[SOLVED] gauge theory [Archive]

Tlas

Apr8-04, 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>Hi everyone,\n\nDoes anybody know what is the physical motivation behind demanding\nlocal gauge invariance? I mean the only argument I have seen is\n"because it works" one. Is there something better based on locality\nfor example?\nThanks for input\n\nT.T.\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>Hi everyone,

Does anybody know what is the physical motivation behind demanding
local gauge invariance? I mean the only argument I have seen is
"because it works" one. Is there something better based on locality
for example?
Thanks for input

T.T.

Hendrik van Hees

Apr9-04, 02:22 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>Tlas wrote:\n\n> Hi everyone,\n>\n> Does anybody know what is the physical motivation behind demanding\n> local gauge invariance? I mean the only argument I have seen is\n> "because it works" one. Is there something better based on locality\n> for example?\n> Thanks for input\n\nIt is a nice idea to build particle theories from symmetry principles.\n\nStart with the Poincare symmetry of space time within special\nrelativity. In fact, you can reconstract the space time model from its\nsymmetries. This idea goes back to Felix Klein\'s "Erlanger Programm",\nwhere he looked on geometries from the standpoint of their\nclassification according to symmetry principles.\n\nImho, this concept is unavoidable when you like to construct a quantum\ntheory without "handwaving arguments" like what is known "canonical\nquantisation", which goes wrong for relatively simple systems (like the\nspinning top).\n\nSo, if you start from Minkowski space and its symmetry group, the\nPoincare group, and like to build a quantum theory, one is lead via the\nWigner Bargmann theorem, which states that instead of the classical\ngroup you have to look on its covering group and its central\nextensions.\n\nThe bottom line of this very formal argument is, that you have to\nsubstitute the Lorentz group (generated by boosts and rotations) by its\ncovering group which is SL(2,C).\n\nThen you can also show that any ray representation is up to equivalence\ninduced by a unitary representation on Hilbert space.\n\nThus, you have to investigate the physically reasonable unitary\nrepresentations of this covering of the Poincare group (in the\nfollowing I write simply Poincare group for it) on a Hilbert space.\n\nEach elementary particle is then described by an irreducible\nrepresentation of the Poincare group, which is merely the definition of\nwhat we understand under an elementary particle.\n\nYou can systematically look for these irreducible representations, which\nwas done first by Wigner in 1939.\n\nIt turns out that there are two rough classes of different\nrepresentations, which make physically sense at the end of the day,\nnamely those describing particles with a rest mass >0 and those\ndescribing particles with 0 mass.\n\nIn the case of massive particles the second important parameter of the\ngroup representation is, how the Hilbert space states for a particle at\nrest behave under rotations. This defines its spin, which can be 0,\n1/2, 1, etc.\n\nFor massless particles the notion of spin doesn\'t make sense, because\nthere is no rest frame for them. In the formalism of the representation\ntheory you\'ll never make this mistake :-)): It turns out that here the\nsocalled helicity is the analogon for the spin. For massive particles\nthe helicity is the projection of the spin angular momentum vector to\nthe direction of the momentum. In the case of massive particles you can\nchange the helicity by "overtaking" the particle (i.e. being faster\nthan the particle), i.e., it is dependent on the frame of reference.\nSince for massless particles this is impossible, the helicity is a\nframe-independent statement. The helicity of massless particles can\nalso take all half integer or integer values, like the spin for massive\nparticles. Unfortunately quite often the physicists also call the\nhelicity of massless particles "spin". You should memorize that this is\nsloppy.\n\nThe formalism of the Poincare group representations tells you further\nthat you have to describe massless particles with helicities >=0 as\ngauge fields if you do not run into trouble with continuously many\nintrinsic degrees of freedom (that\'s what is known as the "null\nrotations" within the little group, which in the standard qft is\ndescribed always trivially in order to prevent the existence of\nparticles with continuously many intrinsic degrees of freedom; I\'m not\nsure whether representations where these null rotations are described\nnon-trivially make any physical sense at all, but perhaps there is\nliterature about it).\n\nThe representation theory of the Poincaregroup is presented nicely by\nWeinberg in his books on qft (Quantum theory of fields, Vol. I).\n\nYou find it also in my qft script, downloadable from my home page.\n\n\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>Tlas wrote:

> Hi everyone,
>
> Does anybody know what is the physical motivation behind demanding
> local gauge invariance? I mean the only argument I have seen is
> "because it works" one. Is there something better based on locality
> for example?
> Thanks for input

It is a nice idea to build particle theories from symmetry principles.

Start with the Poincare symmetry of space time within special
relativity. In fact, you can reconstract the space time model from its
symmetries. This idea goes back to Felix Klein's "Erlanger Programm",
where he looked on geometries from the standpoint of their
classification according to symmetry principles.

Imho, this concept is unavoidable when you like to construct a quantum
theory without "handwaving arguments" like what is known "canonical
quantisation", which goes wrong for relatively simple systems (like the
spinning top).

So, if you start from Minkowski space and its symmetry group, the
Poincare group, and like to build a quantum theory, one is lead via the
Wigner Bargmann theorem, which states that instead of the classical
group you have to look on its covering group and its central
extensions.

The bottom line of this very formal argument is, that you have to
substitute the Lorentz group (generated by boosts and rotations) by its
covering group which is SL(2,C).

Then you can also show that any ray representation is up to equivalence
induced by a unitary representation on Hilbert space.

Thus, you have to investigate the physically reasonable unitary
representations of this covering of the Poincare group (in the
following I write simply Poincare group for it) on a Hilbert space.

Each elementary particle is then described by an irreducible
representation of the Poincare group, which is merely the definition of
what we understand under an elementary particle.

You can systematically look for these irreducible representations, which
was done first by Wigner in 1939.

It turns out that there are two rough classes of different
representations, which make physically sense at the end of the day,
namely those describing particles with a rest mass >0 and those
describing particles with mass.

In the case of massive particles the second important parameter of the
group representation is, how the Hilbert space states for a particle at
rest behave under rotations. This defines its spin, which can be 0,
1/2, 1, etc.

For massless particles the notion of spin doesn't make sense, because
there is no rest frame for them. In the formalism of the representation
theory you'll never make this mistake :-)): It turns out that here the
socalled helicity is the analogon for the spin. For massive particles
the helicity is the projection of the spin angular momentum vector to
the direction of the momentum. In the case of massive particles you can
change the helicity by "overtaking" the particle (i.e. being faster
than the particle), i.e., it is dependent on the frame of reference.
Since for massless particles this is impossible, the helicity is a
frame-independent statement. The helicity of massless particles can
also take all half integer or integer values, like the spin for massive
particles. Unfortunately quite often the physicists also call the
helicity of massless particles "spin". You should memorize that this is
sloppy.

The formalism of the Poincare group representations tells you further
that you have to describe massless particles with helicities >=0 as
gauge fields if you do not run into trouble with continuously many
intrinsic degrees of freedom (that's what is known as the "null
rotations" within the little group, which in the standard qft is
described always trivially in order to prevent the existence of
particles with continuously many intrinsic degrees of freedom; I'm not
sure whether representations where these null rotations are described
non-trivially make any physical sense at all, but perhaps there is
literature about it).

The representation theory of the Poincaregroup is presented nicely by
Weinberg in his books on qft (Quantum theory of fields, Vol. I).

You find it also in my qft script, downloadable from my home page.

--
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

Danny Ross Lunsford

Apr9-04, 04:13 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>Tlas wrote:\n> Hi everyone,\n>\n> Does anybody know what is the physical motivation behind demanding\n> local gauge invariance? I mean the only argument I have seen is\n> "because it works" one. Is there something better based on locality\n> for example?\n> Thanks for input\n>\n> T.T.\n\nYes - the gauge invariance frees up just the right number of degrees of\nfreedom in the "mediating" field, to allow an invariant-theoretic\ndescription of the matter that interacts via this field. Basically, it\nsets up interacting systems that have non-trivial solutions.\n\n-drl\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>Tlas wrote:
> Hi everyone,
>
> Does anybody know what is the physical motivation behind demanding
> local gauge invariance? I mean the only argument I have seen is
> "because it works" one. Is there something better based on locality
> for example?
> Thanks for input
>
> T.T.

Yes - the gauge invariance frees up just the right number of degrees of
freedom in the "mediating" field, to allow an invariant-theoretic
description of the matter that interacts via this field. Basically, it
sets up interacting systems that have non-trivial solutions.

-drl

Arkadiusz Jadczyk

Apr12-04, 10:06 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\n\nOn Thu, 8 Apr 2004 22:39:24 +0000 (UTC), timtlas@hotmail.com (Tlas)\nwrote:\n\n>Hi everyone,\n>\n>Does anybody know what is the physical motivation behind demanding\n>local gauge invariance? I mean the only argument I have seen is\n>"because it works" one. Is there something better based on locality\n>for example?\n>Thanks for input\n\nOne relevant observation is, perhaps, that it comes naturally from\ngeneral coordinate invariance (or diffeomorphism invariance) in\nKaluza-Klein theories, where the gauge fields, instead of being\npostulated, come from a Riemannian metric (or from a vielbein, or a\nconnection) in spaces\nof more than four dimensions.\n\nark\n\n--\n\nArkadiusz Jadczyk\nhttp://www.cassiopaea.org/quantum_future/homepage.htm\n\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 Thu, 8 Apr 2004 22:39:24 +0000 (UTC), timtlas@hotmail.com (Tlas)
wrote:

>Hi everyone,
>
>Does anybody know what is the physical motivation behind demanding
>local gauge invariance? I mean the only argument I have seen is
>"because it works" one. Is there something better based on locality
>for example?
>Thanks for input

One relevant observation is, perhaps, that it comes naturally from
general coordinate invariance (or diffeomorphism invariance) in
Kaluza-Klein theories, where the gauge fields, instead of being
postulated, come from a Riemannian metric (or from a vielbein, or a
connection) in spaces
of more than four dimensions.

ark

--

Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm

--

Danny Ross Lunsford

Apr13-04, 11:29 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\n\nArkadiusz Jadczyk <arkREMOVETHIS@ANDTHIScassiopaea.org> wrote in message news:<scsk70h03vkccca37fen7ivcvrckbn237c@4ax.com>. ..\n\n> One relevant observation is, perhaps, that it comes naturally from\n> general coordinate invariance (or diffeomorphism invariance) in\n> Kaluza-Klein theories, where the gauge fields, instead of being\n> postulated, come from a Riemannian metric (or from a vielbein, or a\n> connection) in spaces\n> of more than four dimensions.\n\nIMO this is not a good correspondence. The gauge symmetry is\nincidental. Any generally covariant theory, including the ones with a\ngauge invariance, but also those without, can be cast in Kaluza\'s\nform. Kaluza-like theories all suffer from being inessential, formal\nunification without dynamical content.\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>Arkadiusz Jadczyk <arkREMOVETHIS@ANDTHIScassiopaea.org> wrote in message news:<scsk70h03vkccca37fen7ivcvrckbn237c@4ax.com>...

> One relevant observation is, perhaps, that it comes naturally from
> general coordinate invariance (or diffeomorphism invariance) in
> Kaluza-Klein theories, where the gauge fields, instead of being
> postulated, come from a Riemannian metric (or from a vielbein, or a
> connection) in spaces
> of more than four dimensions.

IMO this is not a good correspondence. The gauge symmetry is
incidental. Any generally covariant theory, including the ones with a
gauge invariance, but also those without, can be cast in Kaluza's
form. Kaluza-like theories all suffer from being inessential, formal
unification without dynamical content.

