Discrete symmetries [Archive]

Gianni Mantovami

May6-04, 07:52 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>\nI\'m reading Weinberg\'s book on QFT. He says that a projective\nrepresentation of the identity component \\$L\\$ of the Lorentz group\ncorresponds to an ordinary representation of the universal covering\n\\$Sl_2(\\bC)\\$.\nI don\'t want to write phases so I decided to work with \\$Sl_2(\\bC)\\$.\nFor example I write:\n\\begin{equation}\nU(A,a) A \\in Sl_{2}(\\bC), a \\in \\bR^{4}\n\\end{equation}\nwhere Weinberg writes:\n\\begin{equation}\nU( \\Lambda, a) \\Lambda \\in L, a \\in \\bR^{4}\n\\end{equation}\nUnfortunately I\'m not able to introduce the discrete symmetries \\$T\\$\nand \\$P\\$. To be precise Weinberg writes the equations:\n\\begin{equation}\n\\mathsf{P} U(\\Lambda, a) \\mathsf{P^{-1}} = U( P \\Lambda P^{-1}, P a )\n\\end{equation}\n\\begin{equation}\n\\mathsf{T} U(\\Lambda, a) \\mathsf{T^{-1}} = U( T \\Lambda T^{-1}, T a )\n\\end{equation}\nI\'d like to do the previous substitution \\$\\Lambda \\mapsto A\\$, but I\'ve\ngot no definition for \\$P A P^{-1}\\$ and \\$T A T^{-1}\\$.\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>I'm reading Weinberg's book on QFT. He says that a projective
representation of the identity component $L$ of the Lorentz group
corresponds to an ordinary representation of the universal covering
$Sl_2(\bC)$.
I don't want to write phases so I decided to work with $Sl_2(\bC)$.
For example I write:
\begin{equation}U(A,a) A \in Sl_{2}(\bC), a \in \bR^{4}\end{equation}
where Weinberg writes:
\begin{equation}U( \Lambda, a) \Lambda \in L, a \in \bR^{4}\end{equation}
Unfortunately I'm not able to introduce the discrete symmetries $T$
and $P$. To be precise Weinberg writes the equations:
\begin{equation}\mathsf{P} U(\Lambda, a) \mathsf{P^{-1}} = U( P \Lambda P^{-1}, P a )\end{equation}\begin{equation}\mathsf{T} U(\Lambda, a) \mathsf{T^{-1}} = U( T \Lambda T^{-1}, T a )\end{equation}
I'd like to do the previous substitution $\Lambda \mapsto A$, but I've
got no definition for $P A P^{-1}$ and $T A T^{-1}$.

