Classical limit of QED
Steve Carlip
from QED, you look at coherent states that approximate the
classical configurations you're interested in. But could someone
point me toward a reference where this is done in some detail,
with a careful description of how, and to what extent, Maxwell's
equations are recovered?
Steve Carlip
Greg Weeks
Steve Carlip (sjca...@ucdavis.edu) wrote:
: ``Everybody knows'' that to recover classical electrodynamics
: from QED, you look at coherent states that approximate the
: classical configurations you're interested in.
I assume you're referring to interacting QED. In that case, there are no
coherent states (other than those of the asymptotic "in" and "out" fields).
Also, there really is no agreed upon "classical electodynamics". Everyone
agrees on Maxwell's equations and the Lorentz force equation. But there
never was agreement on the nature of charged matter and how it responds to
the Lorentz force.
In the equations of QED, charged matter appears as the Dirac field, which
has no classical analog. In the formal h-->0 limit of the equations of QED
(including the commutation relations), the Dirac field disappears, and
you're left with the classical noninteracting Maxwell field.
[I'm cheating a bit. When I say the Dirac field disappears, I'm applying a
result from normed algebras to a rather ill-defined algebra of unbounded
elements. I have little doubt about the result, though.]
In general, although I kept my eyes open for years, I never saw any
nontrivial theory -- eg, classical electrodynamics or nonrelativistic
quantum mechanics -- derived from QED. (The closest I got was Laurence
Yaffe's presentation in s.p.r some years ago, and I wasn't convinced.)
Greg
Andrea Barbieri
news:10221715...@cswreg.cos.agilent.com
>
> Steve Carlip (sjca...@ucdavis.edu) wrote:
> : ``Everybody knows'' that to recover classical electrodynamics
> : from QED, you look at coherent states that approximate the
> : classical configurations you're interested in.
>
> I assume you're referring to interacting QED. In that case, there
> are no coherent states (other than those of the asymptotic "in" and
> "out" fields).
>
> [...]
>
> In general, although I kept my eyes open for years, I never saw any
> nontrivial theory -- eg, classical electrodynamics or nonrelativistic
> quantum mechanics -- derived from QED. (The closest I got was
> Laurence Yaffe's presentation in s.p.r some years ago, and I wasn't
> convinced.)
>
I have not performed a full calculation, but it seems to me that by
posing
h = epsilon * h (Planck constant)
c = c / epsilon (light speed)
t = epsilon * t (time coordinate)
and letting epsilon --> 0, you get classical EM field in interaction
with nonrelativistic classical electrons:
since mc^2 ---> infty, the Dirac field becomes a non relativistic
Pauli spinor, which in the h ---> 0 limit becomes a nonrelativistic
classical electron.
x^0 = ct, alpha = e^2/hc and E = h*@/@t are unaffected by the
epsilon ---> 0 limit, so that
1) the wave equation for the EM field still holds
2) the strenght of the interaction and the energy of the classical
electrons remain bounded.
If I am not missing something, the limit can be performed on the
expressions of expectation values in path integral form, starting from
the relativistic QED action.
best,
a
--
Posted via Mailgate.ORG Server - http://www.Mailgate.ORG
Bill Frensley
Greg Weeks wrote:
>
.......
> In general, although I kept my eyes open for years, I never saw any
> nontrivial theory -- eg, classical electrodynamics or nonrelativistic
> quantum mechanics -- derived from QED. (The closest I got was Laurence
> Yaffe's presentation in s.p.r some years ago, and I wasn't convinced.)
>
This is a very telling comment. The social hierarchy among physics
disciplines (field theory and particle physics is more "fundamental"
than solid-state, atomic, or chemical physics) is based upon the
implicit
notion that there is a single deductive system that encompasses all of
physical science, with the more "fundamental" results being those closer
to the roots of the system. Thus, nonrelativistic quantum mechanics is
derivable "in principle" from QED. If one cannot make this most obvious
connection in practice, I would suggest that a review of our notions of
fundamentality are long overdue.
