Two-Electron Scattering - A Challenge
maxwell
open challenge to theoretical physicists to consider the case of
classical 'electron' scattering. I believe that the classical Maxwell-
Hertz EM theory has NOT shown that it is capable of solving the
simplest dynamical problem, namely the interaction of two equally
charged point masses (IOW repulsive point-particles with finite mass &
charge). Maxwell's EM theory was developed from his researches based
on a continuous model of the universal EM aether, whose distortions
led to the propagation of waves in this medium across space over a
finite time: an effect James Clerk identified with the phenomenon of
light with a vacuum speed of c. Heinrich Hertz added the feature of
continuous electric charge density. It is in this form that classical
EM theory is still taught today. This theory suffers from two
fundamental flaws that have been carried forward into its modern
formulation as quantum field theory (QED). Firstly, its model of
electricity is based on a differentiable function of electric charge
DENSITY, when we now know that electricity is FINITE with its smallest
example of charge (e) not capable of being subject to the limit
process of 'going to zero'. Secondly, Maxwell continued with the ONE
time model of physics introduced by Newton, mathematically formalized
by Lagrange & given its current form by Sir William Rowan Hamilton. It
was this need to reduce the inherent TWO time (asynchronous)
interaction between electrons to changes at a single point in space
AND time (IOW focusing on the 'field point' where only ONE of the
electrons might exist) that forced Maxwell to adopt the Hamiltonian
approach to the dynamics of real fields. These electric & 'magnetic'
intermediaries must now be considered to co-exist everywhere so that
any ONE space-time point can be the subject of analysis (i.e. the
integrand in Hamilton's energy or Lagrange's density integrals).
My extensive studies in Maxwellian EM have failed to uncover a
SINGLE non-numeric solution to the simplest problem of two-electron
scattering; surely a first step in confirming our confidence in such a
theory before we extend it to quantum phenomena. As a result, I
challenge anyone to provide analytic answers to the following 'simple'
problem:
1) The system consists only of two equally charged 'electrons',
treated as point masses with mass, m & charge, e but with no 'spin' or
other quantum properties, such as quantized action.
2) At some single point in time (treated as the time origin in some
suitable inertial reference frame) there occurs zero relative motion
between the two electrons, when their spatial separation is 'd'.
3) Only Maxwell's EM theory can be used in the solution (IOW no Fokker
Lagrangians, etc).
The solution sought will provide closed-form analytic or algebraic
expressions for:
A) the minimum spatial separation, 'd'.
B) the time 'T' when each electron first achieves a speed, c.
C) the spatial separation 'D' when each electron first achieves a
speed, c.
Gentlemen, I await your response.
======================================= MODERATOR'S COMMENT:
I don't think that all claims made here are sufficiently founded, but I'm courious to read the responses :-)
eugene_st...@usa.net
> As a result, I
> challenge anyone to provide analytic answers to the following 'simple'
> problem:
> 1) The system consists only of two equally charged 'electrons',
> treated as point masses with mass, m & charge, e but with no 'spin' or
> other quantum properties, such as quantized action.
> 2) At some single point in time (treated as the time origin in some
> suitable inertial reference frame) there occurs zero relative motion
> between the two electrons, when their spatial separation is 'd'.
> 3) Only Maxwell's EM theory can be used in the solution (IOW no Fokker
> Lagrangians, etc).
> The solution sought will provide closed-form analytic or algebraic
> expressions for:
> A) the minimum spatial separation, 'd'.
I agree that Maxwell's theory doesn't have an accurate solution for
this simple problem. Even a numerical solution is not available. The
main reason is the radiation reaction problem, which is present any
time charged particles accelerate.
> B) the time 'T' when each electron first achieves a speed, c.
> C) the spatial separation 'D' when each electron first achieves a
> speed, c.
Do you mean that c is the speed of light? Massive electrons can never
achieve the speed of light exacly.
Eugene.
maxwell
>
> Do you mean that c is the speed of light? Massive electrons can never
> achieve the speed of light exacly.
