denote tensor basis by: In general, a tensor of rank n is given by: The geometrical information of the tensor such as transformation law and differential connection are all recorded by basis γ μ , and all representations of rank r, s ð Þ tensor denote the same one practical entity T x ð Þ. T ν⋯ μ⋯ is just a quantity table similar to c μν in (15), but the physical and geometrical meanings of the tensor T are represented by basis γ μ . 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 We call d α connection operator [23]. According to its geometrical meanings, connection operator should satisfy the following conditions: 1. It is a real linear transformation of basis γ μ , 2. It satisfies metric consistent condition dg ¼ 0.
Substituting (42) into we get To understand the meaning of π α μν and T α μν , we examine the influence on geodesic.
The term T α μν v μ v ν ¼ 0 due to T α μν ¼ ÀT α νμ . So the symmetrical part π α μν influences the geodesic, but the antisymmetrical part T α μν only influences spin of a particle. This means π α μν 6 ¼ 0 violates Einstein's equivalent principle. In what follows, we take π α μν ¼ 0. By (42) and (57), we get: Theorem 4 In the case π α μν 0, the absolute differential of vector A is given by in which ∇ α denotes the absolute derivatives of vector defined as follows: where A μ ; α and A μ; α are usual covariant derivatives of vector without torsion. Torsion T μνω ∈ Λ 3 is an antisymmetrical tensor of C 3 n 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]: Γ μ is called spinor connection. Representing γ μ Γ μ ∈ Λ 1 ∪ Λ 3 in the form of (15), we get: where α μ is current operator,p μ is momentum operator, and s μ spin operator. They are defined respectively by: where a are the Pauli matrices in curved space-time.
ϒ μ ∈ Λ 1 is called Keller connection, and Ω μ ∈ Λ 3 is called Gu-Nester potential, which is a pseudo vector [23,26,27]. They are calculated by: where Þfor LU decomposition of metric. In the Hamiltonian of a spinor, we get a spin-gravity coupling potential s μ Ω μ . If the metric of the space-time can be orthogonalized, we have Ω μ 0.
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]: in which U, V, W ð Þis just functions of r, θ ð Þ. As r ! ∞ we have: where m, L ð Þare mass and angular momentum of the star, respectively. For common stars and planets, we always have r ≫ m ≫ L. For example, we have m _ ¼3 km for the sun. The nonzero tetrad coefficients of metric (66) are given by: Substituting it into (65) we get By (70), we find that the intensity of Ω α is proportional to the angular momentum of the star, and its force line is given by: (71) shows that the force lines of Ω α 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].

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 Cℓ p,q , we have the following isomorphism For Cℓ 0,2 , we have C ¼ tI þ xγ 1 þ yγ 2 þ zγ 12 with By (73), we find C is equivalent to a quaternion, that is, we have isomorphic relation Cℓ 0,2 ffi .
Similarly, for Cℓ 2,0 , we have C ¼ tI þ xγ 1 þ yγ 2 þ zγ 12 with By (74), the basis is equivalent to Thus, (75) means Cℓ 2,0 ffi Mat 2,  ð Þ. In geometry and physics, the matrix representation of generators of Clifford algebra is more important and fundamental than the representation of whole algebra. Define γ μ by which forms the generator or grade-1 basis of Clifford algebra Cℓ 1,3 . To denote γ μ by Γ μ m ð Þ is for the convenience of representation of high dimensional Clifford algebra. For any matrices C μ satisfying Cℓ 1, 3 Clifford algebra, we have [20,32]: Theorem 6 Assuming the matrices C μ satisfy anti-commutative relation of Cℓ 1,3 then there is a natural number m and an invertible matrix K, such that K À1 C μ K ¼ Γ μ m ð Þ. This means in equivalent sense, we have unique representation (76) for generator of Cℓ 1,3 . In [20], we derived complex representation of generators of Cℓ p,q based on Theorem 6 and real representations according to the complex representations as follows. Theorem 7 Let Other γ μ , μ ≤ 3 ð Þare given by (76). Then the generators of Clifford algebra Cℓ 1,4 are equivalent to ∀γ μ , μ ¼ 0, 1, 2, 3, 5 ð Þ . In order to express the general representation of generators, we introduce some simple notations. I m stands for m Â m unit matrix. For any matrix A ¼ A ab ð Þ, denote block matrix in which the direct product of matrix is Kronecker product. Obviously, we have I 2 ⊗ I 2 ¼ I 4 , I 2 ⊗ I 2 ⊗ I 2 ¼ I 8 , and so on. In what follows, we use Γ μ m ð Þ defined in (76). For μ ∈ 0, 1, 2, 3 f g , Γ μ m ð Þ is 4m Â 4m matrix, which constitute the generator of Cℓ 1,3 . Similar to the above proofs, we can check the following theorem by method of induction. Theorem 8
The real representation of Cℓ p,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 G c n ð Þ stand for any one set of all complex generators of Cℓ n given in Theorem 8, and set the coefficients before all σ μ andσ μ as 1 or i. Denote G cþ stands for the set of complex generators of Cℓ n,0 and G cÀ for the set of complex generators of Cℓ 0,n . Then, we have: By the construction of generators, we have only two kinds of γ μ 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 σ 2 are imaginary but all other σ μ ∀μ 6 ¼ 2 ð Þare real. Again assume Denote J 2 ¼ iσ 2 , we have J 2 2 ¼ ÀI 2 . J 2 becomes the real matrix representation for imaginary unit i. Using the direct products of complex generators with I 2 , J 2 ð Þ, we can easily construct the real representation of all generators for Cℓ p,q from G cþ as follows.
Theorem 9 1. For Cℓ n,0 , we have real matrix representation of generators as 2. For Cℓ 0,n , we have real matrix representation of generators as 3. For Cℓ p,q , we have real matrix representation of generators as Obviously we have C p n C q n ¼ C p n À Á 2 choices for the real generators of Cℓ p,q from each complex representation. Proof. By calculating rules of block matrix, it is easy to check the following relations: By these relations, Theorem 9 becomes a direct result of Theorem 8. For example, we have 4 Â 4 real matrix representation for generators of Cℓ 0,3 as follows: It is easy to check