- Bill Frensley
Greg Weeks
: If I am not missing something, the limit can be performed on the
: expressions of expectation values in path integral form, starting from
: the relativistic QED action.
The h-->0 limit of vacuum expectation values of products of fields yields
the classical zero-field solution only.
If you want to obtain more of the classical theory, you need expectation
values between coherent states. The states do not survive the h-->0 limit,
but the expectation values do. But this only works with free bosonic
theories.
Greg
John Baez
Bill Frensley <Bill.F...@sbcglobal.net> wrote:
>[...] nonrelativistic quantum mechanics is
>derivable "in principle" from QED. If one cannot make this most obvious
>connection in practice, I would suggest that a review of our notions of
>fundamentality are long overdue.
Actually, the only people interested in making these "obvious"
connections precise are mathematical physicists - people who are mainly
interested in the logical structure of physics, rather than its
applications to any one particular realm. There are lots of open
problems like this that sound easy but are actually very delicate.
The only way to solve them is by lots of hard work and new ideas.
For example, despite work by people from Lorentz and Dirac on down to
some very good present-day mathematical physicists, I don't think anyone
fully understands the theory of classical relativistic charged point
particles interacting with the electromagnetic field. Stephen Parrott's
book on this subject shows how gnarly it is. Does anyone remember the
name of that Polish fellow who recently claimed to have cracked this
chestnut?
Andrea Barbieri
>
> The h-->0 limit of vacuum expectation values of products of fields
> yields the classical zero-field solution only.
>
As regards the EM field, this doesn't surprise me, since a finite
product of fields corresponds to a finite number of quanta, but
what about one-electron states? how can the charge disappear in
the classical limit?
> If you want to obtain more of the classical theory, you need
> expectation values between coherent states. The states do not
> survive the h-->0 limit, but the expectation values do. But this
> only works with free bosonic theories.
>
Sounds interesting... can you point me to some refs?
Maciej Janowicz
> For example, despite work by people from Lorentz and Dirac on down to
> some very good present-day mathematical physicists, I don't think anyone
> fully understands the theory of classical relativistic charged point
> particles interacting with the electromagnetic field. Stephen Parrott's
> book on this subject shows how gnarly it is. Does anyone remember the
> name of that Polish fellow who recently claimed to have cracked this
> chestnut?
Several comments:
1. On the John Baez question.
Try to find papers by Jerzy Kijowski (approximate pron.: Keeyovskee).
I don't know if it is him Professor Baez has in mind, but Kijowski built at
least a very
reasonable Hamiltonian formalism for the classical relativistic charged
particles interacting
with electromagnetic field. I also think that useful additional comments
and references can be found
in the papers by a friend of mine, Dariusz Chruscinski who made his PhD
under Kijowski's supervision
few years ago.
2. On the Greg Weeks comments.
Your statements about algebras are far beyond my grasp, but what do you
think
about Schwinger's construction, nicely described by Bogoliubov and Shirkov
("Introduction to the theory of quantized fields", Ch. 7/42): you construct
a general one-particle
spinor state (by taking the superposition of states resulting from vacuum
after the creation
operators act upon it), and then have the Dirac field operator acting on
this state.
The resulting wave function satisfies the Dirac equation in external field
with the mass replaced, naturally, by the (integral) mass operator. Then
you take $e = 0$
to turn off the self-interaction, the mass operator degenerates just to
mass,
and what remains is the Dirac equation for one particle in the external
classical field.
Isn't it, in a sense, the remaining part of the required transition to
classical regime ?
3. On Steve Carlip's original question.
In a book edited by Klauder and Skagerstam on coherent states, try to find a
paper
dealing with von Neumann lattice states (discrete subsets of coherent states
set which
are just complete). If I remember correctly, they have been applied there
to get semi-classical
limit of dynamics of a non-relativistic electron in external, but quantized
electromagnetic
potential. Also, it seems to me that analogous constructions are contained
in a contribution by Bialynicki-Birula to the book edited by Barut and
published
in 1980 as well as in a book by Bagrov and a co-worker about the exact
solutions
to the relativistic wave equations.
