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You're using an out-of-date version of Internet Explorer. By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. Ashfaque AMIM Throughout the text, from the word go, the reader finds various worked examples to help understand the ideas presented. In summary, the book is divided into six chapters with a set of exercises provided at the end of each one. To lay the foundations, key ideas and concepts from linear algebra are discussed as subsequent ideas are developed on the basis of these.

The notion of the tensor algebra is introduced leading towards the definitions of exterior algebras and Grassmann algebras, which although synonymous, are presented in their own rights respectively. Three different definitions of the geometric algebras or better known as Clifford algebras, are presented in order to highlight the excellent properties of the universal Clifford algebras.

Classification and representations of the Clifford algebras is followed by a study of the groups associated with the Clifford algebras, P in and Spin, and consequently the corresponding Lie algebras associated with these groups are presented with a brief look at one principal application: conformal transformations associated with the Minkowski space-time. The final chapter of the book is devoted to spinors. Three different types of definitions, each with its on set of properties, can be found.

That of an algebraic spinor, of a classical spinor and a spinor operator. Furthermore, the relation amongst the three definitions is explored and the reader finds that under certain set of conditions the three otherwise distinct definitions are equivalent. The chapter ends with an elaborate study of Weyl spinors. The appendix bears witness to the standard two-component spinor formalism in the Van der Waerden framework. As a mathematical physicist, I was fascinated by the chapter on spinors. Roughly speaking spinors are mathematical entities that allow a more general treatment of the notion of invariance under rotations and Lorentz boosts.

Lay: Linear Algebra and its Applications. Instructor's Edition. Addison-Wesley, Reading, MA, Riesz: Cliflord Numbers and Spinors. Reprinted as facsimile eds. Bolinder, P. Lounesto by Kluwer, Dordrecht, The Netherlands, Strang: Introduction to Linear Algebra. To each ordered pair of real numbers x, y there corresponds a unique complex number x i y. The xy-plane, where the complex numbers are represented, is called the complex plane. Its x-axis is the real axis and y-axis the imaginary axis. This is the polar form of z.

The familiar addition rules for the sine and cosine result in the polar form of multiplication, 3 Electrical engineers denote the polar form by r k. Incidentally, Im zlE2 measures the signed area of the parallelogram determined by zl and z2. Thus, the norm of a product is the product of the norms, and the phase-angle of a product is the sum of the phase-angles mod 27r.

In a field numbers can be both added and multiplied. The above rules of addition and multiplication make up the axioms of a field F. It is tempting to regard R as a unique subfield in C. Such extra structure turns the field C into a real algebra C. This 5 The finite fields IFp. This implies distributivity. In other words, distributivity has no independent meaning for an algebra. The above construction of C as the real linear space R 2 brings in more structure tban just the field structure: it makes C an algebra over R. The field C has an infinity of automorphisms.

In contrast, the only automorphisms of the real algebra C are the identity automorphism and complex conjugation. Complex conjugation is the only field automorphism of C which is different from the identity but preserves a fixed subfield R. I The other automorphisms of the field C send a real subfield R onto an isomorphic copy of R , which is necessarily different from the original subfield R. However, any field automorphism of C fixes point-wise the rational subfield Q. It is known that there is a field automorphism of C sending -. Related to each real subfield there is a unique complex conjugation across that subfield, and all such automorphisms of finite order are complex conjugations for some real subfield.

The image a R under such an automorphism a of a distinguished real subfield R is dense in C [in the topology of the metric 1. This is a dense set in C. The above discussion indicates that there is no unique complex conjugation in the field of complex numbers, and that the field structure of C does not fix by itself the subfield R of C. If a privileged real subfield R is singled out in C, it brings along a real linear structure on C, and a unique complex conjugation across R , which then naturally imports a metric structure to.

Our main interest in complex numbers in this book is C as a real algebra, not so much as a field. However, this is not the only linear representation of C in Mat 2,R. We will see that i represents i an oriented plane area in R2, ii a quarter turn of R2. The Euclidean plane R 2 has a quadratic form We introduce an associative product of vectors such that Using distributivity this results in the multiplication rules The element ele2 satisfies and therefore cannot be a scalar or a vector.

It is an example of a bivector, the unit bivector. Complex Numbers 26 2. The basis elements obey the multiplication table The basis elements span the subspaces consisting of 1 el, e 2 el2 l2 R scalars R2 vectors R2 bivectors. The even part is not only a subspace but also a subalgebra. An 28 Complex Numbers 2. The only common point of the two planes is the zero 0.

The two planes are both parts of the same algebra Ce2. Parity considerations show that - complex number times complex number is a complex number, - vector times complex number is a vector, - complex number times vector is a vector, and - vector times vector is a complex number. The above observations can be expressed by the inclusions ce;ce; c ce;, ce; ce; c ce; , ce;ce; c ce; , ce; ce; c ce;.

