Re: Virtual Particle confusion
From: Ranando King (rk_at_magictouchcorp.com)
Date: 03/01/05
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Date: Tue, 1 Mar 2005 10:54:57 -0600
"PD" <pdraper@yahoo.com> wrote in message
news:1109612128.931289.213860@f14g2000cwb.googlegroups.com...
> Ranando King wrote:
> > "PD" <pdraper@yahoo.com> wrote in message
> > news:1109365094.518160.42090@l41g2000cwc.googlegroups.com...
<snipped>
> > > Yes, but diffussion is NOT a reversible process, even if you can
> > > account for each collision microscopically. Why not???
> >
> > See the 2nd law of thermodynamics.... Energy prefers a high entropy
> state.
>
> And in terms of the microscopic, reversible depiction of the substance,
> please explain how to define the high-entropy state.
In this case, it means that each particle in the closed environment is
equally statistically likely to have the same amount of kinetic energy. That
may not actually be the case, but on average, any sampling of the
environment will tend to match any other sampling of the environment in
terms of average kinetic energy/particle.
> >
> > > Moreover, you are wrong in saying there is no difference between
> random
> > > and non-random kinetic energy. The second is associated with, e.g.
> > > macroscopic translational energy. The first is associated with
> > > temperature. There is limited ability to extract the latter from
> the
> > > former -- that limit is what Carnot discovered and marks the line
> in
> > > the sand.
> >
> > Carnot "discovered" what people in the cleaning industry have known
> for
> > centuries. We have a limited ability to clean up a mess when we don't
> know
> > what's in the mess.
>
> Actually that's not what he said. He said a specific fraction of the
> energy is extractable, even theoretically. This is *not* based on
> limited knowledge of the mess.
Think of it this way. If someone made a stain of unknown type on your
carpet, you'd be able to clean some fraction of it up. Depending on your
cleaner, you can even calculate a best case statistical percentage of the
stain you can expect to remove. That's the nature of Carnot's "discovery".
The only difference is that in his case, the stain is heat and the carpet
was water(a cold source). You can statiscally calculate how much of the heat
you'll be able to use in a best case scenario without knowing the full
nature of the heat.
Now consider if you were working on converting heat to mechanical motion.
Heat is motion in seemingly random directions by a myriad of particles. So
in effect, it's already mechanical motion, just undirected. Suppose you knew
the nature and direction of each individual particle. Using that knowledge,
if you could isolate each individual particle in the environment, you should
be able to catch and use the energy from each individual particle when it is
finally headed along the vector most useful to you.
> > To map this onto your previous statement, the mess is
> > the quantum soup that actually comprises the closed environment we're
> trying
> > to extract energy from. Until we understand all possible interactions
> and
> > their results, we cannot possibly hope to extract 100% of the energy
> > contained within that environment. Put another way, until we know
> quantum
> > mechanics well enough to begin dropping the use of statistics to
> define
> > behavior, we will never be able to use the immense pool of enery in
> the
> > universal background radiation to do anything useful.
>
> And this reveals your misunderstanding of thermodynamics. Even
> theoretically, and based on arguments that have *nothing* to do with
> quantum mechanics (Carnot and Gibbs lived long before anyone knew
> anything about QM), not all energy can be extracted from the "immense
> pool" of thermal energy.
I know they didn't know QM back then. That doesn't mean that I'm not free to
apply it now. What I'm saying is that the understandings provided by Carnot
& Gibbs showed us that at the present time, given our limited knowledge of
how things work, we're bound by certain limits on how efficiently we can
convert energy from one form to another without leaving behind some "waste
energy."
This may fall back to philosophy, but if we really have a proper
understanding of even the most minute of details regarding QM and how to
manipulate them, then we should be able to devise devices that are even more
efficient than the limit proposed by the Carnot cycle.
> >
> > > I don't want to push the analogy too far -- the randomness of
> quantum
> > > mechanics is different than the randomness of a thermal system.
> > > However, you said that randomness is *only* a lack of information,
> a
> > > point that I disagree with. Randomness implies physics on its own.
> >
> > Funny how such a minor statement like "random = unknown" can have
> such a
> > large effect on things, isn't it? But again I posit, just because we
> have
> > not yet found a pattern that fits, does that imply that there isn't a
> > pattern?