Also, check a relatively recent paper by Bialynicki-Birula, Rafelski and
others (I'm very
sorry for not having specific references, but you can find them in, and
download
the relevant papers from, the Iwo Bialynicki-Birula home page - you can
reach it after connecting first
to www.cft.edu.pl), which deals with the dynamics of Wigner functions in
full QED.
Wigner functions should be a very good starting point to get the
semi-classical approximation,
e.g. by using the method of multiple scales, as shown in a different
(quantum-optical) context by
Stenholm (a paper in Phys. Rev. A 1993).
With very best regards,
Maciej Janowicz
Greg Weeks
: Greg Weeks wrote:
: > In general, although I kept my eyes open for years, I never saw any
: > nontrivial theory -- eg, classical electrodynamics or nonrelativistic
: > quantum mechanics -- derived from QED.
:
: ... If one cannot make this most obvious
: connection in practice, I would suggest that a review of our notions of
: fundamentality are long overdue.
For the record, I'm content with our notions of fundamentality, and I have
little doubt that the quantum mechanics of slow electrons experiencing
electrostatic forces (with no positrons in the neighborhood) can be derived
from QED. I just wish I could find the derivation.
Greg
John Baez
Maciej Janowicz <M.W.Ja...@sussex.ac.uk> wrote:
>> For example, despite work by people from Lorentz and Dirac on down to
>> some very good present-day mathematical physicists, I don't think anyone
>> fully understands the theory of classical relativistic charged point
>> particles interacting with the electromagnetic field. [...]
>> Does anyone remember the
>> name of that Polish fellow who recently claimed to have cracked this
>> chestnut?
>1. On the John Baez question.
>Try to find papers by Jerzy Kijowski (approximate pron.: Keeyovskee).
That's the one I meant! It's embarrassing: I've met him and talked
to him several times at the Banach Institute, but I'm terrible with names.
>I don't know if it is him Professor Baez has in mind, but Kijowski built at
>least a very reasonable Hamiltonian formalism for the classical relativistic
>charged particles interacting with electromagnetic field. I also think that
>useful additional comments and references can be found in the papers by a
>friend of mine, Dariusz Chruscinski who made his PhD under Kijowski's
>supervision few years ago.
Thanks! I wish someone would write a nice elementary review article
about this and summarize exactly what is known about this problem now,
and what is still open. Parrott's book was unfortunately written before
Kijowski's work. I'll try finding the papers you mention.
Alfred Einstead
> Maciej Janowicz <M.W.Ja...@sussex.ac.uk> wrote:
> >> For example, despite work by people from Lorentz and Dirac on down to
> >> some very good present-day mathematical physicists, I don't think anyone
> >> fully understands the theory of classical relativistic charged point
> >> particles interacting with the electromagnetic field. [...]
> >least a very reasonable Hamiltonian formalism for the classical relativistic
> Thanks! I wish someone would write a nice elementary review article
> about this and summarize exactly what is known about this problem now,
> and what is still open. Parrott's book was unfortunately written before
> Kijowski's work. I'll try finding the papers you mention.
I'm not sure I understand the problem. You can certainly write
down the equations:
For i = 0, 1, 2, ..., N:
div E_i = rho_i/epsilon_0; curl B_i - (1/c)^2 dE_i/dt = mu_0 J_i
div B_i = 0; curl E_i + dB_i/dt = 0
rho_0 = 0, J_0 = 0
For i = 1, 2, ..., N:
rho_i(r,t) = q_i delta(r - r_i(t))
J_i(r,t) = r_i'(t) rho_i(r,t)
For i = 1, 2, ..., N:
p_i(t) = m_i r_i'(t)/sqrt(1 - (r_i'(t)/c)^2)
p_i'(t) = sum q_i (E_j(r_i(t),t) + r_i'(t) x B_j(r_i(t),t))
sum taken over j = 0,1,2,...,i-1,i+1,...,N
Let's see... the boundary data here would be:
E_0(r,0) = E0(r), B_0(r,0) = B0(r)
For i = 1,2,...,N: r_i(0) = Q_i, p_i(0) = P_i
Solutions for (E_i, B_i), for i = 1, 2, ..., N are uniquely
specified in terms of r_i(t) up to free fields (which can all
be absorbed into (E_0, B_0)). So, the result is a unique
solution in terms of the given boundary data for some range
-Delta < t < Delta.