>
theory that precludes this result. It is only Planck's 1907 Proposal
of a CONSTANT mechanical force that acts continuously that generates
the mass-velocity formula that you are alluding to that 'forbids'
electrons reaching light-speed, c.
Peter
> On May 12, 2:35 pm, eugene_stefanov...@usa.net wrote:
> >
> > Do you mean that c is the speed of light? Massive electrons can never
> > achieve the speed of light exacly.
> >
> Yes, that is exactly what I mean. There is nothing in Maxwell's
> theory that precludes this result. It is only Planck's 1907
1906?
> Proposal
> of a CONSTANT mechanical force that acts continuously that generates
> the mass-velocity formula that you are alluding to that 'forbids'
> electrons reaching light-speed, c.
It seems to me that if Maxwell would have lived longer, he would have closed
this small gap in his theory. For, he had the force known as Lorentz force,
and he worked on electromagnetism for moving bodies. Eventually he had
observed that the two sides of
m a = q E + q v x B
transform differently. As his mathematical skills were not less than that of
Lorentz, Einstein and Planck, he had certainly found the Lorentz
transformation for all variables involved. The Lorentz factor on the l.h.s.
had shown him v<c for ponderable matter. This does not mean, of course, that
he had drawn the same conclusions as Einstein did.
Best wishes,
Peter
Peter
> > 1) The system consists only of two equally charged 'electrons',
> > treated as point masses with mass, m & charge, e but with no 'spin' or
> > other quantum properties, such as quantized action.
> > 2) At some single point in time (treated as the time origin in some
> > suitable inertial reference frame) there occurs zero relative motion
> > between the two electrons, when their spatial separation is 'd'.
> > 3) Only Maxwell's EM theory can be used in the solution (IOW no Fokker
> > Lagrangians, etc).
Where is the difference? (I haven't found this term in Jackson and in
Feynman's lectures, vol. II)
> > The solution sought will provide closed-form analytic or algebraic
> > expressions for:
> > A) the minimum spatial separation, 'd'.
>
> I agree that Maxwell's theory doesn't have an accurate solution for
> this simple problem. Even a numerical solution is not available. The
> main reason is the radiation reaction problem, which is present any
> time charged particles accelerate.
This problem (as described in Sommerfeld's lectures) I haven't understood. Is
the coupled system of equations of motion for fields and charges not closed?
> > B) the time 'T' when each electron first achieves a speed, c.
> > C) the spatial separation 'D' when each electron first achieves a
> > speed, c.
The Newton-Maxwell equation of motion for the charges,
m a = q E + q v x B
has to be corrected for the transformation properties, otherwise the theory
would be wrong, and it would be meaningless to calculate somthing from it
(except for v<<c). After that, (B) and (C) make no sense.
Peter
maxwell
> > The Newton-Maxwell equation of motion for the charges,
>
> m a = q E + q v x B
>
> has to be corrected for the transformation properties, otherwise the theory
> would be wrong, and it would be meaningless to calculate somthing from it
> (except for v<<c). After that, (B) and (C) make no sense.
>
result following Maxwell's 1872 formulation as 'stolen' by Lorentz)
please be my guest, at least up to v = c. This equation is valid for
all v, not just v << c. However, you will need to demonstrate this
'impossibility' rather than simply jump to your 'obvious' conclusion.
Come on guys, physics doesn't get any simpler than the two-body
problem!
Peter
> On May 13, 11:52 am, Peter <end...@dekasges.de> wrote:
> > The Newton-Maxwell equation of motion for the charges,
> >
> > m a = q E + q v x B
> >
> > has to be corrected for the transformation properties, otherwise the
> theory
> > would be wrong, and it would be meaningless to calculate somthing from it
> > (except for v<<c). After that, (B) and (C) make no sense.
> If you wish to use this equation (actually this is the Heaviside 1887
> result following Maxwell's 1872 formulation as 'stolen' by Lorentz)
> please be my guest, at least up to v = c.
The equation
m a(t) = q E(r,t) + q v(t) x B(r,t)
is the answer to a purely mechanical question (posed basically by Helmholtz).
