The Anatomy of the Electroweak and Color Forces
Rock Brentwood
> Here's a question that puzzles me...
>
> But what system Drives, Produces or Creates these forces? We know a
> fair bit what produces light(photons), electro-magnetic forces, but do
> we know what is the source for these other forces?
"The object lesson to be drawn from this is that one should not bank
too much on the idea of 'the ever-forthcoming quantum gravity finally
arriving, as raiders on vessels from downstream, to win the day.' In
the most surprising plot twist of all, the corsairs seen coming up the
river have unfurled a banner that bespeaks classical physics, while
quantum gravity vanished in the mists with the aether as nothing more
than an army of ghosts."
-- Quoting from below
The electromagnetic force and weak nuclear force are inextricably
mixed up into the electroweak force. Though it is not widely
advertised as such, the unification does not generalize Maxwell's
equations for the electromagnetic part of the force, but REPLACES them
-- namely, with non-linear field equations.
Overall the strong nuclear force and electroweak force are gauge
forces. These have the same appearance as electromagnetic forces,
except that the components of the electric and magnetic fileld are
each replicated for a gauge force (12-fold, here, for the combined
strong and electroweak force).
Maxwell's equations still hold, but they are non-linear. Moreover,
each component (E_x, E_y, E_z) of the E field and each component (B^x,
B^y, B^z) of the B field is now endowed with 12 degrees of
"complexion", one for each of the 12 modes of the strong & electroweak
field. That leads to 12 copies, each of the E and B vectors.
The relation to the potentials (each component of A = (A_x, A_y, A_z),
also replicated 12-fold) is non-linear. In particular, the Maxwell
equations for the field-potential relations
B = curl A, E = -grad phi - @A/@t (@ denotes partial derivative)
are replaced by
B = curl A + A x A, E = -grad phi - @A/@t + A phi - phi A.
For an "abelian" gauge field, the A x A and (A phi - phi A) terms will
cancel out, since phi, A_x, A_y and A_z will satisfy the commutative
law for multiplication.
Here ()x() denotes vector product and ().() denotes scalar product.
The magnetic force law is no longer homogeneous. In place of the
Maxwell equations
div B = 0, curl E + @B/@t = 0
one has
div B = B.A - A.B, curl E + @B/@t = -(A x E + E x A + B phi - phi
B).
There are magnetic sources for non-Abelian gauge fields.
This includes electromagnetism, since it is now part of the (non-
Abelian) electroweak force. For electromagnetism, the modes that
contribute non-linearly to B and E are those (A^W, phi^W) and (A^W*,
phi^W*) associated with the W and W*. For B, the non-linear terms run
something like B = ... + ie/h-bar (A^W x A^W*), E = ... + ie/h-bar
(A^W phi^W* - A^W* phi^W), though I might have some factors and signs
off. Similarly, the magnetic charge density will involve a non-linear
combination that runs something like ie/h-bar (B^W.A^W* - B^W*.A^W),
where B^W and B^W* are the modes of the B field associated with W and
W*.
The other 2 Maxwell fields, D and H, will also satisfy non-linear
field laws:
div D + A.D - D.A = rho, curl H - @D/@t + AxH + HxA + phi D - D phi
= J.
So, there will be electric currents produced from the non-linear
contributions of the fields, themselves:
charge density: rho' = (D.A - A.D), current density J' = (D phi -
phi D - AxH - HxA).
The current conservation law will no longer be (@rho/@t + div J = 0),
but
@rho/@t + rho phi - phi rho + div J + A.J - J.A = D.E - E.D + B.H -
H.B.
When adding in the contribution from the field currents, it will
generally STILL not equate to 0 on the right, but to
div (rho + rho') + @(J + J')/@t = D.E - E.D + B.H - H.B.
The relation of the D and H fields can no longer be taken as trivial
with respect to E and B (actually, they never could, even in Maxwell's
theory, as even Maxwell himself had so insistently pointed out). The
Lorentz relations D = epsilon_0 E, B = mu_0 H TREMENDOUSLY generalize.
Not only do the vacuum relations become a vector-matrix relation (D =
epsilon E), with epsilon as 12x12 matrix, H = epsilon c^2 B, same
epsilon matrix, but both epsilon and mu = (1/c)^2 epsilon^{-1} are
variable. The variability leads to "form factors" -- an effective
dielectric medium -- surrounding each source.
Under the usual assumption made for Yang-Mills gauge fields, the D.E -
E.D + B.H - H.B terms will be 0.
