Re: Higgs Particles & Scale-Invariant Metrics & Breaking Scale Symmetry

<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>"Hatim Hegab" &lt;htm_00@hotmail.com&gt; wrote:\n&gt; Can someone here give me a good idea about Higgs theory?, or provide me with\n&gt; some sites that can help?\n\nIt\'s a way of explaining away mass dynamically. So the "m"\'s\nin the Lagrangian are then not constants but actually fields --\nin fact, the same field multiplied by different coupling constants\nfor each type of particle. The field is permanently turned on by\na negative potential energy (negative, that is, relative to the\npotential when the field is turned off). So, if H is the magnitude\nof the field, the potential would be something like:\nk (H^2 - v^2)^2.\n\nRelative to the condition where H is turned off, H = 0, in which\nthe potential is\nk v^4,\nthe potential of the field in the on state will be:\nk (H^2 - v^2)^2 - k v^4\n= k H^4 - 2kv^2 H^2.\n\nThe minimum value, of course, is given by H^2 = v^2. So, it\nfluctuates around the value v, taking on the form\nH = v U chi\nwhere U is a unitary matrix and chi a (arbitrarily chosen) unit\nvector.\n\nAll of this should bother people, because General Relativity says\nthat mass is gravitational charge. If the Higgs is producing mass,\nthen isn\'t it also directly linked to the gravitational field?\n\nIn fact, you can answer this in part, by looking at how the Higgs\nwould act if the world were classical instead of quantum theoretic.\n\nMore interesting is what the Higgs mechanism would look like for\nclassical point particles. In this case, the Lagrangian has the\nform:\nL = m/2 v^2 = m/2 sum (g_{ij} u^i u^j\nwhere\nu^i = d(x^i)/dt\ni = 0,1,2,3\nt = proper time\ng_{ij} = spacetime metric.\n(Which, in fact, yields the geodesic law of motion in a\ngeneral relativistic context).\n\nInterestingly, also, this Lagrangian breaks down when the particle\nis massless -- even though its equations of motion reduce to those\nfor a massless particle, when the limit m -&gt; 0 is taken of them.\n\nBut if one starts out assuming, as above, that mass is dynamically\ngenerated by the Higgs, you\'re starting with massless particles.\n\nUsing the prescription outlined above, the mass is treated as\nthe field, itself, up to a coupling constant. So, it is replaced\nby a coupling constant g = (m/v) multiplied by the field H:\nL = 1/2 g sum (H g_{ij} u^i u^j)\n= 1/2 g sum (H_{ij} u^i u^j)\nwhere\nH_{ij}(x) = H(x) g_{ij}(x).\n\nBasically, what this does is introduce an extra degree of symmetry:\nscale symmetry. So, the Higgs field can be thought of as the\nscale part of a scale-invariant metric, the metric g_{ij} being\nthe rest of the metric, and the potential k(H^2 - v^2)^2 forcing\nH to remain in a permanently turned on state around the value\nH = v; thus breaking scale symmetry.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Hatim Hegab" <htm_00@hotmail.com> wrote:
> Can someone here give me a good idea about Higgs theory?, or provide me with
> some sites that can help?

It's a way of explaining away mass dynamically. So the "m"'s
in the Lagrangian are then not constants but actually fields --
in fact, the same field multiplied by different coupling constants
for each type of particle. The field is permanently turned on by
a negative potential energy (negative, that is, relative to the
potential when the field is turned off). So, if H is the magnitude
of the field, the potential would be something like:
$k (H^2 - v^2)^2$.

Relative to the condition where H is turned off, $H = 0,$ in which
the potential is
$k v^4,$
the potential of the field in the on state will be:
$k (H^2 - v^2)^2 - k v^4= k H^4 - 2kv^2 H^2$.

The minimum value, of course, is given by $H^2 = v^2$. So, it
fluctuates around the value v, taking on the form
$H = v U \chi$
where U is a unitary matrix and $\chi a$ (arbitrarily chosen) unit
vector.

All of this should bother people, because General Relativity says
that mass is gravitational charge. If the Higgs is producing mass,
then isn't it also directly linked to the gravitational field?

In fact, you can answer this in part, by looking at how the Higgs
would act if the world were classical instead of quantum theoretic.