How the Newtonian equation of motion looks like for a body that is subject to
both a force that does *not* change its kinetic energy and a 'more general'
force', with the restriction that a conventional Hamilton function, H(p,r,t)
exists. Moreover, it is assumed the the force separates into a geometric and
a body-dependent factor, as for Newton's theory of gravity.
When one ask how the body reacts back onto the sources of E and B, one finds
the microscopic Maxwell equations for E and B. Thus, those can be seen as
conditions for E and B for charged bodies to move according to the laws of
Hamiltonian mechanics.
However, unlike the term m.a, the r.h.s. is not invariant when changing the
inertial system. Moreover, the l.h.s. was established under the silent
assumption that the change of momentum or velocity (m=const) does not depend
on the current value of itself. This has to be corrected as the Eulerian
state variable velocity enters the r.h.s. The result is
m a -> d/dt (\gamma(v) dv/dt)
where \gamma(v) is the Lorentz factor, 1/SQRT(1-v^2/c^2). The resulting
equation is Lorentz invariant.
> This equation is valid for
> all v, not just v << c.
This is not correct, see above.
> However, you will need to demonstrate this
> 'impossibility' rather than simply jump to your 'obvious' conclusion.
\gamma(v)->oo as v->c from below =>, for each finite force, dv/dt->0. For
constant force, explicit integration yields v->c for t->oo. Hence, the line
v=c cannot be passed by bodies starting motion with v<c or v>c.
More details are in the transparencies of a talk I can email at request.
Peter
maxwell
> ======================================= MODERATOR'S COMMENT:
> I don't think that all claims made here are sufficiently founded, but I'm courious to read the responses :-)
challenge. I hope that this moderator's comments have not discouraged
anyone. Perhaps he would like to be more specific in his addendum that
he doesn't think that "all claims made here are sufficiently
founded". This type of remark smacks of the type of vague commentary
used by anonymous referees in refereed journals when they reject
papers that challenge the orthodoxy - see Fred Hoyle's autobiography
"Home is where the Wind blows" pages 160, 288. Perhaps the four
moderators of this group would like to rise to the challenge; I'm sure
they have all taken graduate courses in classical EM. At the very
least, a reference to a published solution to this problem seems
called for.
Oh No
> My extensive studies in Maxwellian EM have failed to uncover a
>SINGLE non-numeric solution to the simplest problem of two-electron
>scattering; surely a first step in confirming our confidence in such a
>theory before we extend it to quantum phenomena. As a result, I
>challenge anyone to provide analytic answers to the following 'simple'
>problem:
>1) The system consists only of two equally charged 'electrons', treated
>as point masses with mass, m & charge, e but with no 'spin' or other
>quantum properties, such as quantized action.
>2) At some single point in time (treated as the time origin in some
>suitable inertial reference frame) there occurs zero relative motion
>between the two electrons, when their spatial separation is 'd'.
>3) Only Maxwell's EM theory can be used in the solution (IOW no Fokker
>Lagrangians, etc). The solution sought will provide closed-form
>analytic or algebraic
>expressions for:
>A) the minimum spatial separation, 'd'.
>B) the time 'T' when each electron first achieves a speed, c.
>C) the spatial separation 'D' when each electron first achieves a
>speed, c.
> Gentlemen, I await your response.
I do not think that these are reasonable constraints. To deal with two
electron scattering as point masses we need to use a relativistic
quantum formulation. This requires that we do introduce spin, and also
photons which may be regarded as massless point particles - we do not
need to introduce Maxwell's equations, the concept of a field as a
prior, a Lagrangian or a classical Hamiltonian. Concepts like minimum
spacial separation 'd' have no meaning in a quantum formulation. I am
unhappy about saying that an electron achieves speed 'c', because this
is also a meaningless idea in a quantum formulation.
All that done, then the problem is a simple one and is solved formally
using a Feynman diagram.
