Mass and strong interaction.

DW

<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,\nSeveral months ago, Frank Wilczek wrote two\narticles about F=ma (appearing in "Physics Today"\nOct and Dec, 2004). He thinks it\'s a culture to\naccept such a formula. It seems to me he tried to\nunderstand the origin inertia mass from the view\npoint of strong interaction. I am just curious whether we\ncan trace the origin of mass back to the interaction\nof elementary particles. In other words, why we need some\nforce to accelerated something (e.g a neutron) in vacuum from\nthe view point of QFT? Is it due to some kind of interaction\npreventing the acceleration of the neutron? Or is it because\nof the structure of space-time? Is this related to Higgs?\n\nThanks!\nDW\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,
Several months ago, Frank Wilczek wrote two
articles about [itex]F=ma[/itex] (appearing in "Physics Today"
Oct and Dec, 2004). He thinks it's a culture to
accept such a formula. It seems to me he tried to
understand the origin inertia mass from the view
point of strong interaction. I am just curious whether we
can trace the origin of mass back to the interaction
of elementary particles. In other words, why we need some
force to accelerated something (e.g a neutron) in vacuum from
the view point of QFT? Is it due to some kind of interaction
preventing the acceleration of the neutron? Or is it because
of the structure of space-time? Is this related to Higgs?

Thanks!
DW

Igor Khavkine

<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>On 2005-06-23, DW <an_eyas@yahoo.com.cn> wrote:\n> Hello,\n> Several months ago, Frank Wilczek wrote two\n> articles about F=ma (appearing in "Physics Today"\n> Oct and Dec, 2004). He thinks it\'s a culture to\n> accept such a formula. It seems to me he tried to\n> understand the origin inertia mass from the view\n> point of strong interaction. I am just curious whether we\n> can trace the origin of mass back to the interaction\n> of elementary particles. In other words, why we need some\n> force to accelerated something (e.g a neutron) in vacuum from\n> the view point of QFT? Is it due to some kind of interaction\n> preventing the acceleration of the neutron? Or is it because\n> of the structure of space-time? Is this related to Higgs?\n\n>From the point of view of QFT, inertial mass is the minimum energy m\nbetween the ground state and an excited state of a quantum field.\nSometimes m is also referred to as a gap. A single particle corresponds\nto a small excitation on top of the ground state. Think first excited\nstate of a harmonic oscillator compared to its ground state.\n\nRelativistic invariance these states be labeled by momentum and the\nenergy be E=sqrt(p^2+m^2) or E^2-p^2 = m^2. This translates into single\nparticle states being described by solutions of a wave equation of the\nform box(phi) = 0, where "box" is the d\'Alambertian. The type of wave\nfunction phi depends on the type of quantum field under consideration.\nLocality (or rather cluster separability) garantees that several\nparticle states containing widely separating wave packets obey the same\nequation locally.\n\nOnce interactions are taken into account, the joint wave equation\ndescribing a multi-particle system can no longer be factored into\nindividual wave equations and becomes coupled (think multiparticle\ninteracting Schroedinger equation).\n\nThe classical limit is obtained by applying the WKB approximation, which\ntells us that the phase of the wave function satisfies the\nHamilton-Jacobi equation. The characteristics of this first order scalar\nPDE are the trajectories of the solution of the relativistic Hamiltonian\ndynamical system with Hamiltonian of the form\n\nH = (E^2-p^2 - m^2) + (other particle) + (interactions) = 0.\n\nIf the non-relativistic limit is taken, the Hamiltonian formulation\nturns into the more familiar\n\nH = p^2/2m + (other particles) + (interactions),\n\nwhere p is now the non-relativistic momentum p = mv. From here Newton\'s\nsecond law F(due to interactions) = dp/dt = m dv/dt is recovered in the\nusual way.\n\nTo review. The m in the F=ma comes from the relativistic relation\nE^2-p^2 = m^2. This relation comes from the wave equation satisfied by\nfew particle states. This wave equation is derived from the spectrum of\nthe excitations of a quantum field. The specific form of the spectrum is\ndue to relativistic invariance and the existence of a minimum excitation\nenergy (the gap).\n\nThe next question is "why the gap"? And the answer is not really known.\nIt is hypothesized that all elementary particles in the standard model\nstart out as massless (no gap), but then acquire a mass of the form\n"(expectation value of Higgs field) x (coupling constant to Higgs field)".\n\nHope this helps.\n\nIgor\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 [itex]2005-06-23, DW[/itex] <an_eyas@yahoo.com.cn> wrote:
> Hello,
> Several months ago, Frank Wilczek wrote two
> articles about [itex]F=ma[/itex] (appearing in "Physics Today"
> Oct and Dec, 2004). He thinks it's a culture to
> accept such a formula. It seems to me he tried to
> understand the origin inertia mass from the view
> point of strong interaction. I am just curious whether we
> can trace the origin of mass back to the interaction
> of elementary particles. In other words, why we need some
> force to accelerated something (e.g a neutron) in vacuum from
> the view point of QFT? Is it due to some kind of interaction
> preventing the acceleration of the neutron? Or is it because
> of the structure of space-time? Is this related to Higgs?

>From the point of view of QFT, inertial mass is the minimum energy m
between the ground state and an excited state of a quantum field.
Sometimes m is also referred to as a gap. A single particle corresponds
to a small excitation on top of the ground state. Think first excited
state of a harmonic oscillator compared to its ground state.

Relativistic invariance these states be labeled by momentum and the
energy [itex]be E=\sqrt(p^2+m^2)[/itex] or [itex]E^2-p^2 = m^2[/itex]. This translates into single
particle states being described by solutions of a wave equation of the
form [itex]box(\phi) = 0,[/itex] where "box" is the d'Alambertian. The type of wave
function [itex]\phi[/itex] depends on the type of quantum field under consideration.
Locality (or rather cluster separability) garantees that several
particle states containing widely separating wave packets obey the same
equation locally.

Once interactions are taken into account, the joint wave equation
describing a multi-particle system can no longer be factored into
individual wave equations and becomes coupled (think multiparticle
interacting Schroedinger equation).

The classical limit is obtained by applying the WKB approximation, which
tells us that the phase of the wave function satisfies the
Hamilton-Jacobi equation. The characteristics of this first order scalar
PDE are the trajectories of the solution of the relativistic Hamiltonian
dynamical system with Hamiltonian of the form

[itex]H = (E^2-p^2 - m^2) +[/itex] (other particle) + (interactions) = .

If the non-relativistic limit is taken, the Hamiltonian formulation
turns into the more familiar

[itex]H = p^2/2m +[/itex] (other particles) + (interactions),

where p is now the non-relativistic momentum [itex]p = mv[/itex]. From here Newton's
second law F(due to interactions) [itex]= dp/dt = m dv/dt[/itex] is recovered in the
usual way.

To review. The m in the [itex]F=ma[/itex] comes from the relativistic relation
[itex]E^2-p^2 = m^2[/itex]. This relation comes from the wave equation satisfied by
few particle states. This wave equation is derived from the spectrum of
the excitations of a quantum field. The specific form of the spectrum is
due to relativistic invariance and the existence of a minimum excitation
energy (the gap).

The next question is "why the gap"? And the answer is not really known.
It is hypothesized that all elementary particles in the standard model
start out as massless (no gap), but then acquire a mass of the form
"(expectation value of Higgs field) x (coupling constant to Higgs field)".

Hope this helps.

