As you know, I have been asking a lot of questions lately about
wavefunctions and Gaussians and the like.

You may have guessed that I am writing a paper about this, and if so you
would be right. ;-)

I believe there is enough developed now, and I think enough of the kinks
are now out, so you all may take a sneak preview. Thus, I have linked
my latest draft at:

http://jayryablon.wordpress.com/file...hwinger-20.pdf

Setting aside the hypothesized connection between the magnetic anomaly
and uncertainty, Sections 4 through 7, which have not been posted in any
form previously, stand completely by themselves, irrespective of this
hypothesis. These sections are strictly mathematical in nature, and
they provide an exact measure for how the uncertainty associated with a
wavefunction varies upwards from hbar/2 as a function of the potential,
and the parameters of the wavefunction itself. The wavefunction
employed is completely general, and the uncertainty relation is driven
by a potential V.

This is still under development, but this should give you a very good
idea of where this is headed.

If the link doesn't work, right click and download instead, then open.

Thanks to KP, George, Eric, et al. for pointers which helped me to get
this far.

Of course, I welcome comment, as always.

Best regards,

Jay.
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

From what I can tell your argument boils down to starting with

\delta x *\delta p \hbar/2

then multiplying through by g_D

g_D\delta x *\delta p/(\hbar/2) g_D

then you replace g_D on the right side with g that comes from QED and
somehow claim this leads to something, but there is no physical,
mathematical or even logical justification for this step. You could
also repeat the above procedure for the fine structure constant, the
masses of particles, the gravitational constant or even for something
as ridiculous as \pi.

Also the measured magnetic moment is one specific value. There are an
infinite number of wave functions an electron could be in, which would
also give an infinite number of uncertainty relations which according
to your claim would give an infinite number of possible values of g.

Unfortunately, I see no physics at all here.

kp

"kp" wrote in message
...
From what I can tell your argument boils down to starting with

\delta x *\delta p \hbar/2

then multiplying through by g_D

g_D\delta x *\delta p/(\hbar/2) g_D

then you replace g_D on the right side with g that comes from QED and
somehow claim this leads to something, but there is no physical,
mathematical or even logical justification for this step. You could
also repeat the above procedure for the fine structure constant, the
masses of particles, the gravitational constant or even for something
as ridiculous as \pi.

Also the measured magnetic moment is one specific value. There are an
infinite number of wave functions an electron could be in, which would
also give an infinite number of uncertainty relations which according
to your claim would give an infinite number of possible values of g.

Unfortunately, I see no physics at all here.

kp

Please take a look at:

http://en.wikipedia.org/wiki/Land%C3%A9_g-factor

The measured magnetic "g-factor" is a function of s=1/2, l and j=l+s.
Thus, depending on state, there are an infinite number of g-factors as
well. The g-factor of ~2 appears to be for the specific case where l=0
and j=s. What I am suggesting is a 1-to-1 correspondence between
g-factors and uncertainty relations. For an electron in a given quantum
sate, there is one specific, exact g-factor and one specific, exact
uncertainty relation.

What you say in the first paragraph about "multiplying through" is true,
but what I am making note of what I believe is this one-to-one
correspondence between uncertainty value and g-factor value, based on
the particular state of an electron. That is, I am suggesting that
g-factors and uncertainty relationships parallel one another, and are in
fact, different ways of viewing the same phenomenon.

Separate question: forgetting about the g-factor connection, do you see
any problem in the manner in which I fleshed out the uncertainty
relationships, separately from my hypothesis connecting them to the
g-factor?

Thanks,

Jay.

Please take a look at:

http://en.wikipedia.org/wiki/Land%C3%A9_g-factor

For one, this is for electrons in an atom, where the wave functions
are not even close to Gaussian.


The measured magnetic "g-factor" is a function of s=1/2, l and j=l+s.
Thus, depending on state, there are an infinite number of g-factors as
well. *The g-factor of ~2 appears to be for the specific case where l=0
and j=s. *

There is a g factor for the coupling of spin to magnetic fields and
one for the orbital angular momentum.
Which one are you referring to? The only anomalous one is for the
spin and thus for a given spin there is only one universal value. You
can see that the orbital part plays no role in your calculations for
the magnetic moment.