This generalizes even further to relations of the form
D = epsilon E + theta B, H = epsilon c^2 B - theta E
where theta is a second matrix. Both matrices diagonalize with respect
to the natural modes of the electroweak and strong force, so that they
involve only 3 independent components each. In the case of epsilon,
the components 1/(c g^2) are the "coupling constants" g associated
with each sector -- g_1 for U(1) (hypercharge), g_2 for SU(2)
(isospin) and g_3 for SU(3) (the color force).
(Strictly speaking, once you add in theta's you're outside of Yang-
Mills in a more general gauge theory).
For the color force SU(3), the value of the "constant" theta is
treated as an extra parameter in the Standard Model. However, it's not
a constant. Even in the electromagnetic sector, there will be a value
for theta roughly equal to 7/2 (epsilon - epsilon_0) c (as seen in the
Heisenberg-Euler effective classifcal field theory for quantum
electrodynamics).
The asymptotic values of theta can, without loss of generality, be
taken as 0, since one can always add multiples (-theta B) to D and
(+theta E) to H if theta is constant, without affecting the non-linear
Maxwell equations for D and H. Hence, a non-trivial theta is something
that will only be seen
(a) at the cosmological level as a slow cosmological drift
or
(b) near concentrated sources.
For a gauge force that conforms to Lorentz symmetry (i.e. Special
Relativity), one can even have quadratic terms in the constitutive
law, D = l (E x E - c^2 B x B) + m (E x B + B x E); H similar, with l,
m being 12x144 matrices each (but only a few independent components:
only 1 independent component for l and m each in the electroweak
sector). Nothing is known about these coefficients, as far as I'm
aware.
The only Lorentz invariants in 4-D are quadratic and cubic, so no
other terms appear in the constitutive relation for the vacuum.
The coefficient epsilon, for electromagnetism, is an ordinary scalar.
Roughly, it might be modelled by a field equation of the form
((@/@t)^2 - del^2) log epsilon = K epsilon (E^2 - B^2 c^2)
for a suitable value of the constant K. For a point-like source, with
charge e, this in fact produces solutions of 3 types. Taking the
source's "fine structure constant" as alpha = e^2/(2 h epsilon c), one
is able to derive a law of the form
d^2(log alpha)/d(1/r)^2 = k a alpha
here a is the Planck area and k a dimensionless constant related to K
(k = 3 for dynamics based on an effective field theory of a Kaluza-
Klein theory with a Einstein-Hilbert Lagrangian in the higher-
dimensional space).
Bear in mind, despite the surprising emergence of a Planck scale unit,
this is not quantum gravity, nor anything else quantum. It's a
classical equation, since the occurrences of Planck's constant on both
sides cancel. The object lesson to be drawn from this is that one
should not bank too much on the idea of "the ever-forthcoming quantum
gravity finally arriving, as raiders on vessels from downstream, to
win the day." In the most surprising plot twist of all, the corsairs
seen coming up the river have unfurled a banner that bespeaks
classical physics, while quantum gravity vanished in the mists with
the aether as nothing more than an army of ghosts.
Its most important solutions are of the form
(1) alpha approaches a constant as the distance r from the source
approaches infinity. As one approaches the source, alpha gets larger
(i.e. the vacuum around the charge is screened). At a finite radius,
alpha goes to infinity. This is known as the Landau Pole.
(2) alpha approaches infinity as r goes to infinity, and does so in
such a way, that the E field, which is E = D/epsilon, approaches a
constant. In this phase, no isolated monopoles can exist. (One can
solve the above equation for dipoles to see how alpha behaves in that
setting, but the differential equation is not elementary and requires
Mathematica). In the other direction, as r -> 0 and one approaches the
source, alpha gets smaller and approaches 0. This is an instance of
what's called "asymptotic freedom". The vacuum surrounding the source
"anti-screens".
(3) alpha approaches a non-zero finite constant as r goes to
infinity, but tends toward 0 as r goes to 0. The vacuum anti-screens.
There is no Landau Pole, nor confinement. The total energy of the
source is finite.
Phase (1) is seen in the electromagnetic sector. Phase (2) is seen in
the color force sector (i.e. quark confinement). I don't know anything
that corresponds to phase (3).
I'll post more (and more precise) details on the mathematics involved
with all this soon. Much of this I was working on while going through
a reanalysis of Hehl's
Dimensions and Units in Electrodynamics
2004 July 6
Friedrich W. Hehl , Yuri N. Obukhov
arXiv:physics/0407022 v1 5 Jul 2004
I redid the whole thing in more general terms as "Dimensions and Units
in Gauge Theory" in which most of the above (and more) appear.
