Planck Units and (slowly) changing Fine-Structure Constant
robert bristow-johnson
Background (so youse guys don't talk over my head): i'm an EE doing
designing DSP algorithms to process audio/music for a living (i'm kinda a
guru on comp.dsp). i'm pretty good at most of the math physicists are good
at but pretty ignorant of deeper current physical theories (anything past
classical, special relativity, intro to quantum mechanics) like
string-theory, etc. i wanted to ask an abbreviated version of this question
on Science-Friday 2 weeks ago, but they didn't take my call.
anyway, about 2 decades ago i have been thinking along some of the same
lines as Planck (although ignorant that he had been thinking about this 80
years previously) about a "Universal" set of physical units that would
basically make the Universal Constants equal to unity if expressed in terms
of those units. of course i come up with the Planck Units (except i also
came up with a unit charge not equal to the electron charge).
the governing basic formulae are:
c = L/T
E = M * c^2
E = Hbar * (1/T)
E = G * M^2 / L ( or F = G * M^2 / L^2 with E = F*L )
E = k * Q^2 / L ( or F = k * Q^2 / L^2 with E = F*L )
since k = 1/(4*pi*e0) and c^2 = 1/(e0*u0) and u0 has a defined value (in
SI it's 4*pi * 10^-7 because an ampere was defined as a current that would
yield a force of 10^-7 Nt/m for a pair of infinitely long wires spaced 1
meter apart), anyway, in that case the last equation (Coulomb static force)
becomes
E = (c^2 * u0/(4*pi)) * Q^2 / L
now if we leave off the question of unit charge for the time being and solve
the first four equations to get the four unknowns (L, M, T, E), you get the
Planck values which is what you decided to pick for your fundamental units.
L = sqrt( Hbar * G / c^3 )
M = sqrt( Hbar * c / G )
T = sqrt( Hbar * G / c^5 )
( and consequently E = sqrt( Hbar * c^5 / G ) )
from my POV, the compelling reason to pick those for our fundamental
universal units is that the other fundamental constants c, Hbar, and G all
become unity in terms of those units.
now here's what i thought was a bit interesting coming from a different POV
than most everyone else: rather than just define the unit charge to be the
electronic charge (e), i wanted the Coulomb electrostatic force constant to
be unity also and chose the unit charge to make that happen:
Q = sqrt( (4*pi/u0) * Hbar / c )
which comes out to be 11.70623764 * e, or the square root of alpha^-1 times
the electronic charge where alpha is the Fine-Structure Constant.
Q/e = sqrt( (4*pi/u0) * Hbar / c )/e = 1/sqrt(alpha)
==> alpha = u0/(4*pi) * e^2 * c / Hbar .
anyway, i thought that this was kinda neat that this relationship between
what i would call the natural unit of charge to the electronic charge (both
very fundamental and universal quantities) would be related this way.
likewise if you choose e to be your fundamental unit (as do most) then you
do have to have a non-unity Coulomb force constant which is 1/alpha (about
137.036), but that isn't so ugly either. it's just a question of which unit
is "more" fundamental and that's a toss up i guess.
Now, finally, it seems that we must perceive reality in terms of the Planck
Units (T, L, M) and perhaps the unit charge Q. if the Planck Length went
from 10^-35 to 10^-32, then we would be 2000 meters tall instead of 2 but
our meter stick would be 1000 meters and we would still call it a "meter"
and the Planck Length would still be about 10^-35. same with time and mass,
but what about charge???
if we perceive reality in terms of the Planck Units and the electronic
charge, e, then a change in the Fine-Structure Constant, alpha, would be
noticed as a change in the Coulomb Force constant since
F = c^2*u0/(4*pi*alpha) * q^2/r^2
where q is in units of e.
and we would also notice a change in the Characteristic Impedance of
radiation in a vacuum since
Z0 = 4*pi*Hbar*alpha/e^2 .
But, if OTOH, we perceived reality in terms of the Planck Units and the
charge Q = e/sqrt(alpha), then a change in alpha would be noticed as a
change in the charge of an electron.
so which is it? or does it make any difference?
i'll try to monitor this newsgroup but feel free to CC: me at
<rob...@wavemechanics.com> if you want to make sure i see a response.
--
r b-j
Wave Mechanics, Inc.
45 Kilburn St.
Burlington VT 05401-4750
tel: 802/951-9700 ext. 207 http://www.wavemechanics.com/
fax: 802/951-9799 rob...@wavemechanics.com
--
Lubos Motl
comp.dsp people have a guru who is not silly. It is fine that you could
re-discover Planck units. By the way, in Planck units - where you put
epsilon0=mu0=1, speaking in SI units -, the charge of the electron is
really equal to -sqrt(4.pi/137.036...) just like you say.
Physicists are however convenient and they usually express the charge in
units of (minus) the charge of the electron so that it is always integer
(or integer over three, for quarks). You cannot say that one choice of
units is objectively more fundamental; it is a matter of taste. Both are
certainly more fundamental than using Coulombs. However it depends on your
feelings. Anyway you can see that you cannot get rid of the number
1/137.036..., the fine structure constant. It is a dimensionless number
without any units. Therefore it does not depend on our choice of the
units. And there should be some explanation for its value!