The only problem I see is that since (E_i, B_i) is being
excluded from the equation for dp_i/dt, then the separation
of (E_i,B_i) -> (solution in terms of r_i(t)) + Free field
yields an ambiguity for dp_i/dt, since dp_i/dt includes the
free field (E_0,B_0). So, you actually get different
inequivalent systems, depending on which convention you
adopt for the separation:
(E_i,B_i) -> (Solution) + (Free field).
t...@rosencrantz.stcloudstate.edu
Alfred Einstead <whop...@csd.uwm.edu> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote:
>> Thanks! I wish someone would write a nice elementary review article
>> about this and summarize exactly what is known about this problem now,
>> and what is still open. Parrott's book was unfortunately written before
>> Kijowski's work. I'll try finding the papers you mention.
>I'm not sure I understand the problem. You can certainly write
>down the equations:
[Most equations deleted. The one I'm most interested in says ...]
> p_i'(t) = sum q_i (E_j(r_i(t),t) + r_i'(t) x B_j(r_i(t),t))
> sum taken over j = 0,1,2,...,i-1,i+1,...,N
So the Lorentz force on each particle is computed using the fields
caused by all the other particles, right? Doesn't that mean that
there's no radiation reaction, and hence that energy-momentum isn't
conserved?
To be specific, consider the case of two particles with opposite
charges q and -q, and let m1 >> m2. The first particle stays at rest
at the origin. Give the second one an initial velocity that puts it
into a circular orbit. Since the second particle feels only the
Coulomb force due to the stationary first charge, the orbit is stable.
Meanwhile, it's radiating electromagnetic energy off to infinity.
I think that John is asking about the status of theories that do account
for radiation reaction.
-Ted
John Baez
>ba...@galaxy.ucr.edu (John Baez) wrote:
> For example, despite work by people from Lorentz and Dirac on down to
> some very good present-day mathematical physicists, I don't think anyone
> fully understands the theory of classical relativistic charged point
> particles interacting with the electromagnetic field. [...]
>I'm not sure I understand the problem.
Very briefly, the problem is that you get in trouble if you
say a charged point particle doesn't feel any force from the
electromagnetic field created by itself... but you also get in
trouble if you say it *does* - because its own electromagnetic
field is infinite right where it is.
A closely related problem is that naive attempts to compute the
energy of a system of charged point particles interacting with
the electromagnetic field give infinite answers, thanks to the
infinite energy of the electromagnetic field right near the particles.
Of course you have to try 17 different ways to solve these problems
before you realize that the 17 most obvious ways don't work.
There's a nice introduction to this subject in Feynman's lectures
on electromagnetism, where he describes the work of Lorentz and
Dirac on this subject. To dig deeper, I strongly recommend Stephen
Parrott's book. Kijowski's proposed solution involves studying
a tube of radius epsilon around the worldline of each particle,
splitting the electromagnetic field in these tubes into a part
"coming from outside" and a part "coming from the particle inside",
dealing with these separately, and then taking a limit as epsilon -> 0.
He claims this avoids some of the crazy stuff that happens in
other approaches.
I'm not really an expert on this, so I wish someone would become
an expert on it and write an updated version of Parrott's book.
Mark
>In article <e58d56ae.02060...@posting.google.com>,
>Alfred Einstead <whop...@csd.uwm.edu> wrote:
>>I'm not sure I understand the problem.
[Equations written down that showed that, in fact, the problem is
both well-defined with a well-defined solution up to the free field
ambiguity]
>Very briefly, the problem is that you get in trouble if you
>say a charged point particle doesn't feel any force from the
>electromagnetic field created by itself... but you also get in
>trouble if you say it *does* - because its own electromagnetic
>field is infinite right where it is.
