A Type Of Wick Rotation Using Geometric Algebra

[Aethelwulf]

Knowing

 

[math]\Delta = \gamma_{\mu} \partial^{\mu}[/math] 

 

One can introduce this next equation as the square root of the spacetime interval

 

[math]i \gamma^{\mu} \Delta \gamma_{\mu} R = 1[/math] 

 

Repeating indices cancel out

 

[math]i \gamma^{\mu} \gamma_{\mu} \partial^{\mu} \gamma_{\mu} R = 1[/math] [3]

 

so what we really have is

 

[math]i \gamma^{\mu}\partial^{\mu} R = 1[/math]

 

Which is the square root of the space time interval.

 

In an antisymmetric product, the spin matrices is given as

 

[math]2 \sigma_{\mu \nu} = \gamma_{\mu} \gamma_{\nu} - \gamma_{\nu} \gamma_{\mu}[/math]

 

This is redundant view in a geometric algebraic-sense, and... Hestenes of course mentions this in his own work.

 

The unit two form is

 

[math]dx \wedge dy[/math]

 

This term is contracted under the sigma, so it already implies the term [math]\sigma_{\mu \nu}[/math]. The reason lies within making sure for instance, [math]\sigma_{\mu} \sigma_{\mu}[/math] is [math]0[/math] and not [math]1[/math]. This means it has a property which allows it to square to a negative [math]-1[/math], indeed, applying this rule we find a rotation in the imaginary axis

 

[math]i^2 \gamma^{\mu} \partial^{\mu} R = -1[/math]

 

It's almost like a special case of a wick rotation.