Dirac spinors under reflections and inversions [Archive]

Erik

Oct13-04, 02:13 PM

<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>\nBasically, I wonder how Dirac spinors transform under spatial\nreflections and inversions. A long-winded elaboration of my question\nfollows.\n\nLet us define the group Gspace as the group of all transformations of\nspatial coordinates given by\n\n(x,y,z) ---> (+/- x, +/- y, +/- z)\n\nThus, Gspace is what is usually denoted D_2h and contains eight\nelements corresponding to the identity (Espace), three 180 degree\nrotations (RXspace, RYspace, RZspace), three reflections (SXYspace,\nSXZspace, SYZspace), and spatial inversion (INVspace).\n\nWhen coordinates are transformed by a transformation in Gspace, one\nmust make a corresponding transformation of one\'s spinors. Let us call\nthe group of these transformations Gspinor. There should be a way to\nmap elements of Gspace to elements of Gspinor, a function\n\nf : Gspace --> Gspinor.\n\nFor example, when components of the Dirac spinor are chosen as\nLarge-spin up (1), Large-spin down (2), Small-spin up (3), Small-spin\ndown (4), the spatial rotations will result in the following spinor\ntransformations\n\n( psi1(x,y,z) ) ( -i psi2(x,-y,-z) )\n( psi2(x,y,z) ) RXspinor ( -i psi1(x,-y,-z) )\n( psi3(x,y,z) ) ----------> ( -i psi4(x,-y,-z) )\n( psi4(x,y,z) ) ( -i psi3(x,-y,-z) )\n\n( psi1(x,y,z) ) ( -psi2(-x,y,-z) )\n( psi2(x,y,z) ) RYspinor ( psi1(-x,y,-z) )\n( psi3(x,y,z) ) ----------> ( -psi4(-x,y,-z) )\n( psi4(x,y,z) ) ( psi3(-x,y,-z) )\n\n( psi1(x,y,z) ) ( -i psi1(-x,-y,z) )\n( psi2(x,y,z) ) RZspinor ( i psi2(-x,-y,z) )\n( psi3(x,y,z) ) ----------> ( -i psi3(-x,-y,z) )\n( psi4(x,y,z) ) ( i psi4(-x,-y,z) )\n\nI am not interested in the transformation properties of the components\npsi1, psi2, psi3, and psi4---only the way they change positions and\nget phase shifts---and therefore take the elements of Gspinor to be\n4x4 matrices. Because RXspinor^2 = -Espinor, while RXspace^2 = Espace,\nthe group Gspinor has between 8 and 16 elements.\n\nMy problem is this: I want to determine the spinor transformations\n(4x4 matrices)\n\nSXYspinor = f(SXYspace)\nSXZspinor = f(SXZspace)\nSYZspinor = f(SYZspace)\nINVspinor = f(INVspace)\n\nBased on how the z-compenent of an angular momentum transforms under\noperations in Gspace, I have deduced that SXZspinor and SYZspinor will\nswap the spin-up and spin-down components, while SXYspinor and\nINVspinor will not. But the phase factors are yet to be determined and\nsince f is not a homomorphism I don\'t know how to do that.\n\nIs there a good book chapter that explains this explicitly (I have\nseen plenty of books that discuss, e.g., the Lorentz and Poincaré\ngroup at a very abstract level, but I feel like I need this explictly\nspelled out)? Or maybe it is easy enough to explain on this list?\n\nThanks in advance,\nErik\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>Basically, I wonder how Dirac spinors transform under spatial
reflections and inversions. A long-winded elaboration of my question
follows.

Let us define the group Gspace as the group of all transformations of
spatial coordinates given by

(x,y,z) ---> (+/- x, +/- y, +/- z)

Thus, Gspace is what is usually denoted D_{2h} and contains eight
elements corresponding to the identity (Espace), three 180 degree
rotations (RXspace, RYspace, RZspace), three reflections (SXYspace,
SXZspace, SYZspace), and spatial inversion (INVspace).