Igor

Eugene Stefanovich

<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\nDW wrote:\n> In other words, why we need some\n> force to accelerated something (e.g a neutron) in vacuum from\n> the view point of QFT? Is it due to some kind of interaction\n> preventing the acceleration of the neutron? Or is it because\n> of the structure of space-time? Is this related to Higgs?\n\nThis is called "Newton\'s 1-st law": a particle does not\naccelerate (moves with a constant velocity) if there are no\nother particles around which can interact with it.\n\nIf there are other particles around, then normally the first\nparticle feels a force, i.e., accelerates. Particle\'s mass\nm connects acceleration with force. Most naturally, parameter\nm (along with spin) appears in the Wigner\'s classification\nscheme of elementary particles as unitary irreducible\nrepresentations of the Poincare group.\n\nEugene Stefanovich.\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>DW wrote:
> In other words, why we need some
> force to accelerated something (e.g a neutron) in vacuum from
> the view point of QFT? Is it due to some kind of interaction
> preventing the acceleration of the neutron? Or is it because
> of the structure of space-time? Is this related to Higgs?

This is called "Newton's [itex]1-st[/itex] law": a particle does not
accelerate (moves with a constant velocity) if there are no
other particles around which can interact with it.

If there are other particles around, then normally the first
particle feels a force, i.e., accelerates. Particle's mass
m connects acceleration with force. Most naturally, parameter
m (along with spin) appears in the Wigner's classification
scheme of elementary particles as unitary irreducible
representations of the Poincare group.

Eugene Stefanovich.

