Imaginary Numbers, Spatial Inversions, and Duals

Here is the situation so far. We have an 8 dimensional algebra which has two conjugate operations. The conjugate operations allow us to split any element of the algebra into 4 sub categories:

• Real Scalar
• Imaginary Scalar
• Real Vector
• Imaginary Vector

We have already determined the physical significance of the scalar vs. vector categories - we use this to encode the space/time split. Can we determine a physical significance for the real and imaginary split as well?

These categories are defined by their behavior when one of the two conjugates are applied. For instance, objects in the scalar category are unaffected by the algebraic conjugate, and objects in the real category are unaffected by the hermitian conjugate.

How do these categories behave when both conjugates are applied at the same time?

• Real Scalar - unchanged
• Imaginary Scalar - flips sign
• Real Vector - flips sign
• Imaginary Vector - unchanged

This behavior can be explained if we state that both conjugates applied simultaneously has the physical meaning of spatial inversion. We should already know that there are 4 different quantities that behave differently under spatial inversion.

• Scalars - unchanged under spatial inversion
• Pseudo-Scalars - flips sign under spatial inversion
• Vectors - flips sign under spatial inversion
• Pseudo-Vectors - unchanged under spatial inversion

If we assign the physical meaning of spatial inversion to the action of both conjugates applied at the same time, then we can determine an actual physical meaning for the imaginary portion of the algebra - namely, purely imaginary quantities are pseudo-quantities.

For instance, we might represent both time and volume with a real number. However, we would expect volume to change signs upon spatial inversion, whereas time should not.

Likewise, the Electric field should change signs upon a spatial inversion, but the Magnetic field should not.

We have known about pseudo-quantities for a very long time. However, it is the usual practice to place these pseudo-quantities in the same 4 dimensional space as the non-pseudo-quantities, with the stipulation that "you can't add vectors with pseudo-vectors". In other words, a physics equation cannot contain both vector and pseudo-vector pieces.

The reason for this is because these pieces are linearly independent. The introduction of the imaginary unit in the algebra helps us distinguish pseudo-quantities, and also enforces the linear Independence of these quantities.

In 3 dimensions, the primary example of a pseudo-vector is the cross product

C = A × B

the example of a pseudo-scalar is the triple product

D = A ∙ (B × C)

With this identification of the physical meaning behind the real/imaginary split, we can safely state that the algebra represents an 8 dimensional space, without needing to postulate any wierd parallel universes. Rather we merely provide 4 dimensions for normal quantities, and 4 linearly independent dimensions for pseudo-quantities.

So if we are using an 8 dimensional space, how come we are using a 4 dimensional metric? The answer is that the "pseudo" half, or imaginary half, of the space uses a metric that is implied, and which can be determined. The pseudo metric is the same as the non-pseudo metric, except for an overall sign change. Remember that we chose the (1, -1, -1, -1) metric. The pseudo metric is then (-1, 1, 1, 1).

Most texts on relativity state that the overall sign of the metric is unimportant. We see here that the sign of the metric distinguishes the pseudo space from the non-pseudo space. So would it have made a difference if we had chosen the (-1, 1, 1, 1) metric to begin with? The answer is no. I might still choose the time component to correspond with the scalar basis element. I would have to choose i as the scalar basis element. I would still be able to derive the rest of the algebra. The only difference would be the assignment of i to pseudo quantities.

In other words, the choice of metric is related to a duality principle, which has much more physical significance than a mere sign convention. In fact, the duality principle that allows us to have a choice of metrics is the same duality principle which allows us to compose a dual electromagnetic field.

In order to find the dual representation of any algebraic element, we need merely multiply by -i. This operation is equivalent to the Hodge dual used in other representations. The only problem with this dual operator is that it doesn't exactly undo itself. Meaning, two applications of -i results in an overall sign change.

I personally like to use the following for a type of dual operator, since it un-does itself nicely:

A' = iA^†
A'' = iA'^† = i(iA^†)^† = A

This duality is a very profound part of nature, or at least in our representation of nature. It seems like an easy thing to overlook, and therefore it probably contains a bounty of rich truths - hidden in plain sight.