When coordinates are transformed by a transformation in Gspace, one
must make a corresponding transformation of one's spinors. Let us call
the group of these transformations Gspinor. There should be a way to
map elements of Gspace to elements of Gspinor, a function

f : Gspace --> Gspinor.

For example, when components of the Dirac spinor are chosen as
Large-spin up (1), Large-spin down (2), Small-spin up (3), Small-spin
down (4), the spatial rotations will result in the following spinor
transformations

( psi1(x,y,z) ) ( -i psi2(x,-y,-z) )
( psi2(x,y,z) ) RXspinor ( -i psi1(x,-y,-z) )
( psi3(x,y,z) ) ----------> ( -i psi4(x,-y,-z) )
( psi4(x,y,z) ) ( -i psi3(x,-y,-z) )

( psi1(x,y,z) ) ( -psi2(-x,y,-z) )
( psi2(x,y,z) ) RYspinor ( psi1(-x,y,-z) )
( psi3(x,y,z) ) ----------> ( -psi4(-x,y,-z) )
( psi4(x,y,z) ) ( psi3(-x,y,-z) )

( psi1(x,y,z) ) ( -i psi1(-x,-y,z) )
( psi2(x,y,z) ) RZspinor ( i psi2(-x,-y,z) )
( psi3(x,y,z) ) ----------> ( -i psi3(-x,-y,z) )
( psi4(x,y,z) ) ( i psi4(-x,-y,z) )I am not interested in the transformation properties of the components
psi1, psi2, psi3, and psi4---only the way they change positions and
get phase shifts---and therefore take the elements of Gspinor to be
4x4 matrices. Because RXspinor^2 = -Espinor, while RXspace^2 = Espace,
the group Gspinor has between 8 and 16 elements.

My problem is this: I want to determine the spinor transformations
(4x4 matrices)

SXYspinor = f(SXYspace)
SXZspinor = f(SXZspace)
SYZspinor = f(SYZspace)
INVspinor = f(INVspace)

Based on how the z-compenent of an angular momentum transforms under
operations in Gspace, I have deduced that SXZspinor and SYZspinor will
swap the spin-up and spin-down components, while SXYspinor and
INVspinor will not. But the phase factors are yet to be determined and
since f is not a homomorphism I don't know how to do that.

Is there a good book chapter that explains this explicitly (I have
seen plenty of books that discuss, e.g., the Lorentz and Poincaré
group at a very abstract level, but I feel like I need this explictly
spelled out)? Or maybe it is easy enough to explain on this list?

Thanks in advance,
Erik

Charles J. Quarra

Oct14-04, 11:07 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\nerite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google. com>...\n> Basically, I wonder how Dirac spinors transform under spatial\n> reflections and inversions. A long-winded elaboration of my question\n> follows.\n\n\nAllow me to add another question to your own batch.\n\nWhy Dirac spinors transform under finite electric-magnetic duality\nrotations as\n\nPsi -> exp( i gamma_5 *w ) Psi\n\nAfaik gamma_5 is a pseudo-scalar built from y1*y2*y3*y4, meaning that\nit gets a -1 phase at any reflection operation ( det(U)=-1), but i\ndont get the relationship with dyon angle rotations\n\nthe background of the question comes from a discussion occurred on\nthis thread\nhttp://groups.google.co.ve/groups?hl=es&lr=&selm=2b93dd16.0408301338.1a0ac84a%40posting.google .com&rnum=14\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>erite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google.com>...
> Basically, I wonder how Dirac spinors transform under spatial
> reflections and inversions. A long-winded elaboration of my question
> follows.

Allow me to add another question to your own batch.