-drl

JW

Apr13-04, 05:47 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>Danny Ross Lunsford wrote:\n\n> Tlas wrote:\n>> Hi everyone,\n>>\n>> Does anybody know what is the physical motivation behind demanding\n>> local gauge invariance? I mean the only argument I have seen is\n>> "because it works" one. Is there something better based on locality\n>> for example?\n>> Thanks for input\n>>\n>> T.T.\n>\n> Yes - the gauge invariance frees up just the right number of degrees of\n> freedom in the "mediating" field, to allow an invariant-theoretic\n\nThis is cyclic reasoning. If you start with a gauge potential, you run into\ntrouble, but gauge invariance will also rescue you.\n\n> description of the matter that interacts via this field. Basically, it\n> sets up interacting systems that have non-trivial solutions.\n\nHow is this a *physical* motivation, i.e. one that can be directly motivated\nfrom experimental evidence? Mind you, gauge invariance is supposed to be\nhidden from us as configurations related by a gauge transformation are\ndemanded to be physically equivalent. That\'s the whole point of gauge\ninvariance.\n\nUsually, in the end, one calculates gauge invariant observables and the\ngauge symmetry is not seen directly. Rather the model is considered valid\n(hence indirect evidence for gauge invariance), because its predictions are\nborn out by experiment.\n\nThe only thing I can think of right now is the Aharanov-Bohm experiment. In\nthis experiment it turns out that the vector potential (a gauge variant\nobject) is more fundamental than the field strength.\n\nbest,\nJeroen\n\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>Danny Ross Lunsford wrote:

> Tlas wrote:
>> Hi everyone,
>>
>> Does anybody know what is the physical motivation behind demanding
>> local gauge invariance? I mean the only argument I have seen is
>> "because it works" one. Is there something better based on locality
>> for example?
>> Thanks for input
>>
>> T.T.
>
> Yes - the gauge invariance frees up just the right number of degrees of
> freedom in the "mediating" field, to allow an invariant-theoretic

This is cyclic reasoning. If you start with a gauge potential, you run into
trouble, but gauge invariance will also rescue you.

> description of the matter that interacts via this field. Basically, it
> sets up interacting systems that have non-trivial solutions.

How is this a *physical* motivation, i.e. one that can be directly motivated
from experimental evidence? Mind you, gauge invariance is supposed to be
hidden from us as configurations related by a gauge transformation are
demanded to be physically equivalent. That's the whole point of gauge
invariance.

Usually, in the end, one calculates gauge invariant observables and the
gauge symmetry is not seen directly. Rather the model is considered valid
(hence indirect evidence for gauge invariance), because its predictions are
born out by experiment.

The only thing I can think of right now is the Aharanov-Bohm experiment. In
this experiment it turns out that the vector potential (a gauge variant
object) is more fundamental than the field strength.

best,
Jeroen

Danny Ross Lunsford

Apr14-04, 03:17 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>JW wrote:\n\n> Danny Ross Lunsford wrote:\n>>Yes - the gauge invariance frees up just the right number of degrees of\n>>freedom in the "mediating" field, to allow an invariant-theoretic\n>>description of the matter that interacts via this field. Basically, it\n>>sets up interacting systems that have non-trivial solutions.\n>\n> This is cyclic reasoning. If you start with a gauge potential, you run into\n> trouble, but gauge invariance will also rescue you.\n\nWell I don\'t see this - the objective is 1) to have a description of\nmatter that leads to the idea of a conserved quantity 2) to allow it to\ninteract with other matter of the same type via some mediating field.\nConservation implies symmetry (of the Lagrangian), and interaction\nimplies sufficient degrees of freedom to carry interaction in the\nmediating field.\n\n> How is this a *physical* motivation, i.e. one that can be directly motivated\n> from experimental evidence?\n\nBecause the physical evidence amounts to conservation laws imputed from\nanalysis of laboratory scattering experiments. Field theory based on\ngauge invariance is then a phenomenological response to finding a\ndescription of these conservation laws.\n\n> Mind you, gauge invariance is supposed to be\n> hidden from us as configurations related by a gauge transformation are\n> demanded to be physically equivalent. That\'s the whole point of gauge\n> invariance.\n\nAgain, the point of gauge invariance is to provide the framework of\nsymmetry that leads to conservation laws.\n\n> The only thing I can think of right now is the Aharanov-Bohm experiment. In\n> this experiment it turns out that the vector potential (a gauge variant\n> object) is more fundamental than the field strength.\n\nIt is often said in electrodynamics textbooks that the potential is\n"inessential" but this is a poor choice of words. Even in classical\nelectrodynamics the potential is very real.\n\nTo see how gauge invariance works on an accessible\nphysical-phenomenological level, I would recommend doing some research\non the old London theory of superconductivity, in comparison to the BCS\nideas.\n\n-drl\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>JW wrote:

> Danny Ross Lunsford wrote:
>>Yes - the gauge invariance frees up just the right number of degrees of
>>freedom in the "mediating" field, to allow an invariant-theoretic
>>description of the matter that interacts via this field. Basically, it
>>sets up interacting systems that have non-trivial solutions.
>
> This is cyclic reasoning. If you start with a gauge potential, you run into
> trouble, but gauge invariance will also rescue you.

Well I don't see this - the objective is 1) to have a description of
matter that leads to the idea of a conserved quantity 2) to allow it to
interact with other matter of the same type via some mediating field.
Conservation implies symmetry (of the Lagrangian), and interaction
implies sufficient degrees of freedom to carry interaction in the
mediating field.

> How is this a *physical* motivation, i.e. one that can be directly motivated
> from experimental evidence?

Because the physical evidence amounts to conservation laws imputed from
analysis of laboratory scattering experiments. Field theory based on
gauge invariance is then a phenomenological response to finding a
description of these conservation laws.

> Mind you, gauge invariance is supposed to be
> hidden from us as configurations related by a gauge transformation are
> demanded to be physically equivalent. That's the whole point of gauge
> invariance.

Again, the point of gauge invariance is to provide the framework of
symmetry that leads to conservation laws.

> The only thing I can think of right now is the Aharanov-Bohm experiment. In
> this experiment it turns out that the vector potential (a gauge variant
> object) is more fundamental than the field strength.

It is often said in electrodynamics textbooks that the potential is
"inessential" but this is a poor choice of words. Even in classical
electrodynamics the potential is very real.

To see how gauge invariance works on an accessible
physical-phenomenological level, I would recommend doing some research
on the old London theory of superconductivity, in comparison to the BCS
ideas.

-drl

Arnold Neumaier

Apr14-04, 08:35 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\n\n\nDanny Ross Lunsford wrote:\n> JW wrote:\n>\n>>Danny Ross Lunsford wrote:\n>>\n>>>Yes - the gauge invariance frees up just the right number of degrees of\n>>>freedom in the "mediating" field, to allow an invariant-theoretic\n>>>description of the matter that interacts via this field. Basically, it\n>>>sets up interacting systems that have non-trivial solutions.\n>>\n>>This is cyclic reasoning. If you start with a gauge potential, you run into\n>>trouble, but gauge invariance will also rescue you.\n>\n> Well I don\'t see this - the objective is 1) to have a description of\n> matter that leads to the idea of a conserved quantity 2) to allow it to\n> interact with other matter of the same type via some mediating field.\n> Conservation implies symmetry (of the Lagrangian), and interaction\n> implies sufficient degrees of freedom to carry interaction in the\n> mediating field.\n>\n>>How is this a *physical* motivation, i.e. one that can be directly motivated\n>>from experimental evidence?\n>\n> Because the physical evidence amounts to conservation laws imputed from\n> analysis of laboratory scattering experiments. Field theory based on\n> gauge invariance is then a phenomenological response to finding a\n> description of these conservation laws.\n\nThis is not quite conclusive. Any continuous symmetry gives a conserved\ncurrent, and it is easy to write actions with a symmetry that don\'t\nhave gauge invariancee (e.g., a complex scalar field).\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>Danny Ross Lunsford wrote:
> JW wrote:
>
>>Danny Ross Lunsford wrote:
>>
>>>Yes - the gauge invariance frees up just the right number of degrees of
>>>freedom in the "mediating" field, to allow an invariant-theoretic
>>>description of the matter that interacts via this field. Basically, it
>>>sets up interacting systems that have non-trivial solutions.
>>
>>This is cyclic reasoning. If you start with a gauge potential, you run into
>>trouble, but gauge invariance will also rescue you.
>
> Well I don't see this - the objective is 1) to have a description of
> matter that leads to the idea of a conserved quantity 2) to allow it to
> interact with other matter of the same type via some mediating field.
> Conservation implies symmetry (of the Lagrangian), and interaction
> implies sufficient degrees of freedom to carry interaction in the
> mediating field.
>
>>How is this a *physical* motivation, i.e. one that can be directly motivated
>>from experimental evidence?
>
> Because the physical evidence amounts to conservation laws imputed from
> analysis of laboratory scattering experiments. Field theory based on
> gauge invariance is then a phenomenological response to finding a
> description of these conservation laws.

This is not quite conclusive. Any continuous symmetry gives a conserved
current, and it is easy to write actions with a symmetry that don't
have gauge invariancee (e.g., a complex scalar field).

Arnold Neumaier

Jeroen

Apr14-04, 11:48 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\nDanny Ross Lunsford wrote:\n\n> JW wrote:\n>\n>> Danny Ross Lunsford wrote:\n>>>Yes - the gauge invariance frees up just the right number of degrees of\n>>>freedom in the "mediating" field, to allow an invariant-theoretic\n>>>description of the matter that interacts via this field. Basically, it\n>>>sets up interacting systems that have non-trivial solutions.\n>>\n>> This is cyclic reasoning. If you start with a gauge potential, you run\n>> into trouble, but gauge invariance will also rescue you.\n>\n> Well I don\'t see this - the objective is 1) to have a description of\n> matter that leads to the idea of a conserved quantity 2) to allow it to\n\nThis doesn\'t point to gauge invariance per se.\n\n>> How is this a *physical* motivation, i.e. one that can be directly\n>> motivated from experimental evidence?\n>\n> Because the physical evidence amounts to conservation laws imputed from\n> analysis of laboratory scattering experiments. Field theory based on\n> gauge invariance is then a phenomenological response to finding a\n> description of these conservation laws.\n\nOf course, but do these experimental facts point to gauge invariance\ndirectly? If I understood the question of the original poster correctly,\nthis is what he wanted to know. The poster understands that gauge theories\nare very successful. But is there compelling evidence that a theory of\nelementary particle physics _must_ have gauge invariance? That is a good\nquestion, imho.\n\nNote that theories with gauge invariance are just theories with a redundant\nset of variables, i.e. you don\'t need all of them. In gauge theories, this\ndescription turns out to be very elegant and successful. However one might\nwonder if it would be possible to construct theories (perhaps less elegant)\nwithout gauge symmetry that lead to the same results.\n\n>> Mind you, gauge invariance is supposed to be\n>> hidden from us as configurations related by a gauge transformation are\n>> demanded to be physically equivalent. That\'s the whole point of gauge\n>> invariance.\n>\n> Again, the point of gauge invariance is to provide the framework of\n> symmetry that leads to conservation laws.\n\nAgain, it is possible to get those conservations laws without using gauge\nsymmetries (think global symmetries). Note that gauge symmetry is somewhat\nof a misnomer. It is not a symmetry like rotational invariance that relates\ndifferent solutions to each other. Rather gauge symmetries connect\nphysically equivalent solution to each other.\n\n>> The only thing I can think of right now is the Aharanov-Bohm experiment.\n>> In this experiment it turns out that the vector potential (a gauge\n>> variant object) is more fundamental than the field strength.\n>\n> It is often said in electrodynamics textbooks that the potential is\n> "inessential" but this is a poor choice of words. Even in classical\n> electrodynamics the potential is very real.\n\nExactly, in theory the classical vector potential is real, the Aharonov-Bohm\nexperiment shows that this is also true in practice ;-)\n\nbest,\nJeroen\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>Danny Ross Lunsford wrote:

> JW wrote:
>
>> Danny Ross Lunsford wrote:
>>>Yes - the gauge invariance frees up just the right number of degrees of
>>>freedom in the "mediating" field, to allow an invariant-theoretic
>>>description of the matter that interacts via this field. Basically, it
>>>sets up interacting systems that have non-trivial solutions.
>>
>> This is cyclic reasoning. If you start with a gauge potential, you run
>> into trouble, but gauge invariance will also rescue you.
>
> Well I don't see this - the objective is 1) to have a description of
> matter that leads to the idea of a conserved quantity 2) to allow it to

This doesn't point to gauge invariance per se.