Uncle Al

May7-04, 06:39 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>Gianni Mantovami wrote:\n>\n> I\'m reading Weinberg\'s book on QFT. He says that a projective\n> representation of the identity component \\$L\\$ of the Lorentz group\n> corresponds to an ordinary representation of the universal covering\n> \\$Sl_2(\\bC)\\$.\n> I don\'t want to write phases so I decided to work with \\$Sl_2(\\bC)\\$.\n> For example I write:\n> \\begin{equation}\n> U(A,a) A \\in Sl_{2}(\\bC), a \\in \\bR^{4}\n> \\end{equation}\n> where Weinberg writes:\n> \\begin{equation}\n> U( \\Lambda, a) \\Lambda \\in L, a \\in \\bR^{4}\n> \\end{equation}\n> Unfortunately I\'m not able to introduce the discrete symmetries \\$T\\$\n> and \\$P\\$. To be precise Weinberg writes the equations:\n> \\begin{equation}\n> \\mathsf{P} U(\\Lambda, a) \\mathsf{P^{-1}} = U( P \\Lambda P^{-1}, P a )\n> \\end{equation}\n> \\begin{equation}\n> \\mathsf{T} U(\\Lambda, a) \\mathsf{T^{-1}} = U( T \\Lambda T^{-1}, T a )\n> \\end{equation}\n> I\'d like to do the previous substitution \\$\\Lambda \\mapsto A\\$, but I\'ve\n> got no definition for \\$P A P^{-1}\\$ and \\$T A T^{-1}\\$.\n\nBe *very* careful with parity. Time reversal can be approximated by a\nTaylor series in which you sneak up on it from behind in a continuous\nmanner. Parity is all or nothing - a reflection of all coordinates\nthrough the origin. Parity is an absolutely discrete symmetry that\ncannot be approximated by a Taylor series or a sum of infinitesimals.\nNoether\'s theorem with its dependence upon smooth Lie groups is\ninappropriate.\n\nA physical system with a Lagrangian invariant with respect to the\nsymmetry transformations of a Lie group has, in the case of a group\nwith a finite (or countably infinite) number of independent\ninfinitesimal generators, a conservation law for each such generator,\nand certain "dependencies" in the case of a larger infinite number of\ngenerators (General Relativity and the Bianchi identities). The\nreverse is true. Parity plays the game, but you have to dance\ndifferently. Noether\'s theorem is not applicable.\n\n--\nUncle Al\nhttp://www.mazepath.com/uncleal/qz.pdf\nhttp://www.mazepath.com/uncleal/eotvos.htm\n(Do something naughty to physics)\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>Gianni Mantovami wrote:
>
> I'm reading Weinberg's book on QFT. He says that a projective
> representation of the identity component $L$ of the Lorentz group
> corresponds to an ordinary representation of the universal covering
> $Sl_2(\bC)$.
> I don't want to write phases so I decided to work with $Sl_2(\bC)$.
> For example I write:
> \begin{equation}
> U(A,a) A \in Sl_{2}(\bC), a \in \bR^{4}
> \end{equation}
> where Weinberg writes:
> \begin{equation}
> U( \Lambda, a) \Lambda \in L, a \in \bR^{4}
> \end{equation}
> Unfortunately I'm not able to introduce the discrete symmetries $T$
> and $P$. To be precise Weinberg writes the equations:
> \begin{equation}
> \mathsf{P} U(\Lambda, a) \mathsf{P^{-1}} = U( P \Lambda P^{-1}, P a )
> \end{equation}
> \begin{equation}
> \mathsf{T} U(\Lambda, a) \mathsf{T^{-1}} = U( T \Lambda T^{-1}, T a )
> \end{equation}
> I'd like to do the previous substitution $\Lambda \mapsto A$, but I've
> got no definition for $P A P^{-1}$ and $T A T^{-1}$.

Be *very* careful with parity. Time reversal can be approximated by a
Taylor series in which you sneak up on it from behind in a continuous
manner. Parity is all or nothing - a reflection of all coordinates
through the origin. Parity is an absolutely discrete symmetry that
cannot be approximated by a Taylor series or a sum of infinitesimals.
Noether's theorem with its dependence upon smooth Lie groups is
inappropriate.

A physical system with a Lagrangian invariant with respect to the
symmetry transformations of a Lie group has, in the case of a group
with a finite (or countably infinite) number of independent
infinitesimal generators, a conservation law for each such generator,
and certain "dependencies" in the case of a larger infinite number of
generators (General Relativity and the Bianchi identities). The
reverse is true. Parity plays the game, but you have to dance
differently. Noether's theorem is not applicable.

--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)

Alfred Einstead

May12-04, 05:05 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>\nUncle Al <UncleAl0@hate.spam.net> wrote:\n> Be *very* careful with parity. Time reversal can be approximated by a\n> Taylor series in which you sneak up on it from behind in a continuous\n> manner. Parity is all or nothing - a reflection of all coordinates\n> through the origin. Parity is an absolutely discrete symmetry that\n> cannot be approximated by a Taylor series or a sum of infinitesimals.\n> Noether\'s theorem with its dependence upon smooth Lie groups is\n> inappropriate.\n\nNonetheless, you can still gauge on a discrete symmetry group.\nThe resulting gauge boson and Yang-Mills Lagrangian for the\nZ_2 group for parity P is just the Higgs and the Higgs sector\nof the Standard Model (quartic potential and all).\n\nThat\'s the essence of the discovery made by Connes and Lott.\n\nThe Standard Model, a\' la Connes-Lott, Kastler et. al.\nhep-th/9412185\n\nA Detailed Account of Alain Connes\' Version Of The Standard Model IV\nKastler, et. al.\nhep-th/9501077\n\nYang-Mills-Higgs vs. Connes-Lott\nIochum & Schuecker\nhep-th/9501042\nAbstract:\n"By suitable choice of variables, we show that every Connes-Lott\nmodel is a Yang-Mills-Higgs model. The contrary is far from\nbeing true. Necessary conditions are given. Our analysis is\npedestrian and is illustrated by examples."\n\nAnomaly Cancellation and the gauge group of the Standard Model\nin [Non-Commutative Geometry]\nAlvarez\nhep-th/9506115\n\nDerives the charge assignments of the U(1) sector, subsuming the\nusual anomaly cancellation argument posed in the Standard Model.\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>Uncle Al <UncleAl0@hate.spam.net> wrote:
> Be *very* careful with parity. Time reversal can be approximated by a
> Taylor series in which you sneak up on it from behind in a continuous
> manner. Parity is all or nothing - a reflection of all coordinates
> through the origin. Parity is an absolutely discrete symmetry that
> cannot be approximated by a Taylor series or a sum of infinitesimals.
> Noether's theorem with its dependence upon smooth Lie groups is
> inappropriate.

