| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
17 May 2006 07:04:35 PM |
| Object: |
Strong and weak nuclear forces |
Are there any formulas for the strong and weak nuclear foces between
two given particles? I know that the strong nuclear force varies
directly with the distance between the particles. Nevertheless, I do
not know the precise relationship, nor that of the weak force.
Please note that I would like to disregard any quantum effects. I
would imagine that there was some kind of pre-quantum theory for the
nuclear forces. I reason by analogy; before quantum electrodynamics,
there was Maxwell's equations. Similarly, there should have been some
explanation of the strong and weak forces that does not involve quantum
theory.
Thank you in advance.
.
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| User: "Rock Brentwood" |
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| Title: Re: Strong and weak nuclear forces |
17 May 2006 07:35:39 PM |
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wrote:
Are there any formulas for the strong and weak nuclear foces between
two given particles?
These are quantizations of Yang-Mills forces. A Yang-Mills field and
current generalizes a Maxwell field and current in that (a) the charge
of the current source now has a complexion in addition to a magnitude
and (b) the field now has non-linear relations -- including non-zero
contributions on the right-hand side of the magnetic field equations.
Each component of the Yang-Mills field has a complexion, as well.
The equation of motion for a charged test body is the Lorentz equation,
the charge-vector is taken as a scalar product with whatever field
components are involved. In addition, there is an equation which
governs the "precession" of the charge vector -- Wong's equation. The
charge of a test body is NOT conserved. But (a) the change in the
charge is compensated for by the field and (b0 the square magnitude of
the charge vector is conserved.
In a quantized version of the classical Yang-Mills theory, the
precession occurs in quantum jumps, these corresponding to "flavor
changing" interactions.
The currents associated with the fields are proportional to the product
of the fields and potentials. There is both an effective electric
current and effective magnetic current.
Notable is the case of electromagnetism. Since this is caught up in the
SU(2) part of the electroweak field, it too has non-linear terms on the
right hand side. The Maxwell equations are superseded by non-linear
equations with non-zero terms for both the electric and magnetic
currents brought about by the field.
Closed form solutions can be written down for the SU(2) field that
comprise magnetic monopoles. This, therefore, includes magnetic
monopoles in the "electromagnetic" sector of the Yang-Mills field.
You may be aware that there is an argument by Dirac which shows that
charge is quantized in the presence of monopoles. In the SU(2) x U(1)
electroweak model there is a somewhat similar argument present which
shows not only that the U(1) part of the force (called the hypercharge)
is quantized, but it actually determines the values of the hypercharge
for each type of particle, up to choice of unit. The electric charge is
closely related to the hypercharge and knowing one means knowing the
other. Thus, there is already present an argument in electroweak theory
that leads to charge quantization.
What relation this has to Dirac's argument (particularly since
electroweak theory now has magnetic monopole solutions even in the
electromagnetic sector) hasn't been looked into, as far as I'm aware.
The Yang-Mills equations written in Maxwell form will look like the
following. In place of the D and H fields are fields D_a, H_a indexed
by the "complexion". For SU(2) a ranges over 1,2,3. For U(1)xSU(2) a
ranges over 1,2,3 for SU(2) and an additional index for U(1). The
U(1)xSU(2) force thus quadruplicates the electromagnetic field
components.
In place of the E, B fields and A, phi potentials are E^a, B^a, A^a,
phi^a, indexed likewise. The current J is indexed J_a, the charge
density rho_a. These represent the components of the charge
"complexion" vector.
In the following, the summation convention is adopted (repeated indices
in each term are summed over).
Derivatives below denote partial derivatives; ().() and ()x()
respectively denote scalar and dot products.
One has, for the field equations:
div D_a = rho_a - f^c_{ab} A^b.D_c
div B^c = -f^c_{ab} A^a.B^b
curl H_a - dD_a/dt = J_a - f^c_{ab} (phi^b D_c - A^b x H_c)
curl E^c + dB^c/dt = -f^c_{ab} (A^a x E^b - phi^a B^b).
The relation of the fields to the potentials:
E^c = -grad phi^c - dA^c/dt + f^c_{ab} phi^a A^b
B^c = curl A^c + 1/2 f^c_{ab} A^a x A^b.
The total currents are
Rho_a = rho_a - f^c_{ab} A^b.D_c
I_a = J_a - f^c_{ab} (rho^b D_c + A^b x H_c)
and satisfy the conservation law
d(Rho_a)/dt + div I_a = 0.
The source currents, however, do not
d(rho_a)/dt + div J_a = f^c_{ab} (phi^b rho_c - A^b.J_c +
B^b.H_c - E^b.D_c).
The fields are assumed to be in such a relation that
f^c_{ab} (B^b.H_c - E^b.D_c) = 0.
A body with charge vector e_a will undergo a precession given by Wong's
equation
de_a/dt = -f^c_{ab} e_c (phi^b - v.A^b),
where v is its velocity and d/dt, here, denotes total time derivative.
The charge satisfies the Lorentz force law, but the charge vector is
dotted into the force vectors,
dp/dt = e_a (E^a + v x B^a); dE/dt = e_a
v.E^a
where p and E are, respectively, the body's momentum and energy.
The fields are normally assumed to be in at least a functional relation
such that
dD_a/dE^b = epsilon_{ab}; dH_a/dB^b =
mu^{-1}_{ab}
with the epsilon and mu matrices symmetric. The other relations are
dD_a/dB^b + dH_a/dE^b = 0.
Under linear relations, these become
D_a = epsilon_{ab} E^b; B^a = mu^{ab}
H_b.
The constants f^c_{ab} are the structure constants of the relevant
gauge group. For SU(2), f^1_{23} = f^2_{31} = f^3_{12} = 1; with
f^a_{bc} = -f^a_{cb}.
The epsilon and mu constants -- when there's a linear relation -- are
just the metric associated with the gauge group, with the following
relations
epsilon_{ab} = k_{ab}/c^2; mu^{ab} = k^{ab}
where k_{ab} and k^{ab} are, respectively, the metric and its dual.
The structure coefficients are assumed to satisfy the defining
relations of a Lie algebra,
f^c_{ab} = -f^c_{ba}
f^d_{ab} f^e_{cd} + f^d_{bc} f^e_{ad} + f^d_{ca}
f^e_{bd} = 0
and the metric is assumed to be "adjoint invariant", with
k_{ab} f^b_{cd} + k_{cb} f^b_{ad} = 0.
For SU(2) this restricts the metric to be of the form k_{ab} = k if a =
b; k_{ab} = 0 else. The constant k is then closely related to the
"coupling" of the SU(2) force -- that is, the charge unit.
For U(1) there is no non-zero structure coefficients. And the compoents
of f^c_{ab} of indices mixed between U(1) and SU(2) are all 0. The
mixed metric components are, likewise, all 0. The U(1) part of the
metric effectively defines a 2nd coupling constant. For SU(3), there
will be a third.
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| User: "" |
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| Title: Re: Strong and weak nuclear forces |
18 May 2006 05:25:02 AM |
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I believe that you misunderstood my question. I did not actually want
a formal analogue of Maxwell's Equations. I was just using that as an
example of a theory that does not involve quantum mechanics, but was
still a (fairly) accurate model for electrodynamics. All I want are
the formulas of the force of the strong force and force of the weak
force between two given particles, disregarding any quantum effects. I
apologize if my first statement of my question was a bit misleading.
Thank you in Advance.
.
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| User: "Rock Brentwood" |
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| Title: Re: Strong and weak nuclear forces |
08 Jun 2006 06:48:05 PM |
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wrote:
I believe that you misunderstood my question. I did not actually want
a formal analogue of Maxwell's Equations.
You do, you just don't realise it. For, ultimately it's the field law
that the force law arises from.
The only problem is that I didn't give the complete answer. The Higgs
is involved in the electroweak field, which effectively gives the weak
nuclear force carriers mass. In turn, that turns the corresponding
"Maxwell equations" into Maxwell-Proca equations.
The potential phi, of each field, is given by the Klein-Gordon
equation. For an electrostatic field, that would give you:
del^2 phi = K^2 phi
where K > 0 and K is proportional to the mass of the force carrier (the
W or Z particle, here).
If spherically symmetric, this becomes
1/r (r phi)'' = K^2 phi
or
(r phi)'' = K^2 r phi.
The solution, in general, is
r phi = A exp(Kr) + B exp(-Kr).
The only bounded solution,
phi = B/r exp(-Kr).
This identifies a source of charge B.
The corresponding force is minus the derivative of the potential,
becoming
E = B/r^2 exp(-Kr) + KB/r exp(-Kr).
The reciprocal 1/K gives you an effective range for the force, since
it's dying off exponentially. Since K is proprotional to the mass of
the underlying particle (W or Z), then 1/K will be inversely
proportional to the mass.
The "electrostatic" field of a Maxwell-Proca field is a Yukawa field,
having the above-mentioned Yukawa potential. The relation between mass
and range originates with Hideki Yukawa.
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| User: "Rock Brentwood" |
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| Title: Re: Strong and weak nuclear forces |
12 Jun 2006 05:26:28 PM |
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Rock Brentwood wrote:
lugita15@gmail.com wrote:
I believe that you misunderstood my question. I did not actually want
a formal analogue of Maxwell's Equations.
You do, you just don't realise it. For, ultimately it's the field law
that the force law arises from.
....
The "electrostatic" field of a Maxwell-Proca field is a Yukawa field,
having the above-mentioned Yukawa potential. The relation between mass
and range originates with Hideki Yukawa.
Also implicit in the first reply is the relation of the charges to one
another. The color charges have associated with them the analogue of
"chromaticity coordinates". Here, these are picked out from the 3rd and
8th generators of the symmetry group SU(3), and I'll call them L3 and
L8, here.
If you plot the L3 and L8 charges for red, amber (= anti-blue), green,
cyan (= anti-red), blue and magenta (= anti-green), they'll be placed
symmetrically on a hexagon about (L3,L8) = (0,0).
