The spin of an entity, an additional (4th/5th) dimension?

[Matth.ew]

4rth/5th coordinate?*

Hi, I'm just wondering: if there is an entity in space, which can be located with Cartesian coordinates in the 3 dimensions of say, 2 right (from a seemingly arbitrary reference point), 2 up, and 2 forwards, then at this point (2, 2, 2), an entity could also be spinning at this point (2, 2, 2), that moved from point (0, 0, 0) and so this spinning movement at the point itself (2, 2, 2) would surely need to be described?

I was wondering what is the conventional way, presumably there is, for describing this movement beyond the typical 3 dimensional Cartesian coordinates and if so, do people consider it to be at a level beyond these 3 coordinates? (Is there a macroscopic and microscopic level, or something that is used?). For even with the movement that is equal to 2x + 2y + 2z, there is additional movement, there is an additional path.

 

[ZapperZ]

If it is located at (0,0,0), do you have a problem in describing its dynamics?

Zz.

 

[ogg]

There are two types (or perhaps "sources" would be a better term) of angular momentum: spin angular momentum and orbital angular momentum (this discrimination is generally only relevant for very small (ie atomic or smaller) systems).

Orbital angular momentum is the intuitive analog in quantum mechanics (QM) of macroscopic angular momentum (ie spinning momentum), so it is easier to understand than the intrinsic property of QM spin. So, you have to be quite careful when using the word "spin" - it might not be interpreted as YOU think it will be.

Elementary particles (the subject of QM) have intrinsic spin which is not related to their motion in space. Their spin is quantized, meaning it can only take on discrete values. An electron, for instance, has both orbital a.m. and spin a.m. when it is in an orbital around an atom. And you are right that to fully characterize the electron both values need to be known, as well as the atom's location in space-time.

Because of the wave-particle dual nature of a (captured) electron, it can't be characterized as having a point-like location, but is a "smeared out" probability distribution. OK, enough Quantum Mechanics. For stuff like a macroscopic object (say a sphere of crystalline silica of micrometer dimensions, or a galaxy (macroscopic covers a LOT of distance!)) then a particle (or system of particles) has a total angular momentum which has "direction" in space.

That is, its momentum can be characterized as a 3-D vector (3 components). Typically, the first thing you consider in elementary Classical Physics are point-like particles (a point can't spin) with location (3 components) and momentum (3 components). If you want to allow for these objects to spin, you'd need another 3 components, but I'm not familiar with any textbooks that add that. You'd need it for orbital dynamics (rocket science).

Typically you start with a particle or two, and then maybe three (because of the 3 body problem) and then jump to N (where N can be any positive integer). In elementary classical dynamics (called Classical Mechanics), a body is considered "fully" characterized by its 6 components of location and momentum (or if mass varies particle to particle, each of their locations, masses, and 3-velocity(xyz velocity components).

So, a system has 6N coordinates but they are NOT all spatial (in the sense of locations, a subtle difference between location (from an origin) and directions (from the point(s)). The "dimensionality" of a problem depends on the number of independent (usually) variables needed. So, a Classical Mechanics system has 6N variables.

There are many cases where physicists consider a problem having "infinite dimension", but this shouldn't be interpreted to mean they think that there are spatial dimensions other than the three we experience. (In string theory, there are more space-like dimensions, some theories use several dozen! but these are yet to be experimentally verified as "real".) So, the bottom line is both "spin" and "dimension" have technical meanings (both have more than one) which are not the same as what's understood outside of physics.(or I should say: "not necessarily" the same...)

 

[Khashishi]

I'm not sure if the OP is talking about quantum spin, but general geometry.

In 3D space, for the rigid objects, maybe you want to describe the full position and orientation of an object using a set of coordinates. You need 6 coordinates to do this. 3 for translation and 3 for rotation. The 3 translational dimensions use the familiar 3D Cartesian space geometry. Rotation is also 3D but behaves differently. It wraps around after 360 degrees and combining rotations is order-dependent. The name for this geometry is SO(3). The full set is 6 dimensional.

Or maybe you want to describe the momentum instead of the position. In 3D space, for rigid objects you have 3 dimensions for linear momentum and 3 dimensions for angular momentum, so 6 dimensions for describing the general momentum. In the non-relativistic classical limit, these form a 6 dimensional Cartesian space for momentum. With relativity, linear momentum becomes a hyperbolic space, and I don't know about angular momentum.

Or you can include them together as a full 12D configuration space.

 

[Matth.ew]

Thanks. Much appreciated. Both very interesting and good to know responses.