Broken symmetry representation [Archive]

L.R.

Oct20-04, 03:18 AM

Igor Khavkine

Oct22-04, 12:40 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\nOn Wed, 20 Oct 2004 08:18:36 +0000, L.R. wrote:\n\n> The fundamental theory of Wigner says that any symmetry transformation can\n> be represented on the Hilbert space of physical state by an operator that\n> is linear and unitary(except the weird time reversal which is anti-unitary\n> and anti-linear), so the unitary operators play a special role in qft, but\n> even the symmetry is broken, like parity or time reversal, we still use a\n> unitary or anti-unitary operator to represent it on the Hilbert space. Is\n> there some justification about this? I mean since parity is now not a\n> symmetry, can we use some non-unitary operator, can we get any new/strange\n> result by doing this?\n\nI can rotate both a sphere and a hammer by using the same operation. The\ndifference between them is that a sphere looks the same after rotation as\nit did before, but the hammer does not.\n\nThere is a difference between "symmetry" and "symmetry operation" or some\nsimilar terms. The sphere is symmetric under rotation, but a hammer is\nnot. Yet in both cases the rotation is the same and is referred to as a\nsymmetry operation, that is it looks like the action of a group with all\nthe implied properties.\n\nWigner arguments apply to symmetry operations, they are independent of\nwhether the state they are acting on is actually symmetric under these\noperations or not.\n\nHope this helps.\n\nIgor\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>On Wed, 20 Oct 2004 08:18:36 +0000, L.R. wrote:

> The fundamental theory of Wigner says that any symmetry transformation can
> be represented on the Hilbert space of physical state by an operator that
> is linear and unitary(except the weird time reversal which is anti-unitary
> and anti-linear), so the unitary operators play a special role in qft, but
> even the symmetry is broken, like parity or time reversal, we still use a
> unitary or anti-unitary operator to represent it on the Hilbert space. Is
> there some justification about this? I mean since parity is now not a
> symmetry, can we use some non-unitary operator, can we get any new/strange
> result by doing this?

I can rotate both a sphere and a hammer by using the same operation. The
difference between them is that a sphere looks the same after rotation as
it did before, but the hammer does not.

There is a difference between "symmetry" and "symmetry operation" or some
similar terms. The sphere is symmetric under rotation, but a hammer is
not. Yet in both cases the rotation is the same and is referred to as a
symmetry operation, that is it looks like the action of a group with all
the implied properties.

Wigner arguments apply to symmetry operations, they are independent of
whether the state they are acting on is actually symmetric under these
operations or not.

Hope this helps.

Igor