The equations which were written down form a well-posed Cauchy
problem, once you specify the Cauchy data for BOTH the (E0(r,0),B0(r,0))
and (E_total(r,0),B_total(r,0)); or equivalently, for each of
(Ei(r,0),Bi(r,0): i=0,1,2,...,N) separately.
I would say "I don't see any issue here", but in fact it's more
correct to say: there simply is no issue here. No trouble at all.
t...@rosencrantz.stcloudstate.edu
>Though not cited, it was secretly John Baez who wrote:
>>Very briefly, the problem is that you get in trouble if you
>>say a charged point particle doesn't feel any force from the
>>electromagnetic field created by itself... but you also get in
>>trouble if you say it *does* - because its own electromagnetic
>>field is infinite right where it is.
>The equations which were written down form a well-posed Cauchy
>problem, once you specify the Cauchy data for BOTH the (E0(r,0),B0(r,0))
>and (E_total(r,0),B_total(r,0)); or equivalently, for each of
>(Ei(r,0),Bi(r,0): i=0,1,2,...,N) separately.
>
>I would say "I don't see any issue here", but in fact it's more
>correct to say: there simply is no issue here. No trouble at all.
The theory you describew is indeed well-defined and well-posed as a
Cauchy problem. But it doesn't include radiation reaction. I think
the question at hand is whether there is a well-defined theory
that does include radiation reaction.
-Ted
Alfred Einstead
> So the Lorentz force on each particle is computed using the fields
> caused by all the other particles, right?
> charges q and -q, and let m1 >> m2.
Such as this:
dp1/dt = q1 (E0(r1,t) + E2(r1,t) + v1 x (B0(r1,t) + B2(r1,t))
dp2/dt = q2 (E0(r2,t) + E2(r2,t) + v2 x (B0(r2,t) + B1(r2,t))
p1 = m1 v1/sqrt(1 - (v1/c)^2); v1 = d(r1)/dt
p2 = m2 v2/sqrt(1 - (v2/c)^2); v2 = d(r2)/dt
div E0 = div B0 = 0; curl E0 + dB0/dt = 0; curl B0 = -(1/c)^2 dE0/dt
div E1 = q1 delta(r - r1); curl E1 = -dB1/dt
div E2 = q2 delta(r - r2); cirl E2 = -dB2/dt
div B1 = 0; curl B1 = (1/c)^2 dE1/dt + q1 v1 delta(r - r1)
div B2 = 0; curl B2 = (1/c)^2 dE2/dt + q2 v2 delta(r - r2)
This is just Maxwell's equations, plus the force law.
> The first particle stays at rest at the origin.
> Give the second one an initial velocity that puts it
> into a circular orbit. Since the second particle [sic] feels
> only the Coulomb force due to the stationary first charge, the
> orbit is stable.
Oh? Where do you get that from, above?
> Meanwhile, it's radiating electromagnetic energy off to infinity.
Number one, it's not doing anything until you specify what the
free field is.
And two, charges don't actually radiate when bound.
The classical "derivation" does not follow from this, except
by making an (unwarranted, and empirically invalid) assumption
about the boundary values E0(r,0), and B0(r,0). (It's
empirically invalid, since the solution "derived" contradicts
empirical fact).
The boundary values are arbitrary, since the solutions for
E1(r,t) and E2(r,t) are only given up to free fields. So,
the equations for E0 and B0 are, in fact, unconstrained,
and E0 and B0 are completely ARBITRARY.
Alfred Einstead
> The theory you describew is indeed well-defined and well-posed as a
> Cauchy problem. But it doesn't include radiation reaction.
Um, no. What was written down was the Maxwell equations themselves,
plus the force law. So, it includes everything.
The only issue that arises there is the free field ambiguity, as
described previously.
t...@rosencrantz.stcloudstate.edu
Alfred Einstead <whop...@csd.uwm.edu> wrote:
>t...@rosencrantz.stcloudstate.edu wrote:
>> So the Lorentz force on each particle is computed using the fields
>> caused by all the other particles, right?