Why Dirac spinors transform under finite electric-magnetic duality
rotations as

\Psi -> \exp( i \gamma_5 *w ) \Psi

Afaik \gamma_5 is a pseudo-scalar built from y1*y2*y3*y4, meaning that
it gets a -1 phase at any reflection operation ( det(U)=-1), but i
dont get the relationship with dyon angle rotations

the background of the question comes from a discussion occurred on
this thread
http://groups.google.co.ve/groups?hl=es&lr=&selm=2b93dd16.0408301338.1a0ac84a%40posting.google .com&rnum=14

Danny Ross Lunsford

Oct14-04, 11:07 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>\nerite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google. com>...\n> Basically, I wonder how Dirac spinors transform under spatial\n> reflections and inversions. A long-winded elaboration of my question\n> follows.\n>\n> Let us define the group Gspace as the group of all transformations of\n> spatial coordinates given by\n>\n> (x,y,z) ---> (+/- x, +/- y, +/- z)\n\n....\n\nThis is too complicated. The important transformation is spacetime\nparity, so you have to bring in y5. Note\n\nym -> ym\' = y5 ym y5\n\nis a legitimate change of basis:\n\n{ym\', yn\'} = y5 {ym, yn} y5 = {ym, yn}\n\nand\n\nym\'^2 = ym^2\n\nThe transformed spinor is\n\npsi\' = y5 psi\n\nand that is what you are seeking. In the Weyl (chiral) basis\n\npsi = (phi, chi) -> (phi, -chi)\n\nIn the Dirac basis\n\npsi = (psi+, psi-) -> (psi-, psi+)\n\nthat is, the "large" and "small" components of psi are interchanged.\nThis essentially means that the roles of matter and antimatter are\nreversed. So spacetime parity is what allows one to bring in\nantimatter as a logically independent substance. Without spacetime\nparity there is no antimatter, and that is why one needs 4-component\nspinors to represent fermions.\n\n-drl\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>erite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google.com>...
> Basically, I wonder how Dirac spinors transform under spatial
> reflections and inversions. A long-winded elaboration of my question
> follows.
>
> Let us define the group Gspace as the group of all transformations of
> spatial coordinates given by
>
> (x,y,z) ---> (+/- x, +/- y, +/- z)

....

This is too complicated. The important transformation is spacetime
parity, so you have to bring in y5. Note

ym ->[/itex] ym' [itex]= y5 ym y5

is a legitimate change of basis:

{ym', yn'} = y5 {ym, yn} y5 = {ym, yn}

and

ym'^2 = ym^2

The transformed spinor is

\psi' = y5 \psi

and that is what you are seeking. In the Weyl (chiral) basis

\psi = (\phi, \chi) -> (\phi, -\chi)

In the Dirac basis

\psi = (\psi+, \psi-) -> (\psi-, \psi+)

that is, the "large" and "small" components of \psi are interchanged.
This essentially means that the roles of matter and antimatter are
reversed. So spacetime parity is what allows one to bring in
antimatter as a logically independent substance. Without spacetime
parity there is no antimatter, and that is why one needs 4-component
spinors to represent fermions.

