Understanding Particle Spin and Dimensional Restrictions

[Spin_Network]

What happens to a particles spin identity, if the particle is decomposed by dimensional restriction?

Does a specific dimension 'fix' a particle's spin?..if certain particles trancend from 4-D to 2-D, then what change occurs in the particles spin atributes, if any?

Is there a transitional spin cut-off at some finite dimensional level?

 

[Hans de Vries]

What happens to a particles spin identity, if the particle is decomposed by dimensional restriction?
Does a specific dimension 'fix' a particle's spin?..if certain particles trancend from 4-D to 2-D, then what change occurs in the particles spin atributes, if any?
Is there a transitional spin cut-off at some finite dimensional level?

Nobody knows the internal mechanism of how spin works, let alone how that
would work with less dimensions.

Most likely it would fail altogether since the fact that the total spin
angular momentum $\sqrt{s(s+1)}\hbar$ is more than the spin's z-component $s\hbar$
needs at least 3 spatial dimensions.

What we know from spin, in relation with the x y and z axis stems basically
from measurements with concatenated Stern-Gerlach apparatus.
The (a) mathematical description found that works leads us to the Pauli matrices.

(Feynman's "Lectures on physics part III" and Sakurai's "Modern Quantum
mechanics")Regards, Hans

 

[i.mehrzad]

why does the particas spin represented by +1/2 and -1/2

 

[Spin_Network]

Nobody knows the internal mechanism of how spin works, let alone how that
would work with less dimensions.

Most likely it would fail altogether since the fact that the total spin
angular momentum $\sqrt{s(s+1)}\hbar$ is more than the spin's z-component $s\hbar$
needs at least 3 spatial dimensions.

What we know from spin, in relation with the x y and z axis stems basically
from measurements with concatenated Stern-Gerlach apparatus.
The (a) mathematical description found that works leads us to the Pauli matrices.

(Feynman's "Lectures on physics part III" and Sakurai's "Modern Quantum
mechanics")


Regards, Hans

Many thanks Hans.

If I give a specific situation for simplistic brevity, I have a 'spinning-top'
3-D of course and is following all the standard laws of spatial dimensional action-reactions. If I grab this top with both hands, it will translate some of its attributes(spin) to my hands, say friction the moment I connect?

Now what if I could grab it from say, a 2-D dimension?..as if it actually collided with my static 2-D hands that happen to be waiting in the 2-Dimensional wings so to speak?

Would say the frictional factor in spacetime, be translated to "other" by the inter-dimensional aspect?..or would there be evidence of some frictional aspect always present?

What I am asking is if "spin" is anhialated completely?

 

[hellfire]

Does a specific dimension 'fix' a particle's spin?

I do not exactly understand your question and your last comment, but may be this helps: Spin is intrinsic angular momentum and is therefore mathematically related to rotations. In three spatial dimensions the rotation group SO(3) has a double cover SU(2) from which two elements correspont to one of SO(3). This means that objects transforming under SO(3) need to rotate a 2π angle to make a complete rotation whereas objects transforming under SU(2) need 4π for a complete rotation. This makes possible that half-integer spins exist in three dimensions. However, for example, no triple cover of SO(3) exists, which would lead to 1/3 spins (three elements of the triple cover corresponding to one of SO(3)). If the number of spatial dimensions is different, this relation between SO(n) and its covering groups change. It would be a nice exercise to see how this generalizes. For example, for two dimensional spaces one may have indeed other fractional spins than half-integer. As far as I know this is called anyonic statistics. Thus, the fact that spin is related to rotations makes it dependent of the dimensionality of space you are considering.

 

[Spin_Network]

I do not exactly understand your question and your last comment, but may be this helps: Spin is intrinsic angular momentum and is therefore mathematically related to rotations. In three spatial dimensions the rotation group SO(3) has a double cover SU(2) from which two elements correspont to one of SO(3). This means that objects transforming under SO(3) need to rotate a 2π angle to make a complete rotation whereas objects transforming under SU(2) need 4π for a complete rotation. This makes possible that half-integer spins exist in three dimensions. However, for example, no triple cover of SO(3) exists, which would lead to 1/3 spins (three elements of the triple cover corresponding to one of SO(3)). If the number of spatial dimensions is different, this relation between SO(n) and its covering groups change. It would be a nice exercise to see how this generalizes. For example, for two dimensional spaces one may have indeed other fractional spins than half-integer. As far as I know this is called anyonic statistics. Thus, the fact that spin is related to rotations makes it dependent of the dimensionality of space you are considering.

Hellfire many thanks, after re-looking at my post it is quite obvious that I had worded it in such a abstract fashion, but...your response is exactly what I had been looking for, amazing!

I am quite lazy in my post fashion, but always hope there is someone outthere who looks a little further, the rotational factor is what I was aiming at,(already knew this but now understand it a great deal more)..my next inquiries are going to be "anyonic statistics"..thanks again.

 

[Spin_Network]

I do not exactly understand your question and your last comment, but may be this helps: Spin is intrinsic angular momentum and is therefore mathematically related to rotations. In three spatial dimensions the rotation group SO(3) has a double cover SU(2) from which two elements correspont to one of SO(3). This means that objects transforming under SO(3) need to rotate a 2π angle to make a complete rotation whereas objects transforming under SU(2) need 4π for a complete rotation. This makes possible that half-integer spins exist in three dimensions. However, for example, no triple cover of SO(3) exists, which would lead to 1/3 spins (three elements of the triple cover corresponding to one of SO(3)). If the number of spatial dimensions is different, this relation between SO(n) and its covering groups change. It would be a nice exercise to see how this generalizes. For example, for two dimensional spaces one may have indeed other fractional spins than half-integer. As far as I know this is called anyonic statistics. Thus, the fact that spin is related to rotations makes it dependent of the dimensionality of space you are considering.

There is more than I need here:http://arxiv.org/abs/hep-th/0511086