What you say in the first paragraph about "multiplying through" is true,
but what I am making note of what I believe is this one-to-one
correspondence between uncertainty value and g-factor value, based on
the particular state of an electron. *That is, I am suggesting that
g-factors and uncertainty relationships parallel one another, and are in
fact, different ways of viewing the same phenomenon.


Again an electron can have an infinite number of possible wave
functions and thus satisfy an infinite number of uncertainty
relations. I don't see anyway to deduce a universal single number
from this.


Separate question: forgetting about the g-factor connection, do you see
any problem in the manner in which I fleshed out the uncertainty
relationships, separately from my hypothesis connecting them to the
g-factor?


I didn't closely check everything, but it seems OK.

kp

On May 2, 12:27 pm, kp wrote:
Please take a look at:

http://en.wikipedia.org/wiki/Land%C3%A9_g-factor

For one, this is for electrons in an atom, where the wave functions
are not even close to Gaussian.



The measured magnetic "g-factor" is a function of s=1/2, l and j=l+s.
Thus, depending on state, there are an infinite number of g-factors as
well. The g-factor of ~2 appears to be for the specific case where l=0
and j=s.

There is a g factor for the coupling of spin to magnetic fields and
one for the orbital angular momentum.
Which one are you referring to? The only anomalous one is for the
spin and thus for a given spin there is only one universal value. You
can see that the orbital part plays no role in your calculations for
the magnetic moment.



What you say in the first paragraph about "multiplying through" is true,
but what I am making note of what I believe is this one-to-one
correspondence between uncertainty value and g-factor value, based on
the particular state of an electron. That is, I am suggesting that
g-factors and uncertainty relationships parallel one another, and are in
fact, different ways of viewing the same phenomenon.

Again an electron can have an infinite number of possible wave
functions and thus satisfy an infinite number of uncertainty
relations. I don't see anyway to deduce a universal single number
from this.

Separate question: forgetting about the g-factor connection, do you see
any problem in the manner in which I fleshed out the uncertainty
relationships, separately from my hypothesis connecting them to the
g-factor?

I didn't closely check everything, but it seems OK.

kp

Is the Schwinger Anomaly when you're dating some old gal and she tells
you she's a Schwinger and you get your hopes up for getting it wet for
once but then she tells you that anomalously she has transformed into
a Roman Catholic Nun for the evening?

"kp" wrote in message
...

.. . .
Again an electron can have an infinite number of possible wave
functions and thus satisfy an infinite number of uncertainty
relations. I don't see anyway to deduce a universal single number
from this.

[Yablon]

But, for all electrons which are, by definition, in a ground state,
i.e., with all quantum numbers equal zero except for the spin 1/2, would
there not then be a single ground state wavefunction? If not, what
would differentiate one ground state electron from another?

Jay.

"kp" wrote in message
...

.. . .
What you say in the first paragraph about "multiplying through" is
true,
but what I am making note of what I believe is this one-to-one
correspondence between uncertainty value and g-factor value, based on
the particular state of an electron. That is, I am suggesting that
g-factors and uncertainty relationships parallel one another, and are
in
fact, different ways of viewing the same phenomenon.


Again an electron can have an infinite number of possible wave
functions and thus satisfy an infinite number of uncertainty
relations. I don't see anyway to deduce a universal single number
from this.

[Yablon]
I will beef up this discussion next pass through. However, one of the
things the uncertainty calculation demonstrates is that the statement
"an electron can have an infinite number of possible wave
functions and thus satisfy an infinite number of uncertainty relations"
is actually false, and includes a supposition which is actually not
borne out by the mathematics. Let me refer to a couple of equations in
http://jayryablon.wordpress.com/file...hwinger-20.pdf
which bear this out.

First, while I refer to V as a "potential," if you go to (4.5) and
(4.7), I should really just refer to V(x) in (4.5) and V(d/dB) in (4.7)
as "polynomial functions" of x and d/dB respectively. To say anything
more is reading in more than is strictly justified. The question then
becomes, a) what effect does this polynomial V have on the uncertainty
of the wavefunction (4.1) which contains this V?, and b) since this
quadratic V enters along with A and B, what effect do A and B have on
the uncertainty?