>
> Of course not, but that's different than saying, "The pattern that
> physicists use makes no causal sense to me, and so there must be a
> different pattern underlying that one that does make causal sense to
> me."
That's not quite what I'm saying. I'm saying that the "pattern that
physicists use" is incomplete, and they're using statistics to patch the
holes. So there must be some hard science capable of explaining these
statisically significant events and patterns. We just haven't discovered it
yet.
> > Same question in different words is: just because Einstein's
> > equations work without considering aether, does that imply aether
> doesn't
> > exist?
>
> Of course not, but the presence of the aether should have testable
> effects that distinguish it from an aetherless description. Many such
> models have indeed been proposed, tested, and shown to not bear on
> reality. You might ask whether an aether can exist *anyway* but be
> *wholly indiscernable* experimentally from an aetherless model. The
> answer there is Occam's razor: if there is no need for it, and there is
> no testable verification of it, then it should not be included in any
> description of nature.
While a perfectly fine thing to do in the face of current knowledge,
consider that trading the Lorentz Ether Relativity theory for General &
Special Relativity led to many mathematical artifacts, clouds of infinite
mass surrounding a core of negative infinite mass in just the right
quantities to leave a net positive mass, mulitiple quantum universes all
coexisting in the exact same reference frame at the same time, virtual
particles existing for a period of time short enough not to violate
Conservation of Energy yet carrying force exchange information, etc...
If this is simpler than dealing with the possibility of aether... well...
> > Or yet another way: just because we cannot connect QM with the
> > standard model, does that mean that there's no solution?
>
> I don't know where you got the idea that QM has a disconnect from the
> Standard Model. The Standard Model is *entirely based on* QM -- there
> is no disconnect. Perhaps you are thinking of gravity, which is *not*
> described by the Standard Model?
Sorry about the wording there. I was getting 2 different ideas mixed up my
mind when typing that. I meant to refer the disconnect between gravity and
the Standard Model.
> >
> > > Temperature is a *statistical* property. If you think otherwise,
> tell
> > > me how to find the temperature of a single particle or a pair of
> > > particles.
> >
> > If you really want the answer, just do this:
> >
> > T = (3868mv^2)/1.602E-20
> >
> > where m is the mass of the particle and v is the speed. The
> statistics
> > involved in taking temperature shortcut the need to know the mass and
> speed
> > of each individual particle and molecule. As I've stated before,
> statistics
> > are used to acquire results when all the info to perform hard
> calculations
> > is not available.
>
> And this reveals your lack of understanding of thermodynamics.
> Temperature is not *defined* for a system that is not in equilibrium.
> One or two particles cannot be in equilibrium. You are taking a formula
> for the *average* kinetic energy of particles in a thermodynamic
> equilibrium and improperly extrapolating that to a system that is not,
> and cannot be, in thermodynamic equilibrium. This is a crucial point in
> thermodynamics.
Actually, it's a logical extrapolation. Consider what the phrase "average
kinetic energy" actually means. The kinetic energy of any particle in a
closed environment is given by
E=(mv^2)/2
where m is the mass of the particle and v is the particle's velocity. The
temperature of the environment, being a measure of the average kinetic
energy of the environment, is given by adding all the E for the individual
particles and dividing by the total # of particles. If the # of particles is
1, then the formula above applies.
Thermodynamic equilibrium simply means that the system is neither gaining or
losing energy to the outside environment. So if that lonely particle is
maintaining the same E, then it is in thermodynamic equilibrium.
> >
> > <snipped>
> > > > The HUP merely states that we change the
> > > > properties of those things we observe as we observe them, so it's
> not
> > > > presently possible to know all of the properties of something
> we're
> > > trying
> > > > to observe.
> > > >
> > > > If you don't understand that this is admitting that we can't know
> all
> > > the
> > > > information, then I feel I'm not the one to explain it to you.
> It's
> > > not
> > > > Heisenberg's *RANDOM* Principle. It's Heisenberg's *Uncertainty*
> > > > Principle.... Uncertainty, as in not sure of. Just because you're
> not
> > > sure
> > > > of a value doesn't mean that the value is actually random.
> > >
> > > Nuh-uh. The quantum correlations between distant electron spins
> points
> > > to the fact that there is more going on here than the Clumsy Thumbs
> > > model of HUP.
> >
> > Exactly, now just apply that same line of thinking to ALL of QM...
