Velocity, density, ect. : air : molecules spinor field :?:?

[Spinnor]

Consider a compressible fluid such as air. Assume we can neglect viscosity. We might describe such a fluid at some small region with a set of numbers. Three numbers would give the components of the velocity vector of the air at that small region and two more numbers would give the density and temperature of the air in that same small region.

Now suppose we have continuous functions of position and time that give the velocity, density, and temperature of air in some large region of interest. If we evaluate these functions at a "point" we must be clear that these functions only make sense if the "point" is in fact a region that is macroscopically small but large in the sense that the region contains many molecules. So we have continuous vector and scalar fields that describe the state of air which on a large scale can be thought of as a continuous compressible substance when in fact air is made up of numerous particles.

In a similar manner can one envision a multitude of discrete "things" (points, lines, or surfaces ect. with extra properties as needed to solve the problem) such that a very small region (say 10^-60 m^3) would contain many of these "things" so that for all practical purposes one would have a continuous field made up of discrete things that sit in spacetime (or are spacetime?) that would be properly described by a spinor field? Can we "build" some "structure" that sits in spacetime and we can visualize that is properly described by spinors?

If there is a small compact extra dimension, does this help solve my problem?

Thank you for any thoughts.

 

[Spinnor]

For a solution to the Dirac equation we need four complex numbers at each point in spacetime?

We need some structure that requires four complex numbers at each small volume of space?

Thank you for any pointers.

 

[Spinnor]

I wrote:

...For a solution to the Dirac equation we need four complex numbers at each point in spacetime?...

Are all four complex numbers independent, that is can we reduce the amount of information needed at each point to describe a solution of the Dirac equation?

Sorry the multiposting here and thank you for any help.

 

[Spinnor]

Maybe this construction will work?

At some point I wanted to think about the 3 dimensional version of the anchored string. See:

https://www.physicsforums.com/showthread.php?t=304257

It is not hard to imagine the 3 dimensional analog of the anchored string. Let us imagine a 3 dimensional solid under "isotropic tension" in the space S^3 let this space have a large radius, R. Let points in the solid not move in the ordinary 3 spatial dimensions of S^3. Just as a point on a string moves in some tangential space we imagine that at each point in our large space S^3 their sits another hidden space, let it again be the space S^3 but with a small radius, r. There is a relationship between SU(2) and S^3 which may solve my problem.

Let us assume for starters that each point of our tensioned solid in our large space S^3 has the same coordinates in our small hidden space S^3, there are no waves.

Now "grab" a single point P of our solid and give it a quick shake, remember movement is allowed only in our small space S^3. Move the point quite quickly along some path in our small hidden space S^3 that returns where it started. We will produce a wave that moves outward from P.

There may be two types of path which might give rise to different waves. A circular path in our small hidden space S^3 whose radius is much smaller then r and the "straight" path which comes back to where it started in our small hidden space S^3?

We describe the "configuration" of each point of our tensioned solid with coordinates in our small space S^3 which can also be done with spinors?

Thank you for any thoughts.