Is there an algebraical connection between SU(2) spinors and O(3) vectors [Archive]

Ilian Peruhov

Oct19-04, 03:29 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nHow can one prove that SU(2) on pairs of complex numbers (spinors) is\nequal to O(3) on triples of real numbers (3-vectors)?\nIs there a definite algebraical connection between a certain\nvector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which\nundergo the transformations O3 and SU2?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>How can one prove that SU(2) on pairs of complex numbers (spinors) is
equal to O(3) on triples of real numbers (3-vectors)?
Is there a definite algebraical connection between a certain
vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which
undergo the transformations O3 and SU2?

John T Lowry

Oct20-04, 03:18 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Ilian Peruhov" <ilper@au-plovdiv.bg> wrote in message\nnews:a6e10bf8.0410180333.4a505c29@posting .google.com...\n>\n>\n> How can one prove that SU(2) on pairs of complex numbers (spinors) is\n> equal to O(3) on triples of real numbers (3-vectors)?\n> Is there a definite algebraical connection between a certain\n> vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which\n> undergo the transformations O3 and SU2?\n\nThere is an intimate connection. SU(2) is the (simply connected)\nuniversal covering group for (multivalued) O(3). For details, see\nGottfried\'s Quantum Mechanics pp. 279 ff and Goldstein\'s Classical\nMechanics pp 109 ff, on Cayley-Klein parameters. Your 3-vector (2,4,8),\nin the 2-by-2 formulation, would be (top row): 8 2-4i, (bottom row):\n2+4i -8.\n\nJohn Lowry\nFlight Physics\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Ilian Peruhov" <ilper@au-plovdiv.bg> wrote in message
news:a6e10bf8.0410180333.4a505c29@posting.google.c om...
>
>
> How can one prove that SU(2) on pairs of complex numbers (spinors) is
> equal to O(3) on triples of real numbers (3-vectors)?
> Is there a definite algebraical connection between a certain
> vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which
> undergo the transformations O3 and SU2?

There is an intimate connection. SU(2) is the (simply connected)
universal covering group for (multivalued) O(3). For details, see
Gottfried's Quantum Mechanics pp. 279 ff and Goldstein's Classical
Mechanics pp 109 ff, on Cayley-Klein parameters. Your 3-vector (2,4,8),
in the 2-by-2 formulation, would be (top row): 8 2-4i, (bottom row):
2+4i -8.

John Lowry
Flight Physics

Daryl McCullough

Oct20-04, 03:18 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Ilian Peruhov says...\n\n>How can one prove that SU(2) on pairs of complex numbers (spinors) is\n>equal to O(3) on triples of real numbers (3-vectors)?\n>Is there a definite algebraical connection between a certain\n>vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which\n>undergo the transformations O3 and SU2?\n\nHere\'s a mapping between a complex 2-component spinor Psi\nand a real 3-component vector V: Let Psi_1 and Psi_2 be the\ntwo components of Psi.\n\n1. Write Psi_1 as\nexp(i X) sqrt(2r) cos(theta/2) exp(i phi/2)\n2. Write Psi_2 as\nexp(i X) sqrt(2r) sin(theta/2) exp(-i phi/2)\n\nThen the spinor Phi gives rise to the 3-vector\nV = (V_x,V_y,V_z) defined by\n\nV_x = r cos(theta) cos(phi)\nV_y = r cos(theta) sin(phi)\nV_z = r sin(theta)\n\n(The phase X is irrelevant).\n\nNote that this mapping is 2-1: Replacing theta by theta + 2 pi\nor replacing phi by phi + 2 pi leaves V alone, but changes the\nsign of Psi.\n\nAnother way to see the relationship between V and Psi is to\nuse the Pauli spin-matrices:\n\nPsi Psi^* = V_x sigma_x + V_y sigma_y + V_z sigma_z + |V| 1\n\nwhere 1 = the 2x2 identity matrix, and |V| = the magnitude of V,\nand Psi^* = the complex conjugate of the transpose of Psi.\n\nThe reverse transformation is:\n\nPsi_1 = square-root(|V|+V_z) square-root(V_x-iV_y)/square-root(|V|)\nPsi_2 = square-root(|V|-V_z) square-root(V_x+iV_y)/square-root(|V|)\n\nwhich is multi-valued because of the square-roots. The phase X is\nnot determined.\n\n--\nDaryl McCullough\nIthaca, NY\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Ilian Peruhov says...

>How can one prove that SU(2) on pairs of complex numbers (spinors) is
>equal to O(3) on triples of real numbers (3-vectors)?
>Is there a definite algebraical connection between a certain
>vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which
>undergo the transformations O3 and SU2?

Here's a mapping between a complex 2-component spinor \Psi
and a real 3-component vector V: Let \Psi_1 and \Psi_2 be the
two components of \Psi.

1. Write \Psi_1 as
\exp(i X) \sqrt(2r) cos(\theta/2) \exp(i \phi/2)
2. Write \Psi_2 as
\exp(i X) \sqrt(2r) sin(\theta/2) \exp(-i \phi/2)

Then the spinor \Phi gives rise to the 3-vector
V = (V_x,V_y,V_z) defined by

V_x = r cos(\theta) cos(\phi)V_y = r cos(\theta) sin(\phi)V_z = r sin(\theta)

(The phase X is irrelevant).

Note that this mapping is 2-1: Replacing \theta by \theta + 2 \pi
or replacing \phi by \phi + 2 \pi leaves V alone, but changes the
sign of \Psi.

Another way to see the relationship between V and \Psi is to
use the Pauli spin-matrices:

\Psi \Psi^* = V_x \sigma_x + V_y \sigma_y + V_z \sigma_z + |V| 1

where 1 = the 2x2 identity matrix, and |V| = the magnitude of V,
and \Psi^* = the complex conjugate of the transpose of \Psi.

The reverse transformation is:

\Psi_1 = square-root(|V|+V_z) square-root(V_x-iV_y)/square-root(|V|)\Psi_2 = square-root(|V|-V_z) square-root(V_x+iV_y)/square-root(|V|)

which is multi-valued because of the square-roots. The phase X is
not determined.

--
Daryl McCullough
Ithaca, NY

Mark

Oct20-04, 03:18 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nilper@au-plovdiv.bg (Ilian Peruhov) wrote in message news:<a6e10bf8.0410180333.4a505c29@posting.google. com>...\n> How can one prove that SU(2) on pairs of complex numbers (spinors) is\n> equal to O(3) on triples of real numbers (3-vectors)?\n> Is there a definite algebraical connection between a certain\n> vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which\n> undergo the transformations O3 and SU2?\n\nO3 is homophic to SU2, so SU2\'s spinor is equal to O3\'s spinor, SU2\'s\nvector is equal to O3\'s vector.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>ilper@au-plovdiv.bg (Ilian Peruhov) wrote in message news:<a6e10bf8.0410180333.4a505c29@posting.google.com>...
> How can one prove that SU(2) on pairs of complex numbers (spinors) is
> equal to O(3) on triples of real numbers (3-vectors)?
> Is there a definite algebraical connection between a certain
> vector(say 2,4,8) and one!! subsequent spinor (4 real numbers), which
> undergo the transformations O3 and SU2?

O3 is homophic to SU2, so SU2's spinor is equal to O3's spinor, SU2's
vector is equal to O3's vector.