>> How is this a *physical* motivation, i.e. one that can be directly
>> motivated from experimental evidence?
>
> Because the physical evidence amounts to conservation laws imputed from
> analysis of laboratory scattering experiments. Field theory based on
> gauge invariance is then a phenomenological response to finding a
> description of these conservation laws.

Of course, but do these experimental facts point to gauge invariance
directly? If I understood the question of the original poster correctly,
this is what he wanted to know. The poster understands that gauge theories
are very successful. But is there compelling evidence that a theory of
elementary particle physics _must_ have gauge invariance? That is a good
question, imho.

Note that theories with gauge invariance are just theories with a redundant
set of variables, i.e. you don't need all of them. In gauge theories, this
description turns out to be very elegant and successful. However one might
wonder if it would be possible to construct theories (perhaps less elegant)
without gauge symmetry that lead to the same results.

>> Mind you, gauge invariance is supposed to be
>> hidden from us as configurations related by a gauge transformation are
>> demanded to be physically equivalent. That's the whole point of gauge
>> invariance.
>
> Again, the point of gauge invariance is to provide the framework of
> symmetry that leads to conservation laws.

Again, it is possible to get those conservations laws without using gauge
symmetries (think global symmetries). Note that gauge symmetry is somewhat
of a misnomer. It is not a symmetry like rotational invariance that relates
different solutions to each other. Rather gauge symmetries connect
physically equivalent solution to each other.

>> The only thing I can think of right now is the Aharanov-Bohm experiment.
>> In this experiment it turns out that the vector potential (a gauge
>> variant object) is more fundamental than the field strength.
>
> It is often said in electrodynamics textbooks that the potential is
> "inessential" but this is a poor choice of words. Even in classical
> electrodynamics the potential is very real.

Exactly, in theory the classical vector potential is real, the Aharonov-Bohm
experiment shows that this is also true in practice ;-)

best,
Jeroen

Danny Ross Lunsford

Apr15-04, 02:57 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>Arnold Neumaier wrote:\n\n>>Because the physical evidence amounts to conservation laws imputed from\n>>analysis of laboratory scattering experiments. Field theory based on\n>>gauge invariance is then a phenomenological response to finding a\n>>description of these conservation laws.\n>\n> This is not quite conclusive. Any continuous symmetry gives a conserved\n> current, and it is easy to write actions with a symmetry that don\'t\n> have gauge invariancee (e.g., a complex scalar field).\n\nIsn\'t the invariance here simply the EM U(1) phase?\n\n-scratches head-\n\n-drl\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:

>>Because the physical evidence amounts to conservation laws imputed from
>>analysis of laboratory scattering experiments. Field theory based on
>>gauge invariance is then a phenomenological response to finding a
>>description of these conservation laws.
>
> This is not quite conclusive. Any continuous symmetry gives a conserved
> current, and it is easy to write actions with a symmetry that don't
> have gauge invariancee (e.g., a complex scalar field).

Isn't the invariance here simply the EM U(1) phase?

-scratches head-

-drl

Danny Ross Lunsford

Apr15-04, 02:57 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>Jeroen wrote:\n\n>>Well I don\'t see this - the objective is 1) to have a description of\n>>matter that leads to the idea of a conserved quantity\n>\n> This doesn\'t point to gauge invariance per se.\n\nAlmost - the other necessary part is the "unrolling" aspect of\nintroducing a potential (this is the real math issue) - that is, the\npossibility of converting a first-order system to a second-order one\nthat can propagate in a manner consistent with the required conservation\nlaw. Gauge invariance guarantees a determined system of equations - for\nexample, the free EM field has 6 components that satisfy 8 equations. If\none wrote down any system of 8 equations for 6 unknowns, the chances of\nfinding actual solutions would be slim.\n\n>>Because the physical evidence amounts to conservation laws imputed from\n>>analysis of laboratory scattering experiments. Field theory based on\n>>gauge invariance is then a phenomenological response to finding a\n>>description of these conservation laws.\n>\n> Of course, but do these experimental facts point to gauge invariance\n> directly? If I understood the question of the original poster correctly,\n> this is what he wanted to know. The poster understands that gauge theories\n> are very successful. But is there compelling evidence that a theory of\n> elementary particle physics _must_ have gauge invariance? That is a good\n> question, imho.\n\nSure, it\'s a great question! I\'ll have to defer to mathematicians, but\nI\'d be surprised to see a physically reasonable theory that wasn\'t.\n\n> Note that theories with gauge invariance are just theories with a redundant\n> set of variables, i.e. you don\'t need all of them.\n\nThis is too broad. The gauge freedom does not represent "redundancy" so\nmuch as it does "consistency" and "integrability". In fact the\ninteresting thing about gauge theories is that they are so thoroughly\nconstrained as to final form (that is, the gauge group itself can\'t be\njust anything).\n\n\n>>Again, the point of gauge invariance is to provide the framework of\n>>symmetry that leads to conservation laws.\n>\n> Again, it is possible to get those conservations laws without using gauge\n> symmetries (think global symmetries).\n\nWell one can\'t have an electron disappear here and "simultaneously"\nreappear in Paris. You must not only have conservation but also\npropagation. This enormously restricts the final form of a valid theory.\n\n> Note that gauge symmetry is somewhat\n> of a misnomer. It is not a symmetry like rotational invariance that relates\n> different solutions to each other. Rather gauge symmetries connect\n> physically equivalent solution to each other.\n\nActually in the original gauge theory (Weyl\'s extension of Riemannian\ngeometry) gauge or "calibration" invariance was on exactly the same\nlogical footing as general coordinate invariance - Weyl\'s idea was to\nstrictly localize both length and direction (as opposed to local\ndirection and global length standard, as in Riemannian geometry proper).\nThis theory is actually superior to GR as an abstract idea, because the\nreducibility of the metric is lifted (9 direction cosines plus the\nvolume element).\n\n-drl\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>Jeroen wrote:

>>Well I don't see this - the objective is 1) to have a description of
>>matter that leads to the idea of a conserved quantity
>
> This doesn't point to gauge invariance per se.

Almost - the other necessary part is the "unrolling" aspect of
introducing a potential (this is the real math issue) - that is, the
possibility of converting a first-order system to a second-order one
that can propagate in a manner consistent with the required conservation
law. Gauge invariance guarantees a determined system of equations - for
example, the free EM field has 6 components that satisfy 8 equations. If
one wrote down any system of 8 equations for 6 unknowns, the chances of
finding actual solutions would be slim.

>>Because the physical evidence amounts to conservation laws imputed from
>>analysis of laboratory scattering experiments. Field theory based on
>>gauge invariance is then a phenomenological response to finding a
>>description of these conservation laws.
>
> Of course, but do these experimental facts point to gauge invariance
> directly? If I understood the question of the original poster correctly,
> this is what he wanted to know. The poster understands that gauge theories
> are very successful. But is there compelling evidence that a theory of
> elementary particle physics _must_ have gauge invariance? That is a good
> question, imho.

Sure, it's a great question! I'll have to defer to mathematicians, but
I'd be surprised to see a physically reasonable theory that wasn't.

> Note that theories with gauge invariance are just theories with a redundant
> set of variables, i.e. you don't need all of them.

This is too broad. The gauge freedom does not represent "redundancy" so
much as it does "consistency" and "integrability". In fact the
interesting thing about gauge theories is that they are so thoroughly
constrained as to final form (that is, the gauge group itself can't be
just anything).

>>Again, the point of gauge invariance is to provide the framework of
>>symmetry that leads to conservation laws.
>
> Again, it is possible to get those conservations laws without using gauge
> symmetries (think global symmetries).

Well one can't have an electron disappear here and "simultaneously"
reappear in Paris. You must not only have conservation but also
propagation. This enormously restricts the final form of a valid theory.

> Note that gauge symmetry is somewhat
> of a misnomer. It is not a symmetry like rotational invariance that relates
> different solutions to each other. Rather gauge symmetries connect
> physically equivalent solution to each other.

Actually in the original gauge theory (Weyl's extension of Riemannian
geometry) gauge or "calibration" invariance was on exactly the same
logical footing as general coordinate invariance - Weyl's idea was to
strictly localize both length and direction (as opposed to local
direction and global length standard, as in Riemannian geometry proper).
This theory is actually superior to GR as an abstract idea, because the
reducibility of the metric is lifted (9 direction cosines plus the
volume element).

-drl

Jeroen

Apr15-04, 04:37 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\nDanny Ross Lunsford wrote:\n>> Of course, but do these experimental facts point to gauge invariance\n>> directly? If I understood the question of the original poster correctly,\n>> this is what he wanted to know. The poster understands that gauge\n>> theories are very successful. But is there compelling evidence that a\n>> theory of elementary particle physics _must_ have gauge invariance? That\n>> is a good question, imho.\n>\n> Sure, it\'s a great question! I\'ll have to defer to mathematicians, but\n> I\'d be surprised to see a physically reasonable theory that wasn\'t.\n\nWhat comes to my mind is S-matrix theory (not that I\'m enthusiastic about\nthat).\n\n>> Note that theories with gauge invariance are just theories with a\n>> redundant set of variables, i.e. you don\'t need all of them.\n>\n> This is too broad. The gauge freedom does not represent "redundancy" so\n> much as it does "consistency" and "integrability". In fact the\n> interesting thing about gauge theories is that they are so thoroughly\n> constrained as to final form (that is, the gauge group itself can\'t be\n> just anything).\n\nMy point was/is: If you want to describe a photon, why not use a field with\njust two components (which is the number of physical degrees of freedom)?\nIt appears to be convenient to use a four component field, which is a\nredundant description. Gauge symmetry is necessary to get rid of this\nredundancy. Agreed, gauge symmetries are so beautiful that there has to be\na truth to it (but that is not a scientific argument, is it?).\n\nbest,\nJeroen\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>Danny Ross Lunsford wrote:
>> Of course, but do these experimental facts point to gauge invariance
>> directly? If I understood the question of the original poster correctly,
>> this is what he wanted to know. The poster understands that gauge
>> theories are very successful. But is there compelling evidence that a
>> theory of elementary particle physics _must_ have gauge invariance? That
>> is a good question, imho.
>
> Sure, it's a great question! I'll have to defer to mathematicians, but
> I'd be surprised to see a physically reasonable theory that wasn't.

What comes to my mind is S-matrix theory (not that I'm enthusiastic about
that).

>> Note that theories with gauge invariance are just theories with a
>> redundant set of variables, i.e. you don't need all of them.
>
> This is too broad. The gauge freedom does not represent "redundancy" so
> much as it does "consistency" and "integrability". In fact the
> interesting thing about gauge theories is that they are so thoroughly
> constrained as to final form (that is, the gauge group itself can't be
> just anything).

My point was/is: If you want to describe a photon, why not use a field with
just two components (which is the number of physical degrees of freedom)?
It appears to be convenient to use a four component field, which is a
redundant description. Gauge symmetry is necessary to get rid of this
redundancy. Agreed, gauge symmetries are so beautiful that there has to be
a truth to it (but that is not a scientific argument, is it?).

best,
Jeroen

Arnold Neumaier

Apr15-04, 05:12 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\nDanny Ross Lunsford wrote:\n> Arnold Neumaier wrote:\n>\n>\n>>>Because the physical evidence amounts to conservation laws imputed from\n>>>analysis of laboratory scattering experiments. Field theory based on\n>>>gauge invariance is then a phenomenological response to finding a\n>>>description of these conservation laws.\n>>\n>>This is not quite conclusive. Any continuous symmetry gives a conserved\n>>current, and it is easy to write actions with a symmetry that don\'t\n>>have gauge invariancee (e.g., a complex scalar field).\n>\n>\n> Isn\'t the invariance here simply the EM U(1) phase?\n\nNot if it is the only field around.\nThere is only a global symmetry, but not a local one. The\nkinetic term in the action does not behave correctly since there\nis no vector potential to correct for the cahnge in the derivative.\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>Danny Ross Lunsford wrote:
> Arnold Neumaier wrote:
>
>
>>>Because the physical evidence amounts to conservation laws imputed from
>>>analysis of laboratory scattering experiments. Field theory based on
>>>gauge invariance is then a phenomenological response to finding a
>>>description of these conservation laws.
>>
>>This is not quite conclusive. Any continuous symmetry gives a conserved
>>current, and it is easy to write actions with a symmetry that don't
>>have gauge invariancee (e.g., a complex scalar field).
>
>
> Isn't the invariance here simply the EM U(1) phase?

