question concerning SU(2) summetry for electron-neutrino doublet [Archive]

Shuang Meng

Feb23-05, 04:01 AM

Cl.Massé

Mar1-05, 02:11 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>"Shuang Meng" <smeng2004@yahoo.com> a écrit dans le message de\nnews:20050222003202.19083.qmail@web61103.mail. yahoo.com...\n\n> In electroweak theory, electron and neutrino form a SU(2) doublet and\n> transform under the SU(2) gauge group.\n>\n> My question is, how you would expect the Dirac equation for electron\n> and neutrino to hold under a local SU(2) transformation. It seems to\n> me that if the transformation coefficients are space-time independent,\n> then the Dirac equation would no longer hold f or the new\n> electron/neutrino fields under such a transformation. Then why SU(2)\n> is considered as a local symmetry for the electron-neutrino doublet?\n>\n> I know this may be a simple question. But I couldn\'t find any book or\n> resource elaborating on this matter. And I hope some experts here can\n> help to shed light on this.\n\nIt\'s true this aspect is rarely well treated.\n\nI suppose you already heard that the isospin is an internal symmetry.\nBut what does that mean? This is easy to see in the case of the Dirac\nequation. Instead of one spinor, you have two spinors that obey\nindependently the Dirac equation. They can be transformed locally in\none another. Concretely, the total spinor has two columns instead of\none, and the SU(2) 2x2 matrix act on it *on the right*.\n\nTo the contrary of a spatial, non-internal symmetry, rotating the frame\nof reference doesn\'t imply a rotation in the isospin space, but implies\nan action of the corresponding SU(2) matrix *on the left*, and therefore\na transformation of the Dirac matrices.\n\nThat said, the gauge interaction run analogously to the electromagnetic\ninteraction. For the latter, imagine that each complex component of the\nspinor in fact is a two dimensional horizontal real vector, and that the\nU(1) 2x2 real matrix acts on it *on the right*. The Dirac equation then\nbecomes two independent real equations for each column.\n\n--\n~~~~ clmasse on free F-country\nLiberty, Equality, Profitability.\n\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>"Shuang Meng" <smeng2004@yahoo.com> a écrit dans le message de
news:20050222003202.19083.qmail@web61103.mail.yaho o.com...

> In electroweak theory, electron and neutrino form a SU(2) doublet and
> transform under the SU(2) gauge group.
>
> My question is, how you would expect the Dirac equation for electron
> and neutrino to hold under a local SU(2) transformation. It seems to
> me that if the transformation coefficients are space-time independent,
> then the Dirac equation would no longer hold f or the new
> electron/neutrino fields under such a transformation. Then why SU(2)
> is considered as a local symmetry for the electron-neutrino doublet?
>
> I know this may be a simple question. But I couldn't find any book or
> resource elaborating on this matter. And I hope some experts here can
> help to shed light on this.

It's true this aspect is rarely well treated.

I suppose you already heard that the isospin is an internal symmetry.
But what does that mean? This is easy to see in the case of the Dirac
equation. Instead of one spinor, you have two spinors that obey
independently the Dirac equation. They can be transformed locally in
one another. Concretely, the total spinor has two columns instead of
one, and the SU(2) 2x2 matrix act on it *on the right*.

To the contrary of a spatial, non-internal symmetry, rotating the frame
of reference doesn't imply a rotation in the isospin space, but implies
an action of the corresponding SU(2) matrix *on the left*, and therefore
a transformation of the Dirac matrices.

That said, the gauge interaction run analogously to the electromagnetic
interaction. For the latter, imagine that each complex component of the
spinor in fact is a two dimensional horizontal real vector, and that the
U(1) 2x2 real matrix acts on it *on the right*. The Dirac equation then
becomes two independent real equations for each column.

--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability.