charge quantization and gravity [Archive]

Martin Lohmann

Apr11-04, 11:44 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\nWhen Dirac showed that the existence magnetic monopole would somehow\nthrough nontrivial homotopy groups imply the quantization of electric\ncharge, many physicists believed that this possibility would be that\none that is realized in nature because there were no other explanation\nfor it and because a topological explanation sounded somewhat\n"beautiful".\nNow, electric charge is, in the Weinberg model of electroweak\ninteractions and in the standard model, linked to weak isospin and\nweak hypercharge. If you fix this quantities for particles, then\nelectric charge is naturally quantized. It now happens that there are\nseveral constraints on the values of weak isospin and weak hypercharge\ndue to anomaly cancellation which, in the standard model without\ngravity, fixes these numbers up to a constant.\nBut if you also include the simplest model of gravity in the standard\nmodel, you get another constraint due to gravitational anomalies which\ntotally fixes the values of isospin and hypercharge. So this extension\nof the standard model leads naturally to a charge quantization, just\nas the magnetic monopole.\nMy question is now, why do physicists not believe in this "extended"\nstandard model, while they did believe in the magnetic monopole? Is it\njust because of renormalization?\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>When Dirac showed that the existence magnetic monopole would somehow
through nontrivial homotopy groups imply the quantization of electric
charge, many physicists believed that this possibility would be that
one that is realized in nature because there were no other explanation
for it and because a topological explanation sounded somewhat
"beautiful".
Now, electric charge is, in the Weinberg model of electroweak
interactions and in the standard model, linked to weak isospin and
weak hypercharge. If you fix this quantities for particles, then
electric charge is naturally quantized. It now happens that there are
several constraints on the values of weak isospin and weak hypercharge
due to anomaly cancellation which, in the standard model without
gravity, fixes these numbers up to a constant.
But if you also include the simplest model of gravity in the standard
model, you get another constraint due to gravitational anomalies which
totally fixes the values of isospin and hypercharge. So this extension
of the standard model leads naturally to a charge quantization, just
as the magnetic monopole.
My question is now, why do physicists not believe in this "extended"
standard model, while they did believe in the magnetic monopole? Is it
just because of renormalization?

Hendrik van Hees

Apr11-04, 12:27 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nMartin Lohmann wrote:\n\n> Now, electric charge is, in the Weinberg model of electroweak\n> interactions and in the standard model, linked to weak isospin and\n> weak hypercharge. If you fix this quantities for particles, then\n> electric charge is naturally quantized. It now happens that there are\n> several constraints on the values of weak isospin and weak hypercharge\n> due to anomaly cancellation which, in the standard model without\n> gravity, fixes these numbers up to a constant.\n\nI always wondered, if there is only this one pattern (of course up to a\nconstant) of charges or whether there exist others. Do you have\nreferences, where these questions are discussed?\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</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>Martin Lohmann wrote:

> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.

I always wondered, if there is only this one pattern (of course up to a
constant) of charges or whether there exist others. Do you have
references, where these questions are discussed?

--
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

Arvind Rajaraman

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\nmartin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google. com>...\n> Now, electric charge is, in the Weinberg model of electroweak\n> interactions and in the standard model, linked to weak isospin and\n> weak hypercharge. If you fix this quantities for particles, then\n> electric charge is naturally quantized. It now happens that there are\n> several constraints on the values of weak isospin and weak hypercharge\n> due to anomaly cancellation which, in the standard model without\n> gravity, fixes these numbers up to a constant.\n> But if you also include the simplest model of gravity in the standard\n> model, you get another constraint due to gravitational anomalies which\n> totally fixes the values of isospin and hypercharge.\n\nThis is not correct. In four dimensions, the gauge and gravitational\nanomalies come from triangle diagrams which involve either one or\nthree chiral currents. The gauge anomalies involve the U(1)\nhypercharge and the SU(2) gauge bosons. The nontrivial anomalies are\nthe ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and\nSU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if\nthe sum of hypercharges in a generation is zero, and the sum of the\ncubes is also zero. These happen to be satisfied.\n\nThe only gravitational anomaly is U(1)-graviton-graviton, which tells\nus again that the sum of hypercharges in a generation is zero. So we\nget no new constraints.\n\nFurthermore, it is clear that multiplying all the hypercharges by a\nconstant cannot affect the vanishing of the anomalies. So we cannot\nget charge quantization from anomaly considerations alone.\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>martin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google.com>...
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.
> But if you also include the simplest model of gravity in the standard
> model, you get another constraint due to gravitational anomalies which
> totally fixes the values of isospin and hypercharge.

This is not correct. In four dimensions, the gauge and gravitational
anomalies come from triangle diagrams which involve either one or
three chiral currents. The gauge anomalies involve the U(1)
hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
the sum of hypercharges in a generation is zero, and the sum of the
cubes is also zero. These happen to be satisfied.

The only gravitational anomaly is U(1)-graviton-graviton, which tells
us again that the sum of hypercharges in a generation is zero. So we
get no new constraints.

Furthermore, it is clear that multiplying all the hypercharges by a
constant cannot affect the vanishing of the anomalies. So we cannot
get charge quantization from anomaly considerations alone.

Igor

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\n\nmartin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google. com>...\n> When Dirac showed that the existence magnetic monopole would somehow\n> through nontrivial homotopy groups imply the quantization of electric\n> charge, many physicists believed that this possibility would be that\n> one that is realized in nature because there were no other explanation\n> for it and because a topological explanation sounded somewhat\n> "beautiful".\n> Now, electric charge is, in the Weinberg model of electroweak\n> interactions and in the standard model, linked to weak isospin and\n> weak hypercharge. If you fix this quantities for particles, then\n> electric charge is naturally quantized. It now happens that there are\n> several constraints on the values of weak isospin and weak hypercharge\n> due to anomaly cancellation which, in the standard model without\n> gravity, fixes these numbers up to a constant.\n> But if you also include the simplest model of gravity in the standard\n> model, you get another constraint due to gravitational anomalies which\n> totally fixes the values of isospin and hypercharge. So this extension\n> of the standard model leads naturally to a charge quantization, just\n> as the magnetic monopole.\n> My question is now, why do physicists not believe in this "extended"\n> standard model, while they did believe in the magnetic monopole? Is it\n> just because of renormalization?\n\n\nWhich "extended" standard model are you referring to? There\'s String\nTheory, Loop Quantum Gravity, M-Theory, or Supergravity, just to name\na few of the competing models that are out there these days. And just\nas a point, not everyone buys into the existence of magnetic\nmonopoles, either. In fact, I think those that do are very much in\nthe minority. The bottom line is that all of these ideas are still\nseverely lacking in any real experimental/observational support.\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>martin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google.com>...
> When Dirac showed that the existence magnetic monopole would somehow
> through nontrivial homotopy groups imply the quantization of electric
> charge, many physicists believed that this possibility would be that
> one that is realized in nature because there were no other explanation
> for it and because a topological explanation sounded somewhat
> "beautiful".
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.
> But if you also include the simplest model of gravity in the standard
> model, you get another constraint due to gravitational anomalies which
> totally fixes the values of isospin and hypercharge. So this extension
> of the standard model leads naturally to a charge quantization, just
> as the magnetic monopole.
> My question is now, why do physicists not believe in this "extended"
> standard model, while they did believe in the magnetic monopole? Is it
> just because of renormalization?

Which "extended" standard model are you referring to? There's String
Theory, Loop Quantum Gravity, M-Theory, or Supergravity, just to name
a few of the competing models that are out there these days. And just
as a point, not everyone buys into the existence of magnetic
monopoles, either. In fact, I think those that do are very much in
the minority. The bottom line is that all of these ideas are still
severely lacking in any real experimental/observational support.

Arvind Rajaraman

Apr13-04, 03:43 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>arvindra72@hotmail.com (Arvind Rajaraman) wrote in message news:<cc21857b.0404112128.387c559a@posting.google. com>...\n> martin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google. com>...\n> > Now, electric charge is, in the Weinberg model of electroweak\n> > interactions and in the standard model, linked to weak isospin and\n> > weak hypercharge. If you fix this quantities for particles, then\n> > electric charge is naturally quantized. It now happens that there are\n> > several constraints on the values of weak isospin and weak hypercharge\n> > due to anomaly cancellation which, in the standard model without\n> > gravity, fixes these numbers up to a constant.\n> > But if you also include the simplest model of gravity in the standard\n> > model, you get another constraint due to gravitational anomalies which\n> > totally fixes the values of isospin and hypercharge.\n>\n> This is not correct. In four dimensions, the gauge and gravitational\n> anomalies come from triangle diagrams which involve either one or\n> three chiral currents. The gauge anomalies involve the U(1)\n> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are\n> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and\n> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if\n> the sum of hypercharges in a generation is zero, and the sum of the\n> cubes is also zero. These happen to be satisfied.\n\nI should make the above statements precise. Actually, the first\nvanishes if the sum of hypercharges of the left handed particles is\nzero, and the second if the sum of the cubes of the left handed\nparticles minus that of the right handed particles is zero. So the\ngravitational anomaly is actually new, it says that the sum of all the\nhypercharges in a generation is zero.\n\n> Furthermore, it is clear that multiplying all the hypercharges by a\n> constant cannot affect the vanishing of the anomalies. So we cannot\n> get charge quantization from anomaly considerations alone.\n\nThis statement remains valid.\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>arvindra72@hotmail.com (Arvind Rajaraman) wrote in message news:<cc21857b.0404112128.387c559a@posting.google.com>...
> martin_lohmann@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.0404110719.482686a4@posting.google.com>...
> > Now, electric charge is, in the Weinberg model of electroweak
> > interactions and in the standard model, linked to weak isospin and
> > weak hypercharge. If you fix this quantities for particles, then
> > electric charge is naturally quantized. It now happens that there are
> > several constraints on the values of weak isospin and weak hypercharge
> > due to anomaly cancellation which, in the standard model without
> > gravity, fixes these numbers up to a constant.
> > But if you also include the simplest model of gravity in the standard
> > model, you get another constraint due to gravitational anomalies which
> > totally fixes the values of isospin and hypercharge.
>
> This is not correct. In four dimensions, the gauge and gravitational
> anomalies come from triangle diagrams which involve either one or
> three chiral currents. The gauge anomalies involve the U(1)
> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
> the sum of hypercharges in a generation is zero, and the sum of the
> cubes is also zero. These happen to be satisfied.

I should make the above statements precise. Actually, the first
vanishes if the sum of hypercharges of the left handed particles is
zero, and the second if the sum of the cubes of the left handed
particles minus that of the right handed particles is zero. So the
gravitational anomaly is actually new, it says that the sum of all the
hypercharges in a generation is zero.

> Furthermore, it is clear that multiplying all the hypercharges by a
> constant cannot affect the vanishing of the anomalies. So we cannot
> get charge quantization from anomaly considerations alone.

This statement remains valid.

Alfred Einstead

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>martin_lohmann@t-online.de (Martin Lohmann) wrote:\n> Now, electric charge is, in the Weinberg model of electroweak\n> interactions and in the standard model, linked to weak isospin and\n> weak hypercharge. If you fix this quantities for particles, then\n> electric charge is naturally quantized. It now happens that there are\n> several constraints on the values of weak isospin and weak hypercharge\n> due to anomaly cancellation which, in the standard model without\n> gravity, fixes these numbers up to a constant.\n\nThe "several constraints" do NOT fix the charge, not even\nassuming generation-independence. They only fix an undetermined\nlinear combination of the quantum numbers\nG = (Baryon-Lepton)/2, and Y = Weak Isospin.\n\nIn order to recover Y, you also need to have the additional\nassumption that:\n* there exists at least one sterile fermion sector.\nThese would be the right-neutrinos and left-antineutrinos.\n\nMore generally, you have both Y and G and (as alluded to elsewhere)\nthere are other regularities that supersede the considerations\nyou\'re rasing and point in an entirely different direction. In\nparticular: (1) the 5 quantum numbers a, b, c, d, e:\nFor U(2):\na, b: Y/g\' - G +/- I3/g\nFor U(3):\nc: G + sqrt(4/3) L3/gs\nd, e: G - sqrt(1/3) L3/gs +/- L8/gs\nall assume only the values +/- 1/2 over the fermion spectrum;\nwith all 32 combinations giving you each of the particles of\neach generation; (2) all flavor-changing interactions are\nof the form\nFor U(2):\na up b down <-> a down b up\nFor U(3):\nc up d down <-> c down d up\nd up e down <-> d down e up\ne up c down <-> e down c up\n\n(3) the 2 invariants are those associated with the U(2) x U(3)\nstructure alluded to above:\nFor U(2):\n3 (Y/g\' - G)^2 + (I1^2 + I2^2 + I3^2)/g^2 = 3/4\nFor U(3):\n6 G^2 + (L1^2 + ... + L8^2)/gs^2 = 3/2\nwhich means it\'s actually Y\' = Y/g\' - G, I3, G, L3, L8 that are\nthe natural quantities to consider, rather than Y, I3, L3 and L8.\nIn particular (4) they are mutually orthogonal over the fermion\nspectrum:\ntr(Y\' I3) = tr(Y\' G) = tr(Y\' L3) = tr(Y\' L8) = 0\ntr(I3 G) = tr(I3 L3) = tr(I3 L8) = 0\ntr(G L3) = tr(G L8) = tr(L3 L8) = 0.\n\nThere\'s a deeper level of quantization at work here -- the 5-fold\nq-bit quantization described above which bespeaks an underlying\nU(2) x U(3) symmetry group -- all of which supersedes the Dirac\nargument.\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>martin_lohmann@t-online.de (Martin Lohmann) wrote:
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.

The "several constraints" do NOT fix the charge, not even
assuming generation-independence. They only fix an undetermined
linear combination of the quantum numbers
G = (Baryon-Lepton)/2, and Y = Weak Isospin.

In order to recover Y, you also need to have the additional
assumption that:
* there exists at least one sterile fermion sector.
These would be the right-neutrinos and left-antineutrinos.

More generally, you have both Y and G and (as alluded to elsewhere)
there are other regularities that supersede the considerations
you're rasing and point in an entirely different direction. In
particular: (1) the 5 quantum numbers a, b, c, d, e:
For U(2):
a, b: Y/g' - G +/- I3/g
For U(3):
c: G + \sqrt(4/3) L3/gs
d, e: G - \sqrt(1/3) L3/gs +/- L8/gs
all assume only the values +/- 1/2 over the fermion spectrum;
with all 32 combinations giving you each of the particles of
each generation; (2) all flavor-changing interactions are
of the form
For U(2):
a up b down <-> a down b up
For U(3):
c up d down <-> c down d up
d up e down <-> d down e up
e up c down <-> e down c up

(3) the 2 invariants are those associated with the U(2) x U(3)
structure alluded to above:
For U(2):
3 (Y/g' - G)^2 + (I1^2 + I2^2 + I3^2)/g^2 = 3/4
For U(3):
6 G^2 + (L1^2 + ... + L8^2)/gs^2 = 3/2
which means it's actually Y' = Y/g' - G, I3, G, L3, L8 that are
the natural quantities to consider, rather than Y, I3, L3 and L8.
In particular (4) they are mutually orthogonal over the fermion
spectrum:
tr(Y' I3) = tr(Y' G) = tr(Y' L3) = tr(Y' L8) =
tr(I3 G) = tr(I3 L3) = tr(I3 L8) =
tr(G L3) = tr(G L8) = tr(L3 L8) = .

There's a deeper level of quantization at work here -- the 5-fold
q-bit quantization described above which bespeaks an underlying
U(2) x U(3) symmetry group -- all of which supersedes the Dirac
argument.

Martin Lohmann

Apr14-04, 05:51 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>> I always wondered, if there is only this one pattern (of course up to a\n> constant) of charges or whether there exist others. Do you have\n> references, where these questions are discussed?\n\nWell, I learned about constraints on these values due to anomaly\ncancellation in "Effective Lagrangians for the Standard Model" by A.\nDobado et al., but this wont help you too much, because it treats only\nthis pattern of cancellation of gravitational anomalies and usual\nstandard mdel anomalies.\nBut I cant really think of another way to limit the number possible\nchoices of these constants, because they are nothing but coupling\nconstants of the SU(2)_L x U(1) electroweak gauge group and are\ntherefore classically not constrained, but on a quantum level only due\nto anomalies (which is, as far as I know, the only striking difference\nbetween classical and quantum symmetries).\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>> I always wondered, if there is only this one pattern (of course up to a
> constant) of charges or whether there exist others. Do you have
> references, where these questions are discussed?

Well, I learned about constraints on these values due to anomaly
cancellation in "Effective Lagrangians for the Standard Model" by A.
Dobado et al., but this wont help you too much, because it treats only
this pattern of cancellation of gravitational anomalies and usual
standard mdel anomalies.
But I cant really think of another way to limit the number possible
choices of these constants, because they are nothing but coupling
constants of the SU(2)_L x U(1) electroweak gauge group and are
therefore classically not constrained, but on a quantum level only due
to anomalies (which is, as far as I know, the only striking difference
between classical and quantum symmetries).