Not if it is the only field around.
There is only a global symmetry, but not a local one. The
kinetic term in the action does not behave correctly since there
is no vector potential to correct for the cahnge in the derivative.

Arnold Neumaier

Arnold Neumaier

Apr15-04, 11:20 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>Tlas wrote:\n> Hi everyone,\n>\n> Does anybody know what is the physical motivation behind demanding\n> local gauge invariance? I mean the only argument I have seen is\n> "because it works" one. Is there something better based on locality\n> for example?\n\nWeinberg\'s book on QFT argues for gauge invariance from\ncausality + masslessness.\n\nArnold Neumaier\n\n\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>Tlas wrote:
> Hi everyone,
>
> Does anybody know what is the physical motivation behind demanding
> local gauge invariance? I mean the only argument I have seen is
> "because it works" one. Is there something better based on locality
> for example?

Weinberg's book on QFT argues for gauge invariance from
causality + masslessness.

Arnold Neumaier

Danny Ross Lunsford

Apr15-04, 11:26 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>Arnold Neumaier wrote:\n\n>>Because the physical evidence amounts to conservation laws imputed from\n>>analysis of laboratory scattering experiments. Field theory based on\n>>gauge invariance is then a phenomenological response to finding a\n>>description of these conservation laws.\n>\n> This is not quite conclusive. Any continuous symmetry gives a conserved\n> current, and it is easy to write actions with a symmetry that don\'t\n> have gauge invariancee (e.g., a complex scalar field).\n\nIsn\'t the invariance here simply the EM U(1) phase? That is phi is a\nmatter field which undergoes a phase change when locally coupled to the\nEM (gauge) field.\n\n-scratches head-\n\n-drl\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:

>>Because the physical evidence amounts to conservation laws imputed from
>>analysis of laboratory scattering experiments. Field theory based on
>>gauge invariance is then a phenomenological response to finding a
>>description of these conservation laws.
>
> This is not quite conclusive. Any continuous symmetry gives a conserved
> current, and it is easy to write actions with a symmetry that don't
> have gauge invariancee (e.g., a complex scalar field).

Isn't the invariance here simply the EM U(1) phase? That is \phi is a
matter field which undergoes a phase change when locally coupled to the
EM (gauge) field.

-scratches head-

-drl

Arnold Neumaier

Apr15-04, 11:40 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>Jeroen wrote:\n\n> My point was/is: If you want to describe a photon, why not use a field with\n> just two components (which is the number of physical degrees of freedom)?\n> It appears to be convenient to use a four component field, which is a\n> redundant description. Gauge symmetry is necessary to get rid of this\n> redundancy. Agreed, gauge symmetries are so beautiful that there has to be\n> a truth to it (but that is not a scientific argument, is it?).\n\nYou can read about this point of view in the work of\nSteven Weinberg\nPhys.Rev.134:B882-B896,1964\nPhys.Rev.135:B1049-B1056,1964\nPhys.Rev.138:B988-B1002,1965\n\n\nArnold Neumaier\n\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>Jeroen wrote:

> My point was/is: If you want to describe a photon, why not use a field with
> just two components (which is the number of physical degrees of freedom)?
> It appears to be convenient to use a four component field, which is a
> redundant description. Gauge symmetry is necessary to get rid of this
> redundancy. Agreed, gauge symmetries are so beautiful that there has to be
> a truth to it (but that is not a scientific argument, is it?).

You can read about this point of view in the work of
Steven Weinberg
Phys.Rev.134:B882-B896,1964
Phys.Rev.135:B1049-B1056,1964
Phys.Rev.138:B988-B1002,1965

Arnold Neumaier

Danny Ross Lunsford

Apr15-04, 11:40 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>Jeroen wrote:\n\n> My point was/is: If you want to describe a photon, why not use a field with\n> just two components (which is the number of physical degrees of freedom)?\n\nWell you can, but the form of the final theory will just be EM in a\nspecific gauge - it won\'t be form-invariant under Lorentz\ntransformations and going to a new frame will require a corresponding\ngauge transformation. Logically, gauge invariance is on the same footing\nas coordinate invariance and going from, say, a ground lab to an\norbiting lab requires both a coordinate and a gauge transformation.\n\nI strongly urge you to study the Weyl theory, which, even though it is\nwrong in 4d (for reasons that are physical and not mathematical),\nillustrates all the key ideas in a way that separates it from the\nambiguous aspects of quantum field theory.\n\n-drl\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>Jeroen wrote:

> My point was/is: If you want to describe a photon, why not use a field with
> just two components (which is the number of physical degrees of freedom)?

Well you can, but the form of the final theory will just be EM in a
specific gauge - it won't be form-invariant under Lorentz
transformations and going to a new frame will require a corresponding
gauge transformation. Logically, gauge invariance is on the same footing
as coordinate invariance and going from, say, a ground lab to an
orbiting lab requires both a coordinate and a gauge transformation.

I strongly urge you to study the Weyl theory, which, even though it is
wrong in 4d (for reasons that are physical and not mathematical),
illustrates all the key ideas in a way that separates it from the
ambiguous aspects of quantum field theory.

-drl

Danny Ross Lunsford

Apr15-04, 11:40 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>Arnold Neumaier wrote:\n\n> Not if it is the only field around.\n> There is only a global symmetry, but not a local one. The\n> kinetic term in the action does not behave correctly since there\n> is no vector potential to correct for the cahnge in the derivative.\n\nWell this is true of any matter field, no?\n\n-drl\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:

> Not if it is the only field around.
> There is only a global symmetry, but not a local one. The
> kinetic term in the action does not behave correctly since there
> is no vector potential to correct for the cahnge in the derivative.

Well this is true of any matter field, no?

-drl

Tlas

Apr16-04, 02:28 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>Thank you all for providing answers to the question. However, it looks\nlike I was somewhat vague in formulating it. What I meant is the\nfollowing:\nWe know that if we postulate that a field is to be invariant under\nsome group of local transformations then we will have to introduce a\nconnection (gauge field) to be able to take covariant derivatives,\ndefine a parallel propagator and so on...\nThe question is : why do we postulate the local invariance? I mean the\nlagrangian that is usually written down is invariant (accidentally?)\nunder some global symmetry, why is it promoted to a local one?\nFinally, it looks like local gauge invariance is a\ngeneralization/special case of general covariance. Unfortunately\ngeneral covariance is much more intuitive since there it is invariance\nunder change of coordinates of space-time (you arrange your clocks and\nrulers differently) while in gauge invariance it is change of\ncoordinates of some internal space which is (I am not sure of this)\ninaccessible to direct measurements. Is there an experiment which\nwill somehow point that the theory should be invariant under\nindependent "rotations" in this internal space at different points?\nThe Aharonov-Bohm one was mentioned above but I couldn\'t figure out\nany precise relationship.\n\nRegards,\nT.T.\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>Thank you all for providing answers to the question. However, it looks
like I was somewhat vague in formulating it. What I meant is the
following:
We know that if we postulate that a field is to be invariant under
some group of local transformations then we will have to introduce a
connection (gauge field) to be able to take covariant derivatives,
define a parallel propagator and so on...
The question is : why do we postulate the local invariance? I mean the
lagrangian that is usually written down is invariant (accidentally?)
under some global symmetry, why is it promoted to a local one?
Finally, it looks like local gauge invariance is a
generalization/special case of general covariance. Unfortunately
general covariance is much more intuitive since there it is invariance
under change of coordinates of space-time (you arrange your clocks and
rulers differently) while in gauge invariance it is change of
coordinates of some internal space which is (I am not sure of this)
inaccessible to direct measurements. Is there an experiment which
will somehow point that the theory should be invariant under
independent "rotations" in this internal space at different points?
The Aharonov-Bohm one was mentioned above but I couldn't figure out
any precise relationship.

Regards,
T.T.

Tlas

Apr16-04, 05:51 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\n\n\nWhere exactly does he do it?\n\nT.T.\n\n\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c5m973\\$1ap\\$1@lfa222122.richmond.edu>...\ n> Tlas wrote:\n> > Hi everyone,\n> >\n> > Does anybody know what is the physical motivation behind demanding\n> > local gauge invariance? I mean the only argument I have seen is\n> > "because it works" one. Is there something better based on locality\n> > for example?\n>\n> Weinberg\'s book on QFT argues for gauge invariance from\n> causality + masslessness.\n>\n> Arnold Neumaier\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>Where exactly does he do it?

T.T.

Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c5m973$1ap$1@lfa222122.richmond.edu>...
> Tlas wrote:
> > Hi everyone,
> >
> > Does anybody know what is the physical motivation behind demanding
> > local gauge invariance? I mean the only argument I have seen is
> > "because it works" one. Is there something better based on locality
> > for example?
>
> Weinberg's book on QFT argues for gauge invariance from
> causality + masslessness.
>
> Arnold Neumaier

Arnold Neumaier

Apr16-04, 05:51 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\n\n\nDanny Ross Lunsford wrote:\n> Arnold Neumaier wrote:\n>\n>> Not if it is the only field around.\n>> There is only a global symmetry, but not a local one. The\n>> kinetic term in the action does not behave correctly since there\n>> is no vector potential to correct for the change in the derivative.\n>\n>\n> Well this is true of any matter field, no?\n\nWhat do you mean by \'this\'?\nYou need a complex field to have a symmetry, and you need the absence\nof a gauge potential with the charges carried by the field for the\nargument to work. If both is included in your \'this\', you are right.\n\n\nArnold Neumaier\n\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>Danny Ross Lunsford wrote:
> Arnold Neumaier wrote:
>
>> Not if it is the only field around.
>> There is only a global symmetry, but not a local one. The
>> kinetic term in the action does not behave correctly since there
>> is no vector potential to correct for the change in the derivative.
>
>
> Well this is true of any matter field, no?

What do you mean by 'this'?
You need a complex field to have a symmetry, and you need the absence
of a gauge potential with the charges carried by the field for the
argument to work. If both is included in your 'this', you are right.

Arnold Neumaier

Danny Ross Lunsford

Apr17-04, 05:08 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>Arnold Neumaier wrote:\n\n> What do you mean by \'this\'?\n> You need a complex field to have a symmetry, and you need the absence\n> of a gauge potential with the charges carried by the field for the\n> argument to work. If both is included in your \'this\', you are right.\n\nWell one can change the phase of a Dirac spinor uncoupled to an EM field\nwithout changing anything because observables are bilinear covariants\nand those are phase-invariant.\n\nGauge invariance requires a gauge field in addition to a matter field.\n\n-drl\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:

> What do you mean by 'this'?
> You need a complex field to have a symmetry, and you need the absence
> of a gauge potential with the charges carried by the field for the
> argument to work. If both is included in your 'this', you are right.

Well one can change the phase of a Dirac spinor uncoupled to an EM field
without changing anything because observables are bilinear covariants
and those are phase-invariant.

Gauge invariance requires a gauge field in addition to a matter field.