> anywhere
> > you find statistics instead of hard calculation.
>
> Nonsense. QM says that the hard calculation *must be done* using the
> statistical nature of objects that small. An alternative that does not
> do that has a testable difference from QM predictions; in tests, QM
> wins.
Did you ever stop to ask why QM depends on statistics, or do you simply
accept it at face value? It's because of the HUP. When they attempt to
measure one property of something that small, they inevitably end up
changing at least one of it's other properties. Using statistics, they can
predict those changes to some degree. But since they do not know with
impunity, they're not ever going to be completely certain until they come to
a better understanding of what's driving those statisitcal patterns.
> >
> > > >
> > > > > Moreover, entropy is a *measure* of the randomness of a
> > > > > system. The 2nd law of thermodynamics is specifically a
> statement
> > > about
> > > > > causal ordering and entropy.
> > > >
> > > > Entropy... Chaos-theory... the basic principle that all things
> seek
> > > to fall
> > > > apart... the idea that things prefer to go from structured states
> > > (high
> > > > energy) to unstructured (low energy) is perfectly valid. It's
> just
> > > the old
> > > > concept that "nature abhores a vacuum" repeated. It's far easier
> for
> > > the
> > > > universe to maintain a state of constant, unilaterally equal
> > > potential (high
> > > > entropy) than it is to maintain clustered, high potential regions
> > > with
> > > > low-to-no potential areas inbetween (low entropy). That follows
> > > common
> > > > sense.
> > > >
> > > > The second law of thermodynamics simply says that you can't move
> > > energy in a
> > > > closed system from high entropy to low entropy.
> > >
> > > Define "high entropy" and "low entropy".
> >
> > High entropy: Highly consistant, unilateral energy state. Think messy
> > bedroom where you're just as likely to find a shirt on the light
> fixture as
> > anywhere else in the room.
> >
> > Low entropy: Highly organized, inconsistent energy state. Think clean
> > bedroom where there's a place for everything and everything is in its
> place.
>
> I'm looking for a more rigorous definition, not something that is based
> on a loose analogy. In particular, I'm looking for your definition that
> works in a microscopically deterministic model (any one).
>
> I'm obviously calling your bluff here. You are asserting that
> statistical definitions are based on lack of information of the "true"
> deterministic model, and yet you say that entropy is a well-defined
> quantity for a true, deterministic model. I'm asking for a clear
> definition from you that is consistent with that.
Wow... crossed up terminology. I never said that entropy "is a well-defined
quantity." I stated that entropy is *Not A Measure Of The Randomness Of A
System*. Entropy is a measure of the equality of energy state distribution
in a given environment. I'm simply saying that what's being measured by
entropy has nothing to do with randomness.
> >
> > > > Treating entropy as a
> > > > measure of randomness is a serious mistake... a mistake large
> enough
> > > to keep
> > > > the obviously deterministic macroscopic universe from being
> > > understood on a
> > > > sub-atomic scale. Entropy is a measure of the lack of isolated
> > > structure.
> > > > Think of it this way. Suppose a universe as large as our own
> existed
> > > having
> > > > no virtual particles, and no other energy outside of a single,
> > > stationary
> > > > electron. That universe would have very low (near 0) entropy
> because
> > > all of
> > > > it's energy would be confined to a relatively very small space.
> > > Suppose now
> > > > that another universe existed that was just like the previous one
> I
> > > > described save for its size being infinitesimally larger than the
> > > volume of
> > > > the electron it contains. That universe would have an
> exceptionally
> > > high
> > > > (near infinite) entropy because its energy would be spread out
> fairly
> > > > consistently across its entire volume.
> > >
> > > And that would be wrong. Your values of entropy in both cases are
> > > wrong. Count allowable states.
> >
> > Ok. :)
> > You got me on semantics with that one, but it's easy to fix the
> example,
> > replace the electron in the first universe with a black hole the size
> of our
> > solar system. Just for the sake of argument, this black hole isn't
> radiating
> > its energy away. Make the second universe to be infinitesimally
> larger than
> > our solar system but scatter all of the energy of the black hole
> around that
> > universe evenly as background radiation.
>
> I'm sorry, I don't follow your example here. And I don't know what you
> mean by "semantics".