-drl

John T Lowry

Oct14-04, 11:07 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"Erik" <erite423@yahoo.se> wrote in message\nnews:d018ba78.0410130657.7ec7339e@posting .google.com...\n>\n> Basically, I wonder how Dirac spinors transform under spatial\n> reflections and inversions. A long-winded elaboration of my question\n> follows.\n>\n> Let us define the group Gspace as the group of all transformations of\n> spatial coordinates given by\n>\n> (x,y,z) ---> (+/- x, +/- y, +/- z)\n>\n> Thus, Gspace is what is usually denoted D_2h and contains eight\n> elements corresponding to the identity (Espace), three 180 degree\n> rotations (RXspace, RYspace, RZspace), three reflections (SXYspace,\n> SXZspace, SYZspace), and spatial inversion (INVspace).\n>\n> When coordinates are transformed by a transformation in Gspace, one\n> must make a corresponding transformation of one\'s spinors. Let us call\n> the group of these transformations Gspinor. There should be a way to\n> map elements of Gspace to elements of Gspinor, a function\n>\n> f : Gspace --> Gspinor.\n>\n> For example, when components of the Dirac spinor are chosen as\n> Large-spin up (1), Large-spin down (2), Small-spin up (3), Small-spin\n> down (4), the spatial rotations will result in the following spinor\n> transformations\n>\n> ( psi1(x,y,z) ) ( -i psi2(x,-y,-z) )\n> ( psi2(x,y,z) ) RXspinor ( -i psi1(x,-y,-z) )\n> ( psi3(x,y,z) ) ----------> ( -i psi4(x,-y,-z) )\n> ( psi4(x,y,z) ) ( -i psi3(x,-y,-z) )\n>\n> ( psi1(x,y,z) ) ( -psi2(-x,y,-z) )\n> ( psi2(x,y,z) ) RYspinor ( psi1(-x,y,-z) )\n> ( psi3(x,y,z) ) ----------> ( -psi4(-x,y,-z) )\n> ( psi4(x,y,z) ) ( psi3(-x,y,-z) )\n>\n> ( psi1(x,y,z) ) ( -i psi1(-x,-y,z) )\n> ( psi2(x,y,z) ) RZspinor ( i psi2(-x,-y,z) )\n> ( psi3(x,y,z) ) ----------> ( -i psi3(-x,-y,z) )\n> ( psi4(x,y,z) ) ( i psi4(-x,-y,z) )\n>\n> I am not interested in the transformation properties of the components\n> psi1, psi2, psi3, and psi4---only the way they change positions and\n> get phase shifts---and therefore take the elements of Gspinor to be\n> 4x4 matrices. Because RXspinor^2 = -Espinor, while RXspace^2 = Espace,\n> the group Gspinor has between 8 and 16 elements.\n>\n> My problem is this: I want to determine the spinor transformations\n> (4x4 matrices)\n>\n> SXYspinor = f(SXYspace)\n> SXZspinor = f(SXZspace)\n> SYZspinor = f(SYZspace)\n> INVspinor = f(INVspace)\n>\n> Based on how the z-compenent of an angular momentum transforms under\n> operations in Gspace, I have deduced that SXZspinor and SYZspinor will\n> swap the spin-up and spin-down components, while SXYspinor and\n> INVspinor will not. But the phase factors are yet to be determined and\n> since f is not a homomorphism I don\'t know how to do that.\n>\n> Is there a good book chapter that explains this explicitly (I have\n> seen plenty of books that discuss, e.g., the Lorentz and Poincaré\n> group at a very abstract level, but I feel like I need this explictly\n> spelled out)? Or maybe it is easy enough to explain on this list?\n>\n> Thanks in advance,\n> Erik\n\nPOSSIBLY Sakurai\'s Advanced Quantum Mechanics, around pp 99-100 would\nhelp. He treats both inversion (turns out to be with Gamma4) and what he\ncalls "mirror reflection" (parity plus a pi-rotation). I feel for you in\nthe general dearth of concrete examples in this stuff. My first Lorentz\nboost took me the better part of a day to tie down all the loose ends.\n\nJohn T. 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>"Erik" <erite423@yahoo.se> wrote in message
news:d018ba78.0410130657.7ec7339e@posting.google.c om...
>
> Basically, I wonder how Dirac spinors transform under spatial
> reflections and inversions. A long-winded elaboration of my question
> follows.
>
> Let us define the group Gspace as the group of all transformations of
> spatial coordinates given by
>
> (x,y,z) ---> (+/- x, +/- y, +/- z)
>
> Thus, Gspace is what is usually denoted D_{2h} and contains eight
> elements corresponding to the identity (Espace), three 180 degree
> rotations (RXspace, RYspace, RZspace), three reflections (SXYspace,
> SXZspace, SYZspace), and spatial inversion (INVspace).
>
> When coordinates are transformed by a transformation in Gspace, one
> must make a corresponding transformation of one's spinors. Let us call
> the group of these transformations Gspinor. There should be a way to
> map elements of Gspace to elements of Gspinor, a function
>
> f : Gspace --> Gspinor.
>
> For example, when components of the Dirac spinor are chosen as
> Large-spin up (1), Large-spin down (2), Small-spin up (3), Small-spin
> down (4), the spatial rotations will result in the following spinor
> transformations
>
> ( psi1(x,y,z) ) ( -i psi2(x,-y,-z) )
> ( psi2(x,y,z) ) RXspinor ( -i psi1(x,-y,-z) )
> ( psi3(x,y,z) ) ----------> ( -i psi4(x,-y,-z) )
> ( psi4(x,y,z) ) ( -i psi3(x,-y,-z) )
>
> ( psi1(x,y,z) ) ( -psi2(-x,y,-z) )
> ( psi2(x,y,z) ) RYspinor ( psi1(-x,y,-z) )
> ( psi3(x,y,z) ) ----------> ( -psi4(-x,y,-z) )
> ( psi4(x,y,z) ) ( psi3(-x,y,-z) )
>
> ( psi1(x,y,z) ) ( -i psi1(-x,-y,z) )
> ( psi2(x,y,z) ) RZspinor ( i psi2(-x,-y,z) )
> ( psi3(x,y,z) ) ----------> ( -i psi3(-x,-y,z) )
> ( psi4(x,y,z) ) ( i psi4(-x,-y,z) )
>
> I am not interested in the transformation properties of the components
> psi1, psi2, psi3, and psi4---only the way they change positions and
> get phase shifts---and therefore take the elements of Gspinor to be
> 4x4 matrices. Because RXspinor^2 = -Espinor, while RXspace^2 = Espace,
> the group Gspinor has between 8 and 16 elements.
>
> My problem is this: I want to determine the spinor transformations
> (4x4 matrices)
>
> SXYspinor = f(SXYspace)
> SXZspinor = f(SXZspace)
> SYZspinor = f(SYZspace)
> INVspinor = f(INVspace)
>
> Based on how the z-compenent of an angular momentum transforms under
> operations in Gspace, I have deduced that SXZspinor and SYZspinor will
> swap the spin-up and spin-down components, while SXYspinor and
> INVspinor will not. But the phase factors are yet to be determined and
> since f is not a homomorphism I don't know how to do that.
>
> Is there a good book chapter that explains this explicitly (I have
> seen plenty of books that discuss, e.g., the Lorentz and Poincaré
> group at a very abstract level, but I feel like I need this explictly
> spelled out)? Or maybe it is easy enough to explain on this list?
>
> Thanks in advance,
> Erik