In other words, there are two complex numbers which produce the same final result but via different actions. Complex Numbers rZ 2. The rotation group SO 2 is isomorphic to the unitary group U 1. The fact that two opposite elements of the spin group Spin 2. A position of the hands of your watch corresponds t o two positions of the Sun. A rotating mirror turns half t h e angle of the image.

Circulating a coin one full turn around another makes t h e coin turn twice around its center. Is Exercise 6 History Imaginary numbers first appeared around , when Tartaglia and Cardano expressed real roots of a cubic equation in terms of conjugate complex numbers. The first one to represent complex numbers by points on a plane was a Norwegian surveyor, Caspar Wessel, in He posited an imaginary axis perpendicular to the axis of real numbers. Complex numbers got their name from Gauss, and their formal definition as pairs of real numbers is due to Hamilton in first published Compute 1 - zl l - zz.

Show that the field C cannot be ordered. The identity automorphism is similar only to itself. Determine all the rational rotations of R2. Show that a 2-dimensional real algebra with unity 1 is both commutative and associative. In an ordered field non-zero numbers have positive squares, and the sum of such squares is positive, and therefore non-zero. Consequently, it is impossible to order the field C. Bibliography L. Ahlfors: Complex Analysis. McGraw-Hill, New York, Cartan expos6 d'aprks l'article allemand de E.

Study : Nombres complexes; pp. Molk red. Reprinted in E. Cartan: CEuvres complktes, Partie Gauthier-Villars, Paris, , pp. Churchill, J. Brown, R. Verhey: Complex Variables and Applications. Ebbinghaus et al. Springer, New York, Springer, Berlin, For instance, a moment of a force, angular velocity of a rotating body, and magnetic induction can be described with bivectors. In three dimensions bivectors are dual to vectors, and their use can be circumvented. Scalars, vectors, bivectors and the volume element span the exterior algebra AIR3, which provides a multivector structure for the Clifford algebra C13 of the Euclidean space R3.

Two bivectors have the same direction if they are on parallel planes the same attitude and are similarly oriented the same sense of rotation. Vector directed line segment 1. The area or norm of a bivector A is denoted by IAl. Two bivectors A and B in parallel planes have the same attitude, and we write A 11 B. The shape of the area is irrelevant. Representing a bivector as an oriented parallelogram suggests that a bivector can be thought of as a geometrical product of vectors along its sides. With this in mind we introduce the exterior product a A b of two vectors a and b as the bivector obtained by sweeping b along a.

This can be simply expressed by writing 3. In three dimensions this is always possible because any two planes will either be parallel or intersect along a common line. This is evident in the equality 3. The construction of bivectors calls only for a linear structure, and no metric is needed. This is no longer true in four dimensions; for instance el A e? The bivector can be expressed as a 'determinant' el A e 2. The usual volume is IVI. In spite of the resemblance between the determinant expressions for the exterior product a A b and the cross product a x b there is a difference: the exterior product does not require a metric while the cross product requires or induces a metric.

The metric gets involved in positioning the vector a x b perpendicular to the bivector a A b. This gives rise to the Clifford dual defined as u e 1 2 3 for u E C e 3. Bivectors and the Exterior Algebra 40 3. The product of two elements u and v in the Clifford algebra Ce3 of the Euclidean space IW3 is denoted by juxtaposition, uv, to distinguish it from the exterior product u A v. Clifford in Bivectors and the Exterior Algebra a.

Such a property of an algebra is usually referred to by saying that the algebra is graded over the index group Z. The Clifford products of even and odd subspaces satisfy the inclusion relations ce; ce; c ce; ce, ceg c ce; , , ce; ce; c ce; , ce; ce; c ce;. The Clifford algebra Ce3 is not graded over Z. However, we can reconstruct the exterior product from the Clifford product in a unique manner. We shall refer to the dimension grading of the associated exterior algebra by saying that the Clifford algebra has a multivector structure.

Recall that R and R3 have, by definition, unique copies in Ce3. The exterior product of a vector and a bivector can be depicted as an oriented volume: aAB BAa The orientation is obtained by putting the arrows in succession. Consider a vector x tilted by an angle cp out of the plane of a bivector B. Let a be the orthogonal projection of x in the plane of B. The left contraction can be obtained from the exterior product and the Clifford product as follows: This means that the left contraction is dual to the exterior product. Exercises 1. Find the area of the triangle with vertices 1, -4, -6 , O,O, 0.

Show that el23 commutes with e l , eg, eg. Compute the perpendicular and parallel components of a in the plane of B. Reconstruct the dot product a. Show that where, for a k-vector u E Determine a. Determine p. Bibliography R. Deheuvels: Formes quadratiques et groupes classiques. Presses Universitaires de France, Paris, Dieudonnk: The tragedy of Grassmann.