Eugene Stefanovich

<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>Igor Khavkine wrote:\n> On 2005-06-23, DW <an_eyas@yahoo.com.cn> wrote:\n>\n>>Hello,\n>> Several months ago, Frank Wilczek wrote two\n>>articles about F=ma (appearing in "Physics Today"\n>>Oct and Dec, 2004). He thinks it\'s a culture to\n>>accept such a formula. It seems to me he tried to\n>>understand the origin inertia mass from the view\n>>point of strong interaction. I am just curious whether we\n>>can trace the origin of mass back to the interaction\n>>of elementary particles. In other words, why we need some\n>>force to accelerated something (e.g a neutron) in vacuum from\n>>the view point of QFT? Is it due to some kind of interaction\n>>preventing the acceleration of the neutron? Or is it because\n>>of the structure of space-time? Is this related to Higgs?\n>\n>\n>>From the point of view of QFT, inertial mass is the minimum energy m\n> between the ground state and an excited state of a quantum field.\n> Sometimes m is also referred to as a gap. A single particle corresponds\n> to a small excitation on top of the ground state. Think first excited\n> state of a harmonic oscillator compared to its ground state.\n>\n> Relativistic invariance these states be labeled by momentum and the\n> energy be E=sqrt(p^2+m^2) or E^2-p^2 = m^2. This translates into single\n> particle states being described by solutions of a wave equation of the\n> form box(phi) = 0, where "box" is the d\'Alambertian. The type of wave\n> function phi depends on the type of quantum field under consideration.\n> Locality (or rather cluster separability) garantees that several\n> particle states containing widely separating wave packets obey the same\n> equation locally.\n>\n> Once interactions are taken into account, the joint wave equation\n> describing a multi-particle system can no longer be factored into\n> individual wave equations and becomes coupled (think multiparticle\n> interacting Schroedinger equation).\n>\n> The classical limit is obtained by applying the WKB approximation, which\n> tells us that the phase of the wave function satisfies the\n> Hamilton-Jacobi equation. The characteristics of this first order scalar\n> PDE are the trajectories of the solution of the relativistic Hamiltonian\n> dynamical system with Hamiltonian of the form\n>\n> H = (E^2-p^2 - m^2) + (other particle) + (interactions) = 0.\n>\n> If the non-relativistic limit is taken, the Hamiltonian formulation\n> turns into the more familiar\n>\n> H = p^2/2m + (other particles) + (interactions),\n>\n> where p is now the non-relativistic momentum p = mv. From here Newton\'s\n> second law F(due to interactions) = dp/dt = m dv/dt is recovered in the\n> usual way.\n>\n> To review. The m in the F=ma comes from the relativistic relation\n> E^2-p^2 = m^2. This relation comes from the wave equation satisfied by\n> few particle states. This wave equation is derived from the spectrum of\n> the excitations of a quantum field. The specific form of the spectrum is\n> due to relativistic invariance and the existence of a minimum excitation\n> energy (the gap).\n>\n> The next question is "why the gap"? And the answer is not really known.\n> It is hypothesized that all elementary particles in the standard model\n> start out as massless (no gap), but then acquire a mass of the form\n> "(expectation value of Higgs field) x (coupling constant to Higgs field)".\n\nExpanding on my previous post let me suggest a different logic of\nintroducing mass, force, and acceleration in QFT.\n\n1. Elementary particles are described by unitary irreducible\nrepresentations of the Poincare group. They are characterized by\ntwo parameters (according to two Casimir operators of the Poincare\ngroup): m (mass) and s (spin). Wigner, 1939.\n\n2. Particle position is described by the Newton-Wigner (1949) operator\nr. In a multiparticle non-interacting system, the commutators\nof r and p (momentum) with the Hamiltonian H_0 are\n\n[r, H_0] = iv\n[p, H_0] = 0\n\nso that\n\nr(t) = exp(iHt) r(0) exp(-iHt) = r + vt\np(t) = p(0)\n\ni.e., each particle moves with constant velocity\n(v = p/sqrt(p^2 + m^2)), which is Newton\'s 1st law.\n\n3. In the presence of interaction, the multiparticle Hamiltonian\nis H = H_0 + V (Dirac, 1949), where V depends on positions\nand momenta of all\nparticles in the system (ignore for a moment creation/annihilation\nterms, they don\'t change anything, in principle, see below).\nThe time derivative of p then becomes\n\ndp/dt = -i [p, H] = dV/dr (1)\n\nThe operator F = dV/dr is called "force" (by definition).\n\n4. In the non-relativistic limit p = mv, and eq. (1) yields\n\na = dv/dt = F/m\n\nwhich is the 2nd Newton\'s law (a is acceleration).\n\n5. Creation and annihilation of particles are described by\nterms in V that do not commute with the particle number operator.\nThey can be added to eq. (1). The most prominent effect would\nbe the appearance of "radiation reaction" forces due to emission\nof photons by accelerated charges.\n\nEugene.\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>Igor Khavkine wrote:
> On [itex]2005-06-23, DW[/itex] <an_eyas@yahoo.com.cn> wrote:
>
>>Hello,
>> Several months ago, Frank Wilczek wrote two
>>articles about [itex]F=ma[/itex] (appearing in "Physics Today"
>>Oct and Dec, 2004). He thinks it's a culture to
>>accept such a formula. It seems to me he tried to
>>understand the origin inertia mass from the view
>>point of strong interaction. I am just curious whether we
>>can trace the origin of mass back to the interaction
>>of elementary particles. In other words, why we need some
>>force to accelerated something (e.g a neutron) in vacuum from
>>the view point of QFT? Is it due to some kind of interaction
>>preventing the acceleration of the neutron? Or is it because
>>of the structure of space-time? Is this related to Higgs?
>
>
>>From the point of view of QFT, inertial mass is the minimum energy m
> between the ground state and an excited state of a quantum field.
> Sometimes m is also referred to as a gap. A single particle corresponds
> to a small excitation on top of the ground state. Think first excited
> state of a harmonic oscillator compared to its ground state.
>
> Relativistic invariance these states be labeled by momentum and the
> energy [itex]be E=\sqrt(p^2+m^2)[/itex] or [itex]E^2-p^2 = m^2[/itex]. This translates into single
> particle states being described by solutions of a wave equation of the
> form [itex]box(\phi) = 0,[/itex] where "box" is the d'Alambertian. The type of wave
> function [itex]\phi[/itex] depends on the type of quantum field under consideration.
> Locality (or rather cluster separability) garantees that several
> particle states containing widely separating wave packets obey the same
> equation locally.
>
> Once interactions are taken into account, the joint wave equation
> describing a multi-particle system can no longer be factored into
> individual wave equations and becomes coupled (think multiparticle
> interacting Schroedinger equation).
>
> The classical limit is obtained by applying the WKB approximation, which
> tells us that the phase of the wave function satisfies the
> Hamilton-Jacobi equation. The characteristics of this first order scalar
> PDE are the trajectories of the solution of the relativistic Hamiltonian
> dynamical system with Hamiltonian of the form
>
> [itex]H = (E^2-p^2 - m^2) +[/itex] (other particle) + (interactions) = .
>
> If the non-relativistic limit is taken, the Hamiltonian formulation
> turns into the more familiar
>
> [itex]H = p^2/2m +[/itex] (other particles) + (interactions),
>
> where p is now the non-relativistic momentum [itex]p = mv[/itex]. From here Newton's
> second law F(due to interactions) [itex]= dp/dt = m dv/dt[/itex] is recovered in the
> usual way.
>
> To review. The m in the [itex]F=ma[/itex] comes from the relativistic relation
> [itex]E^2-p^2 = m^2[/itex]. This relation comes from the wave equation satisfied by
> few particle states. This wave equation is derived from the spectrum of
> the excitations of a quantum field. The specific form of the spectrum is
> due to relativistic invariance and the existence of a minimum excitation
> energy (the gap).
>
> The next question is "why the gap"? And the answer is not really known.
> It is hypothesized that all elementary particles in the standard model
> start out as massless (no gap), but then acquire a mass of the form
> "(expectation value of Higgs field) x (coupling constant to Higgs field)".