Regards
--
Charles Francis
moderator sci.physics.foundations.
substitute charles for NotI to email
Peter
surprised about this posting. However, as the moderation policy is concerned,
few remarks are in order.
maxwell <sp...@shaw.ca> writes:
> On May 12, 1:52 pm, maxwell <s...@shaw.ca> wrote:
> > [snip]
> > ======================================= MODERATOR'S COMMENT:
> > I don't think that all claims made here are sufficiently founded, but
> I'm courious to read the responses :-)
> After almost a week there has yet to be a real response to this public
> challenge.
> I hope that this moderator's comments have not discouraged
> anyone.
Why? "I'm courious" means that I expect an interesting discussion.
> Perhaps he would like to be more specific in his addendum that
> he doesn't think that "all claims made here are sufficiently
> founded".
Not at all. The negation of 'all is correct' is not 'nothing is correct', but
'something is correct and others is not correct'.
> This type of remark smacks of the type of vague commentary
> used by anonymous referees in refereed journals when they reject
> papers that challenge the orthodoxy - see Fred Hoyle's autobiography
> "Home is where the Wind blows" pages 160, 288.
This type was and is not intended; moreover, your posting was approved.
> Perhaps the four
> moderators of this group would like to rise to the challenge; I'm sure
> they have all taken graduate courses in classical EM. At the very
> least, a reference to a published solution to this problem seems
> called for.
???
I think we should take this as a misunderstanding. However, I see that short
Moderator's Comments turn often out to be confusing rather than stimulating
and will stop it.
Best wishes,
Peter
maxwell
> Thus spake maxwell <s...@shaw.ca>
up on the word 'electron'. I was simply using it as short-hand for a
point-particle with discrete mass & electric charge. The classical
problem as I presented it still remains & I continue to expect you to
respond 'classically' using only Heaviside's Duplex ("Maxwell's")
Equations without 'spin' or 'photons' etc.
However, if you can also solve the corresponding quantum problem with
a simple Feynman diagram, please be my guest, but formal solutions in
terms of integrals that can only be evaluated numerically will not be
considered adequate.
Indeed, to satisfy those who believe that an ultimate speed of 'c' is
not achievable then I am prepared to modify this problem to one where
the final speed is 99% of c. Since QFT is built from Lagrangians that
explicitly include the locations of the electrons e.g. x(t) perhaps
you can elaborate on why a closest distance of approach (d) has "no
meaning in a quantum formulation"? What is wrong with calculating the
expected value of the separation at time zero i.e. <{x1(0) -
x2(0)}**2>?
maxwell
> In the meanwhile, several replies have been posted, so that I'm quite
> surprised about this posting. However, as the moderation policy is concerned,
> few remarks are in order.
>
generate some activity on this challenge that I regard as fundamental
to progress in EM.
I did not mean to imply that 'nothing is correct in EM' so I plead
innocent of the counter charge "The negation of 'all is correct' is
not 'nothing is correct', but 'something is correct and others is not
sand' if it cannot solve this 'simple' problem that ties Maxwell-
Lorentz theory back to reality. Experimentalists have no difficulty
'shooting' electrons at each other. Additionally you did not elaborate
as I just requested when I referred back to your initial moderator's
comments "all claims made here are not sufficiently founded". I was
only intending to pose this classical challenge & appreciate your
interest in meeting it. Apart from a brief response from Eugene & now
a QFT 'diversion' from Charles, it seems that only you & I todate are
really interested.
My latest request for a 'published reference' is a sub-challenge to
anyone who knows of any published work on this problem. I was hoping
that the tens of thousands of physics students who have taken graduate
courses in classical EM might have read something that was relevant
or, at least, attempted it as a classroom exercise.
I agree with you about the misunderstanding & I offer my apologies for
anything that I might have done that contributed to this. I do agree
with your ending remarks on moderator's possibly confusing brief
comments. As an alternative, I think an immediate response by the
moderator to a posting offers much more room for clarity on all
occasions.
Oh No
>Thank you, Charles for your partial response. However don't get hung
>up on the word 'electron'. I was simply using it as short-hand for a
>point-particle with discrete mass & electric charge.