-drl

Alfred Einstead

Apr17-04, 05:08 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>timtlas@hotmail.com (Tlas) wrote:\n> Does anybody know what is the physical motivation behind demanding\n> local gauge invariance? I mean the only argument I have seen is\n> "because it works" one.\n\nLocal, as opposed to global: global gauge invariance is already\nincluded in local gauge invariance as a special case. The extra\nassumptions needed to get global, instead of just local, amount\nto stating to the effect that the whole universe is conspiring\nto affect each point in space and time.\n\nFor gauge invariance: as long as the matter fields have phases\nand anything else of the like, you have gauge invariance just\nfrom that. A phase is just a fancy way of saying U(1) gauge\ninvariance; and a set of independent phases is just a way of\nsaying U(1)^n gauge invariance -- which just means gauge\ninvariance in general with respect to an abelian group;\nand each corresponding gauge field gives you a copy of either\na Maxwell or Proca interaction.\n\nSo the extra element is the non-abelianness of gauge\ninvariance; which amounts to the additional element of\ninteractions that are also self-interacting. That\'s\nalready seen with the weak nuclear force carriers W+, W- and Z;\nfor instance.\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>timtlas@hotmail.com (Tlas) wrote:
> Does anybody know what is the physical motivation behind demanding
> local gauge invariance? I mean the only argument I have seen is
> "because it works" one.

Local, as opposed to global: global gauge invariance is already
included in local gauge invariance as a special case. The extra
assumptions needed to get global, instead of just local, amount
to stating to the effect that the whole universe is conspiring
to affect each point in space and time.

For gauge invariance: as long as the matter fields have phases
and anything else of the like, you have gauge invariance just
from that. A phase is just a fancy way of saying U(1) gauge
invariance; and a set of independent phases is just a way of
saying U(1)^n gauge invariance -- which just means gauge
invariance in general with respect to an abelian group;
and each corresponding gauge field gives you a copy of either
a Maxwell or Proca interaction.

So the extra element is the non-abelianness of gauge
invariance; which amounts to the additional element of
interactions that are also self-interacting. That's
already seen with the weak nuclear force carriers W+, W- and Z;
for instance.

Arnold Neumaier

Apr17-04, 05:08 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>Tlas wrote:\n\n> We know that if we postulate that a field is to be invariant under\n> some group of local transformations then we will have to introduce a\n> connection (gauge field) to be able to take covariant derivatives,\n> define a parallel propagator and so on...\n> The question is : why do we postulate the local invariance? I mean the\n> lagrangian that is usually written down is invariant (accidentally?)\n> under some global symmetry, why is it promoted to a local one?\n\nIt is needed to make vector fields renormalizable.\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>Tlas wrote:

> We know that if we postulate that a field is to be invariant under
> some group of local transformations then we will have to introduce a
> connection (gauge field) to be able to take covariant derivatives,
> define a parallel propagator and so on...
> The question is : why do we postulate the local invariance? I mean the
> lagrangian that is usually written down is invariant (accidentally?)
> under some global symmetry, why is it promoted to a local one?

It is needed to make vector fields renormalizable.

Arnold Neumaier

Hendrik van Hees

Apr18-04, 03:50 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>Tlas wrote:\n\n> We know that if we postulate that a field is to be invariant under\n> some group of local transformations then we will have to introduce a\n> connection (gauge field) to be able to take covariant derivatives,\n> define a parallel propagator and so on...\n> The question is : why do we postulate the local invariance? I mean the\n> lagrangian that is usually written down is invariant (accidentally?)\n> under some global symmetry, why is it promoted to a local one?\n\nThe heuristic idea is to localise symmetries, i.e., to make the theory\ninvariant under symmetry transformations, which can be different at any\nspace time point. I find this argument not very convincing, since in\nthe standard model are a lot of global symmetries left which lead, for\ninstance, to lepton or baryon number conservation.\n\nI find another argument much more reasonable. We start from classical\nspace-time descriptions to build up the quantum theory of particles,\ni.e., we look for mathematical structures, which are consistent with\nthe space-time geometry we use to describe space and time.\n\nThen we define "elementary particle" as those particles, which are\ndescribed by the irreducible representations of the symmetry group of\nthe space-time model. In the case of special relativity the geometry is\nthe Minkowski geometry of space time and the symmetry group is the\nPoincare group.\n\nNow come some subtleties of the quantum theoretical formalism into the\ngame: States are described by rays in Hilbert space, so that we have to\nextend the symmetry group of space time to what\'s called a central\nextension of its covering group and look for unitary (or antiunitary)\nrepresentations of this extended group. That is known as the\nWigner-Bargmann theorem.\n\nIn the case of the Poincare group this is relatively simple, because\nthere exists up to isomorphy only one central extension and the\ncovering group is given by taking SL(2,C) instead of SO(1,3) for the\nhomogeneous Poincare transformations (=Lorentz transformations).\n\nThen one investigates, which unitary or antiunitary representations\nmight exist and which make physically sense.\n\nFrom the demand of a stable ground state (vacuum), i.e., the boundedness\nof the Hamilton operator from below, which is defined to generate the\ntime evolution of the system (as a symmetry operator it\'s the generator\nfor time translations), one finds that time reversal has to be\ndescribed by antiunitary transformations while all others have to be\nunitary.\n\nThen one looks at the part of the group which is connected continuously\nto the group identity, which is the proper orthochronous Poincare group\n(POG). Representations of the full Poincare group (including time and\nspace reflections) can be built from these.\n\nIt turns out that the POG has two big classes of representations which\nlead to sensible descriptions of particles, one class describes massive\nparticles, the other massless.\n\nIn the massive case, the representation o the POG is further given by\nthe transformation properties of the states, which describe particles\nat rest, under rotations. These rotations are generated by the spin\noperators. Each irreducible representation of the rotation group (or\nhere, in the quantum case, its covering group SU(2)) is given by the\nspin quantum number s which can be 0,1/2,1,...\n\nIn the massless case the things are a little more subtle. The\nrepresentation theory tells us, that we need to find the irreducible\nrepresentations of the subgroup of the Lorentz group which leaves the\nstates of one standard momentum fixed, the socalled "little group". In\nthe case of massive particles we have chosen zero momentum as the\nstandard momentum and then the "little group" was the rotation group\nleading to spin as an intrinsic degree of freedom of particles. In the\nmassless case there is no rest frame, but we can chose any momentum as\nthe standard momentum, usually one takes a momentum in z-direction.\n\nThe little group is again three-dimensional. It consists of all Lorentz\ntransformations which leave the four-vector (1,0,0,1) invariant. Of\ncourse, one subgroup is given by the rotations around the z-axis, but\nthere are also socalled null-rotations which make the little group\nnon-compact. The little group is in fact isomorphic to the isomorphy\ngroup of the euclidean plane, i.e., the rotations and the translations\nwithin a plane.\n\nThe "translation subgroup" has now only representations with the full\nR^2 as group parameters. This would mean that we had particles with\ncontinuous "spin-like" intrinsic parameters, which has never been\nobserved yet.\n\nThus, we take only those representations where the "translations" (or in\nthe original Lorentz representation picture the "null rotations") are\nreprented trivially, so that there is no observable connected with\nthem.\n\nWhat is left is the rotations around the z-axis. The rotations around\nthe z-axis are an Abelian group, so that we could describe them by any\ncovering of the group U(1), but since we like to have a representation\nof the full rotation group (resp. its covering group), when we extend\nthe representation to all other states of the particle, not only those\nof fixed "standard momentum", we have to take the coverings to the\nnumbers 0,1/2,1,... Since these rotations are around the momentum\ndirection of the particle, that\'s called helicity, and for each of the\nabove numbers s (often also called "spin quantum number", which should\nalways be taken in the here described meaning), there are thus two\nsorts of massless particles, one with helicity +s and one with helicity\n-s.\n\nWhen we like to extend the representation of the POG to those of the\nfull Poincare group, for instance to take into account space reflection\nsymmetry (the electromagnetic and the strong interactions obey this\nsymmetry), in general one has to take both particles. That\'s the\nreason, why the electromagnetic field has two degrees of freedom, one\nwith helicity 1, one with helicity -1 (which are the self-dual or\nanti-self dual fields in the free case).\n\nIf we "switch on interactions" now, we have to be careful not to destroy\nthe fact that the "null rotations" of the massless particles are still\nrepresented trivially, and it turns out that this is possible only if\nthe interactions respect gauge invariance, because for the free fields\nthe triviality of null rotations is described by the fact that all\nfields, connected only by a gauge transformation have to be identified,\ni.e., when we try to represent the massless representations of the POG\non a linear function space of fields this does not work, but we have to\ntake the quotient space of fields modulo of its gauge transformations.\nE.g., for helicity \\pm 1-fields (for instance the photon field of QED)\nwe have to identify A_{\\mu} and A_{\\mu}+\\partial_{\\mu} \\chi with an\narbitrary scalar field \\chi.\n\nThis property must be kept intact when taking into account interactions.\nIn a "minimal" way this is realised by coupling the helicity \\pm\n1-fields to conserved currents, which leads to the idea of minimal\nsubstitution.\n\nIn principle one could write down any gauge-invariant interaction, but\nwhen one likes to stick to renormalisable models we are again at the\nminimal substitution description.\n\nOne of the great discoveries of the 20th century was the idea that the\ngauge symmetry can also be realised by not to stick to the simple U(1)\ngauge invariance, demanded by space-time symmetry, but also use other\nnon-Abelian groups. In the context of particle physics this was\nrediscovered by Yang and Mills in 1954 (50th anniversery this year!).\nThe general gauge principle was in fact known much earlier (Klein 1938,\nwho even had the idea to describe weak interactions with a non-abelian\ngauge theory in those early days, but it was forgotten for a while!).\n\n> Finally, it looks like local gauge invariance is a\n> generalization/special case of general covariance. Unfortunately\n> general covariance is much more intuitive since there it is invariance\n> under change of coordinates of space-time (you arrange your clocks and\n> rulers differently) while in gauge invariance it is change of\n> coordinates of some internal space which is (I am not sure of this)\n> inaccessible to direct measurements. Is there an experiment which\n> will somehow point that the theory should be invariant under\n> independent "rotations" in this internal space at different points?\n> The Aharonov-Bohm one was mentioned above but I couldn\'t figure out\n> any precise relationship.\n\nOf course, gauge invariance is a rather restrictive demand for model\nbuilding, and thus it has observable consequences. For instance, LEP\nwas sensitive to higher-order perturbative results and showed that the\ndata are indeed consistent with local gauge invariance in the\nSalam-Glashow-Weinberg theory of electroweak interactions.\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>Tlas wrote:

> We know that if we postulate that a field is to be invariant under
> some group of local transformations then we will have to introduce a
> connection (gauge field) to be able to take covariant derivatives,
> define a parallel propagator and so on...
> The question is : why do we postulate the local invariance? I mean the
> lagrangian that is usually written down is invariant (accidentally?)
> under some global symmetry, why is it promoted to a local one?

The heuristic idea is to localise symmetries, i.e., to make the theory
invariant under symmetry transformations, which can be different at any
space time point. I find this argument not very convincing, since in
the standard model are a lot of global symmetries left which lead, for
instance, to lepton or baryon number conservation.

I find another argument much more reasonable. We start from classical
space-time descriptions to build up the quantum theory of particles,
i.e., we look for mathematical structures, which are consistent with
the space-time geometry we use to describe space and time.

Then we define "elementary particle" as those particles, which are
described by the irreducible representations of the symmetry group of
the space-time model. In the case of special relativity the geometry is
the Minkowski geometry of space time and the symmetry group is the
Poincare group.

Now come some subtleties of the quantum theoretical formalism into the
game: States are described by rays in Hilbert space, so that we have to
extend the symmetry group of space time to what's called a central
extension of its covering group and look for unitary (or antiunitary)
representations of this extended group. That is known as the
Wigner-Bargmann theorem.

In the case of the Poincare group this is relatively simple, because
there exists up to isomorphy only one central extension and the
covering group is given by taking SL(2,C) instead of SO(1,3) for the
homogeneous Poincare transformations (=Lorentz transformations).

Then one investigates, which unitary or antiunitary representations
might exist and which make physically sense.

From the demand of a stable ground state (vacuum), i.e., the boundedness
of the Hamilton operator from below, which is defined to generate the
time evolution of the system (as a symmetry operator it's the generator
for time translations), one finds that time reversal has to be
described by antiunitary transformations while all others have to be
unitary.