Expanding on my previous post let me suggest a different logic of
introducing mass, force, and acceleration in QFT.

1. Elementary particles are described by unitary irreducible
representations of the Poincare group. They are characterized by
two parameters (according to two Casimir operators of the Poincare
group): m (mass) and s (spin). Wigner, 1939.

2. Particle position is described by the Newton-Wigner (1949) operator
r. In a multiparticle non-interacting system, the commutators
of r and p (momentum) with the Hamiltonian [itex]H_0[/itex] are

[r, [itex]H_0] = iv[/itex]
[p, [itex]H_0] =[/itex]

so that

r(t) [itex]= \exp(iHt) r(0) \exp(-iHt) = r + vtp(t) = p(0)[/itex]

i.e., each particle moves with constant velocity
[itex](v = p/\sqrt(p^2 + m^2)),[/itex] which is Newton's 1st law.

3. In the presence of interaction, the multiparticle Hamiltonian
is [itex]H = H_0 + V[/itex] (Dirac, 1949), where V depends on positions
and momenta of all
particles in the system (ignore for a moment creation/annihilation
terms, they don't change anything, in principle, see below).
The time derivative of p then becomes

[tex]dp/dt = -i[/itex] [p, [itex]H] = dV/dr[/itex] (1)

The operator [itex]F = dV/dr[/itex] is called "force" (by definition).

4. In the non-relativistic limit p = mv, and eq. (1) yields

[itex]a = dv/dt = F/m[/tex]

which is the 2nd Newton's law (a is acceleration).

5. Creation and annihilation of particles are described by
terms in V that do not commute with the particle number operator.
They can be added to eq. (1). The most prominent effect would
be the appearance of "radiation reaction" forces due to emission
of photons by accelerated charges.

Eugene.