Linear and Multilinear Algebra 8 , Greub: Multilinear Algebra, 2"d edn. Helmstetter: Alg6bres de Clifford et algkbres de Weyl. Cahiers Math. Sobczyk: Vector Calculus with Complex Variables. Stewart: Hermann Grassmann was right. Nature , 1 May , Sturmfels: Algorithms of Invariant Theory. Springer, Wien, The Schrodinger equation explains all atomic phenomena except those involving magnetism and relativity. The spin interacts with the magnetic field, and the electron goes up or down according as the spin is parallel or opposite to the vertical magnetic field.

Pauli introduced the spin into quantum mechanics by adding a new term into the Schrijdinger equation. Pauli replaced n2 by a'. B operates on two-component column matrices with entries in C. The wave function sends space-time points to Pauli spinors that is, it has values in the complex linear space C2. The multiplication rules of the Pauli spin matrices al, a 2 , a3 E Mat 2, C imply the matrix identity. The length of the representative of a vector 2 is preserved under a similarity transformation U. In this way, not only vectors but also rotations become represented within the matrix algebra Mat 2,C.

The idempotent is primitive and the left ideal is minimal. I 53 Orthogonal unit vectors, orthonormal basis SO 3 have representatives in Ct3, but also spinor spaces or spinor representations of the rotation group SO 3 can be constructed within the Clifford algebra Ce3. In the notation of the Clifford algebra Ce3 we could describe Pauli's achievement by saying that he replaced?

All the arguments and functions now have values in one algebra, which will facilitate numerical computations. In this chapter we shall study more closely the Clifford algebra Cea and the spin group Spin 3 , and reformulate once more the Schrodinger-Pauli equation in terms of Ct3. An arbitrary element in Ct3 is a sum of a scalar, a vector, a bivector and a volume element, and can be written as a a be Belz3, where a,,B E R and a, b E R3. Compute the product elze The spinor representations cannot be reached by tensor methods, as irreducible components of tensor products of antisymmetric powers of R3.

Mostly we shall regard this set a s a real algebra with scalar multiplication taken over the real numbers in R although the matrix entries are in the complex field C. The essential difference between the Clifford algebra CC3 and its matrix image Mat 2, C is that in the Clifford algebra CC3 we will, in its definition, distinguish a particular subspace, the vector space R3, 4.

No such distinguished subspace has been singled out in the definition of the matrix algebra Mat 2, C. Instead, we have chosen the traceless Hermitian matrices to represent R3, and thereby added extra structure to Mat 2, C. These matrices are Pauli Spin Matrices and Spinors 56 which is closed under multiplication. The even subalgebra is isomorphic to the division ring of quaternions W, as can be seen by the following correspondences: R e m a r k. The Clifford algebra Ct3 contains two subalgebras, isomorphic to [the center] and W [the even subalgebra], in such a way that [temporarily we denote these subalgebras by their isomorphic images] 1.

C13 is generated as a real algebra by and W, 3. These three observations can be expressed as 4. The correspondences ol E e l , a 2 2 ez, 03 2. We recognize that the reverse fi is represented by the Hermitian conjugate ut and the Clifford-conjugate G by the matrix u-' det u E Mat 2, R [for an invertible u]. The conjugation can be used to determine the inverse of u E Ct3, ufi 0.

The opposite of ara-l, the vector is obtained by reflecting r across the mirror perpendicular to a [reflection across the plane aelZ3]. The perpendicular component r l remains invariant under both the reflections while the two successive reflections together rotate the parallel component rll in the plane of a and b by twice the angle between a and b.

The group Spin 3 , called the spin group, is a two-fold covering group of the rotation group SO 3. This can be depicted by a sequence which is exact, that is, the image of a homomorphism coincides with the kernel of the successive homomorphism. The spin group Spin 3 is a universal cover of the rotation group S 0 3 , that is, the Lie group Spin 3 is simply connected.

The group SO 3 is doubly connected. Each rotation, la1 60 Pauli Spin Matrices and Spinors 4. We shall regard the correspondences e l N ul, e2 N u2, e3 cz u3 as an identification between Ct3 and Mat 2, C. This left ideal S of Ct3 contains no left ideal other than S itseif and the zero ideal 0. Such a left ideal is called minimal in Ct3. Thus, IF is a division ring with unity f [this follows from the idempotent f being primitive in Ce3].

As a 2dimensional real division algebra IF must be isomorphic to C. I The minimal left ideal has a natural right IF-linear structure defined by We shall provide the minimal left ideal S with this right IF-linear structure, and call it a spinor space. Employing the basis i f o , -f2 for the IF-linear space S, the elements r e l , 7 e2 ,r e3 will be represented by the matrices ul ,u2, u3. In this way the Pauli matrices are reproduced. In this case, the element does not appear in the division ring?!