Nonetheless, you can still gauge on a discrete symmetry group.
The resulting gauge boson and Yang-Mills Lagrangian for the
Z_2 group for parity P is just the Higgs and the Higgs sector
of the Standard Model (quartic potential and all).

That's the essence of the discovery made by Connes and Lott.

The Standard Model, a' la Connes-Lott, Kastler et. al.
http://www.arxiv.org/abs/hep-th/9412185

A Detailed Account of Alain Connes' Version Of The Standard Model IV
Kastler, et. al.
http://www.arxiv.org/abs/hep-th/9501077

Yang-Mills-Higgs vs. Connes-Lott
Iochum & Schuecker
http://www.arxiv.org/abs/hep-th/9501042
Abstract:
"By suitable choice of variables, we show that every Connes-Lott
model is a Yang-Mills-Higgs model. The contrary is far from
being true. Necessary conditions are given. Our analysis is
pedestrian and is illustrated by examples."

Anomaly Cancellation and the gauge group of the Standard Model
in [Non-Commutative Geometry]
Alvarez
http://www.arxiv.org/abs/hep-th/9506115

Derives the charge assignments of the U(1) sector, subsuming the
usual anomaly cancellation argument posed in the Standard Model.

Arnold Neumaier

May12-04, 06:36 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\nAlfred Einstead wrote:\n\n> Nonetheless, you can still gauge on a discrete symmetry group.\n> The resulting gauge boson and Yang-Mills Lagrangian for the\n> Z_2 group for parity P is just the Higgs and the Higgs sector\n> of the Standard Model (quartic potential and all).\n\nAnd what would you get if you gauge C or T?\n\n\nArnold Neumaier\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>Alfred Einstead wrote:

> Nonetheless, you can still gauge on a discrete symmetry group.
> The resulting gauge boson and Yang-Mills Lagrangian for the
> Z_2 group for parity P is just the Higgs and the Higgs sector
> of the Standard Model (quartic potential and all).

And what would you get if you gauge C or T?

Arnold Neumaier

Hendrik van Hees

May12-04, 01:37 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>Uncle Al wrote:\n\n> Be *very* careful with parity. Time reversal can be approximated by a\n> Taylor series in which you sneak up on it from behind in a continuous\n> manner. Parity is all or nothing - a reflection of all coordinates\n> through the origin. Parity is an absolutely discrete symmetry that\n> cannot be approximated by a Taylor series or a sum of infinitesimals.\n> Noether\'s theorem with its dependence upon smooth Lie groups is\n> inappropriate.\n\nIn any case, it\'s good advice to be very careful with the discrete part of\nthe full Poincare group, but could you comment in more detail on your\nsuggestion to represent time reversal by a Taylor series?\n\nIn the Lorentz group\'s fundamental representation, at least, it cannot be\ncontinuously connected to the identity, since its determinant is -1, and\nthe determinant of Lorentz transformation matrices is \\pm 1.\n\nFurther, in the quantum context, we have to represent time reversal as an\nantiunitary rather than a unitary tranformation, since there does not exist\na unitary one due to the fact that we have only positive energies. It\'s one\nof the fundamental assumptions of qft to have a Hamiltonian, which is\nbounded from below, i.e., to have a stable ground state.\n>\n> A physical system with a Lagrangian invariant with respect to the\n> symmetry transformations of a Lie group has, in the case of a group\n> with a finite (or countably infinite) number of independent\n> infinitesimal generators, a conservation law for each such generator,\n> and certain "dependencies" in the case of a larger infinite number of\n> generators (General Relativity and the Bianchi identities). The\n> reverse is true. Parity plays the game, but you have to dance\n> differently. Noether\'s theorem is not applicable.\n\nThat\'s surely true. But, if your suggestion is right (which I doubt for the\nabove given reasons), and time reversal symmetry can be seen as\ninfinitesimally generated, what is then the meaning of the conserved\nNoether current due to this generator?\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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>Uncle Al wrote:

> Be *very* careful with parity. Time reversal can be approximated by a
> Taylor series in which you sneak up on it from behind in a continuous
> manner. Parity is all or nothing - a reflection of all coordinates
> through the origin. Parity is an absolutely discrete symmetry that
> cannot be approximated by a Taylor series or a sum of infinitesimals.
> Noether's theorem with its dependence upon smooth Lie groups is
> inappropriate.

In any case, it's good advice to be very careful with the discrete part of
the full Poincare group, but could you comment in more detail on your
suggestion to represent time reversal by a Taylor series?

In the Lorentz group's fundamental representation, at least, it cannot be
continuously connected to the identity, since its determinant is -1, and
the determinant of Lorentz transformation matrices is \pm 1.

Further, in the quantum context, we have to represent time reversal as an
antiunitary rather than a unitary tranformation, since there does not exist
a unitary one due to the fact that we have only positive energies. It's one
of the fundamental assumptions of qft to have a Hamiltonian, which is
bounded from below, i.e., to have a stable ground state.
>
> A physical system with a Lagrangian invariant with respect to the
> symmetry transformations of a Lie group has, in the case of a group
> with a finite (or countably infinite) number of independent
> infinitesimal generators, a conservation law for each such generator,
> and certain "dependencies" in the case of a larger infinite number of
> generators (General Relativity and the Bianchi identities). The
> reverse is true. Parity plays the game, but you have to dance
> differently. Noether's theorem is not applicable.

That's surely true. But, if your suggestion is right (which I doubt for the
above given reasons), and time reversal symmetry can be seen as
infinitesimally generated, what is then the meaning of the conserved
Noether current due to this generator?

--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366

Jeroen

May12-04, 05:23 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>Arnold Neumaier wrote:\n\n>\n>\n> Alfred Einstead wrote:\n>\n>> Nonetheless, you can still gauge on a discrete symmetry group.\n>> The resulting gauge boson and Yang-Mills Lagrangian for the\n>> Z_2 group for parity P is just the Higgs and the Higgs sector\n>> of the Standard Model (quartic potential and all).\n>\n> And what would you get if you gauge C or T?\n>\n\nGauging C leads to so-called Alice electrodynamics and Ceshire charges\n(afaik). Can\'t tell you much more.\n\nbest,\nJeroen\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>Arnold Neumaier wrote:

>
>
> Alfred Einstead wrote:
>
>> Nonetheless, you can still gauge on a discrete symmetry group.
>> The resulting gauge boson and Yang-Mills Lagrangian for the
>> Z_2 group for parity P is just the Higgs and the Higgs sector
>> of the Standard Model (quartic potential and all).
>
> And what would you get if you gauge C or T?
>

Gauging C leads to so-called Alice electrodynamics and Ceshire charges
(afaik). Can't tell you much more.

