Para-Vectors, Bipara-Vectors, and Spinors

We can think of a para-vector as the linear combination of a real scalar with a real vector. The para-vector is the equivalent to the four-vector in tensor based formalisms. If we were using tensor language, the four-vector would have been defined in terms of its transformation properties. Likewise, we can define a para-vector from it's transformation properties.

For instance, I can say that an object is a para-vector without making reference to the state of the components, but merely by noting the way that it transforms. Para-vectors transform like this:

A' = LAL^†

As previously stated, a bipara-vector is the product of 2 paravectors. However, relativity dictates that any physically significant quantity is covariant, or able to maintain the same form after a lorentz transformation. Consider what will happen if we merely multiply two para-vectors together

C = AB

Now let's apply a lorentz transformation to the equation

C' = LAL^†LBL^†

Due to the L^†L factor that ends up getting sandwiched in between A and B, we see that C is not a covariant quantity - it changes it's form after the transformation. However, consider the definition

C = AB

C' = LAL^†L^† B L

C' = LAB L

C' = LCL

In the third line we use the fact that LL = 1. This allows the sandwich factor to be "disolved", and allows C' to take the same form as C. Thus, according to this definition, C is a covariant quantity. We have also derived the transformation rule for a bipara-vector. We can use this transformation law as the definition of a bipara-vector.

Note that in both cases, the para-vector transforms to a para-vector, and the bipara-vector transforms to a bipara-vector. Both transformation laws also preserve the "group" property of the transformation. In other words, if I were to perform 2 transformations in a row, the result could be equivalent to some single overall transformation.

If we must have a transformation law in order for a quantity to be physically significant, we ask: if L itself is physically significant, is it a para-vector, or a bipara-vector? In order to answer this, consider the action of two sequential transformations

A' = L2LAL^†L2^†

B' = L2LBL L2

In both of these cases L acts before L₂. This set of sequential transformations would have been equivalent to a single transformation by a composite L'.

L' = L₂L

We can consider here that L has been transformed by L₂. If we consider this to be a valid transformation law, then we see that L does not transform like a para-vector, or a bipara-vector. We call quantities that transform in this way "Spinors". Spinors transform with only a single application of the transformation like this:

S' = LS

Spinors behave like basic building blocks of relativistically significant quantities. For instance, we can use two spinors to construct a para-vector.

A = S₁S₂^†

This quantity obeys the transformation law of a para-vector.

We can also use two spinors to construct a bipara-vector

B = S₁S₂

This quantity obeys the transformation law of a bipara-vector.

To recap, here are our 3 relativistically significant quantities

S' = LS ; spinor

A' = LAL^† ; para-vector

B' = LBL ; bipara-vector