Next we will replace such passive spinors by active spinor operators. It should be emphasized that not only did we get all the components of the spin vector s at one stroke, but we also got the entity s as a whole.

Clifford Algebra

The above mapping should not be confused with the 'Cartan map', see Cartan p. I Pauli Spin Matrices and Spinors 64 In operator form the Schrodinger-Pauli equation shows explicitly the quantization direction e3 of the spin. Compute the squares of i 1 e3 f 1 - es e Find all the four square roots of cos p e l 2 sin p. Find the exponentials of f5 1 - e3 e Compute ufi.

Hint: ufi is of the form z yelas, x, Y E R. Hint: compute a be '. Compute xyz l and xyz 3. Hint: use reversion. Birkhauser, Boston, MA, A Bethe, E. Cartan: The Theory of Spinors. The M. Press, Cambridge, MA, Charlier, A. Bkrard, M. Charlier, D. Fristot: Tensors and the Clifiord A! Marcel Dekker, New York, Feynman, R. Leighton, M. III, Quantum Mechanics. Hestenes: Space-Time Algebra. Gordon and Breach, New York, , , Keller, S.

Rodriguez-Romo: Multivectorial generalization of the Cartan map. Merzbacher: Quantum Mechanics. Wiley, New York, Pauli: Zur Quantenmechanik des magnetischen Elektrons. Physik 4 3 , Riesz: Clifford Numbers and Spinors. Quaternions We saw in the chapter on Complex Numbers that it is convenient to use the real algebra of complex numbers C to represent the rotation group SO 2 of the plane R2.

In this chapter we shall study rotations of the 3-dimensional space R3. The composition of spatial rotations is no longer commutative, and we need a non-commutative multiplication to represent the rotation group SO 3. This can be done within the real algebra of 3 x 3-matrices Mat 3, R , or by the real algebra of quaternions, W,invented by Hamilton. Similarly, the algebra of quaternions W may be used to represent rotations of the 3-dimensional space JR3.

It will turn out that quaternions are also convenient to represent the rotations of the Cdimensional space JR4. Note that the multiplication is by definition non-commutative. One can show that quaternion multiplication is associative.

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The generalized imaginary units will be denoted either by i , j , k or by i ,j, k. They have two different roles: they act as generators of rotations, that is, they are bivectors, or translations, that is, they are vectors. This distinction is not clear-cut since bivectors are dual to vectors in JR3 5.

The cross product satisfies two rules the latter being called the Jacobi identity; this makes R3 with the cross product a Lie algebra. In particular, the cross product is not associative, a x b x c a x b x c. The center is of course closed under multiplication. The center of the division ring W is isomorphic to the field of real numbers R. In contrast to the case of the complex field , the real axis in W is the unique: subfield which is the center of W. A rotation has three parameters in dimension 3. In other words, SO 3 and S3 are 3-dimensional manifolds. The three parameters are the angle of rotation and the two direction cosines of the axis of rotation.

Note that a a. The sense of the rotation is counter-clockwise when regarded from the arrow-head of a. A rotation has two completely orthogonal invariant planes; both the invariant planes can turn arbitrarily; this takes two parameters. Fixing a plane in R4 takes the remaining four parameters: three parameters for a unit vector in S3, plus two parameters for another orthogonal unit vector in S2,minus one parameter for rotating the pairs of such vectors in the plane.

Any real 4 x 4-matrix is a linear combination of matrices of the form LORb. Because of the non-commutativity of W it is, however, necessary to distinguish between two types of linear spaces over W, namely right linear spaces and left linear spaces. A right linear space over W consists of an additive group V and a map such that the usual distributivity and unity axioms hold and such that, for all A, p E W and x E V, xX In the matrix form the above definition means that 74 Quaternions R e m a r k.

However, many generalizations are uninteresting, the classes of functions are too small or too large. In the following we will first eliminate the uninteresting generalizations. We conclude that the set of quaternion differentiable functions reduces to a small and uninteresting set. This set is too big to be interesting. Fourth, we could consider functions which are conformal almost everywhere in R4. This already shows that the last alternative results in an interesting class of new functions, to some extent analogous to the class of holomorphic functions of a complex variable.

Historical survey Hamilton invented his quaternions in when he tried to introduce a product for vectors in R3 similar to the product of complex numbers in 6. Hamilton tried to find an algebraic system which would do for the space R3 the same thing as complex numbers do for the plane R2.