markwh04@yahoo.com

<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>DW wrote:\n> I am just curious whether we can trace the origin of\n> mass back to the interaction of elementary particles.\n\nYes.\n\nIf you start a quantum field theory out with a Lagrangian that has no\nmass terms, it will acquire such terms in the higher order radiative\ncorrections.\n\nFor fermions, for instance, labelling fermions as y\' and y; gauge\nbosons as B; and contractions by []; then the graph corresponding to\n[y(u) y\'(v)] [B(u) B(v)] e^2 y(v) y\'(u)\ngives you an effective mass term proportional to e^2.\n\nIn this context, it\'s of interest to note that for the Leptons, the\ntrace Tr(M_L) of the mass matrix M_L is, to a high degree of accuracy,\njust\nTr(M_L) = v e^2\nwhere v is the Higgs vacuum potential.\n\nScharf et. al. took the general idea one step further in the mid to\nlate 1990\'s. Implementing a 2nd quantized version of gauge invariance,\ncalled perturbative gauge invariance:\n[Q, T[exp(iA)]] = del_{nu} T[d_Q^{nu}(A(x)) exp(iA)]\nwith Q being an appropriately defined coboundary operator; they managed\nin formulating a already-renormalized definition of the T[] operator\nwhich when applied to the formal solution\nS = T[exp(i H_I)]\nstarting from a general interactive Hamiltonian H_I, gives you (in the\ncase of a massive SU(2)xU(1) gauge theory) EXACTLY the right\nconstraints found in the Standard Model, and (as counter-terms) even\nderives a Higgs field out of the blue with the right expression for the\npotential.\n\nThough their method starts out with a massive field in the initial\nLagrangian, I don\'t see any obstacle against removing it from the\ninitial Lagrangian and deriving it as one of the instances of a\nuniversal Lagrangian which envelopes the standard model as one of its\ninstances.\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>DW wrote:
> I am just curious whether we can trace the origin of
> mass back to the interaction of elementary particles.

Yes.

If you start a quantum field theory out with a Lagrangian that has no
mass terms, it will acquire such terms in the higher order radiative
corrections.

For fermions, for instance, labelling fermions as y' and y; gauge
bosons as B; and contractions by []; then the graph corresponding to
[y(u) y'(v)] [B(u) B(v)] [itex]e^2 y(v) y'(u)[/itex]
gives you an effective mass term proportional to [itex]e^2[/itex].

In this context, it's of interest to note that for the Leptons, the
trace [itex]Tr(M_L)[/itex] of the mass matrix [itex]M_L[/itex] is, to a high degree of accuracy,
just
[itex]Tr(M_L) = v e^2[/itex]
where v is the Higgs vacuum potential.

Scharf et. al. took the general idea one step further in the mid to
late 1990's. Implementing a 2nd quantized version of gauge invariance,
called perturbative gauge invariance:
[Q, [itex]T[\exp(iA)]] = del_{\nu} T[d_Q^{\nu}(A(x)) \exp(iA)][/itex]
with Q being an appropriately defined coboundary operator; they managed
in formulating a already-renormalized definition of the T[] operator
which when applied to the formal solution
[itex]S = T[\exp(i H_I)][/itex]
starting from a general interactive Hamiltonian [itex]H_I,[/itex] gives you (in the
case of a massive SU(2)xU(1) gauge theory) EXACTLY the right
constraints found in the Standard Model, and (as counter-terms) even
derives a Higgs field out of the blue with the right expression for the
potential.

Though their method starts out with a massive field in the initial
Lagrangian, I don't see any obstacle against removing it from the
initial Lagrangian and deriving it as one of the instances of a
universal Lagrangian which envelopes the standard model as one of its
instances.

Al.Rivero@gmail.com

<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\nmarkwh04@yahoo.com ha escrito:\n\n>\n> In this context, it\'s of interest to note that for the Leptons, the\n> trace Tr(M_L) of the mass matrix M_L is, to a high degree of accuracy,\n> just\n> Tr(M_L) = v e^2\n> where v is the Higgs vacuum potential.\n\nHello Marc, do you have a clue about who or when did this observation?\nI first noticed it from Jay R. Yablon some months ago, and I did not\nremember to see it in any publication. Do you have better pointers?\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>markwh04@yahoo.com ha escrito:

>
> In this context, it's of interest to note that for the Leptons, the
> trace [itex]Tr(M_L)[/itex] of the mass matrix [itex]M_L[/itex] is, to a high degree of accuracy,
> just
> [itex]Tr(M_L) = v e^2[/itex]
> where v is the Higgs vacuum potential.

Hello Marc, do you have a clue about who or when did this observation?
I first noticed it from Jay R. Yablon some months ago, and I did not
remember to see it in any publication. Do you have better pointers?

\

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