Skip it then. Every time I've tried to explain myself in a situation like
this, I inevitably wind up sounding sarcastic and/or demeaning. So I'll just
save myself the embarassment and you the emotional trauma. ;) If you want
me to try anyway, just ask and I'll do my best.
<snipped>
> > > Not sure I know what you mean by this. Where is causality hidden?
> > > The quantum interference is part of the causality!
> >
> > Quantum interference is part of the math, not the causality. There is
> no
> > reasonable, logical, or rational way to assume that all the quantum
> states
> > of each possible permutation of that particle interfere with each
> other to
> > form some coherent result without assuming that the particle can
> exist in
> > every possible state simultaneously.
>
> The statement, "There is no reasonable, logical, or rational way to
> assume..." is precisely what I question. That is EXACTLY what QM
> posits. To assert it is impossible is unjustified unless you can point
> to
> a) a theoretical self-inconsistency (the theory is not mathematically
> or conceptually self-consistent);
> b) a clear contradiction with experimental facts.
> Violation of common sense is NOT sufficient grounds for discarding.
> Many people thought that constant velocity without an applied
> propulsive force was incompatible with common sense. Many people
> thought that the Earth orbiting the sun was a violation of common
> sense. Many people thought that observer dependence of simultaneity was
> a violation of common sense.
I don't doubt that failure of "common sense" alone is insufficient reason.
However, let's take a look at Schrodinger and his quasi-dead cat.
Fast-forward the experiment to where the halflife of the radioactive element
has already elapsed so that now the cat is both 100% alive and 100% dead at
the same time (bologna, but that's the experiment). Much like subatomic
particles which are supposed to be able to interfere with alternate quantum
versions of itself, the 100% alive state of the cat will almost invariably
interfere with the 100% dead state of the cat because cat's don't like being
trapped. So if Schrodinger's cat had survived, you could expect to find the
cat trying to escape the box...or at least meowing.
The really obsurd conclusion here is that if a cat meows in a Schrodinger
experiment and nothing is around to observe it, it might have been a dead
cat meowing! Not only is this a violation of common sense, it's a laughable
absurdity. The point here is that even though you either did not or cannot
observe something's state for whatever reason, that does not imply that the
unobserved object occupies every possible state simultaneously.
Halflives, even moreso than temperature, are a statistical property. Just
because a material has a half-life of 10 minutes doesn't mean that given 2
atoms of the material, in 10 minutes one of the atoms must have decayed. It
means that at least one of the atoms is likely to have decayed. But given
only 2 atoms, it's not significantly less probable that no or both atoms
will have decayed. That does not, however, place the atoms in some quantum
dual state until someone observes it. It mearly means that the atoms are in
some state that we do not presently know.
> > That implies that the particle would
> > have to have the energy of every possible state simultaneously. Not
> > Possible. (I know, qm states that it is.)
>
> And why is it not possible? Note that energy is an observable quantity.
> You have to be sure you understand how to calculate the energy (or for
> that matter of any observable quantity) of a sum over all histories --
> it is NOT the algebraic sum of the energies of every possible state.
> This may be what's standing in your way.
It's not possible because the sum of the energies of a number of states has
to sum up to the energy of the state of the particle as it is when the
particle is observed. It doesn't matter that it's not an algebraic sum. The
rules of math reveal an inconsistency here. If the letters a to e represent
the 5 possible well known energy states of a particular unobserved in its
multi-state, f is the summation function, and a is the observed state of
that particle, then:
a = f(a, b, c, d, e)
If a, b, c, d, and e, are all known to be different state values then if the
experiment is repeated exactly so that nothing has changed in the input to
the experiment but the output is now b, then:
b = f(a, b, c, d, e)
Since the values of the individual states are well known and known to be
different, then:
a != b
Therefore
f(a,b,c,d,e) != f(a,b,c,d,e)
and that's quite impossible if a,b,c,d, and e are known values. The only way
this is even remotely possible is if a,b,c,d, and e are functions
themselves. If indeed they are functions, then something is going on
underneath the statistics that must be understood.
> >
> > > >
> > > > > > If that's true, then
> > > > > > there's got to be a fundamental reason behind that tendancy.