Oh. You mean an electron then? :-)
> The classical problem as I presented it still remains
I don't consider so. There is no classical description of such an
object.
>& I continue to expect you to respond 'classically' using only
>Heaviside's Duplex ("Maxwell's") Equations without 'spin' or 'photons'
>etc.
Strange expectation. I have other problems to solve.
>However, if you can also solve the corresponding quantum problem with a
>simple Feynman diagram, please be my guest,
You can get the solution from a book. What you can't get from a book is
this way of interpreting qed.
> but formal solutions in terms of integrals that can only be evaluated
>numerically will not be considered adequate.
Why on earth not. Probably the majority of integrals one encounters in
real world problems can only be evaluated numerically. There is nothing
wrong with that.
>Indeed, to satisfy those who believe that an ultimate speed of 'c' is
>not achievable then I am prepared to modify this problem to one where
>the final speed is 99% of c. Since QFT is built from Lagrangians that
>explicitly include the locations of the electrons e.g. x(t) perhaps you
>can elaborate on why a closest distance of approach (d) has "no meaning
>in a quantum formulation"?
One does not have positions except in measurement of positiion.
Otherwise only probability amplitudes. They are somewhat different.
>What is wrong with calculating the expected value of the separation at
>time zero i.e. <{x1(0) - x2(0)}**2>?
>
If you are not actually doing a measurement, then what you get is a
charge density, as appears in classical electrodynamics, not a pointlike
charge. If you are doing a measurement, you are also messing up the
scattering.
Peter
The problem was stated within CEM, thus, I will stay there. With the Maxwell-
Lorenz force, we have
m d(g(v1).v1))/dt = e(E2 + v1 x B2)
m d(g(v2).v2))/dt = e(E1 + v2 x B1)
g(v) is the Lorentz factor, introduced for symmetry reasons (but it can be
derived separately). E1/2 and B1/2 are given functions of x1/2, v1/2, a1/2
and t (Heaviside, Feynman; see Jackson, § 6.5). Is this problem
mathematically ill posed or solvable for certain domains of initial locations
and velocities only? This are the only motivations for your challenge I can
see at this moment.
For the simple case that their initial velocities are (anti) parallel,
B1=B2=0. If v2(0)=-v1(0), the problem further simplifies to
m d(g(v1).v1))/dt = e E1
It's a 2nd-order ODE for x(t), therefore, you cannot expect a closed form
solution, x(t)=...
Lack of time doesn't allow me to program this equation in Mathcad and to
explore numerical solutions. For existence and uniqueness issues, it needs
much more maths than I command :-(
Looking forward,
Peter
maxwell
> Thus spake maxwell <s...@shaw.ca>
>
best the experts can come up with then I will have to consider re-
issuing this challenge to the 'multitudes' who inhabit the
Electromagnetism Group on Google.
Oh No
>Sorry, Charles, I view your answers as a 'cop-out'. If that's the best
>the experts can come up with then I will have to consider re- issuing
>this challenge to the 'multitudes' who inhabit the Electromagnetism
>Group on Google.
If you formulate the problem as a classical problem of point particles
then you are stuck with the fact that the energy contained in the field
of a charged point particle is infinite. Your problem is not well
defined, so the only answer is to cop out.
eugene_st...@usa.net
> The problem was stated within CEM, thus, I will stay there. With the Maxwell-
> Lorenz force, we have
>
> m d(g(v1).v1))/dt = e(E2 + v1 x B2)
> m d(g(v2).v2))/dt = e(E1 + v2 x B1)
>
> g(v) is the Lorentz factor, introduced for symmetry reasons (but it can be
> derived separately). E1/2 and B1/2 are given functions of x1/2, v1/2, a1/2
> and t (Heaviside, Feynman; see Jackson, § 6.5). Is this problem
> mathematically ill posed or solvable for certain domains of initial locations
> and velocities only?