Hamilton also tried to find a generalized complex number system in three dimensions. However, no such associative hypercomplex numbers exist in three dimensions. Quaternions form a division ring which we have denoted by W in honor of Hamilton. Does an involutory automorphism of the real algebra Mat 2, W necessarily send a diagonal matrix of the form i d where a E W to a diagonal matrix? Suppose A R is a simple real associative algebra of dimension 5 4 with center R. Show that A is W or Mat 2,R. Solutions 2.

Bibliography S. Altmann: Rotations, Quaternions, and Double Groups. Oxford University Press, Oxford, Kluwer, Dordrecht, The Netherlands, Giirlebeck, W. Akademie-Verlag, Berlin, Birkhauser, Basel, Hamilton: Elements of Quaternions. Longmans Green, London, Chelsea, New York, Hankins: Sir William Rowan Hamilton. Porteous: Topological Geometry. Van Nostrand Reinhold, London, Cambridge University Press, Cambridge, Sudbery: Quaternionic Analysis.

Cambridge Philos. The purpose is to help readers to get a solid view, or as solid a view as possible, of the first dimension beyond our ability to visualize. This is an important intermediate step in scrutinizing higher dimensions. We start by reviewing regular figures in lower dimensions. We shall also call them a 3-cell, 4cell,5-cell, As p grows toward infinity, we get in the limit an co-cell, where the line is divided into line segments of equal length. As a degenerate case we get a 2-cell, which is bounded by 2 line segments in the same place.

The number of regular p-gons meeting a t a vertex is the same, say q ; it satisfies because the sum of angles of faces meeting a t a vertex cannot exceed 21r. The above inequality can also be written in the form 6. The above inequality is a consequence of the Euler formula and the equation A regular polyhedron p, q 2 3 must satisfy the foregoing inequality, and so only a few pairs p, q are possible.

There are five Platonic solids. A dihedron is bounded by two regular polygons positioned in the same place. These regular tilings are called tessellations. A vertex is regular, if a plane cuts off a regular polygon whose central normal passes through the vertex. A regular vertex A polytope is a higher-dimensional analog of a polyhedron. A polytope is regular if its faces and vertices are regular. A 4-dimensional regular polytope with p, q-cells as faces must have q, r-cells as vertices. The sum of the solid angles of the faces meeting at a vertex cannot exceed 4n. As a consequence, there remain six possible combinations of p, q and q, r.

The 3-dimensional space can be filled with cubes, a configuration with 6. A hypersphere with radius r in R4 has 3-dimensional surface 2n2r3 and 4-dimensional hypervolume ;n2r4. For lower-dimensional spheres we have the following table: n I surface I volume If the volume of the sphere in Rn is denoted by wnrn then its surface is nwnrn-I. Then the matrix eA represents a rotation of the 4-dimensional Euclidean space R4. In general, a rotation of R4 has two invariant planes which are completely orthogonal; in particular they have only one point in common.

The antisymmetric matrix A has imaginary eigenvalues, say ficr and fiP, the eigenvalues of the rotation matrix eA are unit complex numbers e f and e f " , and the invariant planes turn by angles cr and P under e A. In fact, every point of IR4 stays in some invariant plane, but not every plane of R4 is an invariant plane of e A.

If a rotation U of IR4 has rotation angles a and ,8 we shall denote it by U o , p. This equivalence class together 6. A pair of completely orthogonal planes, both with a fixed sense of rotation, induces a pair of senses of rotations for all pairs of completely orthogonal planes. R4 A rotating ball in IW3 has an axis of rotation, like the axis going through the North and South Poles, and a plane of rotation, like the plane of the equator.

A rotating ball in IK4 has two lanes of rotation, which are completely orthogonal to each other in the sense that they have only one point in common. Let the angular velocities in these planes be bivectors w l and wz. The Clifford algebra Ct4 is isomorphic to the real algebra of 2 x 2-matrices Mat 2, E-JI with quaternions as entries, 6. Instead, bivectors are 6. In the 3-dimensional space IW3 there are only simple bivectors, that is, all the bivectors represent a plane. In the 4-dimensional space R4 this is not the case any more.

An Introduction to Clifford Algebras and Spinors

I If the square of a bivector is real, then it is simple. There is an exception to this uniqueness, crucial to the study of four dimensions: If the simple components of a bivector have equal squares, that is equal norms, then the decomposition to a sum of simple components is not unique. The bivector ele2 e3e4 can also be decomposed into a sum of two completely orthogonal bivectors as follows: 6. In other words, the Lie algebras 1 Although the square of a 3-vector is real, it need not be simple. This is an interesting and non-trivial question in dimension 4.