Then one looks at the part of the group which is connected continuously
to the group identity, which is the proper orthochronous Poincare group
(POG). Representations of the full Poincare group (including time and
space reflections) can be built from these.

It turns out that the POG has two big classes of representations which
lead to sensible descriptions of particles, one class describes massive
particles, the other massless.

In the massive case, the representation o the POG is further given by
the transformation properties of the states, which describe particles
at rest, under rotations. These rotations are generated by the spin
operators. Each irreducible representation of the rotation group (or
here, in the quantum case, its covering group SU(2)) is given by the
spin quantum number s which can be 0,1/2,1,...

In the massless case the things are a little more subtle. The
representation theory tells us, that we need to find the irreducible
representations of the subgroup of the Lorentz group which leaves the
states of one standard momentum fixed, the socalled "little group". In
the case of massive particles we have chosen zero momentum as the
standard momentum and then the "little group" was the rotation group
leading to spin as an intrinsic degree of freedom of particles. In the
massless case there is no rest frame, but we can chose any momentum as
the standard momentum, usually one takes a momentum in z-direction.

The little group is again three-dimensional. It consists of all Lorentz
transformations which leave the four-vector (1,0,0,1) invariant. Of
course, one subgroup is given by the rotations around the z-axis, but
there are also socalled null-rotations which make the little group
non-compact. The little group is in fact isomorphic to the isomorphy
group of the euclidean plane, i.e., the rotations and the translations
within a plane.

The "translation subgroup" has now only representations with the full
R^2 as group parameters. This would mean that we had particles with
continuous "spin-like" intrinsic parameters, which has never been
observed yet.

Thus, we take only those representations where the "translations" (or in
the original Lorentz representation picture the "null rotations") are
reprented trivially, so that there is no observable connected with
them.

What is left is the rotations around the z-axis. The rotations around
the z-axis are an Abelian group, so that we could describe them by any
covering of the group U(1), but since we like to have a representation
of the full rotation group (resp. its covering group), when we extend
the representation to all other states of the particle, not only those
of fixed "standard momentum", we have to take the coverings to the
numbers 0,1/2,1,... Since these rotations are around the momentum
direction of the particle, that's called helicity, and for each of the
above numbers s (often also called "spin quantum number", which should
always be taken in the here described meaning), there are thus two
sorts of massless particles, one with helicity +s and one with helicity
-s.

When we like to extend the representation of the POG to those of the
full Poincare group, for instance to take into account space reflection
symmetry (the electromagnetic and the strong interactions obey this
symmetry), in general one has to take both particles. That's the
reason, why the electromagnetic field has two degrees of freedom, one
with helicity 1, one with helicity -1 (which are the self-dual or
anti-self dual fields in the free case).

If we "switch on interactions" now, we have to be careful not to destroy
the fact that the "null rotations" of the massless particles are still
represented trivially, and it turns out that this is possible only if
the interactions respect gauge invariance, because for the free fields
the triviality of null rotations is described by the fact that all
fields, connected only by a gauge transformation have to be identified,
i.e., when we try to represent the massless representations of the POG
on a linear function space of fields this does not work, but we have to
take the quotient space of fields modulo of its gauge transformations.
E.g., for helicity \pm 1-fields (for instance the photon field of QED)
we have to identify A_{\mu} and A_{\mu}+\partial_{\mu} \chi with an
arbitrary scalar field \chi.

This property must be kept intact when taking into account interactions.
In a "minimal" way this is realised by coupling the helicity \pm
1-fields to conserved currents, which leads to the idea of minimal
substitution.

In principle one could write down any gauge-invariant interaction, but
when one likes to stick to renormalisable models we are again at the
minimal substitution description.

One of the great discoveries of the 20th century was the idea that the
gauge symmetry can also be realised by not to stick to the simple U(1)
gauge invariance, demanded by space-time symmetry, but also use other
non-Abelian groups. In the context of particle physics this was
rediscovered by Yang and Mills in 1954 (50th anniversery this year!).
The general gauge principle was in fact known much earlier (Klein 1938,
who even had the idea to describe weak interactions with a non-abelian
gauge theory in those early days, but it was forgotten for a while!).

> Finally, it looks like local gauge invariance is a
> generalization/special case of general covariance. Unfortunately
> general covariance is much more intuitive since there it is invariance
> under change of coordinates of space-time (you arrange your clocks and
> rulers differently) while in gauge invariance it is change of
> coordinates of some internal space which is (I am not sure of this)
> inaccessible to direct measurements. Is there an experiment which
> will somehow point that the theory should be invariant under
> independent "rotations" in this internal space at different points?
> The Aharonov-Bohm one was mentioned above but I couldn't figure out
> any precise relationship.

Of course, gauge invariance is a rather restrictive demand for model
building, and thus it has observable consequences. For instance, LEP
was sensitive to higher-order perturbative results and showed that the
data are indeed consistent with local gauge invariance in the
Salam-Glashow-Weinberg theory of electroweak interactions.

--
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

Danny Ross Lunsford

Apr19-04, 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>Hendrik van Hees wrote:\n\n[a fine research program..]\n\nUnfortunately a simple question will remain. Why does the momentum\noperator take the form dm + ig Tk Akm? Like it or not matter seems to be\n"connected". Both GR and particle theory, insofar as they are successful\ndescriptions of matter, are based on connections. So the fine research\nprogram is going to end up at the beginning unless you can explain this.\n\nIt should be noted than in conformal spaces the covariant derivative has\njust the right form. The coupling is related to the conformal weight.\nMasses are clearly related to "scaling". These facts cannot be accidental.\n\n-drl\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>Hendrik van Hees wrote:

[a fine research program..]

Unfortunately a simple question will remain. Why does the momentum
operator take the form dm + ig Tk Akm? Like it or not matter seems to be
"connected". Both GR and particle theory, insofar as they are successful
descriptions of matter, are based on connections. So the fine research
program is going to end up at the beginning unless you can explain this.

It should be noted than in conformal spaces the covariant derivative has
just the right form. The coupling is related to the conformal weight.
Masses are clearly related to "scaling". These facts cannot be accidental.

-drl

Arnold Neumaier

Apr19-04, 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>Tlas wrote:\n>\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c5m973\\$1ap\\$1@lfa222122.richmond.edu>...\ n\n>>Does anybody know what is the physical motivation behind demanding\n>>>local gauge invariance? I mean the only argument I have seen is\n>>>"because it works" one. Is there something better based on locality\n>>>for example?\n>>\n>>Weinberg\'s book on QFT argues for gauge invariance from\n>>causality + masslessness.\n\n\n> Where exactly does he do it?\n\n\nI don\'t have Vol. 1 here, so I can\'t point you to the exact page.\nBut he discusses massless fields in Chapter 5, and observes\n(probably there, or in the beginning of Chapter 8 on QED)\nroughly the following:\n\nSince massless spin 1 fields have only two degrees of freedom,\nthe 4-vector one can make from them does not transform correctly\nbut only up to a gauge transformation making up for the missing\nlongitudinal degree of freedom. Since sufficiently long range\nelementary fields (less than exponential decay) are necessarily\nmassless, they must either have spin <=1/2 or have gauge behavior.\n\nTo couple such gauge fields to matter currents, the latter\nmust be conserved, which means (given the known conservation laws)\nthat the gauge fields either have spin 1 (coupling to a conserved\nvector current), or spin 2 (coupling to the energy-momentum tensor).\n[Actually, he does not discuss this for Fermion fields,\nso spin 3/2 (gravitinos) is perhaps another special case.]\n\nSpin 1 leads to standard gauge theories, while spin 2 leads\nto general covariance (and gravitons) which, in this context,\nis best viewed also as a kind of gauge invariance.\n\nThere are some assumptions in the derivation, which you\'ll find\nout when you read the details.\n\n\nFor more details, see Weinberg\'s papers\nPhys.Rev.134:B882-B896,1964\nPhys.Rev.135:B1049-B1056,1964\nPhys.Rev.138:B988-B1002,1965\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>Tlas wrote:
>
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c5m973$1ap$1@lfa222122.richmond.edu>...

>>Does anybody know what is the physical motivation behind demanding
>>>local gauge invariance? I mean the only argument I have seen is
>>>"because it works" one. Is there something better based on locality
>>>for example?
>>
>>Weinberg's book on QFT argues for gauge invariance from
>>causality + masslessness.

> Where exactly does he do it?

I don't have Vol. 1 here, so I can't point you to the exact page.
But he discusses massless fields in Chapter 5, and observes
(probably there, or in the beginning of Chapter 8 on QED)
roughly the following:

Since massless spin 1 fields have only two degrees of freedom,
the 4-vector one can make from them does not transform correctly
but only up to a gauge transformation making up for the missing
longitudinal degree of freedom. Since sufficiently long range
elementary fields (less than exponential decay) are necessarily
massless, they must either have spin <=1/2 or have gauge behavior.

To couple such gauge fields to matter currents, the latter
must be conserved, which means (given the known conservation laws)
that the gauge fields either have spin 1 (coupling to a conserved
vector current), or spin 2 (coupling to the energy-momentum tensor).
[Actually, he does not discuss this for Fermion fields,
so spin 3/2 (gravitinos) is perhaps another special case.]

Spin 1 leads to standard gauge theories, while spin 2 leads
to general covariance (and gravitons) which, in this context,
is best viewed also as a kind of gauge invariance.

There are some assumptions in the derivation, which you'll find
out when you read the details.

For more details, see Weinberg's papers
Phys.Rev.134:B882-B896,1964
Phys.Rev.135:B1049-B1056,1964
Phys.Rev.138:B988-B1002,1965

Arnold Neumaier

Doug Sweetser

Apr19-04, 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>Hello Arnold:\n\n> Weinberg\'s book on QFT argues for gauge invariance from\n> causality + masslessness.\n\nWow! I realize that everyone accepts mass is generated from spontaneous\nsymmetry breaking using the Higgs mechanism. This justifies the\nbillion dollar bet at CERN. Yet from a logical perspective, if a\nsystem is no longer gauge invariant, then an implication would be it\nmay have mass. One of those technical details I don\'t know is does the\nHiggs mechanism effectively break gauge symmetry? I remember -\nimprecisely - that the Higgs is a way to preserve the symmetry of the\nstandard model Lagrange density, but alter ... oops, can\'t quite\nrecall.\n\n\ndoug\nquaternions.com\n\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>Hello Arnold:

> Weinberg's book on QFT argues for gauge invariance from
> causality + masslessness.

Wow! I realize that everyone accepts mass is generated from spontaneous
symmetry breaking using the Higgs mechanism. This justifies the
billion dollar bet at CERN. Yet from a logical perspective, if a
system is no longer gauge invariant, then an implication would be it
may have mass. One of those technical details I don't know is does the
Higgs mechanism effectively break gauge symmetry? I remember -
imprecisely - that the Higgs is a way to preserve the symmetry of the
standard model Lagrange density, but alter ... oops, can't quite
recall.

doug
quaternions.com

Arnold Neumaier

Apr21-04, 04:22 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>Doug Sweetser wrote:\n> Hello Arnold:\n>\n>\n>>Weinberg\'s book on QFT argues for gauge invariance from\n>>causality + masslessness.\n>\n>\n> Wow! I realize that everyone accepts mass is generated from spontaneous\n> symmetry breaking using the Higgs mechanism.\n\n\nMy statement is about the abstract Lagrangian setting --\nconstructing gauge invariant Lagrangians from fields --\nwhereas broken symmetry is a matter of particular quantum states.\nThe Lagrangian of a theory with broken symmetry is still fully symmetric,\nbut the ground states, on the basis of which perturbation theory is done,\nis less symmetric.\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>Doug Sweetser wrote:
> Hello Arnold:
>
>
>>Weinberg's book on QFT argues for gauge invariance from
>>causality + masslessness.
>
>
> Wow! I realize that everyone accepts mass is generated from spontaneous
> symmetry breaking using the Higgs mechanism.