We understand this number in terms of more fundamental constants of the
electroweak theory (g and g'), measured at higher energies (instead of
zero energies - as alpha), but a complete calculation yielding
1/137.036... is still missing. String theory is believed to be capable to
derive its value one day.
Particle physicists usually measure the charge so that the electron has
Q=-1. But then they must include the fine structure constant into the
definition of the energy. The energy density - or the Lagrangian (which is
something related that has the same dimension) - is defined as 1/g^2 times
E^2/2 etc. in the conventions where Q=-1 for the electron.
Either you say that the minimal charge is some strange number (instead of
1), or you can say that it is one but the energy density is not defined as
E^2/2 but this times a strange constant. You cannot get rid of the
constant at both places simultaneously. In fact, both conventions are used
by particle physicists sometimes. It causes a lot of confusion but there
are more difficult problems in the world. ;-)
> Now, finally, it seems that we must perceive reality in terms of the Planck
> Units (T, L, M) and perhaps the unit charge Q. if the Planck Length went
> from 10^-35 to 10^-32, then we would be 2000 meters tall instead of 2 but
It is correct that you invite us to perceive reality in Planck units, but
you do not do it yourself. If you did so, you would rather say: if one
meter was defined not as 10^35 Planck lengths but only 10^32 Planck
lengths, then we would be 2000 meters tall instead of 2 (anyway, two is
also too much) :-) - because everyone knows that a human being must be
about 2.10^35 Planck lengths tall in order to have the right number of
atoms.
> our meter stick would be 1000 meters and we would still call it a "meter"
> and the Planck Length would still be about 10^-35. same with time and mass,
> but what about charge???
And you would also realize that you can say the same sentence with the
charge, too. Your problem is that you omitted the units. The Planck length
is not 10^-35. The Planck length is 10^-35 meters. And a meter is a random
consequence of history; a practical unit in everyday life but a silly unit
without any depth; 10^35 Planck lengths, in a more fundamental language.
You can play the same game with Coulombs instead of meters and the result
is similar; you should remember that you are redefining meters and
Coulombs, not Planck length etc. Planck length is equal to one in natural
units and cannot be redefined.
However maybe you did not want to talk about Coulombs but the two
different conventions for the fundamental unit of charge. Well, if you
changed the number 1/137.036 (contained in the ratio of your two
"fundamental" units) to something else, the world would certainly change
dramatically! In fact, life would be killed if you changed the number by
less than one per cent.
The fine structure constant can be seen at many places. For example, the
(squared) speed of electrons in the hydrogen atom is roughly 1/137 of the
(squared) speed of light. As a consequence of this, the spectrum of the
hydrogen atoms have the famous lines with energies 1/n^2, but if you look
at the lines with a better resolution, you find out that they are
separated to several sublines; they form the so-called fine structure of
the Hydrogen spectrum. The distance between the main lines of the spectrum
is - if I simplify - 137 times bigger than the distance between the lines
in the fine structure; therefore the name. If 137 was replaced by 10, the
spectrum would look completely different, most known nuclei would decay
radioactively (because proton repel each other electromagnetically and
this force would be stronger than the extra "chromostatic" attraction
between quarks - in our world, the electromagnetism is weaker and the
attraction by gluons wins). Simply, it would be a different world. A
couple of dimensionful numbers can be changed without changing the world
(at most, their number can equal to the number of independent units); it
just corresponds to redefining your units. However you cannot change
dimensionless numbers. One is always one (for example, it is equal to its
square) and cannot be redefined to be three. On the contrary, there are
three definitions of quarks and leptons and you cannot redefine this
number to five. :-)
Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
robert bristow-johnson
> Dear Robert,
>
> comp.dsp people have a guru who is not silly.
Cf:
http://groups.google.com/groups?threadm=3695AA17.2F32%40viconet.com
http://groups.google.com/groups?threadm=36A5935D.136E%40viconet.com
they were mad at me for saying that if the dependent variable of the
dirac-delta function must be dimensionally in reciprocal units of the
independent variable since the integral must be 1 (dimensionless).
> It is fine that you could
> re-discover Planck units. By the way, in Planck units - where you put
> epsilon0=mu0=1, speaking in SI units -, the charge of the electron is
> really equal to -sqrt(4.pi/137.036...) just like you say.
actually, i would put mu0 = 4*pi and epsilon0 = 1/(4*pi) so that the
simple Coulomb force equation has a constant = 1. same for the
gravitational force equation.
> Physicists are however convenient and they usually express the charge in
> units of (minus) the charge of the electron so that it is always integer
> (or integer over three, for quarks).
which is one reason (caveat: i don't really know diddley about quarks)
because of this e/3 charge thing that i didn't like about setting the
unit charge to be e.
> You cannot say that one choice of
> units is objectively more fundamental; it is a matter of taste. Both are
> certainly more fundamental than using Coulombs. However it depends on your
> feelings. Anyway you can see that you cannot get rid of the number
> 1/137.036..., the fine structure constant.
of course not. it's just a matter about where one wishes to see it.