best,
Jeroen

Uncle Al

May13-04, 05:25 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>\nAlfred Einstead wrote:\n>\n> Uncle Al <UncleAl0@hate.spam.net> wrote:\n> > Be *very* careful with parity. Time reversal can be approximated by a\n> > Taylor series in which you sneak up on it from behind in a continuous\n> > manner. Parity is all or nothing - a reflection of all coordinates\n> > through the origin. Parity is an absolutely discrete symmetry that\n> > cannot be approximated by a Taylor series or a sum of infinitesimals.\n> > Noether\'s theorem with its dependence upon smooth Lie groups is\n> > inappropriate.\n>\n> Nonetheless, you can still gauge on a discrete symmetry group.\n> The resulting gauge boson and Yang-Mills Lagrangian for the\n> Z_2 group for parity P is just the Higgs and the Higgs sector\n> of the Standard Model (quartic potential and all).\n>\n> That\'s the essence of the discovery made by Connes and Lott.\n>\n> The Standard Model, a\' la Connes-Lott, Kastler et. al.\n> hep-th/9412185\n>\n> A Detailed Account of Alain Connes\' Version Of The Standard Model IV\n> Kastler, et. al.\n> hep-th/9501077\n>\n> Yang-Mills-Higgs vs. Connes-Lott\n> Iochum & Schuecker\n> hep-th/9501042\n> Abstract:\n> "By suitable choice of variables, we show that every Connes-Lott\n> model is a Yang-Mills-Higgs model. The contrary is far from\n> being true. Necessary conditions are given. Our analysis is\n> pedestrian and is illustrated by examples."\n>\n> Anomaly Cancellation and the gauge group of the Standard Model\n> in [Non-Commutative Geometry]\n> Alvarez\n> hep-th/9506115\n>\n> Derives the charge assignments of the U(1) sector, subsuming the\n> usual anomaly cancellation argument posed in the Standard Model.\n\nThanks for the references!\n\nI have continuously maintained that there is no lack of theory arguing\nall sides of gravitation, including straddling it lengthwise. It is\ntherefore of paramount importance that an extremal (e.g., in\nalpha-quartz) parity Eotvos experiment be performed. We don\'t need\nmore theory, we need severe empirical pruning of what we already have.\n\nParity is the universe\'s "gotcha!" Fundamental physics would be\nremarkably clean except for parity violations. Parity violations are\nendemic in weak (and, obviously, Weak) interactions. Parity effects\nhave historically been a very reluctant admission every step of the\nway. Gravitation is ripe for the "gotcha!"\n\n--\nUncle Al\nhttp://www.mazepath.com/uncleal/qz.pdf\nhttp://www.mazepath.com/uncleal/eotvos.htm\n(Do something naughty to 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>Alfred Einstead wrote:
>
> Uncle Al <UncleAl0@hate.spam.net> wrote:
> > Be *very* careful with parity. Time reversal can be approximated by a
> > Taylor series in which you sneak up on it from behind in a continuous
> > manner. Parity is all or nothing - a reflection of all coordinates
> > through the origin. Parity is an absolutely discrete symmetry that
> > cannot be approximated by a Taylor series or a sum of infinitesimals.
> > Noether's theorem with its dependence upon smooth Lie groups is
> > inappropriate.
>
> Nonetheless, you can still gauge on a discrete symmetry group.
> The resulting gauge boson and Yang-Mills Lagrangian for the
> Z_2 group for parity P is just the Higgs and the Higgs sector
> of the Standard Model (quartic potential and all).
>
> That's the essence of the discovery made by Connes and Lott.
>
> The Standard Model, a' la Connes-Lott, Kastler et. al.
> http://www.arxiv.org/abs/hep-th/9412185
>
> A Detailed Account of Alain Connes' Version Of The Standard Model IV
> Kastler, et. al.
> http://www.arxiv.org/abs/hep-th/9501077
>
> Yang-Mills-Higgs vs. Connes-Lott
> Iochum & Schuecker
> http://www.arxiv.org/abs/hep-th/9501042
> Abstract:
> "By suitable choice of variables, we show that every Connes-Lott
> model is a Yang-Mills-Higgs model. The contrary is far from
> being true. Necessary conditions are given. Our analysis is
> pedestrian and is illustrated by examples."
>
> Anomaly Cancellation and the gauge group of the Standard Model
> in [Non-Commutative Geometry]
> Alvarez
> http://www.arxiv.org/abs/hep-th/9506115
>
> Derives the charge assignments of the U(1) sector, subsuming the
> usual anomaly cancellation argument posed in the Standard Model.

Thanks for the references!

I have continuously maintained that there is no lack of theory arguing
all sides of gravitation, including straddling it lengthwise. It is
therefore of paramount importance that an extremal (e.g., in\alpha-quartz) parity Eotvos experiment be performed. We don't need
more theory, we need severe empirical pruning of what we already have.

Parity is the universe's "gotcha!" Fundamental physics would be
remarkably clean except for parity violations. Parity violations are
endemic in weak (and, obviously, Weak) interactions. Parity effects
have historically been a very reluctant admission every step of the
way. Gravitation is ripe for the "gotcha!"

--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)