Thomas Larsson
Jan11-05, 03:48 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>very_cryptic@hotmail.com wrote:\n\n>This would suggest, if follow canonical quantization, that all physical\n>states transform trivially under gauge symmetries. However, we are not\n>forced to quantize canonically. In that case, is it possible to make\n>gauge symmetries more like global symmetries by having physical unitary\n>representations of the gauge symmetry which does not transform\n>trivially?\n\nShort answer: If one sticks with the gauge symmetry proper, no. Because\nthe group of gauge transformations does not have any non-trivial\nphysical unitary representations at all.\n\nCaveat: The word "physical" is assumed to mean "of lowest-energy type",\nwhich is the type of representations relevant in quantum theory. There\nare classical unitary representations, acting on fields valued in\nmodules of the global gauge group. Already in CFT, one makes the\ndistinction between primary fields (classical) and lowest-weight reps\n(quantum). There might also be a gauge analogue of the so-called\nAshtekar-Isham-Lewandowski representation of the diffeomorphism group\nhttp://www.arxiv.org/abs/gr-qc/0303074 .\n\nProof: Consider for simplicity the group of gauge transformations on an\nN-dimensional torus. The restriction of a unitary rep of the torus group\nto any loop yields a unitary rep of the corresponding loop group. Every\nsuch restriction is trivial, since it is well-known that loop groups do\nnot have non-trivial proper unitary representations. Only the trivial\ntorus rep has only trivial loop restrictions. QED.\n\nEvasion: The loop group does have non-trivial unitary reps provided that\none allows for a central extension, i.e. a gauge anomaly. Hence the\ntorus group must have an extension, too. Unlike conventional gauge\nanomalies proportional to the third Casimir, this Kac-Moody-like anomaly\nis proportional to the second Casimir, and hence conventional wisdom\nabout inconsistency of gauge anomalies does not apply. We must of course\nnot treat an anomalous gauge symmetry as gauge, but rather as a\nconventional global symmetry.\n\nNote: Conventional anomalies proportional to the third Casimir are\ninconsistent. Proof again by restriction to all loops. Loop algebras\ndon\'t have third Casimir extensions, hence the extension must vanish,\nand no non-trivial unitaries exist.\n\nLong answer: http://www.arxiv.org/abs/hep-th/0501043\n\n[PS: Is it only me who has problems with Google? When I try to respond, it says\nUnable to retrieve message 1104504766.379442.139380@z14g2000cwz.googlegroups. 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>very_cryptic@hotmail.com wrote:
>This would suggest, if follow canonical quantization, that all physical
>states transform trivially under gauge symmetries. However, we are not
>forced to quantize canonically. In that case, is it possible to make
>gauge symmetries more like global symmetries by having physical unitary
>representations of the gauge symmetry which does not transform
>trivially?
Short answer: If one sticks with the gauge symmetry proper, no. Because
the group of gauge transformations does not have any non-trivial
physical unitary representations at all.
Caveat: The word "physical" is assumed to mean "of lowest-energy type",
which is the type of representations relevant in quantum theory. There
are classical unitary representations, acting on fields valued in
modules of the global gauge group. Already in CFT, one makes the
distinction between primary fields (classical) and lowest-weight reps
(quantum). There might also be a gauge analogue of the so-called
Ashtekar-Isham-Lewandowski representation of the diffeomorphism group
http://www.arxiv.org/abs/http://www.arxiv.org/abs/gr-qc/0303074 .
Proof: Consider for simplicity the group of gauge transformations on an
N-dimensional torus. The restriction of a unitary rep of the torus group
to any loop yields a unitary rep of the corresponding loop group. Every
such restriction is trivial, since it is well-known that loop groups do
not have non-trivial proper unitary representations. Only the trivial
torus rep has only trivial loop restrictions. QED.
Evasion: The loop group does have non-trivial unitary reps provided that
one allows for a central extension, i.e. a gauge anomaly. Hence the
torus group must have an extension, too. Unlike conventional gauge
anomalies proportional to the third Casimir, this Kac-Moody-like anomaly
is proportional to the second Casimir, and hence conventional wisdom
about inconsistency of gauge anomalies does not apply. We must of course
not treat an anomalous gauge symmetry as gauge, but rather as a
conventional global symmetry.
Note: Conventional anomalies proportional to the third Casimir are
inconsistent. Proof again by restriction to all loops. Loop algebras
don't have third Casimir extensions, hence the extension must vanish,
and no non-trivial unitaries exist.
Long answer: http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0501043
[PS: Is it only me who has problems with Google? When I try to respond, it says
Unable to retrieve message 1104504766.379442.139380@z14g2000cwz.googlegroups. com]
>This would suggest, if follow canonical quantization, that all physical
>states transform trivially under gauge symmetries. However, we are not
>forced to quantize canonically. In that case, is it possible to make
>gauge symmetries more like global symmetries by having physical unitary
>representations of the gauge symmetry which does not transform
>trivially?
Short answer: If one sticks with the gauge symmetry proper, no. Because
the group of gauge transformations does not have any non-trivial
physical unitary representations at all.
Caveat: The word "physical" is assumed to mean "of lowest-energy type",
which is the type of representations relevant in quantum theory. There
are classical unitary representations, acting on fields valued in
modules of the global gauge group. Already in CFT, one makes the
distinction between primary fields (classical) and lowest-weight reps
(quantum). There might also be a gauge analogue of the so-called
Ashtekar-Isham-Lewandowski representation of the diffeomorphism group
http://www.arxiv.org/abs/http://www.arxiv.org/abs/gr-qc/0303074 .
Proof: Consider for simplicity the group of gauge transformations on an
N-dimensional torus. The restriction of a unitary rep of the torus group
to any loop yields a unitary rep of the corresponding loop group. Every
such restriction is trivial, since it is well-known that loop groups do
not have non-trivial proper unitary representations. Only the trivial
torus rep has only trivial loop restrictions. QED.
Evasion: The loop group does have non-trivial unitary reps provided that
one allows for a central extension, i.e. a gauge anomaly. Hence the
torus group must have an extension, too. Unlike conventional gauge
anomalies proportional to the third Casimir, this Kac-Moody-like anomaly
is proportional to the second Casimir, and hence conventional wisdom
about inconsistency of gauge anomalies does not apply. We must of course
not treat an anomalous gauge symmetry as gauge, but rather as a
conventional global symmetry.
Note: Conventional anomalies proportional to the third Casimir are
inconsistent. Proof again by restriction to all loops. Loop algebras
don't have third Casimir extensions, hence the extension must vanish,
and no non-trivial unitaries exist.
Long answer: http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0501043
[PS: Is it only me who has problems with Google? When I try to respond, it says
Unable to retrieve message 1104504766.379442.139380@z14g2000cwz.googlegroups. com]