- Hello Matti,

> One can get *integer* spins as states of 4D- rigid body. I am not sure

[DL] Neither do I. But I have the feeling that there should be *integers*

> whether one can to get spin half odd integer states for which matrix

> elements of rotation matrices are two valued, perhaps.

of h/2 and not of h. Therefore fermions are "odd integers" and bosons are

"even integers".

> On the other hand, spin half results naturally from purely geometric

[DL] Penrose?

> construction. Mathematicians speak of spinor structure.

> I agree. Have you ever encountered orientation entanglement relation.

[DL] The book "Gravitation"?

> It can be found from Misner, Thorne, Wheeler.

> It provides

[DL] What you describe below sounds like what Milo describes when he

> the only concrete representation that I know for what happens in

> rotations of 2pi and 4pi of fermion.

>

speaks about "spherical rotation". So, I guess it works. I will try it too.

> Take cube inside larger cube and

[DL] Take a 3D body, mark the OX, OY and OZ axes, then rotate the body

> connect its vertices to the corresponding vertices of bigger cube.

> 2pi rotation entangles the threads. So does 4pi but you can de-entangle

> the mess. I still do not understand what is involved and how Dirac

> invented it.

>

around the OZ axis.

The ZOX axis is sweeping the XOY plane, describing a PLANAR angle. This

angle increases from

zero to 2Pi.

Now, if you take a 4D body, mark the four axes, then rotate the body around

the OT axis,

what will you see? The TOX plane is sweeping the XYZ space, describing a

SOLID angle.

This angle increases from zero to 4Pi.

Exactly as the plane XOY transforms into itself during the regular rotation,

the space XYZ transforms

into itself during the 4D rotation.

Therefore, 2D beings can perform regular rotations inside their own world,

and similarly we are able

to perform 4D rotations inside our world!!! But, indeed the output looks

weird!

More than that: in 4D you cannot get threads (wires) entangled; you cannot

knit; you cannot have

chains and so on.

Cheers,

Daniel

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<TITLE>RE: The physical origin of the electron spin (1)</TITLE>

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<P><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Hello Matti,</FONT>

</P>

<P><FONT SIZE=2 FACE="Arial">One can get *integer* spins as states of 4D-

rigid body. I am not sure</FONT>

<BR><FONT SIZE=2 FACE="Arial">whether one can to get spin half odd

integer states for which matrix</FONT>

<BR><FONT SIZE=2 FACE="Arial">elements of rotation matrices are two valued,

perhaps.</FONT>

<BR><B><I><FONT COLOR="#0000FF" SIZE=2

FACE="Arial">[DL]</FONT></I></B><I></I> <FONT COLOR="#0000FF" SIZE=2

FACE="Arial"> Neither do I. But I have the feeling that there should be

*integers*</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">of h/2 and not of h. Therefore

fermions are "odd integers" and bosons are "even

integers".</FONT>

</P>

<P><FONT SIZE=2 FACE="Arial">On the other hand, spin half results naturally

from purely geometric</FONT>

<BR><FONT SIZE=2 FACE="Arial">construction. Mathematicians speak of spinor

structure. </FONT>

<BR><B><I><FONT COLOR="#0000FF" SIZE=2

FACE="Arial">[DL]</FONT></I></B><I></I> <FONT COLOR="#0000FF" SIZE=2

FACE="Arial"> Penrose?</FONT>

</P>

<P><FONT SIZE=2 FACE="Arial">I agree. Have you ever encountered

orientation entanglement relation.</FONT>

<BR><FONT SIZE=2 FACE="Arial">It can be found from Misner, Thorne,

Wheeler. </FONT>

<BR><B><I><FONT COLOR="#0000FF" SIZE=2

FACE="Arial">[DL]</FONT></I></B><I></I> <FONT COLOR="#0000FF" SIZE=2

FACE="Arial"> The book "Gravitation"? </FONT>

</P>

<P><FONT SIZE=2 FACE="Arial">It provides</FONT>

<BR><FONT SIZE=2 FACE="Arial">the only concrete representation that I

know for what happens in</FONT>

<BR><FONT SIZE=2 FACE="Arial">rotations of 2pi and 4pi of fermion. </FONT>

</P>

<P><B><I><FONT COLOR="#0000FF" SIZE=2

FACE="Arial">[DL]</FONT></I></B><I></I> <FONT COLOR="#0000FF" SIZE=2

FACE="Arial"> What you describe below sounds like what Milo describes when

he</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">speaks about "spherical