Transformation of Clifford algebra
Assume V is the base vector space of Cℓ p,q , then Clifford algebra has the following global properties [22,33,34]: Cℓ p,q ffi Cℓ þ p,qþ1 : Cℓ p,q is a ℤ 2 -graded superalgebra, and Cℓ þ p,q is a subalgebra of Cℓ p,q . We have:

Definition 6
The conjugation of element in Cℓ p,q is defined by The main involution of element is defined by The norm and inverse of element are defined by By the definition, it is easy to check

Definition 7
The Pin group and Spin group of Cℓ p,q are defined by The transformation x ↦ α g À Á xg * is called sandwich operator. Pin or Spin group consists of two connected components with N g À Á For ∀g ∈ Pin p,q , x ∈ V, the sandwich operator is a linear transformation for vector in V, In all transformations of vector, the reflection and rotation transformations are important in geometry. Here, we discuss the transformation in detail. Let m ∈ Λ 1 be a unit vector in V, then the reflection transformation of vector X ∈ Λ 1 with respect to n À 1 dimensional mirror perpendicular to m is defined by [35]: Let m ¼ γ a m a , X ¼ γ a X a , substituting it into (106) and using (21), we have: Eq. (107) clearly shows the geometrical meaning of reflection. By (106), we learn reflection transformation belongs to Pin p,q group (Figure 3).
The rotation transformation R ∈ Spin p,q , The group elements of elementary transformation in Λ 2 are given by [22,36]: The total transformation can be expressed as multiplication of elementary transformations as follows: (111) has 1 2 n À 1 ð Þn generating elements like SO n ð Þ. In (111), we have commutative relation as follows: in which γ ab ⊙ γ cd ¼ η bc γ ad À η ac γ bd þ η ad γ bc À η bd γ ac ∈ Λ 2 : If a 6 ¼ b 6 ¼ c 6 ¼ d, the right hand terms vanish, and then two elementary transformations commute with each other.
R forms a Lie Group of 1 2 n À 1 ð Þn paraments. In the case Cℓ n,0 or Cℓ 0,n , R is compact group isomorphic to SO n ð Þ. Otherwise, R is noncompact one similar to Lorentz transformation. The infinitesimal generators of the corresponding Lie group is γ ab , and the Lie algebra is given by: Thus, Λ 2  p,q ð Þis just the Lie algebra of proper Lorentz transformation of the space-time  p,q .

Application in classical geometry
Suppose the basic space of projective geometry is n-dimensional Euclidean space π (see Figure 4), and the basis is γ a ja ¼ 1, 2, ⋯, n f g . The coordinate of point x is Reflection transformation X 0 ¼ X ⊥ À X ∥ .
given by x ¼ γ a x a . The projective polar is P, and its height from the basic space π is h. The total projective space is n þ 1 dimensional, and an auxiliary basis γ nþ1 ¼ γ p is introduced. The coordinate of the polar P is p ¼ γ μ p μ . In this section, we use Greek characters for n þ 1 indices. Assume the unit directional vector of the projective ray is t ¼ γ μ t μ , the unit normal vector of the image space π 0 is n ¼ γ μ n μ , coordinate in π 0 is y ¼ γ μ y μ , and the intercept of π 0 with the n þ 1 coordinate axis is a. Then, we have: The equation of projective ray is given by: where λ is parameter coordinate of the line. In the basic space π, we have s nþ1 ¼ 0 and λ ¼ Àh=t p , so the coordinate of the line in π reads Let s ¼ y and substitute (117) into (116) we get image equation as follows: In the above equation t ⊙ n 6 ¼ 0, 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 y $ x is nonlinear. In (119), only the parameters a, n ð Þ are related to image space π 0 ; so, all geometric variables independent of two parameters a, n ð Þ are projective invariants. In what follows we prove the fundamental theorems of projective geometry by Clifford algebra.
Theorem 10 For 4 different points y 1 , y 2 , y 3 , y 4 È É on a straight line L, the following cross ratio is a projective invariant 12; 34 ð Þ |y 1 À y 3 | |y 2 À y 3 | Á |y 2 À y 4 | |y 1 À y 4 | : (120) This proves the simple ratio 12, 13 ð Þis an affine invariant. Likewise, we can prove 12, 23 ð Þand 13, 23 ð Þare 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 þ 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  1,1 is attached to n dimensional Euclidean space  n ,  1,1 has an orthonormal basis e þ , e À f g, which has the following properties: e 2 þ ¼ 1, e 2 À ¼ À1, e þ ⊙ e À ¼ 0: In practical application, e þ , e À f gis replaced by null basis e 0 , e f g e 0 ¼ 1 2 e À À e þ ð Þ , e ¼ e À þ e þ : They satisfy e 2 0 ¼ e 2 ¼ 0, e ⊙ e 0 ¼ À1: A unit pseudo-scalar E for  1,1 is defined by: In conformal geometric algebra, we work with  nþ1,1 ¼  n ⊕  1,1 . Define the horosphere of  n by: N n e is a homogeneous model of  n . The powerful applications of conformal geometry come from this model. By calculation, for ∀x ∈  n we have: 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 d μ γ ν 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 γ μ ∇ μ 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 ¼ g μν γ μ ⊗ γ ν and T ¼ T μνω γ μνω . 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!