Exterior algebra of space

1 of 7 Blog post 22-10-24 In this post we will construct the Grassmann (or exterior) algebra of a 3D real vector space. It will be eightdimensional. Its basis will have eight elements. The basis of Λ(V) has eight elements With this post we will start a new chapter of our spin chronicles with another approach - via Clifford algebra the Clifford algebra of space. What is space? For us space will be a three-dimensional affine Euclidean space, let's call it M. The fact that it is affine, means that there is a 3-dimensional real vector space, let us call V (this is not the same V as in previous posts, this is the V of the new chapter), and we can translate any point x of M by a vector a in V, to make another point x+a. The fact that M is Euclidean means that in V we have a positive definite scalar product that we will denote aꞏb. In the following we will deal exclusively with V, so we will use also letters x,y etc. for vectors in V. And to ease the notation we will write x,y, ... instead of x,y,.... In V we will restrict ourselves to orthonormal bases ei, that is we will require eiꞏej = δij. Any two such bases are related by a unique orthogonal matrix, and element of the group O(3) of 3x3 matrices R such that RT R = R RT = I: e'i = ej Rji. Note: I will be using lower indices to number vector components, and upper indices to number basis vectors. Every orthonormal matrix R has determinant +1 or -1. Base connected by R of determinant 1 are said to be of the same parity, those connected by R of determinant -1 are said to be of opposite parity. In the following we will restrict ourselves to one parity, which, by convention, we will call positive. This restricts transformation between bases to the subgroup SO(3) of O(3) - special orthogonal group in three (real) dimensions, consisting of orthogonal matrices R with det(R) = 1. But all this will be relevant only in the next post. To define Λ(V) we do not need any scalar product in V. We need only its vector space structure. So Grassmann algebra is a pre-metric construction. We consider the space of multi-vectors. They will form the Grassmann (or "exterior") algebra Λ(V). There will be scalars (these form the sub-algebra, essentially the one-dimensional algebra of real numbers R), 22/10/2024, 19:30 clifford algebra 1 2 of 7 file:///D:/Downloads/clifford algebra 1.html vectors (they make our V), bi-vectors, and three-vectors. No more. Sometimes we may use the term "rank". So, scalars are of rank 0, vectors of rank 1, bi-vectors of rank 2, and three-vectors, of course, of rank 3. No higher rank multi-vectors for three-dimensional V. We know what are vectors, let us introduce bi-vectors. In any basis of V, a vector v is represented by its components vi. A bi-vector f is represented by an anti-symmetric matrix fij = -fji. Similarly a three-vector is represented by a totally anti-symmetric matrix fijk = - fjik =.-fikj. Since i,j,k can only take values 1,2,3, every 3-vector is of the form fijk = c εijk, where c is a real number, and εijk is the totally anti-symmetric Levi-Civita tensor taking values 0,1,-1, with ε123 = 1. It will be convenient to use Kronecker delta symbols. One of them, δij, is well known. Then we have (writing on a web page lower indices directly under upper indices is too complicated for me, so my formulas look differently than in a "real" math text) δklij = δikδjl - δil,δjk, and δlmn ijk = δil δmnjk - δim δlnjk + δin δlmjk. It is easy to see the pattern. We can verify that the following identities hold for contractions over repeated indices: δlmn123 = εlmn, δ123lmn = εlmn, δimnijk = δmnjk , (1/2!) δijnijk = δkn . δklij and δlmnijk are, by construction, anti-symmetric with respect to lower indices, and also with respect to upper indices. They are equal to +1 if lower indices are an even permutation of upper indices, -1 for odd permutations, and zero otherwise. We will use them to define exterior product of multi-vectors. We will denote it by "∧". Multiplication by scalars is the normal one: we just multiply any multi-vector by the real number, from the left or from the right - it is the same Multiplication by vectors is defined as follows. If v,w are two vectors, then v∧w is a bi-vector with components: 22/10/2024, 19:30 clifford algebra 1 3 of 7 file:///D:/Downloads/clifford algebra 1.html (v∧w)ij = δijkl vkwl = viwj - vjwi. Notice that v∧w = - w∧v. In particular v∧v = 0 for every vector v. Multiplication of vectors with bi-vectors is defined in a similar way. If v is a vector and f is a bi-vector, then v∧vis a three-vector with components (v∧f)ijk = δijkklm vkflm. Similarly from the right (f∧w)ijk = δijkklm fklvlm. Since δijkklm = δijkmkl , we have v∧f = f∧v. Thus vectors commute with bi-vectors. Exercise 1: Show that for any bi-vector f we have: (1/2!) δijkl fkl = fij, and for any three-vector f we have: (1/3!) δijklmn flmn = fijk. Exercise 2: Show that if f is a three-vector, then it commutes with every element of the algebra. Finally a three-vector multiplied by a vector or two-vector gives 0. This way we have defined the algebra of multi-vectors Λ(V), known also as the Grassmann algebra of V. One can verify that the product is associative, The unit element of this algebra is the scalar 1. As a vector space Λ(V) is of dimension 1+3+3+1 = 8 = 23. This happens to be twice the dimension of the algebra of quaternions. Later on we will see that there are reasons for it. If ei is an arbitrary basis of V, we introduce the multi-vectors of rank 0, 1,2,3 respectively, defined by their components: The components of these basic multi-vectors can be written as rank 0: scalar 1 rank 1: (ei)j = δij, rank 2: (eij)kl = (1/2!) δijkl ek∧el rank 3: (eijk)lmn = (1/3!) δijklmn el∧em∧en. The basis of Λ(V) consists then of 8 elements: 22/10/2024, 19:30 clifford algebra 1 4 of 7 file:///D:/Downloads/clifford algebra 1.html 1, e1, e2, e3, e12 = e1∧ e2, e23 = e2∧ e3, e31 = e3∧e1 e123 = e1∧e2∧e3. We do not need, for example, e21, because e21 = - e12. Similarly, we do not need, for example, e231, since e231 = e123. We can easily figure out the multiplication table of these eight basic vectors, for instance e1∧ e23 = e123, e12 ∧ e1 = 0, e23 ∧ e1 = e123, etc. We can also check that the product is associative, for instance (e1∧ e2)∧ e3 = e1 ∧ (e2∧ e3). Every bi-vector f is then: f = Σi<j fij eij, and every three-vector f is then: f = Σi<j<k fijk eijk. Note: In mathematics Grassmann algebra is defined in a different way, without indices, as a quotient of the infinite dimensional tensor algebra by an infinite dimensional ideal, to end up with a finite-dimensional space. It has its advantages. Here I have chosen a computer-friendly, constructive approach. Note: Ultimately we will need a separate Grassmann (and Clifford) algebra at each point of our space (a field of algebras). This will lead us to infinite number of dimensions of the field. But let us deal with just one point at a time. Infinity of the exterior algebra field 22/10/2024, 19:30 clifford algebra 1 5 of 7 file:///D:/Downloads/clifford algebra 1.html In the next post we will deform the product in Λ(V) to obtain a new algebra structure on the same space - that will be the Clifford geometric algebra Cl(V) of V. So far, defining the multiplication, we never used the scalar product aꞏb of V . That will change with the geometric algebra product.The scalar product will be used in the formula defining the deformation. P.S. Here are two relevant pages from Справочник По Математике Корн Г, Корн Т 1974 22/10/2024, 19:30 clifford algebra 1 6 of 7 file:///D:/Downloads/clifford algebra 1.html P.S. Reading "Unreal Probabilities: Partial Truth with Clifford Numbers" by Carlos. C. Rodriguez. There, at the beginning: "The main motivation for this article has come from realizing that the derivations in Cox [4] still apply if real numbers are replaced by complex numbers as the encoders of partial truth. This was first mentioned by Youssef [12] and checked in more detail by Caticha [2] who also showed that non-relativistic Quantum theory, as formulated by Feynman [5], is the only consistent calculus of probability amplitudes. By measuring propositions with Clifford numbers we automatically include the reals, complex, quaternions, spinors and any combination of them (among others) as special cases." 22/10/2024, 19:30 clifford algebra 1 7 of 7 file:///D:/Downloads/clifford algebra 1.html And at the end: "Comments and conclusion What the hell is this all about and what it may be likely to become...." My answer: It is about fields of Clifford algebras and Clifford algebra-valued "measures". Reading . "Intelligent machines in the twenty-first century: foundations of inference and inquiry" by Kevin H. Knuth. There "Complex numbers and quaternions also conform to Cox’s consistency requirements (Youssef 1994; also S. Youssef (2001), unpublished work), as do the more general Clifford algebras (Rodrıguez 1998), which are multivectors in the geometric algebra (Hestenes & Sobczyk 1984) described in Lasenby et al . (2000). Furthermore, Caticha (1998) has derived the calculus of wave function amplitudes and the Schrodinger equation entirely by constructing a poset of experimental set-ups and using the consistency requirements with degrees of inclusion represented with complex numbers. This leads to a very satisfying description of quantum mechanics in terms of measurements, which explains how it looks like probability theory—yet is not. We expect that the generalizations of lattice theory described here will not only identify unrecognized relationships among disparate fields, but also allow new measures to be developed and understood at a very fundamental level." A partial truth value can be a multi-vector! 22/10/2024, 19:30