Martin Lohmann

Apr15-04, 11:19 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>> This is not correct. In four dimensions, the gauge and gravitational\n> anomalies come from triangle diagrams which involve either one or\n> three chiral currents. The gauge anomalies involve the U(1)\n> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are\n> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and\n> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if\n> the sum of hypercharges in a generation is zero, and the sum of the\n> cubes is also zero. These happen to be satisfied.\n>\n> The only gravitational anomaly is U(1)-graviton-graviton, which tells\n> us again that the sum of hypercharges in a generation is zero. So we\n> get no new constraints.\n>\n> Furthermore, it is clear that multiplying all the hypercharges by a\n> constant cannot affect the vanishing of the anomalies. So we cannot\n> get charge quantization from anomaly considerations alone.\n\n\n\nAs far as I know, there are the anomalies due to SU(3), SU(2)_L and\nU(1) gauge symmetry and due to gravitational effects (to be more\nprecise, the gravitational anomalies are present if you take the\nordinary standard model lagrangian and make it covariant under general\nspacetime diffeomorphisms by replacing all derivatives with their\ngeneral relativistic counterparts; quantizing this is another\nproblem). SU(3) anomaly cancellation implies that\n\n0 = SUM y_L - y_R\n\nwhere the sum goes over all quark families and L and R denote the\nleft-and right- handed components of hypercharge. From SU(2)_L we get\n\n0 = 3 x SUM_1 y_L + SUM_2 y_L\n\nwhere SUM_1 goes over quark families and SUM_2 over lepton families.\nThe U(1) anomaly gives\n\n0 = 3 x SUM_1 y_L^3 - y_R^3 + SUM_2 y_L^3 - y_R^3.\n\nThese conditions do determine the hypercharges up to a constant which\ncan be obtained by setting the electrons electric charge equal to -1.\nBut there is another anomaly, namely the gravitational anomaly, which\ngives another constraint,\n\n0 = 3 x SUM_3 y_L - y_R + SUM_4 y_L - y_R\n\nwhere SUM_3 is over all quarks and SUM_4 is over all leptons. This\nequations fully determines the hypercharges. Of course, it seems\nstrange to me how one can determine the electrons charge by purely\ntheoretical methods, but when I read this, it seemed consistent\n(please correct me if I am wrong).\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>> This is not correct. In four dimensions, the gauge and gravitational
> anomalies come from triangle diagrams which involve either one or
> three chiral currents. The gauge anomalies involve the U(1)
> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
> the sum of hypercharges in a generation is zero, and the sum of the
> cubes is also zero. These happen to be satisfied.
>
> The only gravitational anomaly is U(1)-graviton-graviton, which tells
> us again that the sum of hypercharges in a generation is zero. So we
> get no new constraints.
>
> Furthermore, it is clear that multiplying all the hypercharges by a
> constant cannot affect the vanishing of the anomalies. So we cannot
> get charge quantization from anomaly considerations alone.

As far as I know, there are the anomalies due to SU(3), SU(2)_L and
U(1) gauge symmetry and due to gravitational effects (to be more
precise, the gravitational anomalies are present if you take the
ordinary standard model lagrangian and make it covariant under general
spacetime diffeomorphisms by replacing all derivatives with their
general relativistic counterparts; quantizing this is another
problem). SU(3) anomaly cancellation implies that

= SUM y_L - y_R

where the sum goes over all quark families and L and R denote the
left-and right- handed components of hypercharge. From SU(2)_L we get

= 3 x SUM_1 y_L + SUM_2 y_L

where SUM_1 goes over quark families and SUM_2 over lepton families.
The U(1) anomaly gives

= 3 x SUM_1 y_L^3 - y_R^3 + SUM_2 y_L^3 - y_R^3[/itex].

These conditions do determine the hypercharges up to a constant which
can be obtained by setting the electrons electric charge equal to -1.
But there is another anomaly, namely the gravitational anomaly, which
gives another constraint,

[itex]= 3 x SUM_3 y_L - y_R + SUM_4 y_L - y_R

where SUM_3 is over all quarks and SUM_4 is over all leptons. This
equations fully determines the hypercharges. Of course, it seems
strange to me how one can determine the electrons charge by purely
theoretical methods, but when I read this, it seemed consistent
(please correct me if I am wrong).