>> To be specific, consider the case of two particles with opposite
>> charges q and -q, and let m1 >> m2.
>
>Such as this:
>dp1/dt = q1 (E0(r1,t) + E2(r1,t) + v1 x (B0(r1,t) + B2(r1,t))
>dp2/dt = q2 (E0(r2,t) + E2(r2,t) + v2 x (B0(r2,t) + B1(r2,t))
>p1 = m1 v1/sqrt(1 - (v1/c)^2); v1 = d(r1)/dt
>p2 = m2 v2/sqrt(1 - (v2/c)^2); v2 = d(r2)/dt
>div E0 = div B0 = 0; curl E0 + dB0/dt = 0; curl B0 = -(1/c)^2 dE0/dt
>div E1 = q1 delta(r - r1); curl E1 = -dB1/dt
>div E2 = q2 delta(r - r2); cirl E2 = -dB2/dt
>div B1 = 0; curl B1 = (1/c)^2 dE1/dt + q1 v1 delta(r - r1)
>div B2 = 0; curl B2 = (1/c)^2 dE2/dt + q2 v2 delta(r - r2)
Fine. Now let me impose the initial conditions E0(r,0) = B0(r,0) = 0,
and let particle 1 initially be at rest at the origin. With these
assumptions, and m1 >> m2, particle 1 will stay at rest at the origin.
Then E1 is a Coulomb field and B1 = 0, right?
Then clearly a solution for the motion of particle 2 is a stable
circular orbit (or an elliptical orbit, or any other solution to the
Kepler problem).
Let me stop there and ask: is this a correct solution to the equations
of your theory? If not, why not?
If it is, then this theory does not include radiation reaction. That
is, it does not conserve energy. Let this particle orbit stably for
a long time, and then integrate the Poynting vector over a large
sphere centered on the origin. You'll find that there's
always positive energy flux out through the sphere. You can take
this system home and use it to build a perpetual-motion machine
in your basement.
>>From what you say below, it looks to me like I'm supposed to exclude
E0 = B0 = 0 as an initial condition. (It's "empirically invalid" and
the like.) I'll comment on this when I get there.
>This is just Maxwell's equations, plus the force law.
>
>> The first particle stays at rest at the origin.
>> Give the second one an initial velocity that puts it
>> into a circular orbit. Since the second particle [sic] feels
By the way, what's that "[sic]" doing there? It's not at all uncommon
for me to make errors in posts that justify a "[sic]", but I've
reread that sentence a couple of dozen times and I can't see what
error you think I've made.
>> only the Coulomb force due to the stationary first charge, the
>> orbit is stable.
>
>Oh? Where do you get that from, above?
Is it clear now?
>> Meanwhile, it's radiating electromagnetic energy off to infinity.
>
>Number one, it's not doing anything until you specify what the
>free field is.
OK. It's zero. Sorry I didn't say that before.
>And two, charges don't actually radiate when bound.
I'm at a loss to guess what you mean by this. Do you mean something
quantum mechanical (electrons in stationary quantum states of a
hydrogen atom don't radiate)? If so, that's irrelevant, since we're
trying to construct a classical theory here. Classically, an
accelerated charge certainly does radiate. And indeed, the solution
I've described above in your theory does radiate (in the sense
that the Poynting vector integrated over a sphere is positive).
>The classical "derivation" does not follow from this, except
>by making an (unwarranted, and empirically invalid) assumption
>about the boundary values E0(r,0), and B0(r,0). (It's
>empirically invalid, since the solution "derived" contradicts
>empirical fact).
Now you've lost me. I thought I was allowed to choose the initial
conditions. (Surely that's the way initial conditions usually work!)
You seem to be saying that only certain sets of initial conditions
lead to physically valid solutions, and the rest lead to
unphysical solutions. Can you give me a rule for telling which
sets of initial conditions lead to physically meaningful solutions
and which don't?