This, of course, touches on what I REALLY wanted to respond to. In
just the past day or so, a major "setback" came in the US funding of
science programs and this was brought up in an article "Is physics in
America dead?" linked to Yahoo:
"In the FY-2008 omnibus spending bill, the US Congress has decided to
zero out all funding for ITER, the international fusion reactor to be
built in Cadarache, France. While unilateral withdrawal of US funding
for international organisations is hardly news, this still raises a
lot of big question marks over many planned international science
projects."
Full Story: www.scientificblogging.com
In response...
The next great innovation in Physics will not come from the received
wisdom, refined over 80 year (of non-results) in unifying quantum
theory and relativity into "quantum gravity", and not through the Big
Science projects of the post War era; but from the lonely
independently employed or unemployed unknown who shows it all up
(again), resolving the key issues of the field in an almost prescient
manner, without the need for hints from the findings of big budget
programmes. It will not be quantum gravity, as it is conceived of
today (else someone would have done it by now, with the best and
brightest futuiley having tried for nearly a century), nor will it
conform to the received party line that had become almost a dead
cliche'(or else it, too, would have panned out by now), but will
involve something that lies hiding in plain sight, but yet which will
blindside everyone.
It's time to get away from the parasitic mode of ever-larger
programmes compensating for the lack of imagination and starting doing
REAL science for a change -- Einstein-style. Loner-style, not 50-
authors-names-on-a-paper military-industrial-complex team-based style.
Even the likes of Penrose and Baez have started to reiterate similar
points in recent times about the need for the outside loner to make
his appearance.
Rock Brentwood
> On Jan 6, 8:51 pm, TheGh...@haunted.hill (Martha's Ghost) wrote:
> > Here's a question that puzzles me...
>
> > But what system Drives, Produces or Creates these forces? We know a
> > fair bit what produces light(photons), electro-magnetic forces, but do
> > we know what is the source for these other forces?
[Somewhat detailed account of the forces as gauge forces and their
"Maxwell equations" + editorial on latest science news]
> I'll post more (and more precise) details on the mathematics involved
> with all this soon. Much of this I was working on while going through
> a reanalysis of Hehl's
> Dimensions and Units in Electrodynamics
> 2004 July 6
> Friedrich W. Hehl , Yuri N. Obukhov
> arXiv:physics/0407022 v1 5 Jul 2004
>
> I redid the whole thing in more general terms as "Dimensions and Units
> in Gauge Theory" in which most of the above (and more) appear.
http://federation.g3z.com/Physics/index.htm#Hehl1
Dimensions and Units in Gauge Theory (PDF, 316k)
Supplementary article: The Anatomy of the Electroweak and Color Gauge
Forces (PDF, 70k)
Based on the original, "Dimensions and Units in Electrodynamics" (Hehl
& Obukhov, arXiv:physics/0407022 v1 5 Jul 2004), this is a substantial
rewriting, with changes in the language, notation, layout and graphics
of the original. I have added a large amount of material which
dovetails into Hehl's works, but goes substantially beyond it,
particularly with the generalization of everything here to non-Abelian
gauge fields. This includes electromagnetism, as it is now understood,
since it is an integral part of a non-Abelian gauge field.
It also includes gravity, when its kinematics are formulated as a
gauge field for the general affine group GA(n).
Hehl;'s axiomatics have been modified to fit the more general context.
There are major changes required.
The most significant oversight by Hehl is that with electroweak
unification, the Maxwell equations are no longer valid, but have been
replaced by non-linear equations that possess non-zero magnetic
currents. All 3 of the Hehl axioms are false for electromagnetism, in
virtue of the unification. Moreover, the dimensional analysis is seen
to be incomplete when electromagnetism is viewed in the broader
context of gauge theory; in particular, the appropriate dimension for
the field is [F] ppp 1, while Hehl's "absolute" dimension is, itself,
actually a "relative" dimension: [Fa] ppp H/Q, where a is the gauge
index.