> > > Quantum
> > > > > physics
> > > > > > merely states that there is a tendancy and shows how strong
> that
> > > > > tendancy
> > > > > > is. However, it doesn't even come close to explaining why
> that
> > > > > tendancy
> > > > > > exists. Therefore, we're missing information, information
> that
> > > would
> > > > > likely
> > > > > > lead to the elimination of the dependency on statistics in
> > > quantum
> > > > > physics.
> > > > >
> > > > > You are espousing what's called "hidden variable" model of
> quantum
> > > > > mechanics, which may make intuitive sense to you, but you're
> far
> > > from
> > > > > the first to think of it. But intuitive good sense does not
> make it
> > > > > correct, let alone unavoidably correct. Hidden variable
> theories
> > > have
> > > > > definite experimental predictions that are testable. Read the
> > > > > literature on this subject. The experimental tests fail. Nature
> > > does
> > > > > NOT need hidden variables and in fact is inconsistent with that
> > > > > picture, as counterintuitive as that may seem.
> > > >
> > > > I know I'm not the first to have this idea, and I won't be the
> last.
> > > Think
> > > > of it this way. Quantum physics cannot currently be used to
> describe
> > > the
> > > > macroscopic universe.
> > >
> > > And on what basis do you make THAT statement? Is a laser a
> macroscopic
> > > object or not?
> >
> > The laser is, the photons that comprise it aren't.
>
> Careful. Photons are only one part of the laser. (A box full of photons
> cannot be made to lase.) Regardless, a laser is a macroscopic object
> whose properties are described (in fact, based on) quantum mechanics.
> You made a broad statement that QM cannot currently be used to describe
> the macroscopic universe. I gave you an example of one where it can.
> You now give me a few examples of cases where it cannot, aside from
> purely gravitational phenomena.
Sorry, but eliminating gravitational phenomena, fairly well eliminates much,
if not most of the macroscopic universe. Try this one. Use QM to explain
*why* the variant mass of an object traveling at near light speeds *DOES
NOT* accompany an increase in gravitational potential. It's a
non-gravitational phenomena, just as you requested. If you want me to
exclude any question regarding gravity in its entirety then try this one.
Explain *why* I can neither find nor create magnetic monopoles.
Here's a microscopic question, too. How is it that the point electron is not
a black hole if it's mass is all contained within a single point in space,
aka a singularity? Compound this with the notion of renormalization where
the electron core itself has an infinite negative mass (a white hole)
encased in a cloud of virtual particles of positive infinite mass (a black
hole).
> >
> > > >Every attempt to do so to date has failed.
> > >
> > > What attempts are you referring to?
>
> Unanswered.
Unanswered because it was answered before it was asked by the statements
that preceeded it.
> > >
> > > > What does
> > > > that mean? It means there's something we're not yet
> understanding,
> > > right?
> > >
> > > Well, THAT's true, but it doesn't say that QM is full of hokum.
> >
> > QM isn't "full of hokum", it's full of statistical uncertainties.
>
> No, it's full of statistical quantities, which produce extremely
> precise predictions which are not verified experimentally and not
> predicted by any deterministic model.
You forgot the words "to date" at the end of that statement.
> >
> > > > String theory was invented to try and find that something.
> M-Theory
> > > was
> > > > invented to try and find that something. QED, QCD, and probably
> other
> > > > theories were invented to try and find that something. All have
> > > failed to
> > > > date.
> > >
> > > QED and QCD have failed? How so?
> >
> > They were both part of the attempt to produce a Unified Theory.
>
> That is incorrect. Neither were produced as an attempt to produce a
> Unified Theory. QED is a quantum mechanical (strictly speaking, quantum
> field) description of electromagnetism, pure and simple. Nothing more
> is claimed of it. QCD is a quantum mechanical (quantum field)
> description of the strong nuclear force, pure and simple. Nothing more
> is claimed of it. To date, QED and QCD are both extremely well-tested
> models.
>
> Since then, there have been some suggestions that an underlying theory
> that is also quantum mechanical can absorb QED, QCD, weak and
> gravitational interactions should be discoverable. It hasn't been found
> yet. That doesn't mean that QCD or QED or quantum mechanics are
> failures -- far from it.
At the same rate, before the invention of the telescope, saying the earth
was the center of the universe also explained lots of things quite well. I
know from years of experience programming that a bad solution can still
handle 99.9% of all problems thrown at it. But eventually, because the
solution is a bad one, there will be an irrecoverable error that cannot be
made to fit the desired results. In the case of QM, that error is the
inability to include gravity. In such cases, the core of the program has to
be examined and rewritten. In this case, that would mean altering the basic
premise of QM and changing the model accordingly.