My statement is about the abstract Lagrangian setting --
constructing gauge invariant Lagrangians from fields --
whereas broken symmetry is a matter of particular quantum states.
The Lagrangian of a theory with broken symmetry is still fully symmetric,
but the ground states, on the basis of which perturbation theory is done,
is less symmetric.

Arnold Neumaier

Hendrik van Hees

Apr22-04, 04:23 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>Doug Sweetser wrote:\n\n> Wow! I realize that everyone accepts mass is generated from\n> spontaneous\n> symmetry breaking using the Higgs mechanism.\n\nThe bulk of the mass around us is not generated from the Higgs mechanism\nbut from the anomalous breaking of conformal invariance of nearly\nmassless QCD.\n\nNevertheless, this argument with the Higgs mechanism does not spoil the\narguments given so far, since in the case of non-abelian gauge fields\nthe only mechanism to get a consistent theory (be it renormalisable or\nan effective non-renormalisable theorye) with massive gauge fields is\nthe Higgs mechanism. The equations of motion are completely gauge\ninvariant, while the ground states transform non-trivially under gauge\ntransformations.\n\nThe abelian case is an exception, because there you can have a massive\nvector field without an extra Higgs field by using Stueckelberg\'s trick\nto introduce a bosonic "Stueckelberg ghost" (in addition to the usual\nfermionic Faddeev Popov ghosts). At the end the Stueckelberg ghost and\nthe FP ghosts decouple as usual from all other fields and can be\nabsorbed in the overall constant normalisation of the generating\nfunctional Z, which is physically not observable.\n\nAt finite temperature all the ghosts in the free Lagrangian are crucial\nto get the correct and gauge independent counting of field degrees of\nfreedom for the (free) massive vector field, namely 3 bosonic degrees\nof freedom (4 from the gauge field +1 from the Stueckelberg ghost -2\nfrom the FP ghosts, which makes 3 :-)).\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>Doug Sweetser wrote:

> Wow! I realize that everyone accepts mass is generated from
> spontaneous
> symmetry breaking using the Higgs mechanism.

The bulk of the mass around us is not generated from the Higgs mechanism
but from the anomalous breaking of conformal invariance of nearly
massless QCD.

Nevertheless, this argument with the Higgs mechanism does not spoil the
arguments given so far, since in the case of non-abelian gauge fields
the only mechanism to get a consistent theory (be it renormalisable or
an effective non-renormalisable theorye) with massive gauge fields is
the Higgs mechanism. The equations of motion are completely gauge
invariant, while the ground states transform non-trivially under gauge
transformations.

The abelian case is an exception, because there you can have a massive
vector field without an extra Higgs field by using Stueckelberg's trick
to introduce a bosonic "Stueckelberg ghost" (in addition to the usual
fermionic Faddeev Popov ghosts). At the end the Stueckelberg ghost and
the FP ghosts decouple as usual from all other fields and can be
absorbed in the overall constant normalisation of the generating
functional Z, which is physically not observable.

At finite temperature all the ghosts in the free Lagrangian are crucial
to get the correct and gauge independent counting of field degrees of
freedom for the (free) massive vector field, namely 3 bosonic degrees
of freedom (4 from the gauge field +1 from the Stueckelberg ghost -2
from the FP ghosts, which makes 3 :-)).

--
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

Hendrik van Hees

Apr22-04, 04:24 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>Danny Ross Lunsford wrote:\n\n> Hendrik van Hees wrote:\n>\n> [a fine research program..]\n>\n> Unfortunately a simple question will remain. Why does the momentum\n> operator take the form dm + ig Tk Akm? Like it or not matter seems to\n> be "connected". Both GR and particle theory, insofar as they are\n> successful descriptions of matter, are based on connections. So the\n> fine research program is going to end up at the beginning unless you\n> can explain this.\n\nThe momentum operator is defined to be the generator of space\ntranslations and this is what comes out when you investigate the gauge\ninvariant Lagrangian with help of Noether\'s theorem.\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>Danny Ross Lunsford wrote:

> Hendrik van Hees wrote:
>
> [a fine research program..]
>
> Unfortunately a simple question will remain. Why does the momentum
> operator take the form dm + ig Tk Akm? Like it or not matter seems to
> be "connected". Both GR and particle theory, insofar as they are
> successful descriptions of matter, are based on connections. So the
> fine research program is going to end up at the beginning unless you
> can explain this.

The momentum operator is defined to be the generator of space
translations and this is what comes out when you investigate the gauge
invariant Lagrangian with help of Noether's theorem.

--
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

Doug Sweetser

Apr24-04, 12:18 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>Hello Arnold:\n\n> Since massless spin 1 fields have only two degrees of freedom,\n> the 4-vector one can make from them does not transform correctly\n> but only up to a gauge transformation making up for the missing\n> longitudinal degree of freedom.\n\nI thought there was a longitudinal and scalar mode of emission, and they\nare set up to exactly cancel always.\n\n> To couple such gauge fields to matter currents, the latter\n> must be conserved, which means (given the known conservation laws)\n> that the gauge fields either have spin 1 (coupling to a conserved\n> vector current), or spin 2 (coupling to the energy-momentum tensor).\n> [Actually, he does not discuss this for Fermion fields,\n> so spin 3/2 (gravitinos) is perhaps another special case.]\n>\n> Spin 1 leads to standard gauge theories, while spin 2 leads\n> to general covariance (and gravitons) which, in this context,\n> is best viewed also as a kind of gauge invariance.\n\nThe spin 2 graviton field will also represent 2 degrees of freedom\nbecause they travel at the speed of light no? Based on the field\nequations of general relativity, these are transverse modes of\nemission.\n\ndoug\nquaternions.com\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>Hello Arnold:

> Since massless spin 1 fields have only two degrees of freedom,
> the 4-vector one can make from them does not transform correctly
> but only up to a gauge transformation making up for the missing
> longitudinal degree of freedom.

I thought there was a longitudinal and scalar mode of emission, and they
are set up to exactly cancel always.

> To couple such gauge fields to matter currents, the latter
> must be conserved, which means (given the known conservation laws)
> that the gauge fields either have spin 1 (coupling to a conserved
> vector current), or spin 2 (coupling to the energy-momentum tensor).
> [Actually, he does not discuss this for Fermion fields,
> so spin 3/2 (gravitinos) is perhaps another special case.]
>
> Spin 1 leads to standard gauge theories, while spin 2 leads
> to general covariance (and gravitons) which, in this context,
> is best viewed also as a kind of gauge invariance.

The spin 2 graviton field will also represent 2 degrees of freedom
because they travel at the speed of light no? Based on the field
equations of general relativity, these are transverse modes of
emission.

doug
quaternions.com

Arnold Neumaier

Apr28-04, 02:46 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>Doug Sweetser wrote:\n> Hello Arnold:\n>\n>\n>>Since massless spin 1 fields have only two degrees of freedom,\n>>the 4-vector one can make from them does not transform correctly\n>>but only up to a gauge transformation making up for the missing\n>>longitudinal degree of freedom.\n>\n>\n> I thought there was a longitudinal and scalar mode of emission, and they\n> are set up to exactly cancel always.\n\nNo; there is no cancellation. In a momentum representation, the\n4-vector A(p) is orthogonal to p, which implies that there is no\nscalar part, and (at least the way Weinberg treats everything)\nA_0=0, which implies that only a 2D space is present.\nThe gauge transform is needed to remove the 0-component of the field\nobtained when a Lorentz transformation is applied.\n\n\n\n>>To couple such gauge fields to matter currents, the latter\n>>must be conserved, which means (given the known conservation laws)\n>>that the gauge fields either have spin 1 (coupling to a conserved\n>>vector current), or spin 2 (coupling to the energy-momentum tensor).\n>>[Actually, he does not discuss this for Fermion fields,\n>>so spin 3/2 (gravitinos) is perhaps another special case.]\n>>\n>>Spin 1 leads to standard gauge theories, while spin 2 leads\n>>to general covariance (and gravitons) which, in this context,\n>>is best viewed also as a kind of gauge invariance.\n>\n>\n> The spin 2 graviton field will also represent 2 degrees of freedom\n> because they travel at the speed of light\n\nyes.\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>Doug Sweetser wrote:
> Hello Arnold:
>
>
>>Since massless spin 1 fields have only two degrees of freedom,
>>the 4-vector one can make from them does not transform correctly
>>but only up to a gauge transformation making up for the missing
>>longitudinal degree of freedom.
>
>
> I thought there was a longitudinal and scalar mode of emission, and they
> are set up to exactly cancel always.

No; there is no cancellation. In a momentum representation, the
4-vector A(p) is orthogonal to p, which implies that there is no
scalar part, and (at least the way Weinberg treats everything)
A_0=0, which implies that only a 2D space is present.
The gauge transform is needed to remove the 0-component of the field
obtained when a Lorentz transformation is applied.

>>To couple such gauge fields to matter currents, the latter
>>must be conserved, which means (given the known conservation laws)
>>that the gauge fields either have spin 1 (coupling to a conserved
>>vector current), or spin 2 (coupling to the energy-momentum tensor).
>>[Actually, he does not discuss this for Fermion fields,
>>so spin 3/2 (gravitinos) is perhaps another special case.]
>>
>>Spin 1 leads to standard gauge theories, while spin 2 leads
>>to general covariance (and gravitons) which, in this context,
>>is best viewed also as a kind of gauge invariance.
>
>
> The spin 2 graviton field will also represent 2 degrees of freedom
> because they travel at the speed of light

yes.

Arnold Neumaier

Doug Sweetser

Apr29-04, 08:13 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>\nHello Arnold:\n\nSo this is how Weinberg handles it:\n\n> No; there is no cancellation. In a momentum representation, the\n> 4-vector A(p) is orthogonal to p, which implies that there is no\n> scalar part, and (at least the way Weinberg treats everything)\n> A_0=0, which implies that only a 2D space is present.\n> The gauge transform is needed to remove the 0-component of the field\n> obtained when a Lorentz transformation is applied.\n\nIn the original paper by Gupta, he does talk about cancelation. I also\nhave trouble understanding how to make sure the scalar part is always\nzero. Usually a change in inertial reference frame will change that\nsituation, so such an analysis would require a special choice for the\nframe.\n\ndoug\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>Hello Arnold:

So this is how Weinberg handles it:

> No; there is no cancellation. In a momentum representation, the
> 4-vector A(p) is orthogonal to p, which implies that there is no
> scalar part, and (at least the way Weinberg treats everything)
> A_0=0, which implies that only a 2D space is present.
> The gauge transform is needed to remove the 0-component of the field
> obtained when a Lorentz transformation is applied.

In the original paper by Gupta, he does talk about cancelation. I also
have trouble understanding how to make sure the scalar part is always
zero. Usually a change in inertial reference frame will change that
situation, so such an analysis would require a special choice for the
frame.

doug

Arnold Neumaier

Apr30-04, 11:34 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>Doug Sweetser wrote:\n\n>>No; there is no cancellation. In a momentum representation, the\n>>4-vector A(p) is orthogonal to p, which implies that there is no\n>>scalar part, and (at least the way Weinberg treats everything)\n>>A_0=0, which implies that only a 2D space is present.\n>>The gauge transform is needed to remove the 0-component of the field\n>>obtained when a Lorentz transformation is applied.\n>\n>\n> In the original paper by Gupta, he does talk about cancelation. I also\n> have trouble understanding how to make sure the scalar part is always\n> zero. Usually a change in inertial reference frame will change that\n> situation, so such an analysis would require a special choice for the\n> frame.\n\nI don\'t know about Gupta.\n\nNo matter what the reference frame, one can for any given 4-vector A\nalways choose a gauge transformation to make A_0=0. Of course, the gauge\ntransformation needed is frame dependent.\n\n\nArnold Neumaier\n\n\n\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>Doug Sweetser wrote:

>>No; there is no cancellation. In a momentum representation, the
>>4-vector A(p) is orthogonal to p, which implies that there is no
>>scalar part, and (at least the way Weinberg treats everything)
>>A_0=0, which implies that only a 2D space is present.
>>The gauge transform is needed to remove the 0-component of the field
>>obtained when a Lorentz transformation is applied.
>
>
> In the original paper by Gupta, he does talk about cancelation. I also
> have trouble understanding how to make sure the scalar part is always
> zero. Usually a change in inertial reference frame will change that
> situation, so such an analysis would require a special choice for the
> frame.