In other words, how can I actually use this theory to compute
anything? If I want to take a heavy charged particle, put a light
charged particle into a circular orbit about it, and watch the orbit
decay as the charge radiates away its energy (wich is what a "good"
classical theory of charged particles should predict), can I do that
using your theory by some clever choice of E0 and B0? If so, how?
Anyway, here's my bottom line: I claim that your theory has solutions
that fail to conserve energy. It may also have solutions that do
conserve energy, but it's not at all clear to me how one is supposed
to pick out those solutions and exclude the "bad" ones like the one I
exhibited above. Without such a recipe, I don't think you can claim
that
: I would say "I don't see any issue here", but in fact it's more
: correct to say: there simply is no issue here. No trouble at all.
-Ted
observer
Maxwell equations are "well defined in continuum" charge and current
density
are diferential functions. Particles in this model are treated as
parts of continuum sorounded with other particles.
(You can define some boundary cases isolated surface, line or point
like distribution of charge, but then you must use integral form of
equations instead od differential to deal with singularities.) Maxwell
equations describe field resulting from charges + free field. From
that field we can construct energy stress tensor and calculate Lorentz
force on a rigid body defined by its enclosing surface. How to define
suface of a isolated point particle?
Maciej Janowicz
The statement (of A.E.) that the bounnd charges do not radiate is
definitely not textbook-like. But there is a subtlety here: a
bound charge does indeed not radiate if you use, e.g., the
half-sum of the retarded and advanced Green function to eliminate
the field from the equations of motion of the charge. And
obviously the choice of the Green function is arbitrary; Dirac
took the half-difference of the above two. It has been even told
me that this arbitrariness was used in an unsucsessful attempt to
reconcile the old (Bohr-Sommerfeld) quantum mechanics with
classical electrodynamics. But these were pure speculations, and
one really has to have extremely strong arguments to question the
textbook knowledge about radiation of accelerated charges.
The above-mentioned ambiguity is related to the ambiguity in the
choice of initial data for E_0 and B_0 in the equation by Alfred
Einstead, but I think it also shows that he underestimates how
ugly and dangerous face this ambiguity has.
To better judge what is the physical content of the Alfred Einstead
electrodynamics of moving bodies, may I ask him
1. To write down his action principle and the Hamiltonian when only
one particle is present ?
I have two additional requests addressed to him:
2. Please do read :
H.P. Gittel, J. Kijowski and E. Zeidler,
Commun. Math. Phys. 198, 711 (1998).
I should be very glad to see your comments on this paper.
3. Could you please explain whether there is a difference
between Mark William Hopkins and Alfred Einstead ? Interestingly
-if you allow for a joke - the latter's name is so similar to
that of the great musicologist, the author of that excellent
monograph about Mozart...
II. On unification of the classical particle theory of matter
with electromagnetism.
I think that the difficulties are related to the fact that we
have a very dualistic description in the Lorentz-Dirac, Rohrlich
and related theories: particle-like for matter and wave-like for
the radiation. I would speculate that the troubles can be
overcome in an easier and more elegant way by abandoning the
dualism and thinking on the unification on three different
levels.
1) Particle level. Charged matter consists of particles, and the
radiation is represented as Newton wanted, namely as particles or
arrows, perhaps the geometrical optics could be a starting point
of description. Matter-radiation interactions happen due to the
classical scattering of ("hard-ball -like, perhaps) particles.
Such a bizarre theory doesn't exist, as I think, and I wonder if
it is really possible, but I believe it is conceivable.
2) Wave level. We just have coupled Maxwell-Dirac equations
for *classical* fields, and, strictly speaking, only the
electromagnetic field has physical interpretation. But,
mathematically, everything should be well defined. There is a
book by Moshe Flato and coworkers with some nice statements about
the existence of global solutions to the Maxwell-Dirac equations
under, it seems, reasonable conditions on initial data. (By the
way, maybe someone can recommend a review paper containing
users-friendly description of the whole nonlinear group
representation program of Flato and coworkers ?) In addition, we
have some beautiful (up to the energy sign), solitary-wave
numerical solutions to the Maxwell-Dirac system as, e.g., that
found by A. Garrett Lisi in the J. Phys. A around 1995. In
principle, semiclassical quantization can be performed to get
something closer to quantum reality.