The charge-space is, like color space, 2-dimensional, with L3
effectively giving you the blue-red tint, and L8 the green vs. blue-red
tint.
The same rule "opposites attract, likes repel" applies here. But it's
more general. Opposites in electricity can be thought of as 180 degrees
apart in complexion. Here, complexion has an entire circle along the
color wheel to move along. Red and green are 120 degrees apart, red and
amber are only 60 degrees apart. The strength of attraction or
repulsion is proportional to the cosine of the angle between the two
complexions.
Thus, red will attract cyan with the same strength that it repels red.
It will attract green and blue with the same strength it repels amber
and magenta; the red-green and red-blue attraction is about 1/2 the
strength of the red-cyan attraction.
The combinations that are mutually attractive are the couples,
red-cyan, green-magenta and blue-amber (the mesons); and the menage a'
trois triples, red-green-blue, cyan-amber-magenta (the baryons and
anti-baryons).
The expression of the field law in the form of analogue Maxwell
equations also brings forth another aspect of the field theory. The
relation between the (D,H) and (B,E) fields in electromagnetism is
governed by the constitutive laws between the two sets of fields.
Normally regarded as a mere phenomenology of materials at the
macroscopic level, this is, in fact, something that also exists at the
microscopic level, even in vacuuo near charge sources. The constitutive
relations describe the dielectric and polarizing structure of the
vacuum, itself.
In the Yang-Mills equations, that's what was captured by the epsilon
and mu tensors alluded to when describing the "gauge metric".
A quantum version of the field theory spawns "corrections" in the field
law, which amounts to posing non-trivial relations between (D,H) and
(B,E). For instance, in electromagnetism, the effective field that
captures part of the interaction of the electron's matter field and the
electromagnetic field (called the Euler-Heisenberg Lagrangian)
transforms the relation (D = epsilon_0 E) seen in electrodynamics to
one of the form
D = epsilon_0 (E + a (E^2 - B^2) E + b E.B B)
where a and b are constants proportional to Planck's constant and
proportional to the fourth power of the electron mass. For H, the same
constants are involved in a similar relation
H = (1/mu_0) (B + a (E^2 - B^2) B - b E.B E).
(Might have to add other factors to keep the units consistent).
More generally, all the nice formulae expressions that pertain to
classical field theory (Wong's equations, the Yukawa potential and
force, etc.) receive quantum corrections that primarily account for the
dielectric structure of the vacuum, particularly in the presence of
charge sources.
The vacuum -- as Maxwell himself had hypothesized -- is a polarizing
medium which is even capable of dielectric breakdown (known in the
modern times as matter/anti-matter creation) and charge
[anti-]screening.
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| User: "PD" |
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| Title: Re: Strong and weak nuclear forces |
18 May 2006 01:16:15 PM |
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wrote:
I believe that you misunderstood my question. I did not actually want
a formal analogue of Maxwell's Equations. I was just using that as an
example of a theory that does not involve quantum mechanics, but was
still a (fairly) accurate model for electrodynamics. All I want are
the formulas of the force of the strong force and force of the weak
force between two given particles, disregarding any quantum effects. I
apologize if my first statement of my question was a bit misleading.
Thank you in Advance.
There is no simple "force goes like this power of the distance" law for
the strong and weak forces. There have been attempts to model it (see,
for example, the "Yukawa potential"), but those models break too easily
to be of any real use.
Now perhaps your follow-up question will be:
WHY does the electromagnetic interaction have such a simple formula for
a force law, and the strong and weak forces do not?
PD
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| User: "FrediFizzx" |
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| Title: Re: Strong and weak nuclear forces |
19 May 2006 12:52:47 AM |
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"PD" <TheDraperFamily@gmail.com> wrote in message
news:1147976175.845534.260780@g10g2000cwb.googlegroups.com...
|
| wrote:
| > I believe that you misunderstood my question. I did not actually
want
| > a formal analogue of Maxwell's Equations. I was just using that as
an
| > example of a theory that does not involve quantum mechanics, but was
| > still a (fairly) accurate model for electrodynamics. All I want are
| > the formulas of the force of the strong force and force of the weak
| > force between two given particles, disregarding any quantum effects.
I
| > apologize if my first statement of my question was a bit misleading.
| >
| > Thank you in Advance.
|
| There is no simple "force goes like this power of the distance" law
for
| the strong and weak forces. There have been attempts to model it (see,
| for example, the "Yukawa potential"), but those models break too
easily
| to be of any real use.
|
| Now perhaps your follow-up question will be:
| WHY does the electromagnetic interaction have such a simple formula
for
| a force law, and the strong and weak forces do not?
I don't think the OP cares much about the "breaking" but was after a
simple approximation. The strong force between nucleons is not too hard
to model in case of deuteron. Just take the binding energy over a
certain distance. However, the weak "force" is certainly more
difficult.
FrediFizzx
http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.pdf
or postscript
http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.ps
http://www.vacuum-physics.com
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