CHAPTER 1
Multivectors and Multiforms

1.1 Vectors and One-Forms

Let us consider two four-dimensional (4D) linear spaces, that of vectors, and that of one-forms . The elements of are most generally denoted by boldface lowercase Latin letters,

while the elements of are most generally denoted by boldface lowercase Greek letters

The space of scalars is denoted by or and its elements are in general represented by nonboldface Latin or Greek letters a, b, c, ., a, ß, ?, ..

Exceptions are made for quantities with established conventional notation. For example, the electric and magnetic fields are one-forms which are respectively denoted by the boldface uppercase Latin letters E and H.

1.1.1 Bar Product |

The product of a vector a and a one-form yielding a scalar is denoted by the "bar" sign | as . The product is assumed symmetric,

Because of the sign, it will be called as the bar product. In the past it has also been known as the duality product or the inner product. The bar product should not be confused with the dot product. The dot product can be defined for two vectors as a · b or two one-forms as and it depends on a particular metric dyadic as will be discussed later.

1.1.2 Basis Expansions

A set of four vectors, e1, e2, e3, e4, is called a basis if any vector a can be expressed as

in terms of some scalars ai. Similarly, any one-form can be expanded in a basis of one-forms, as

The expansion of the bar product yields

The vector and one-form bases are called reciprocal to one another if they satisfy

with

In this case the scalar coeffients in (1.4) and (1.5) satisfy

and the bar product can be expanded as

From here onwards we always assume that when the two bases are denoted by ei and , they are reciprocal.

Vectors can be visualized as yardsticks in the 4D spacetime, and they can be used for measuring one-forms. For example, measuring the electric field one-form by a vector a yields the voltage U between the endpoints of the vector

provided E is constant in space or a is small in terms of wavelength.

The bar product is a bilinear function of a and . Thus, can be conceived as a linear scalar-valued function of for a given vector a. Conversely, any linear scalar-valued function can be expressed as a bar product in terms of some vector a. To prove this, we express in a basis and apply linearity, whence we have

in terms of the reciprocal vector basis {ei}. Thus, the vector a can be defined as

1.2 Bivectors and Two-Forms

1.2.1 Wedge Product ?

The antisymmetric wedge product ? between two vectors a and b yields a bivector, an element of the space of bivectors,

This implies

for any vector a. In general, bivectors are denoted by boldface uppercase Latin letters,

and they can be represented by a sum of wedge products of vectors,

Similarly, the wedge product of two one-forms and produces a two-form

Two-forms are denoted by boldface uppercase Greek letters whenever it appears possible,

and they are linear combinations of wedge products of one-forms,

A bivector which can be expressed as a wedge product of two vectors, in the form

is called a simple bivector. Similarly, two-forms of the special form

are called simple two-forms.

For the 4D vector space as considered here, the bivectors form a space of six dimensions as will be seen below. It is not possible to express the general bivector in the form of a simple bivector.

1.2.2 Basis Expansions

Expanding vectors in a vector basis {ei} induces a basis expansion of bivectors where the basis bivectors can be denoted by eij = ei?ej. Because eii = 0 and six of the remaining twelve bivectors are linearly dependent of the other six,

the space of bivectors is six dimensional. Actually, the bivector basis need not be based on any vector basis. Any set of six linearly independent bivectors could do.

A bivector can be expanded in the bivector basis as

Here, J = ij is a bi-index containing two indices i, j taken in a suitable order. In the following we will apply the order

Similarly, a basis of two-forms can be built upon the basis of one-forms as .

It helps in memorizing if we assume that the index 4 corresponds to the temporal basis element and 1, 2, 3 to the three spatial elements. In this case the spatial indices appear in cyclical order 1 2 3 1 in J while the index 4 occupies the last position.

It is useful to define an operation K(J) yielding the complementary bi-index of a given bi-index J as

Obviously, the complementary index operation satisfies

The basis expansion (1.24) can be used to show that any bivector can be expressed as a sum of two simple bivectors, in the form

Such a representation is not unique. As an example, assuming in (1.24), we can write

Thus, any bivector can be expressed in the form

where the vectors ai are spatial, that is, they satisfy . a1?a2 is called the spatial part of A and a3?e4 its temporal part. Similar rules are valid for two-forms. In particular, any two-form can be expanded in terms of spatial and temporal one-forms as

1.2.3 Bar Product

We can extend the definition of the bar product of a vector and a one-form to that of a bivector and a two-form, . Starting from a simple bivector a?b and a simple one-form the bar product is a quadrilinear scalar function of the two vectors and two one-forms and it can be expressed in terms of the four possible bar products of vectors and one-forms as

Such an expansion follows directly from the antisymmetry of the wedge product and assuming orthogonality of the basis bivectors and two-forms as

by assuming ordered indices. Equation (1.33) can be memorized from the corresponding rule for three-dimensional (3D) Gibbsian vectors denoted by ,

Relations of multivectors and multiforms to Gibbsian vectors are summarized in Appendix B.

As examples of spatial two-forms we may consider the electric and magnetic flux densities, for which we use the established symbols D and B. Bivectors can be visualized as surface regions with orientation (sense of rotation). They can be used to measure the flux of a two-form through the surface region. For example, the magnetic flux F (a scalar) of the magnetic spatial two-form B through the bivector a?b is obtained as

For more details on geometric interpretation of multiforms see, for example, [3, 4].

1.2.4 Contraction Products ? and ?

Considering a bivector a?b and a two-form , the bar product can be conceived as a linear scalar-valued function of the vector a. Thus, there must exist a one-form in terms of which we can express

Obviously, the one-form is a linear function of both b and so that we can express it as a product of the vector b and the two-form and denote it either

or

The operation denoted by the multiplication sign ? or ? will be called contraction, because the two-form is contracted ("shortened") by the vector b from the left or from the right to yield a one-form. Thus, the contraction product obeys the simple rules

Contraction of a bivector A by a one-form can be defined similarly. Applied to...