The Spin Chronicles (Part 5): Exterior algebra of space

In this post we will construct the Grassmann (or exterior) algebra of a 3D real vector space. It will be eight-dimensional. 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 a·b. 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 eⁱ·e^j = δ^ij. Any two such bases are related by a unique orthogonal matrix, and element of the group O(3) of 3x3 matrices R such that R^T R = R R^T = I:

e'ⁱ = e^j R_jⁱ.

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.

The Grassmann (or exterior) algebra Λ(V)

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), vectors (they make our V), bi-vectors, and three-vectors. No more. Sometimes we may use the term "rank" od "degree". So, scalars are of rank (or degree) 0, vectors of rank 1, bi-vectors of rank 2, and three-vectors, of course, of rank 3. All multi-vectors of rank higher than 3 are automatically zero for  three-dimensional V. Therefore we do not include them in Λ(V).

We know what are vectors, let us introduce bi-vectors (or two-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  f_ij = -f_ji. Similarly a three-vector is represented by a totally anti-symmetric matrix f_ijk = - f_jik =.-f_ikj. Since i,j,k can only take values 1,2,3, every 3-vector is of the form

f_ijk = c ε_ijk,

where c is a real number, and ε_ijk is the totally anti-symmetric Levi-Civita tensor taking values 0,1,-1, with ε₁₂₃ = 1.

Kronecker deltas

It will be convenient to use Kronecker delta symbols. One of them, δⁱ_j, 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)

δ_kl^ij = δⁱ_kδ^j_l - δⁱ_l,δ^j_k,

and

δ_lmn ^ijk = δⁱ_l δ_mn^jk - δⁱ_m δ_ln^jk + δⁱ_n δ_lm^jk.

It is easy to see the pattern. We can verify that the following identities hold for contractions (summation) over repeated indices (a convention we will always use):

δ_lmn¹²³ = ε_lmn, δ₁₂₃^lmn = ε^lmn,

δ_imn^ijk = δ_mn^jk ,

(1/2!) δ_ijn^ijk =  δ^k_n .

δ_kl^ij  and δ_lmn^ijk  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.

Exterior product

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:

(v∧w)_ij = δ_ij^kl v_kw_l = v_iw_j - v_jw_i.

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∧f is a three-vector with components

(v∧f)_ijk = δ_ijk^klm v_kf_lm.

Similarly from the right

(f∧w)_ijk = δ_ijk^klm f_klv_m.

Since δ_ijk^klm = δ_ijk^mkl ,

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!) δ_ij^kl f_kl =  f_ij,

and for any three-vector f we have:

(1/3!) δ_ijk^lmn f_lmn =  f_ijk.

Exercise 2: Show that if f is a three-vector, then it commutes with every element of the algebra.

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Finally a three-vector multiplied by a vector or bi-vector gives 0. The same for the product of two bi-vectors. 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 = δⁱ_j,

rank 2: (e^ij)_kl = (1/2!) δ^ij_kl e^k∧e^l

rank 3: (e^ijk)_lmn = (1/3!) δ^ijk_lmn e^l∧e^m∧eⁿ.

The basis of Λ(V) consists then of 8 elements:

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  = Σ_i<j  f_ij e^ij,

and every three-vector f is then:

f  = Σ_i<j<k  f_ijk e^ijk.

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.

The future

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 a·b of V . That will change with the geometric algebra product.The scalar product will be used in the formula defining the deformation.

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

P.S. Here are two relevant and useful pages dealing with Kronecker deltas from

Справочник По Математике Корн Г, Корн Т 1974

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 propo-

sitions with Clifford numbers we automatically include the reals, complex,

quaternions, spinors and any combination of them (among others) as special

cases."

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!