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RE: The physical origin of the electron spin (1)

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  • cyrano@aqua.ocn.ne.jp
    Hello Matti, ... [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
    Message 1 of 3 , Aug 1 4:44 PM
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      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
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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>

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    • Matti Pitkanen by way of cyrano@aqua.oc
      ... The notion of spinor structure is standard geometry. In standard QFT the geometric character of gamma matrices is not obvious because metric is flat.
      Message 2 of 3 , Aug 1 5:25 PM
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        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?

        The notion of spinor structure is standard geometry. In standard
        QFT the geometric character of gamma matrices is not obvious because
        metric is flat. Penrose has introduced twistors which
        is more specific geometric structure.

        >
        > > 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.

        Yes this is true. 2-spheres however can get knotted and linked in
        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
        >
      • Matti Pitkanen by way of cyrano@aqua.oc
        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
        Message 3 of 3 , Aug 2 2:48 PM
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          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
          >
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