Here follows the answer. In dimension 6, 1 Spin 6. The denominator of U is a multiple of the identity I. Summary There are three different kinds of rotations in four dimensions depending on the values of the rotation angles a , ,!? In general, a, b , c, d are linearly independent, that is, a A b A c A d 0. In the case of a simple rotation with ,f? A simple bivector multiplied by one of the idempotents i l f corresponds to an isoclinic rotation. An isoclinic rotation has an infinity of rotation planes, and in fact, each vector is in some invariant rotation plane of an isoclinic rotation.

The two-fold cover Spin 4 of SO 4 has three different subgroups isomorphic to Spin 3 , each with a Lie algebra There is an automorphism of Spin 4 which swaps the last two copies of Spin 3 , but there is no automorphism of Spin 4 swapping the first copy of Spin 3 with either of the other two copies. Compute the squares of i 1 el2 e34 f 2. Show that B a B E R4. Compute exp ae12 pe Determine i 1 e A and i 1 - e B, and show that these bivectors commute.

In two dimensions we can place 4 circles of radius r inside a square of side 4r, and put a circle of radius fi - l r in the middle of the 4 circles. In three dimensions we can place 8 spheres of radius r inside a cube of side 4r, and put a sphere of radius - l r in the middle of the 8 circles. In n dimensions we can place 2n spheres of radius r inside a hypercube of - 1 r in the middle of the 2n side 4r, and put a sphere of radius fi spheres. Let the dimension be progressively increased.

In some dimension the middle sphere actually emerges out of the hypercube. In some dimension the middle sphere becomes even bigger than the hypercube, in the sense that its volume is larger than the volume of the hypercube. Determine those dimensions. In dimension 9 the middle sphere touches the surface of the hypercube, and in dimension 10 it emerges out of the hypercube. In dimension the volume of the middle sphere is larger than the volume of the hypercube.

Coxeter: Regular Polytopes, Methuen, London, Hilbert, S. Geometry and the Imagination, Chelsea, New York, The Cross Product The cross product is useful in many physical applications. It also gives the force moving a t velocity ii in a magnetic field 2. The usefulness of the cross product in three dimensions suggests the following questions: Is there a higher-dimensional analog of the cross product of two vectors in R3? If an analog exists, is it unique? The first question is usually responded to by giving an answer to a modified question by explaining that there is a higher-dimensional analog of the cross product of n- 1 vectors in Rn.

However, such a reply not only does not answer the original question, but also gives an incomplete answer to the modified question. In this chapter we will give a complete answer to the above questions and their modifications. The '7.

An Introduction to Clifford Algebras and Spinors - Jr Vaz Jayme - Häftad () | Bokus

The symmetric bilinear scalar valued product gives rise to the quadratic form which makes the linear space R3 into a quadratic space R3. The quadratic form is positive definite, that is, a. The real linear space R3 with a positive definite quadratic form on itself is called a Euclidean space IR3. The cross product is uniquely determined by a x b.

The antisymmetry of the cross product has a geometric meaning: the lack of '7. The cross product is not associative, a x b x c a x b x c, which results in an inconvenience in computation, because parentheses cannot be omitted. The oriented volume of the 4-dimensional parallelepiped with a, b, c , d as edges is the scalar multiplied by the unit oriented volume e In a similar manner we can introduce in n dimensions a cross product of n - 1 factors. The result is a vector orthogonal to the factors, and the length of the vector is equal to the hypervolume of the parallelepiped formed by the factors.

The cross product of two vectors in R7 can be defined in terms of an orthonormal basis e l , ez,. This cross product of vectors in R7 satisfies the usual properties, that is, a x b. Unlike the 3-dimensional cross product, the 7-dimensional cross product does not satisfy the Jacobi identity, a x b x c b x c x a c x a x b 0, and so it does not form a Lie algebra. However, the 7-dimensional cross product satisfies the Malcev identity, a generalization of Jacobi, see Ebbinghaus et al.

The 3-dimensional cross product is invariant under all rotations of S 0 3 , while the 7-dimensional cross product is not invariant under all of S 0 7 , but only under the exceptional Lie group G 2 , a subgroup of SO 7. When we let a and b run through all of R7, the image set of the simple bivectors a A b is a manifold of dimension 2. So the mapping is not a one-to-one correspondence, but only a method of associating a vector to a bivector. The octonion product in turn is given by 3 This expression is also valid for a, b E R 3 C C e 3 , but the element 1 - e l 2 3 does not pick up an ideal of C e 3.

Recall that C e 3 is simple, that is, it has no proper two-sided ideals. If we were looking for a vector valued product of k factors in Rn, then we should first formalize our problem by modifying the Pythagorean theorem, a candidate being the Gram determinant. A natural thing to do is to consider a vector valued product a1 x a 2 x. The solution to this problem is that there are vector valued cross products in 3 dimensions with 2 factors 7 dimensions with 2 factors n dimensions with n - 1 factors 3 factors 8 dimensions with and no others - except if one allows degenerate solutions, when there would also be in all even dimensions n, n E 22, a vector product with only one factor and in one dimension an identically vanishing cross product with two factors.