POSSIBLY Sakurai's Advanced Quantum Mechanics, around pp 99-100 would
help. He treats both inversion (turns out to be with Gamma4) and what he
calls "mirror reflection" (parity plus a \pi-rotation). I feel for you in
the general dearth of concrete examples in this stuff. My first Lorentz
boost took me the better part of a day to tie down all the loose ends.

John T. Lowry
Flight Physics

Geometric Algebra Rocks!

Oct14-04, 11:07 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\nOn Wed, 13 Oct 2004 19:13:41 +0000 (UTC), erite423@yahoo.se (Erik)\nwrote:\n>\n>Basically, I wonder how Dirac spinors transform under spatial\n>reflections and inversions.\n>\n\nForget spinors! They are simply a subset of Geometric Algebra and\nshould be treated as such. Start here to learn:\n\nhttp://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf\n\nAt the end, you\'ll see spinors are not worth the paper they are\nwritten on.\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>On Wed, 13 Oct 2004 19:13:41 +0000 (UTC), erite423@yahoo.se (Erik)
wrote:
>
>Basically, I wonder how Dirac spinors transform under spatial
>reflections and inversions.
>

Forget spinors! They are simply a subset of Geometric Algebra and
should be treated as such. Start here to learn:

http://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf

At the end, you'll see spinors are not worth the paper they are
written on.