rotation". So, I guess it works. I will try it too.</FONT>

</P>

<P><FONT SIZE=2 FACE="Arial">Take cube inside larger cube and</FONT>

<BR><FONT SIZE=2 FACE="Arial">connect its vertices to the corresponding

vertices of bigger cube.</FONT>

<BR><FONT SIZE=2 FACE="Arial">2pi rotation entangles the threads. So does

4pi but you can de-entangle</FONT>

<BR><FONT SIZE=2 FACE="Arial">the mess. I still do not understand what

is involved and how Dirac</FONT>

<BR><FONT SIZE=2 FACE="Arial">invented it. </FONT>

</P>

<P><B><I><FONT COLOR="#0000FF" SIZE=2

FACE="Arial">[DL]</FONT></I></B><I></I> <FONT COLOR="#0000FF" SIZE=2

FACE="Arial"> Take a 3D body, mark the OX, OY and OZ axes, then rotate the

body around the OZ axis. </FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">The ZOX axis is sweeping the

XOY plane, describing a PLANAR angle. This angle increases from</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">zero to 2Pi.</FONT>

</P>

<P><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Now, if you take a 4D body,

mark the four axes, then rotate the body around the OT axis,</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">what will you see? The TOX

plane is sweeping the XYZ space, describing a SOLID angle. </FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">This angle increases from zero

to 4Pi.</FONT>

</P>

<P><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Exactly as the plane XOY

transforms into itself during the regular rotation, the space XYZ

transforms</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">into itself during the 4D

rotation. </FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Therefore, 2D beings can

perform regular rotations inside their own world, and similarly we are able

</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">to perform 4D rotations inside

our world!!! But, indeed the output looks weird!</FONT>

</P>

<P><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">More than that: in 4D you

cannot get threads (wires) entangled; you cannot knit; you cannot have</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">chains and so on.</FONT>

</P>

<P><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Cheers,</FONT>

<BR><FONT COLOR="#0000FF" SIZE=2 FACE="Arial">Daniel</FONT><FONT SIZE=2

FACE="Arial"> </FONT>

</P>

</BODY>

</HTML> - On Wed, 2 Aug 2000 daniel.lapadatu@... wrote:

> Hello Matti,

The notion of spinor structure is standard geometry. In standard

>

> > One can get *integer* spins as states of 4D- rigid body. I am not sure

> > whether one can to get spin half odd integer states for which matrix

> > elements of rotation matrices are two valued, perhaps.

> [DL] Neither do I. But I have the feeling that there should be *integers*

> of h/2 and not of h. Therefore fermions are "odd integers" and bosons are

> "even integers".

>

> > On the other hand, spin half results naturally from purely geometric

> > construction. Mathematicians speak of spinor structure.

> [DL] Penrose?

QFT the geometric character of gamma matrices is not obvious because

metric is flat. Penrose has introduced twistors which

is more specific geometric structure.

>

rotation,

> > I agree. Have you ever encountered orientation entanglement relation.

> > It can be found from Misner, Thorne, Wheeler.

> [DL] The book "Gravitation"?

>

> > It provides

> > the only concrete representation that I know for what happens in

> > rotations of 2pi and 4pi of fermion.

> >

> [DL] What you describe below sounds like what Milo describes when he

> speaks about "spherical rotation". So, I guess it works. I will try it too.

>

> > Take cube inside larger cube and

> > connect its vertices to the corresponding vertices of bigger cube.

> > 2pi rotation entangles the threads. So does 4pi but you can de-entangle

> > the mess. I still do not understand what is involved and how Dirac

> > invented it.

> >

> [DL] Take a 3D body, mark the OX, OY and OZ axes, then rotate the body

> around the OZ axis.

> The ZOX axis is sweeping the XOY plane, describing a PLANAR angle. This

> angle increases from

> zero to 2Pi.

>

> Now, if you take a 4D body, mark the four axes, then rotate the body around

> the OT axis,

> what will you see? The TOX plane is sweeping the XYZ space, describing a

> SOLID angle.

> This angle increases from zero to 4Pi.