This can be accomplished by Exercises 1. Express the rotation matrix e A in terms of I,A and A2. Express the rotated vector e A r as a linear combination of r, a x r and a. Solutions A A2 2. Harvey: Spinors and Calibrations. Academic Press, San Diego, Massey: Cross products of vectors in higher dimensional Euclidean spaces. Monthly 90 , lo, Schafer: On the algebras formed by the Cayley-Dickson process. In these advanced formalisms the Maxwell equations become more uniform and easier to manipulate; for instance, relativistic covariance is more apparent.

However, the cost of the convenience is that one has to master new concepts in addition to scalars and vectors; and antisymmetric tensors have to be untangled for physical interpretation. Ampitre had developed a mathematical formulation for producing magnetism by electricity, a phenomenon detected by 0rsted in , but his law is not valid in a time-varying situation: take the divergence of both sides to obtain ' v.

T which violates charge conservation. Maxwell corrected this equation into the - V. Hertz in , when he radiated electromagnetic waves by a dipole antenna. The electromagnetic field is now described by the Maxwell equations These equations are linear, and the last two equations with a vanishing righthand side are homogeneous. If E,p are constants, so that they do not depend on position, then the medium is homogeneous. If E , p are scalars, and not matrices or tensors, then In a medium that is uniform in space, i.

In an explosion E and p are time dependent. Electromagnetism expressed in terms of E and I? Minkowski combined the two vectors l? If this equation is substituted into Faraday's law, we get This curl-free quantity is up to a sign the gradient of a scalar, called the electric potential V, We have shown that and as follows: A E and 2?

The change of potentials is called a gauge transformation. In quantum electrodynamics gauge invariance is used t o deduce the existence of a zero-mass carrier for the electromagnetic field. Now we shall find out conditions imposed on the potentials by the remaining Maxwell equations. Substitute l? Lorentz, who demonstrated covariance of the Maxwell equations under Lorentz transformations in See 3.

Clifford algebras automatically take care of the manipulation of indices. The Clifford algebra approach allows various degrees of abstraction which gradually become more and more distant from classical vector analysis. In the Euclidean space R3 we shall deal with the vector l? B e, and you will get 0 4 v. The Clifford algebra Ce3,1 is isomorphic, as an associative algebra, with the algebra of real 4 x 4-matrices Mat 4,R.

F are 8. Hestenes p. In theoretical physics one applies differential forms to electromagnetism, but in electrical engineering one uses almost exclusively the vector analysis of Gibbs and Heaviside. In a curved space-time it is not possible to differentiate vector valued functions, only differential forms can be differentiated [in general relativity vectors are differentiated covariantly]. Similarly, we can regard I? These equations are invariant under the general linear group GL 4,R , and the solutions are independent of the choice of metric. Here E , a , p and pW1 are 3 x 3-matrices.

To find the rules imposed on them, write the above equations in coordinate form: Then, if XnXpv is an irreducible tensor, l4 we must have where the brackets [ ] mean complete alternation of indices. In chiral media the tensor x need not be irreducible, and the number of components may rise to For instance, magnetic saturation and hysteresis are not expressible with a tensor X. The electromagnetic bivectors F and G are replaced by 2-forms F and G.

The current vector J is replaced by a I-form J. The exterior differential raises the degree. In differential forms the Maxwell equations look like 8. This makes the equations independent of any coordinate system. The metric gets involved by the constitutive relations of the medium and the Hodge dual. This form of the Maxwell equations is not only relativistically covariant, 15 The exterior differential is usually denoted by d j. This general linear covariance of the Maxwell equations, and their independence of metriclmedium, were recognized by Weyl , Cartan and van Dantzig Riesz Replace l?

The conformal transformations are not linear in general, that is, they are not in GL 4,R. Electrical engineers use the pairs l? The constitutive relations sending l? Solutions la. The space-component is l? T h e time-component is tii. Bibliography W. Birkauser, Boston, Bolinder: Unified microwave network theory based on Clifford algebra in Lorentz space, pp. Bolinder: Clifford algebra: What is it? Cheng: Field and Wave Electromagnetics. Addison-Wesley, Reading, MA, , Deschamps: Electromagnetics and differential forms.

IEEE, 69 , Hestenes: Space- Time Algebra. Gordon and Breach, Philadelphia, PA, , , Jackson: Classical Electrodynamics.