In addition, there is extra material provided here on the stress
tensor; on the constitutive law; on the link of the permittivity
coefficient to the gauge group metric and "coupling constants", and on
the classical dynamics of the "running of the coupling" that emerges
as an effective theory of a Kaluza-Klein model. The relation between
the permittivity, the g55 scalar and dilaton (which Hehl began to
allude to) are clarified here. Finally, a key oversight of the
original paper is remedied: in spite of the title "Dimensions and
Units ...", no reference was made in the original to the dimensional
analysis that had already been carried out in Maxwell's treatise in
Part IV, Chapter X. The analysis provided by Hehl is just a reworking
of Maxwell's (particularly the table in §623 of the treatise). Other
long-forgotten elements of Maxwell's treatise that had been
rediscovered on numerous occasions (including here in the original
article) are brought back into the light.
Related articles are found immdiately below this one:
The "Maxwell Equations" for Non-Abelian Gauge Fields (PDF, 222k)
The following is based on part 2, by Hehl, of the "Two Lectures on
Fermions and Gravity" article (Hehl, Lemke and Miekle), which was cast
solely in terms of the electromagnetic field, by the original author
(Hehl). A large number of additions have been made; the analysis has
been reworked in the more generalized context of non-Abelian gauge
theory, and errors present in the original (particularly the
derivation of the field Lagrangian) have been corrected.
The most significant oversight in the original has to do with
electromagnetism, in the context of gauge theory. There are essential
elements in gauge theory, highly relevant to the question of duality
even in the more restricted context of electromagnetism, which remain
hidden when focusing only on electromagnetism. Hehl has missed the
full extent of the general idea he's posing here.
This goes one step further than
"Immediately after Einstein's fundamental 1915 paper on General
Relativity and even before his big survey paper on General Relativity
would appear, Einstein observed that Maxwell's equations can be put in
a general covariant form by picking suitable field variables. This
meant, unnoticed even today, more than 80 years later, by most
elementary particle physicists, the reanimation of the D and the B of
Maxwell (or, with Lorentz's choice, of E and B)."
and does more than merely mandate the "reanimation" of D (and H): it
puts the spotlight on the entire issue of duality, itself. With this
in mind, parts 2 and 3 have been completely reworked.
When seen in the full context of gauge theory it becomes immediately
apparent that the fields D and E are not even the same type of object!
In particular, the (D, H) and (E, B) fields cannot be related by a
mere spacetime duality, since their gauge indexes are in different
positions - (Da, Ha) and (Ea, Ba). The constitutive law involves a
metric. That metric is none other than the permittivity, in disguise!
Thus, we further the analysis of Hehl by establishing a link to a
geometric interpretation. It is through this interpretation,
ultimately, that one can understand how the permittivity, dilatons,
and quintessence scalars are all connected to one another.
Much of sections 1 and 2 are kept intact, but with numerous additions.
Section 3, however, is almost completely rewritten, with the analysis
contained therein completely redone. The references of the original
are kept intact, but are not listed here. They may be found in the
"Two Lectures" article.
Gauge Field Equations in Maxwell Form (PDF, 176k)
The theory of classical gauge fields is rendered in Maxwell form. The
gauge group metric kab, in the process, is identified as the
generalization of the vacuum permittivity, thus providing a direct
physical interpretation of the scaling of the gauge group in terms of
a non-trivial dielectric structure of the vacuum, itself, and
recovering arguments posed originally by Maxwell regarding the
existence of a dielectric structure within the vacuum. This formalism
generalizes Yang-Mills theory in treating the two sets of Maxwell
fields (D, H) and (B, E) independently, even in vacuuo. A notable
feature of the generalization is that the stress tensor for the
(generalized) Maxwell field need not be trace free.
The generalization factors out the constitutive relations linking the
two sets of fields from the more fundamental diffeomorphic theory. One
application is that gauge theory can be formulated in non-Lorentzian
manifolds, including the Galilean limit of Newton-Cartan spacetimes.
Another application is that one can discuss alternatives where the
constitutive relations linking the two sets of fields are no longer
linear with fixed coefficients (i.e., where the gauge metric is
variable). In such theories, though Lorentz covariance is still
respected, the Green's functions no longer need be singular on the
light cone and the ultraviolet divergence need no longer be present.
The Gauge-Scalar Fields in Maxwell Form (PDF, 167k)
Expanding on the previous article, the theory of classical coupled
scalar and gauge fields is rendered in Maxwell form. Again, this
generalizes on what is normally defined as the classical field theory
in that both the fields and their duals are treated independently, and
the constitutive relations need not be linear with constant
coefficients.
This includes a detailed development of everything related: equations
of motion for test gauge/scalar charge, Hamiltonian and Poisson
brackets, stress tensor, field laws, constitutive laws, etc. All of
this is much more general that you'll usually see, because the
constitutive laws are signficantly generalized.