Now, fast forward to (7.7). This is the calculated uncertainty of the
wavefunction (4.1). It is independent of the "magnetic anomaly
hypothesis," and "seems OK" to you based on the degree to which you have
checked. (And your posts, I might add, suggest you have a pretty good
nose ;-) Because of the polynomial V in (4.1), which can really
accommodate any function one wishes, this means that wavefunction (4.1)
can be anything one wishes it to be: all one needs to do is change the
polynomial, i.e., change V. This polynomial V is the parameter we use
to accommodate an "infinite number of possible wave functions," no more
and no less than a Taylor or MacLauren series can be used to accommodate
an infinite number of functions. So, we need to look at (7.7) to tell
us in more precise terms, about the range of possible uncertainty
relationships as a function of the range of possible wavefunctions.

First, we see that if V=0, which leaves behind only the first and second
order term in x and d/dB and omits any higher order terms, we simply
have

delta x delta p = 1/2 hbar. (1)

This the Heisenberg as an equality, and is because of the Gaussian
nature of (4.1) when the generalized polynomial V=0. It is when there
is a non-zero V, that A and B and V and d^2V/dB^2 all affect the value
of the uncertainty. The point is this: only a non-zero value of V
affects the uncertainty.

So, what we might do with (7.7) is ask such questions as:

1) Would V (the polynomial) and A and B all be the same for all ground
state electrons? If so,then all ground state electrons would have a
single uncertainty value, and one could consider relating them to
something else which is single valued, such as the spin-based g-factor.

2) If the hypothesis I have made about the anomaly were to be true,
what else would have to be true? From (8.2) where I left off, to get
into the ballpark with Schwinger (that is, to be no further from the
observed magnetic moments than Schwinger was), one would be required to
set:

2V = alpha/2pi (2)

Thus, the polynomial V would, in fact, simply be an alternative way of
expressing the running coupling alpha -- 1/137.036. Let's flip this on
its head:

Let's simply go back to wavefunction (4.1), and substitute alpha/4pi for
V. What my proof demonstrates is that for a wavefunction (4.1) in which
(2) is employed, the uncertainty relationship WILL rise above 1/2 hbar
in precisely the same proportion that the observed spin-based g-factor
rises above 2, at least up to the level of Schwinger. So, you can say
what else you want about the hypothesis, but that is a mathematical
fact, which I will restate as such:

MATHEMATICAL FACT / THEOREM (Unless my uncertainty derivation is in
error):

Start with a wavefunction:

psi = exp [-.5Ax^2 + Bx - alpha/4pi], (3)

Where alpha is the running coupling. The ratio:

delta x delta p / (hbar/2)

is then identical to the ratio:

g/2

up the the degree of precision of Schwinger, and irrespective of A and
B. No two ways about it -- that is what the math says. From there, you
and me any anyone else can make what they wish of that fact. I will
claim no more, and no less.

END MATHEMATICAL FACT / THEOREM

Then, returning to (7.7), the deviation up or down from Schwinger's
ratio now relies on the second derivative of V (or alpha) with respect
to B. I don't know yet how to use this, but it is there.

One thing I will note and it is experimental: the electron g is less
than Schwinger's g, yet the mu and tau g are greater than the Schwinger
g. We still don't know "who ordered this" for the higher generations,
but we do know this particular experimental oddity. And, since making
sense of the higher generations, to this day, still entails reading tea
leaves, this is one clear experimental "tea leaf" of interest.

Because, what (7.7) says is that d^2V/dB^2 must be positive to drop
below Schwinger and nail the electron, while it must be *negative* to
rise up above Schwinger and nail the mu and the tau (assuming A and B
are the same for all three leptons).

So, the little bit we learn about the mu and tau versus the electron
from this tea leaf, is that they are on different sides of an
"inflection" point in the running coupling alpha. I don't know yet how
to blow on this little flame and turn it into a bonfire, but this
certainly ekes out a little bit more of an understanding than we had
before, of the higher generation leptons.

I would love to relate this to the masses of the three leptons, but
don't (yet) see that connection here.

Separate question: forgetting about the g-factor connection, do you
see
any problem in the manner in which I fleshed out the uncertainty
relationships, separately from my hypothesis connecting them to the
g-factor?


I didn't closely check everything, but it seems OK.

[Yablon]
Well, if it is OK, then all I said above is true.

Happy physics hunting!

Jay.

kp