Danny Ross Lunsford

Oct15-04, 01:55 PM

<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\ndisposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0410140632.79bf66f@posting.google.c om>...\n> erite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google. com>...\n> > Basically, I wonder how Dirac spinors transform under spatial\n> > reflections and inversions. A long-winded elaboration of my question\n> > follows.\n>\n>\n> Allow me to add another question to your own batch.\n>\n> Why Dirac spinors transform under finite electric-magnetic duality\n> rotations as\n>\n> Psi -> exp( i gamma_5 *w ) Psi\n>\n> Afaik gamma_5 is a pseudo-scalar built from y1*y2*y3*y4, meaning that\n> it gets a -1 phase at any reflection operation ( det(U)=-1), but i\n> dont get the relationship with dyon angle rotations\n>\n> the background of the question comes from a discussion occurred on\n> this thread\n> http://groups.google.co.ve/groups?hl=es&lr=&selm=2b93dd16.0408301338.1a0ac84a%40posting.google .com&rnum=14\n\n\nThe proper definition of y5 is\n\n(1/24) eps mnab ym yn ya yb\n\nThe epsilon symbol requires a factor of sqrt(det g) to be a tensor,\nand that brings in a factor of i, so\n\ny5 = i y1 y2 y3 y4\n\n(Or i y0 y1 y2 y3 if you like, which is the negative of the "natural"\ndefinition.)\n\nAs shown in another post, y5 psi has matter and antimatter\ninterchanged. Your exponential gets an additional factor of i and so\nis unitary. That transformation represents an angular mixing of matter\nand antimatter,\n\nexp( 1/2 a iy5) psi\n\nso in the Dirac basis\n\npsi+ -> cos 1/2 a (psi+) + i sin 1/2 a (psi-)\n\nThe y\'s transform as\n\nym\' = exp (1/2 a iy5) ym exp (-1/2 a iy5) = exp(ia y5) ym\n\nwith a=pi corresponding to inverting matter and antimatter. So this is\nin a sense a continuous form of matter <-> antimatter.\n\nWe can now gauge this transformation - the Dirac-like equation becomes\n\n[ ym (dm + ig y5 Bm) + im ] psi = 0\n\nwhere Bm is a pseudovectorial gauge field and g a coupling constant.\nNote\n\npsibar [ ym (dm - ig y5 Bm) - im ] = 0\n\nThe Bm term changes sign three times - once from i on Hermitian\nconjugation, once from commuting y5 with ym_dagger, and once from\npulling through y4 to make psibar. So, analogous to the ordinary Dirac\nequation, there is a conserved current\n\nKm = psibar ym psi\n\nbut the coupling is to the pseudovector:\n\nBm (psibar ym y5 psi)\n\nwhich is still a scalar because Bm is also a pseudovector.\n\nThe form of the coupling shows that this is *not* precisely the\nelectromagnetic duality - on adding an overall phase and gauging that,\nwe have an odd gauge field that lives in the Dirac algebra\n\ne Am + g y5 Bm\n\nElectromagnetic duality with a second "dual" potential corresponding\nto magnetic charges would have led one to expect\n\ne Am + g iy5 Bm\n\nwhich would imply that Bm is purely imaginary.\n\nIf we wish to maintain a connection with electromagnetic duality, we\nhave to think of the potential itself as taking complex values. Thus,\ngauging the symmetry of matter and antimatter seems to lead to the\nidea of an intrinsically complex vector potential. You\'re welcome to\npush this as far as you like - something interesting may emerge. The\nobjective would be to understand poles as an intermediate stage\nbetween matter and antimatter:\n\nelectron -> pole -> positron -> antipole -> electron\n\nas\n\n0 -> pi/2 -> pi -> 3pi/2 -> 2pi\n\n-drl\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>disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0410140632.79bf66f@posting.google.com>...
> erite423@yahoo.se (Erik) wrote in message news:<d018ba78.0410130657.7ec7339e@posting.google.com>...
> > Basically, I wonder how Dirac spinors transform under spatial
> > reflections and inversions. A long-winded elaboration of my question
> > follows.
>
>
> Allow me to add another question to your own batch.
>
> Why Dirac spinors transform under finite electric-magnetic duality
> rotations as
>
> \Psi -> \exp( i \gamma_5 *w ) \Psi
>
> Afaik \gamma_5 is a pseudo-scalar built from y1*y2*y3*y4, meaning that
> it gets a -1 phase at any reflection operation ( det(U)=-1), but i
> dont get the relationship with dyon angle rotations
>
> the background of the question comes from a discussion occurred on
> this thread
> http://groups.google.co.ve/groups?hl=es&lr=&selm=2b93dd16.0408301338.1a0ac84a%40posting.google .com&rnum=14