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Wiley, New York, , Jancewicz: Multivectors and Clifiord Algebra in Electrodynamics. World Scientific, Singapore, Juvet, A. Schidlof: Sur les nombres hypercomplexes de Clifford et leurs applications B l'analyse vectorielle ordinaire, B 1'6lectromagnetisme de Minkowski et B la thkorie de Dirac. Lindell: Methods for Electromagnetics Field Analysis. Clarendon Press, Oxford, Lindell, A. Sihvola, S. Tretyakov, A. Artech House, Boston, MA, Lindell, P. Lounesto: Difierentiaalimuodot sahkomagnetiikassa. Helsinki University of Technology, Electromagnetics Laboratory, Lorrain, D.

Freeman, San Francisco, Mercier: Expression des e'quations de l'e'lectromagne'tisme au moyen des nombres de Clifiord.

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Thesis, Universit6 de Genkve, Post: Formal Structure of Electromagnetics. North-Holland, Amsterdam, Reprinted with comments as facsimile by E. Lounesto eds. There is no privileged inertial frame or absolute rest for moving bodies, but time is preserved, that is, time is absolute. The Galilean principle or invariance does not govern all of physics, most notably electromagnetism and in particular light propagation. The wave equation is instead invariant under another transformation, named after H.

In , Einstein took the constancy of the velocity of light as a postulate, and showed that this postulate, together with the principle of relativity, is sufficient for deriving the kinematical formulas of Lorentz. In so doing, Einstein had to revise the notion of time, and abandon the concept of absolute time. We require that y is the same in both equations since the inverse transformation should be identical to the direct one except for a change of v to -v.

In computing y we use the observation of equal velocity of light.

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Divide the two equations which gives the factor 1 Next, compute the transformation of the time coordinate of events. In particular, time and space are orthogonal. If we draw space-time coordinates x, ct on paper so that the time-axis is 'perpendicular' to the space and perform a Lorentz transformation, then the transformed coordinate-axes x', ct' are no longer 'rectangular' but they are orthogonal, by definition.

A boost leaves untouched the perpendicular component Fl, but alters the parallel component Sll. The subgroup with positive determinant, is called the special Lorentz group. The special Lorentz group S O 3 , l has two components. The Lorentz group 0 3 , 1 has four components; these form three twocomponent subgroups preserving space orientation, time orientation or spacetime orientation.

Time-orientation-preserving Lorentz transformations form the orthochronous Lorentz group Ot 3, 1. The Lorentz transformations, which stabilize a space-like vector, form a subgroup O 2, I , the small Lorentz group of R The Lorentz transformations, which stabilize a light-like vector, form a subgroup isomorphic to the group of rigid movements of the Euclidean plane R2.

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Boosts and rotations are special cases of simple Lorentz transformations. If there are two light-like eigenvectors, then they span a time-like eigenplane, which is preserved by the Lorentz transformation; there is also a space-like eigenplane, which is completely orthogonal to the time-like eigenplane. A special orthochronous Lorentz transformation is called simple, if it turns vectors only in one eigenplane, leaving the other eigenplane point-wise invariant.

An elliptic Lorentz transformation is a rotation of the Euclidean space R3, which is orthogonal to an observer, whose time-axis is orthogonal to the space-like eigenplane of the Lorentz transformation. A non-parabolic Lorentz transformation can be written as a product of two commuting simple transformations, one hyperbolic and the other elliptic.

Lorentz transformations can be described within the Clifford algebras C. For two x2y2 vectors x , y in completely orthogonal planes, the scalar product x. A special orthochronous Lorentz transformation can be uniquely decomposed into a product of a boost and a rotation, called the polar decomposition. On pp. Let the light-like eigenvectors of the corresponding Lorentz transformation be l1 and 12, and choose The bivector l1 A 12 anticommutes with a A b and c A d, that is, it is the unique 'normal' to aA b and c Ad. I Jancewicz I Hestenes The author gives on pp. Michelson carried out, for the first time, measurements intended to determine the motion of the Earth relative to an absolute, imaginary 'light medium'.

For this purpose he measured the velocity of light in different directions. Voigt in was the first to recognize that the wave equation is invariant with respect to the change of variables where also time is transformed. In l o Einstein supplemented the principle of relativity by postulating the principle of independence of the velocity of light of the motion of the source. These two principles led Einstein to a revision of the notion of time and enabled him to deduce the kinematical transformation laws of Lorentz; his predecessors had obtained the transformation laws by considering transformations which do not change the form of the Maxwell equations.

Einstein: Zur Elektrodynamik bewegter Korper. Physik 17 , In this paper Einstein compared the same phenomenon when observed in two different frames: a magnet moving near a closed conductor and a closed conductor moving near a magnet.

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In other words, the principle of constancy of the velocity of light becomes superfluous as an amendment to the principle of relativity. The principle of relativity and knowledge of the Maxwell equations are enough to deduce the transformation laws of Lorentz. Nowadays the terms 'relativistic' and 'relativity' almost invariably refer to the Einsteinian principle.