The Lorentz-Dirac theory mixes the levels 1) and 2). For me,
even if the difficulties with infinities and self-acceleration
are overcome, the theory will remain awkward and ugly.
3) Quantum field level... Well, I haven't been aware of the
existence of Scharf's book, thank you for this reference. But, in
addition, Mark (Alfred ?)says the Heisenberg equations are well
defined for quantum fields within the Colombeau theory of
distributions. Mark, could you offer any specific statement of a
theorem ? And a proof ? Are they contained in Landsman's book
you've mentioned ? That would be quite something, I would regain
my faith in physics...
Colombeau's own book is unfortunately out of print. Again, does
anybody know a reference to a readable review paper ? Thank you
very much.
With best regards,
Maciej Janowicz
t...@rosencrantz.stcloudstate.edu
Alfred Einstead <whop...@csd.uwm.edu> wrote:
>t...@rosencrantz.stcloudstate.edu wrote:
>> The theory you describew is indeed well-defined and well-posed as a
>> Cauchy problem. But it doesn't include radiation reaction.
>Um, no. What was written down was the Maxwell equations themselves,
>plus the force law. So, it includes everything.
Can you justify this statement? Heuristically, radiation reaction
comes from the interaction of a charge with *its own* electromagnetic
field, which you've explicitly excluded in your theory.
To change that statement from a heuristic one to a precise one is
difficult -- indeed, that's the whole point here! One approach, for
instance, is to calculate the force on a little ball of charge of some
radius epsilon due to the ball's own electromagnetic field as well as
any external fields. You have to be careful to account for the
stresses that hold the ball together against its own electrostatic
repulsion, and you have to do a mass renormalization, but you can do
it. The result is that the ball does *not* move in the way you'd get
by considering only the Lorentz force due to the external fields; it
does respond to its own field. Moreover, the corrections do not go
away for small epsilon. That's the radiation reaction force.
It seems obvious to me that that's not included in your theory.
I tried to indicate explicitly a solution where it should show
up but doesn't. Let me give you a variant on it that you might
like better.
Start with two charges of masses m1 and m2, with m1 >> m2. Initially
m1 is at rest at the origin and m2 is very very far away moving with
some initial velocity. Since the charges are far apart, that initial
velocity should be constant to an excellent approximation.
In your formalism, I would choose as my initial conditions
E0 = 0
B0 = 0
E1 = Coulomb field
B1 = 0
(E2,B2) = Lorentz-boosted coulomb fields.
Since particle 2 feels only the Coulomb field E1 and no magnetic
field, its equation of motion is precisely the Kepler problem. So it
zooms in on a hyperbolic orbit and zooms back out to infinity. Its
final speed equals its initial speed. But particle 2 will have
emitted acceleration radiation as it whipped around particle 1. That
radiation wanders off to infinity, creating energy out of nothing.
Of course, in a realistic theory, the particle 2 would end up with a
final speed that was less than its initial speed (as indeed
accelerated particles really do in a situation like this in the
classical regime).
If the above is not a mathematically valid solution in your theory,
please explain why not. If it is a valid solution, please explain why
this doesn't bother you!
The only possibility I can think of is that your theory has both
physical and unphysical solutions and that what I've described is a
mathematically valid but unphysical solution. (This is more or less
the normal state of affairs for classical theories of charged
particles, dating back to the Lorentz-Dirac equation.) In other
words, if I'd chosen a nonzero free field to begin with and adjusted
E1 and/or E2 accordingly, I'd've gotten a solution in which the
particle finished with a slower final speed than it started with in
such a way that energy would be conserved. It's certainly not obvious
to me that this is possible, but I suppose it might be.
If so, then what's the recipe for choosing the initial free field
in order to pick out the physical solution in any given situation?
Without that, I can't see how this theory is of any use.
Even if you can't give me a recipe for doing this in any particular
situation, can you show that a "physical" (energy-conserving) solution
even exists in a case like this?
-Ted