Grassmann Algebra

ExploringCliffordAlgebra.nb 36
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12.13 Clifford Algebra
Generating Clifford algebras
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Up to this point we have concentrated on the definition and properties of the Clifford product.
We are now able to turn our attention to the algebras that such a product is capable of generating
in different spaces.
Broadly speaking, an algebra can be constructed from a set of elements of a linear space which
has a product operation, provided all products of the elements are again elements of the set.
In what follows we shall discuss Clifford algebras. The generating elements will be selected
subsets of the basis elements of the full Grassmann algebra on the space. The product operation
will be the Clifford product which we have defined in the first part of this chapter in terms of the
generalized Grassmann product. Thus, Clifford algebras may viewed as living very much within
the Grassmann algebra, relying on it for both its elements and its operations.
In this development therefore, a particular Clifford algebra is ultimately defined by the values of
the scalar products of basis vectors of the underlying Grassmann algebra, and thus by the metric
on the space.
In many cases we will be defining the specific Clifford algebras in terms of orthogonal (not
necessarily orthonormal) basis elements.
Clifford algebras include the real numbers, complex numbers, quaternions, biquaternions, and
the Pauli and Dirac algebras.
Real algebra
All the products that we have developed up to this stage, the exterior, interior, generalized
Grassmann and Clifford products possess the valuable and consistent property that when applied
to scalars yield results equivalent to the usual (underlying field) product. The field has been
identified with a space of zero dimensions, and the scalars identified with its elements.
Thus, if a and b are scalars, then:
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