I don't know about Gupta.

No matter what the reference frame, one can for any given 4-vector A
always choose a gauge transformation to make A_0=0. Of course, the gauge
transformation needed is frame dependent.

Arnold Neumaier

Alfred Einstead

Apr30-04, 11:36 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>Doug Sweetser <sweetser@alum.mit.edu> wrote:\n> In the original paper by Gupta, he does talk about cancelation. I also\n> have trouble understanding how to make sure the scalar part is always\n> zero. Usually a change in inertial reference frame will change that\n> situation, so such an analysis would require a special choice for the\n> frame.\n\nA Lorentz transformation is coupled with a gauge transformation which\nrestores the desired condition. The two have to be done in\nconcert.\n\nThat means you\'re actually working within the larger symmetry group:\n(Poincare\') X (Gauge)\nand not just Poincare\' itself; and are only working within a specific\ncross-section of this group which is not "horizontal" in the usual\nsense of leaving the gauge part of the group stationary.\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>Doug Sweetser <sweetser@alum.mit.edu> wrote:
> In the original paper by Gupta, he does talk about cancelation. I also
> have trouble understanding how to make sure the scalar part is always
> zero. Usually a change in inertial reference frame will change that
> situation, so such an analysis would require a special choice for the
> frame.

A Lorentz transformation is coupled with a gauge transformation which
restores the desired condition. The two have to be done in
concert.

That means you're actually working within the larger symmetry group:
(Poincare') X (Gauge)
and not just Poincare' itself; and are only working within a specific
cross-section of this group which is not "horizontal" in the usual
sense of leaving the gauge part of the group stationary.

Gentil Correa

May2-04, 05:52 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>timtlas@hotmail.com (Tlas) wrote in message news:<856a4a19.0404071431.7714c28a@posting.google. com>...\n> Hi everyone,\n>\n> Does anybody know what is the physical motivation behind demanding\n> local gauge invariance? I mean the only argument I have seen is\n> "because it works" one. Is there something better based on locality\n> for example?\n> Thanks for input\n>\n> T.T.\n\nHi, T.T.\n\nMaybe I am becoming old and rusty, but isn\'t "because it works" the\nbest possible argument? Isn\'t special relativity entirely based on\nthis argument? Who on earth would think of replacing old newtonian and\ncomfortable time with this complicated einsteinian one, were it not\nfor the fact that "it works"? Isn\'t this phrase ("because it works")\nthe key of the experimental method?\n\nThat said, consider that the concepts of gauge theory were obtained,\nin totally\ndifferent contexts, and in complete independence, by physicists and\nmathematicians (which give them the name of Ehresmann connections. I\nhappen to\nhave known Charles Ehresmann. He couldn\'t be less interested, or\nconversant,\nwith Particle Physics). The famous meeting which put together two\neminent Chinese scientists, S.S. Chern, a great differential geometer,\nand C.N. Yang, a great particle physicist and discoverer of\nnon-abelian gauge theories, and the surprise of each of them at the\nfact that the other "knew those things", is a great moment of the\nhistory of science. If this is not a strong argument for the fact that\ngauge concepts are natural, I am missing something.\n\nBest regards,\n\nGentil.\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>timtlas@hotmail.com (Tlas) wrote in message news:<856a4a19.0404071431.7714c28a@posting.google.com>...
> Hi everyone,
>
> Does anybody know what is the physical motivation behind demanding
> local gauge invariance? I mean the only argument I have seen is
> "because it works" one. Is there something better based on locality
> for example?
> Thanks for input
>
> T.T.

Hi, T.T.

Maybe I am becoming old and rusty, but isn't "because it works" the
best possible argument? Isn't special relativity entirely based on
this argument? Who on earth would think of replacing old newtonian and
comfortable time with this complicated einsteinian one, were it not
for the fact that "it works"? Isn't this phrase ("because it works")
the key of the experimental method?

That said, consider that the concepts of gauge theory were obtained,
in totally
different contexts, and in complete independence, by physicists and
mathematicians (which give them the name of Ehresmann connections. I
happen to
have known Charles Ehresmann. He couldn't be less interested, or
conversant,
with Particle Physics). The famous meeting which put together two
eminent Chinese scientists, S.S. Chern, a great differential geometer,
and C.N. Yang, a great particle physicist and discoverer of
non-abelian gauge theories, and the surprise of each of them at the
fact that the other "knew those things", is a great moment of the
history of science. If this is not a strong argument for the fact that
gauge concepts are natural, I am missing something.

Best regards,

Gentil.

Hendrik van Hees

May3-04, 06:33 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>Alfred Einstead wrote:\n\n> A Lorentz transformation is coupled with a gauge transformation which\n> restores the desired condition. The two have to be done in\n> concert.\n\nI just like to add, that from the representation theory of the Poincare\ngroup, there is no way out of this feature. The field operators of a\nmassless helicity \\pm 1-field, which both together define the photon\nfield, Lorentz transform *not* as vectors but als vectors modulo a\ngauge transformation.\n\nThis is different from the case of massive particles, because Wigner\'s\nlittle group is different for the massless case. While for massive\nrepresentations the little group is the rotation group (or more\nprecisely its covering group SU(2)), for massless representation its a\ngroup, generated by rotations around the standard three-momentum (U(1))\nand socalled "null rotations". This group is equivalent to ISO(2), the\nsymmetry group of the euclidean affine plane (i.e., the rotations in\nthe plane, corresponding to the above mentioned U(1) and the\ntranslations in the plane, corresponding to the "null rotations".\n\nThe important difference is, that in the massive case the little group\nis compact and thus has (up to equivalence) unitary finite-dimensional\nrepresentations. It\'s (up to equivalence) the action of rotations on\nthe states of resting particles (when you choose the standard momentum\nof the representations to 0, which is the most convenient and\nphysically intuitive choice; other choices lead not to different but\nequivalent representations of the Poincare group, so that you need not\nbother with them). These are the well-known representations of angular\nmomenta, which in this case of a resting particle are of course the\nspin angular momenta, belonging to the representations of spin quantum\nnumber 0,1/2,1,...\n\nIn the massless case the symmetry group is of course not compact\n(because of the translations which are a subgroup contained in ISO(2)).\nThus, there exist no finite dimensional representations, as you know\nfrom the introductory quantum mechanics lecture, when you look at the\ngeneralised momentum eigenstates. The generalised eigenvalues of each\nmomementum component make up the whole real axis.\n\nNow this means that the null rotations correspond to continuous\nspin-like intrinsic quantum numbers of massless particles. Since we\nnever have observed such particles, we think, that these\nrepresentations are not relevant for particle physics.\n\nThus for massless particles we demand that the null rotations of the\nlittle group are represented trivially, i.e., each null rotation\ncorresponds to the unity operator in Hilbert space, while the rotations\naround the direction of the standard momementum (usually chosen to be\ndirected in z-direction) can be represented by the coverings of U(1)\nwith helicity quantum number 0,\\pm 1/2, \\pm 1,...\n\nIt is easy to show that the demand for the null rotations to be\nrepresented trivially cannot be realised on usual function spaces,\nrepresenting local fields, but one has to introduce a quotient space,\ni.e., in the case of helicities \\pm 1 the linear function space where\nall fields, which differ only by the gradient of a massless scalar\nfield, are identified.\n\nOf course, it is much better, not to look on representations of the\nfields in terms of function spaces, but construct the Hilbert spaces\nfrom the irreps of the Poincare group first and then to build the local\nfields from the creation and annihilation operators. It comes of course\nout that the local field operators transform under Lorentz transforms\nlike a vector field plus the gradient of a scalar field. Thus the\ntheory has to be necessarily a gauge theory, which becomes particularly\nimportant, when the massless helicity \\pm 1-fields are coupled to\n"matter fields", which then leads, for renormalisable theories with\ncertain additional assumptions about symmetries, inevitably to QED-like\ntheories.\n\nOne can, of course, extend this idea to non-abelian gauge theories as\nwell, because all these theories fulfil of course also the abelian\ngauge condition, and couple to conserved currents as the photon field.\n\n>\n> That means you\'re actually working within the larger symmetry group:\n> (Poincare\') X (Gauge)\n> and not just Poincare\' itself; and are only working within a specific\n> cross-section of this group which is not "horizontal" in the usual\n> sense of leaving the gauge part of the group stationary.\n\nFor photons it\'s simply the Poincare group, as explained above.\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>Alfred Einstead wrote:

> A Lorentz transformation is coupled with a gauge transformation which
> restores the desired condition. The two have to be done in
> concert.

I just like to add, that from the representation theory of the Poincare
group, there is no way out of this feature. The field operators of a
massless helicity \pm 1-field, which both together define the photon
field, Lorentz transform *not* as vectors but als vectors modulo a
gauge transformation.

This is different from the case of massive particles, because Wigner's
little group is different for the massless case. While for massive
representations the little group is the rotation group (or more
precisely its covering group SU(2)), for massless representation its a
group, generated by rotations around the standard three-momentum (U(1))
and socalled "null rotations". This group is equivalent to ISO(2), the
symmetry group of the euclidean affine plane (i.e., the rotations in
the plane, corresponding to the above mentioned U(1) and the
translations in the plane, corresponding to the "null rotations".

The important difference is, that in the massive case the little group
is compact and thus has (up to equivalence) unitary finite-dimensional
representations. It's (up to equivalence) the action of rotations on
the states of resting particles (when you choose the standard momentum
of the representations to 0, which is the most convenient and
physically intuitive choice; other choices lead not to different but
equivalent representations of the Poincare group, so that you need not
bother with them). These are the well-known representations of angular
momenta, which in this case of a resting particle are of course the
spin angular momenta, belonging to the representations of spin quantum
number 0,1/2,1,...

In the massless case the symmetry group is of course not compact
(because of the translations which are a subgroup contained in ISO(2)).
Thus, there exist no finite dimensional representations, as you know
from the introductory quantum mechanics lecture, when you look at the
generalised momentum eigenstates. The generalised eigenvalues of each
momementum component make up the whole real axis.

Now this means that the null rotations correspond to continuous
spin-like intrinsic quantum numbers of massless particles. Since we
never have observed such particles, we think, that these
representations are not relevant for particle physics.

Thus for massless particles we demand that the null rotations of the
little group are represented trivially, i.e., each null rotation
corresponds to the unity operator in Hilbert space, while the rotations
around the direction of the standard momementum (usually chosen to be
directed in z-direction) can be represented by the coverings of U(1)
with helicity quantum number 0,\pm 1/2, \pm 1,...

It is easy to show that the demand for the null rotations to be
represented trivially cannot be realised on usual function spaces,
representing local fields, but one has to introduce a quotient space,
i.e., in the case of helicities \pm 1 the linear function space where
all fields, which differ only by the gradient of a massless scalar
field, are identified.

Of course, it is much better, not to look on representations of the
fields in terms of function spaces, but construct the Hilbert spaces
from the irreps of the Poincare group first and then to build the local
fields from the creation and annihilation operators. It comes of course
out that the local field operators transform under Lorentz transforms
like a vector field plus the gradient of a scalar field. Thus the
theory has to be necessarily a gauge theory, which becomes particularly
important, when the massless helicity \pm 1-fields are coupled to
"matter fields", which then leads, for renormalisable theories with
certain additional assumptions about symmetries, inevitably to QED-like
theories.

One can, of course, extend this idea to non-abelian gauge theories as
well, because all these theories fulfil of course also the abelian
gauge condition, and couple to conserved currents as the photon field.

>
> That means you're actually working within the larger symmetry group:
> (Poincare') X (Gauge)
> and not just Poincare' itself; and are only working within a specific
> cross-section of this group which is not "horizontal" in the usual
> sense of leaving the gauge part of the group stationary.

For photons it's simply the Poincare group, as explained above.

--
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