>

> Exactly as the plane XOY transforms into itself during the regular

> the space XYZ transforms

Yes this is true. 2-spheres however can get knotted and linked in

> into itself during the 4D rotation.

> Therefore, 2D beings can perform regular rotations inside their own world,

> and similarly we are able

> to perform 4D rotations inside our world!!! But, indeed the output looks

> weird!

>

> More than that: in 4D you cannot get threads (wires) entangled; you cannot

> knit; you cannot have

> chains and so on.

4D. Difficult to imagine what it looks like when two two-spheres

get linked and desperately try to get rid of each other!

>

> Cheers,

> Daniel

> - I finally understood what happens in orientation entanglement and

why it visualizes spin one half.

What makes spinors possible is that the first homotopy group of SO(3) is

Z2, two-element group. This means that their are closed loops (say orbit

of 2*pi rotation) which cannot be contracted to a point in SO(3).

The loop corresponding to orbit of 4*pi rotation can be contracted to

a point. This implies that rotation by 2*pi corresponds to

either -1 or +1 and -1 gives rise to fermions, +1 to bosons.

The configuration space of rigid body with one point fixed

is rotation group SO(3). Cube is simplest example of rigid

body. Thus the rotations of cube might make possible to visualize

the the nontriviality of the homotopy group of SO(3).

How to then make the homotopy group visible? Connect the vertices

of cube to the vertices of the larger cube by threads! Entanglement

of threads represents the homotopy corresponding

to continuous rotation of 2*pi! That the entanglement

cannot be straightened out corresponds to the impossibility

of contracting the closed loop in SO(3)!

In spacetime Lorentz 'rotations' of 4-cube connected in similar

manner to larger 4-cube by threads should visualize the homotopy

group of Lorentz group, which is also Z_2 (otherwise we would not

have relativistic spin one half particles).

It is NOT possible to have spin half odd integer states in case of rigid

body: the reason is that wave function would be two-valued in

congifugration space. Half odd integer spin is NOT reducible to

spatial geometry but is something more genereal.

MP

On Wed, 2 Aug 2000 daniel.lapadatu@... wrote:

> Hello Matti,

>

> > One can get *integer* spins as states of 4D- rigid body. I am not sure

> > whether one can to get spin half odd integer states for which matrix

> > elements of rotation matrices are two valued, perhaps.

> [DL] Neither do I. But I have the feeling that there should be *integers*

> of h/2 and not of h. Therefore fermions are "odd integers" and bosons are

> "even integers".

>

> > On the other hand, spin half results naturally from purely geometric

> > construction. Mathematicians speak of spinor structure.

> [DL] Penrose?

>

> > I agree. Have you ever encountered orientation entanglement relation.

> > It can be found from Misner, Thorne, Wheeler.

> [DL] The book "Gravitation"?

>

> > It provides

> > the only concrete representation that I know for what happens in

> > rotations of 2pi and 4pi of fermion.

> >

> [DL] What you describe below sounds like what Milo describes when he

> speaks about "spherical rotation". So, I guess it works. I will try it too.

>

> > Take cube inside larger cube and

> > connect its vertices to the corresponding vertices of bigger cube.

> > 2pi rotation entangles the threads. So does 4pi but you can de-entangle

> > the mess. I still do not understand what is involved and how Dirac

> > invented it.

> >

> [DL] Take a 3D body, mark the OX, OY and OZ axes, then rotate the body

> around the OZ axis.

> The ZOX axis is sweeping the XOY plane, describing a PLANAR angle. This

> angle increases from

> zero to 2Pi.

>

> Now, if you take a 4D body, mark the four axes, then rotate the body around

> the OT axis,

> what will you see? The TOX plane is sweeping the XYZ space, describing a

> SOLID angle.

> This angle increases from zero to 4Pi.

>

> Exactly as the plane XOY transforms into itself during the regular rotation,

> the space XYZ transforms

> into itself during the 4D rotation.

> Therefore, 2D beings can perform regular rotations inside their own world,

> and similarly we are able

> to perform 4D rotations inside our world!!! But, indeed the output looks

> weird!

>

> More than that: in 4D you cannot get threads (wires) entangled; you cannot

> knit; you cannot have

> chains and so on.

>

> Cheers,

> Daniel

>