Introduction to Clifford Algebra

It is advantageous to use Clifford algebra, because it gives a unified view of things that otherwise would need to be understood separately:

• The real numbers are a subalgebra of Clifford algebra: just throw away all elements with grade > 0. Alas this doesn’t tell us much beyond what we already knew.
• Ordinary vector algebra is another subalgebra of Clifford algebra. Alas, again, this doesn’t tell us much beyond what we already knew.
• The complex numbers are another subalgebra of Clifford algebra, as discussed in. This gives useful insight into complex numbers and into rotations in two dimensions.
• Quaternions can be understood in terms of another subalgebra of Clifford algebra, namely the subalgebra containing just scalars and bivectors. This is tremendously useful for describing rotations in three or more dimensions (including four-dimensional spacetime). See. Note that the Pauli spin matrices are isomorphic to quaternions.

There is a cut-and-dried procedure for replacing cross products with the corresponding wedge products, as discussed in

. It is advantages to make the change, because the wedge product is more powerful and more well-behaved:

The cross product only makes sense in three dimensions.

The wedge product is well behaved in any number of dimensions, from zero on up.

The cross product is defined in terms of a “right hand rule”.

A wedge product is defined without any notion of handedness, without any notion of chirality. This is discussed in more detail in. This is more important than it might seem, because it changes how we perceive the apparent symmetry of the laws of physics, as discussed in.

The cross product only applies when multiplying one vector by another.

The wedge product can multiply any combination of scalars, vectors, or higher-grade objects.

The wedge product of two vectors is antisymmetric, and involves the sine of the angle between two vectors ... and the same can be said of the cross product.

In all of physics, whenever you see an idea expressed as the cross product of vectors, you will usually be much better off if you re-express the idea in terms of a wedge product. Help stamp out cross products!

• It is traditional to write down four Maxwell equations. However, by using Clifford algebra, we can express the same meaning in just one very compact, elegant equation:

  ∇ F =

  1 c є 0

  J ()

  It is worth learning Clifford algebra just to see this equation. For details, see.

  Also: In their traditional form, the Maxwell equations seem to be not left/right symmetric, because they involve cross products. However, we believe that classical electromagnetism does have a left/right symmetry. By rewriting the laws using geometric products, as in, it becomes obvious that no right-hand rule is needed. A particularly pronounced example of this is Pierre’s puzzle , as discussed in.

  Similarly: The traditional form of the Maxwell equations is not manifestly invariant with respect to special relativity, because it involves a particular observer’s time and space coordinates. However, we believe the underlying physical laws are relativistically invariant. Rewriting the laws using geometric products makes this invariance manifest, as in.

  As an elegant application of the basic idea that the electromagnetic field is a bivector,explains why a field that is purely an electric field in one reference frame must be a combination of electric and magnetic fields when observed in another frame.

  As a more mathematical application of,calculates the field surrounding a long straight wire.

• The ideas of torque, angular momentum, and gyroscopic precession are particularly easy to understand when expressed in terms of bivectors, as mentioned in. See also.
• You can calculate volume using wedge products, as discussed in. This is much preferable to the so-called triple scalar product ( A · B × C ).

Help Stamp Out Cross Products

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