The proper definition of y5 is

(1/24)[/itex] eps mnab ym yn ya yb

The \epsilon symbol requires a factor of \sqrt(det g) to be a tensor,
and that brings in a factor of i, so

y5 = i y1 y2 y3 y4

(Or i y0 y1 y2 y3 if you like, which is the negative of the "natural"
definition.)

As shown in another post, y5 \psi has matter and antimatter
interchanged. Your exponential gets an additional factor of i and so
is unitary. That transformation represents an angular mixing of matter
and antimatter,

\exp( 1/2 a iy5) \psi

so in the Dirac basis

\psi+ -> cos 1/2 a (\psi+) + i sin 1/2 a (\psi-)

The y's transform as

ym' = \exp (1/2 a iy5) ym \exp (-1/2 a iy5) = \exp(ia y5) ym

with a=\pi corresponding to inverting matter and antimatter. So this is
in a sense a continuous form of matter <-> antimatter.

We can now gauge this transformation - the Dirac-like equation becomes

[ ym (dm + ig y5 Bm) + im ] \psi =

where Bm is a pseudovectorial gauge field and g a coupling constant.
Note

psibar [ ym (dm - ig y5 Bm) - im ] =

The Bm term changes sign three times - once from i on Hermitian
conjugation, once from commuting y5 with ym_dagger, and once from
pulling through y4 to make psibar. So, analogous to the ordinary Dirac
equation, there is a conserved current

Km = psibar [itex]ym \psi

but the coupling is to the pseudovector:

Bm (psibar ym y5 \psi)

which is still a scalar because Bm is also a pseudovector.

The form of the coupling shows that this is *not* precisely the
electromagnetic duality - on adding an overall phase and gauging that,
we have an odd gauge field that lives in the Dirac algebra

e Am + g y5 Bm

Electromagnetic duality with a second "dual" potential corresponding
to magnetic charges would have led one to expect

e Am + g iy5 Bm

which would imply that Bm is purely imaginary.

If we wish to maintain a connection with electromagnetic duality, we
have to think of the potential itself as taking complex values. Thus,
gauging the symmetry of matter and antimatter seems to lead to the
idea of an intrinsically complex vector potential. You're welcome to
push this as far as you like - something interesting may emerge. The
objective would be to understand poles as an intermediate stage
between matter and antimatter:

electron -> pole -> positron -> antipole -> electron

as

-> \pi/2 -> \pi -> 3pi/2 -> 2pi

-drl

Thomas Mautsch

Oct15-04, 01:55 PM

<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>\nIn news:<9r4rm0du8vn1f87d5ok5mavhieos8huu86@4ax.com>\ nschrieb "Geometric Algebra Rocks!" <root@bubba.eldosales.com>:\n> On Wed, 13 Oct 2004 19:13:41 +0000 (UTC), erite423@yahoo.se (Erik) wrote:\n>>\n>>Basically, I wonder how Dirac spinors transform under spatial\n>>reflections and inversions.\n>\n> Forget spinors! They are simply a subset of Geometric Algebra and\n> should be treated as such. Start here to learn:\n>\n> http://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf\n>\n> At the end, you\'ll see spinors are not worth the paper they are\n> written on.\n\nIf spinors are "a subset of Geometric Algebra",\nhow come your primer on Geometric Algebra mentions\nthe word spinor only at one place, on page 10:\n\n"(Note: this representation of complex numbers in a real GA\nis a special case of spinors for 3d)."\n^^^^^^^^^^^^^^\n\nWould you mind explaining in more detail,\nhow spinors fit into Geometric Algebra.\n\n\nBy the way, the word "inversion" does not occur\nin the reference you gave, either, so your answer\ndoes not look very useful with regard to Erik\'s question.\n\nRegards,\nThomas\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>In news:<9r4rm0du8vn1f87d5ok5mavhieos8huu86@4ax.com>
schrieb "Geometric Algebra Rocks!" <root@bubba.eldosales.com>:
> On Wed, 13 Oct 2004 19:13:41 +0000 (UTC), erite423@yahoo.se (Erik) wrote:
>>
>>Basically, I wonder how Dirac spinors transform under spatial
>>reflections and inversions.
>
> Forget spinors! They are simply a subset of Geometric Algebra and
> should be treated as such. Start here to learn:
>
> http://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf
>
> At the end, you'll see spinors are not worth the paper they are
> written on.

If spinors are "a subset of Geometric Algebra",
how come your primer on Geometric Algebra mentions
the word spinor only at one place, on page 10:

"(Note: this representation of complex numbers in a real GA
is a special case of spinors for 3d)."
^^^^^^^^^^^^^^

Would you mind explaining in more detail,
how spinors fit into Geometric Algebra.

By the way, the word "inversion" does not occur
in the reference you gave, either, so your answer
does not look very useful with regard to Erik's question.

Regards,
Thomas

Erik

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>\nJohn T Lowry wrote:\n> POSSIBLY Sakurai\'s Advanced Quantum Mechanics, around pp 99-100 would\n> help. He treats both inversion (turns out to be with Gamma4) and what he\n> calls "mirror reflection" (parity plus a pi-rotation). I feel for you in\n> the general dearth of concrete examples in this stuff. My first Lorentz\n> boost took me the better part of a day to tie down all the loose ends.\n>\n> John T. Lowry\n> Flight Physics\n\nThanks. Those pages by Sakurai were very useful in that they gave the\ntransformation matrix for inversion and explicitly informed that the\nphase factor must be chosen arbitrarily. I remain a bit uncertain\nabout reflections, though, because the mapping f that I introduced\npreviously satisfies\n\nSXYspinor = f(SXYspace) = f(INVspace * RZspace) = +/- f(INVspace) *\nf(RZspace)\n\nand I\'m not sure if one choice of sign is necessary or conventional.\n\nRegards,\nErik\n\nPS. Thanks also to Danny Ross Lunsford for another useful answer. DS.\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>John T Lowry wrote:
> POSSIBLY Sakurai's Advanced Quantum Mechanics, around pp 99-100 would
> help. He treats both inversion (turns out to be with Gamma4) and what he
> calls "mirror reflection" (parity plus a \pi-rotation). I feel for you in
> the general dearth of concrete examples in this stuff. My first Lorentz
> boost took me the better part of a day to tie down all the loose ends.
>
> John T. Lowry
> Flight Physics

Thanks. Those pages by Sakurai were very useful in that they gave the
transformation matrix for inversion and explicitly informed that the
phase factor must be chosen arbitrarily. I remain a bit uncertain
about reflections, though, because the mapping f that I introduced
previously satisfies

SXYspinor = f(SXYspace) = f(INVspace * RZspace) = +/- f(INVspace) *
f(RZspace)

and I'm not sure if one choice of sign is necessary or conventional.

Regards,
Erik

PS. Thanks also to Danny Ross Lunsford for another useful answer. DS.