> > To date, no
> > unified theory that covers all known science is available.
>
> And that makes the pieces that are known a failure?
The fact that we can explain it all to a high degree of precision, but only
in pieces that do not fit with each other, reveals that those pieces fail to
accurately describe the universe as a whole. Some of the pieces may be
correct. In fact, there may only be 1 errant piece, but until we know for
sure, all of it is suspect.
> Does the fact that Darwinian evolutionary theory doesn't predict
> radioactive decay make evolutionary theory a failure?
No. Different subject. Excuse my obtuse terminology above. I meant to say
"all known physics". As for evolution, the fact that the statistics predict
that it would have taken significantly longer than the age of the universe
to evolve humans makes that theory a failure.
> > Gravity is a hard
> > target to hit.
>
> Indeed, no argument. What's your point?
One misplaced piece in a jigsaw puzzle fouls the entire puzzle. It doesn't
matter how good the piece looks there or how well it fits, if that's not the
place where the piece belongs, the puzzle cannot be properly completed.
> >
> > > >Does that mean that nature does not need the physics describing
> > > > sub-atomic stuff to be consistent with the physics for
> macrosocpic
> > > stuff?
> > > > That's the kind of argument you just fed me.
> > >
> > > Nonsense. The macroscopic limit of every well-defined QM theory
> > > reproduces what we see classically.
> >
> > So we've successfully explained gravity with QM? Last I checked,
> quantum
> > gravity was inconsistent.
>
> Inconsistent? No. If that were demonstrated, then there wouldn't be so
> many people attempting to make a quantum theory of gravity.
>
> Has it been found? No, certainly not.
>
> Does failing to account for gravity make QM a failure at describing
> anything in the macroscopic universe? Certainly not. Does QM presently
> account for everything in the macroscopic universe? No, but this is not
> grounds for dismissing QM.
Gravity is a major factor in explaining the macroscopic universe. If you
don't account for the effects of gravity, all calculations on real objects
will be off. The amount of that error may be "negligable" in the opinion of
many, but it's still an error that can be avoided by proper accounting.
> >
> > <snipped>
> > > > > Moreover, your prescription still does not work. Take two
> particles
> > > A
> > > > > and B which have central mass values of, say 140 MeV and 142
> MeV,
> > > > > respectively, but which are otherwise completely distinct. The
> > > > > invariant mass of a virtual state is measured to be 139.5 MeV.
> > > > > According to your prescription, this would *always* be assigned
> to
> > > > > particle A. However, it is quite likely that the particle will
> > > instead
> > > > > be identified as particle B, based on conserved quantum
> numbers.
> > > This
> > > > > is routinely seen in experiment, but is inconsistent with your
> > > scheme.
> > > >
> > > > Unlike those of you that believe "random" has a place in physics,
> > > that just
> > > > leads me to believe that there's another term to be considered
> that
> > > is a
> > > > factor of the original contributing masses. My first guess would
> be
> > > to
> > > > examine the charges involved to see if there is a pattern.
> There's
> > > likely
> > > > something in the properties of the original masses involved that
> > > creates an
> > > > additional limiting factor. It might be that particle A has no
> charge
> > > and
> > > > particle B does. So when the interaction occurs, if there is a
> net
> > > charge
> > > > left behind with the 139.5 MeV, only the 142 MeV charged particle
> is
> > > a
> > > > viable option.
> > > >
> > > > The bottom line: Just because a pattern hasn't been found doesn't
> > > imply that
> > > > there isn't one.
> > > >
> > > > R.
> > >
> > > So you are saying that charge and lepton number and QCD color and
> > > baryon number contribute to mass? How so? How does your model work
> to
> > > put that together?
> > >
> >
> > Not so much that they contribute to mass but rather constrain the
> list of
> > available virtual particles that a given invariant mass can become.
>
> Ah, then we agree after all! And that's precisely what current theory
> says.
>
If we go with the assumption that QED and QCD aren't completely wrong, then
there's no doubt that we'd agree on certain things. I'm still having
difficulty with renormalization. Is this supposed to represent a real
process? Or is it just a matter of trying to fix a mathematical
inconsistency?
R
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