More interesting is what the Higgs mechanism would look like for
classical point particles. In this case, the Lagrangian has the
form:
$L = m/2 v^2 = m/2$ sum $(g_{ij} u^i u^j$
where
$u^i = d(x^i)/dti = 0,1,2,3$
t = proper time
$g_{ij} =$ spacetime metric.
(Which, in fact, yields the geodesic law of motion in a
general relativistic context).

Interestingly, also, this Lagrangian breaks down when the particle
is massless -- even though its equations of motion reduce to those
for a massless particle, when the limit $m ->$ is taken of them.

But if one starts out assuming, as above, that mass is dynamically
generated by the Higgs, you're starting with massless particles.

Using the prescription outlined above, the mass is treated as
the field, itself, up to a coupling constant. So, it is replaced
by a coupling constant $g = (m/v)$ multiplied by the field H:
$L = 1/2 g$ sum $(H g_{ij} u^i u^j)= 1/2 g$ sum $(H_{ij} u^i u^j)$
where
$H_{ij}(x) = H(x) g_{ij}(x)$.

Basically, what this does is introduce an extra degree of symmetry:
scale symmetry. So, the Higgs field can be thought of as the
scale part of a scale-invariant metric, the metric $g_{ij}$ being
the rest of the metric, and the potential $k(H^2 - v^2)^2$ forcing
H to remain in a permanently turned on state around the value
$H = v;$ thus breaking scale symmetry.

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whopkins@csd.uwm.edu (Alfred Einstead) writes: > All of this should bother people, because General Relativity says > that mass is gravitational charge. This is _not_ what GR says. GR says that energy and momentum are the source of gravity. > If the Higgs is producing mass, > then isn't it also directly linked to the gravitational field? Only as far as the Higgs field itself has energy and momentum. -- Esa Peuha student of mathematics at the University of Helsinki http://www.helsinki.fi/~peuha/



Esa A E Peuha wrote: > whopkins@csd.uwm.edu (Alfred Einstead) writes: > > All of this should bother people, because General Relativity says > > that mass is gravitational charge. > > This is [sic] _not_ what GR says. GR says that energy and momentum are > the source of gravity. More correctly, mass = gravitational charge is exactly what the Principle of Equivalence says, and that _therefore_ that energy and momentum are the source of gravity. > > If the Higgs is producing mass, > > then isn't it also directly linked to the gravitational field? > > Only as far as the Higgs field itself has energy and momentum. Non-sequitur. The question was posed as a sequey to what followed it, which already resolved the matter.

Re: Higgs Particles & Scale-Invariant Metrics & Breaking Scale Symmetry

<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>whopkins@csd.uwm.edu (Alfred Einstead) writes:\n\n&gt; More correctly, mass = gravitational charge is exactly what the Principle\n&gt; of Equivalence says, and that _therefore_ that energy and momentum are the\n&gt; source of gravity.\n\nThe Equvalence Principle says that inertial mass equals gravity-affected\nmass, but it says absolutely nothing about the relation between\ngravity-affected mass and gravity-generating mass (and it had better\nnot, since there isn\'t any gravity-generating mass in GR). Anyway, you\ncan\'t possibly hope to derive the fact that the electromagnetic field\ngenerates gravity from that argument, since photons are massless.\n\n--\nEsa Peuha\nstudent of mathematics at the University of Helsinki\nhttp://www.helsinki.fi/~peuha/\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>whopkins@csd.uwm.edu (Alfred Einstead) writes:

> More correctly, mass = gravitational charge is exactly what the Principle
> of Equivalence says, and that _therefore_ that energy and momentum are the
> source of gravity.

The Equvalence Principle says that inertial mass equals gravity-affected
mass, but it says absolutely nothing about the relation between
gravity-affected mass and gravity-generating mass (and it had better
not, since there isn't any gravity-generating mass in GR). Anyway, you
can't possibly hope to derive the fact that the electromagnetic field
generates gravity from that argument, since photons are massless.

--
Esa Peuha
student of mathematics at the University of Helsinki
http://www.helsinki.fi/~peuha/



"Esa A E Peuha" wrote in message news:86p3c5zctts.fsf@sirppi.helsinki.fi... > > > The Equvalence Principle says that inertial mass equals gravity-affected > mass, but it says absolutely nothing about the relation between > gravity-affected mass and gravity-generating mass (and it had better > not, since there isn't any gravity-generating mass in GR). Anyway, you > can't possibly hope to derive the fact that the electromagnetic field > generates gravity from that argument, since photons are massless. > But photons are (at least from one viewpoint) localized energy and energy gravitates too, Norm