# Spinors in Classical Mechanics

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I asked this on stack exchange but got no response, perhaps it's better suited here.

I'm trying to deepen my understanding of spinors by looking at applications in simple problems, preferably unrelated to quantum mechanics. For this purpose I'd like to refrain from discussing definitions, generalizations, and all manner of group theory jargon as much as possible, unless there is a demonstration with a good example.

To motivate discussion and give a flavor of what I'm looking for, I want to consider two problems in "spinor" coordinates.

Consider $\mathbb{R}^2$ with $x$ and $y$ as Cartesian coordinates. We can define a complex number $\xi$ as:

$$\xi=(x^2+y^2)^{1/4}e^{i\phi/2}$$

Where:

$$\phi=\tan^{-1}\left(\frac{y}{x}\right)$$

And see that it's essentially just a change of coordinates. We can go back to $x$ and $y$ by using:

$$x=\frac{1}{2}\left(\xi^2+{\xi^*}^2\right)$$

$$y=\frac{1}{2i}\left(\xi^2-{\xi^*}^2\right)$$

Which has the funny property that both $\xi$ and $-\xi$ map to the same point of $x$ and $y$. For the arclength of a line, we know that:

$$s=\int\text{d}t\sqrt{\dot{x}^2+\dot{y}^2}$$

In "spinor" coordinates this becomes:

$$s=2\int\text{d}t\sqrt{\xi\dot{\xi}\xi^*\dot{\xi^*}}$$

It's unclear to me whether writing this offers any advantages or interesting properties. Is there any? What if I want to make local transformations to the curve that keep it's total arclength invariant?

We also know a particle moving in free space is given by:

$$\ddot{x}=0$$
$$\implies x(t)=x(0)+\dot{x}(0)t$$

And likewise for $y$.

However, converting to spinor coordinates in the Lagrangian and solving for equations of motion yield:

$$\ddot{\xi}+\frac{\dot{\xi}^2}{\xi}=0$$

Which is solved by:

$$\xi(t)=\sqrt{\xi(0)^2+2\xi(0)\dot{\xi}(0)t}$$

And likewise for the conjugate.

Is there any insight to be gained from this? In polar coordinates certain problems become much simpler, so I think coordinate transformations in general can lead to insights. Is there any advantages of applications of spinor coordinates in classical mechanics? Any literature you can point me to that explores this further?

asked Sep 27, 2018 in math-ph

Do you call $\sqrt{z}$ a spinor (where $z=x+\text{i}y$)?

Yes, it's related to $z$ in that way, so I suppose someone with good knowledge of holomorphic functions or complex analysis could have something interesting to contribute.  I put "spinor" in quotes above because I think the more correct terminology is to call $\chi^T=(\xi, \xi^*)$ a spinor, since it has the canonical relation to the coordinates through the Pauli matrices $x=\frac{1}{2}\chi^{\dagger}\sigma^x\chi$ and $y=\frac{1}{2}\chi^{\dagger}\sigma^y\chi$.

It can be written simpler: $x=(z+z^*)/2$ and $y=(z-z^*)/2\text{i}$.

I am aware, though I'm looking for a situation where spinors will be somehow advantageous.

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