Hi Peter,
does your system of equations takes into account the "radiation
reaction"? When the electron 1 accelerates (due to the action of the
electron 2) it emits radiation and experiences a back reaction from
this radiation. So, I would expect to see some additional forces on
the right hand sides of your equations, which are functions of a1 and
a2, respectively.
Eugene.
Peter
> On May 19, 1:25 pm, Peter <end...@dekasges.de> wrote:
>
> > The problem was stated within CEM, thus, I will stay there. With the
> Maxwell-Lorenz force, we have
> >
> > m d(g(v1).v1))/dt = e(E2 + v1 x B2)
> > m d(g(v2).v2))/dt = e(E1 + v2 x B1)
> >
> > g(v) is the Lorentz factor, introduced for symmetry reasons (but it can
> be derived separately). E1/2 and B1/2 are given functions of x1/2, v1/2,
> a1/2 and t (Heaviside, Feynman; see Jackson, § 6.5). Is this problem
> > mathematically ill posed or solvable for certain domains of initial
> locations and velocities only?
> Hi Peter,
> does your system of equations takes into account the "radiation
> reaction"? When the electron 1 accelerates (due to the action of the
> electron 2) it emits radiation and experiences a back reaction from
> this radiation. So, I would expect to see some additional forces on
> the right hand sides of your equations, which are functions of a1 and
> a2, respectively.
>
> Eugene.
Hi Eugene,
Thank you for pointing out that. Having in mind exactly Maxwell's expression
for the force (as requested by the challenger), I had forgotten about the
radiation reaction. But do I need it, when the electrical field is calculated
self-consistently?
The power balance for the eqs. above reads
mv1 d(g(v1).v1))/dt + mv2 d(g(v2).v2))/dt = e(E2.v1 + E1.v2)
For the simple case of central impact with equal speeds we have v2=-v1 and
E2=-E1. Thus, e(E2.v1 + E1.v2)<>0. However, the same arguing would apply to
pure Coulomb interaction as well ...
Then, let me ask this way: Is the minimum coupling (it yields just the force
written above, correct?) incomplete?
Best,
Peter
Ken S. Tucker
Hi Peter and all.
My understanding of GR1916 Eq.(65...a) have
the f = 0 in
f = q (E + v x B) reducing to,
qE = qB x v .
Doing a scalar product with "v" yields,
qE.v = q(B x v).v = 0.
The term qE.v = 0 is the basis for Quantum Theory.
That stops a "continuous" transmission of power,
characterized by force (dot) velocity as an electron
would spiral into a nucleus, something classical
theory predicts, but QT excludes.
So our friend "Mr. maxwell", in his benign challenge
is really asking about GR in EM fields. It's deeper
because one needs to show the emission of photons
and EMR, that bridges QT with CEM.
That's doable, so I'm happy to try to provide the means
to do so, from my understanding of GR in steps.
Regards
Ken S. Tucker
Cl.Massé
1179597795....@k79g2000hse.googlegroups.com
> Since QFT is built from Lagrangians that
> explicitly include the locations of the electrons e.g. x(t)
No, it includes space, not as a location of a particle. The particle is
described by a wave function of space.
> perhaps
> you can elaborate on why a closest distance of approach (d) has "no
> meaning in a quantum formulation"?
Because it isn't "well defined", that is the wave function of distance isn't
a delta function.
> What is wrong with calculating the expected value of the separation at
time zero i.e. <{x1(0) -
> x2(0)}**2>?
Nothing. Precisely, it is but an "expectation value".
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability.
Richard Saam
> The solution sought will provide closed-form analytic or algebraic
> expressions for:
We are talking near nuclear dimensions
> A) the minimum spatial separation, 'd'.
~1.2E-12 cm
> B) the time 'T' when each electron first achieves a speed, c.
~1E-22 sec
> C) the spatial separation 'D' when each electron first achieves a
> speed, c.
~sqrt(3)*1.2E-12 cm
maxwell
Very interesting results, Richard. These agree with some of my own
calculations. However the challenge was to provide analytic
expressions for these values before applying accepted values for the
constants. What are your expressions for these values in terms of e,
m etc & how did you arrive at your results?