> It is a dimensionless number
> without any units. Therefore it does not depend on our choice of the
> units. And there should be some explanation for its value!
we would wish so, but i hadn't really had any hope about it (how about
1/(exp(0.5*pi^2) - 2) ? - off by 70 ppm). anyway, my curiousity comes
from (being a layman) just hearing that alpha has been measured to
have changed by about 10 ppm (out of 4 ppbillion uncertainty) in the
12 billion years it took for the big-bang background radiation to
befall us. and i'm wondering (if there were a much larger change in
alpha) how that would be noticed. as a change in e? or a change in
epsilon0 and z0? or something else? or is it moot?
> We understand this number in terms of more fundamental constants of the
> electroweak theory (g and g'), measured at higher energies (instead of
> zero energies - as alpha), but a complete calculation yielding
> 1/137.036... is still missing. String theory is believed to be capable to
> derive its value one day.
that'll be interesting.
<snippage of which i understood maybe 1/2)
> > Now, finally, it seems that we must perceive reality in terms of the Planck
> > Units (T, L, M) and perhaps the unit charge Q. if the Planck Length went
> > from 10^-35 to 10^-32, then we would be 2000 meters tall instead of 2 but
>
> It is correct that you invite us to perceive reality in Planck units,
i meant it more as an (naive) observation rather than an invitation.
> but
> you do not do it yourself. If you did so, you would rather say: if one
> meter was defined not as 10^35 Planck lengths but only 10^32 Planck
> lengths, then we would be 2000 meters tall instead of 2 (anyway, two is
> also too much) :-) - because everyone knows that a human being must be
> about 2.10^35 Planck lengths tall in order to have the right number of
> atoms.
well, i tried to say it as such. however our height not only depends
on the number of atoms but their size and the Rydberg constant (or
more precisely, its reciprocal), which depends on e, seems to have
something to say about that.
i would normally think of it as this: we perceive reality (for me it's
just 3D space and time) in terms of, or relative to, the speed of
light, the gravitational constant, Planck's constant, and perhaps the
charge of the electron. so, it seems to me that we cannot really
perceive a change in the speed of light because our sense of length
and time is relative to that. that's why i've always thought that
those "thought experiments" asking "what if the speed of light was 30
miles per hour? what would life be like?" are similar to asking how
many angels dance on the head of a pin.
anyway, if our perception of reality *is* in terms of c, G, and hBar,
then our perception of length, time, and mass must be in terms of the
Planck Units which is a natural reason to use them for theoretical
thinking.
for my "taste", the charge of an electron becomes more secondary being
that it is more of an "object" in the universe and not a parameter of
the universe itself. it seems more logical or "natural" to first
observe the nature of forces of the universe on objects in general,
select appropriate units that would normalize the constants of
proportionality (of the simplest, most basic equations) to one, and
then secondly start looking at some objects (such as atoms and
sub-atomic particle). we sorta do that with Newton's 2nd law: we
don't say that Force is proportional to mass times acceleration
(although it is for |v| << c), we choose our unit of force so that
force *is* mass times acceleration. i would do this for charge also
so that:
E = m (not m * c^2 since we're normalizing c = 1)
E = omega (not hBar*omega for the same reason)
F = m1*m2 / r^2 (not G*m1*m2 / r^2)
and
F = q1*q2 / r^2 (not k*q1*q2 / r^2)
to satisfy the first three, you need to measure length, time, and mass
in units of Planck. to satisfy the fourth in addition, you need to
measure charge in units of e/sqrt(alpha), not e.
> > our meter stick would be 1000 meters and we would still call it a "meter"
> > and the Planck Length would still be about 10^-35. same with time and mass,
> > but what about charge???
>
> And you would also realize that you can say the same sentence with the
> charge, too. Your problem is that you omitted the units. The Planck length
> is not 10^-35. The Planck length is 10^-35 meters. And a meter is a random
> consequence of history; a practical unit in everyday life but a silly unit
> without any depth; 10^35 Planck lengths, in a more fundamental language.
> You can play the same game with Coulombs instead of meters and the result
> is similar;
yes. and the unit charge is not e but would be e/sqrt(alpha),
correct?
> you should remember that you are redefining meters and
> Coulombs, not Planck length etc. Planck length is equal to one in natural
> units and cannot be redefined.
agreed! it just seems to me that it is not consistent to call the
"Planck charge" (i dunno if the term is really used in your biz) e.
it seems much more consistent to me to call the Planck charge such a
charge that (this is hypothetically since the distances are wildly
small, even for electrons) when two such charges are placed one Planck
length apart, you get one Planck unit of force.
you do that definition first, *then* you do some kind of Miliken
experiment and observe that the charge of an electron appears to be
sqrt(alpha) times your unit charge.
> However maybe you did not want to talk about Coulombs but the two
> different conventions for the fundamental unit of charge. Well, if you
> changed the number 1/137.036 (contained in the ratio of your two
> "fundamental" units) to something else, the world would certainly change
> dramatically! In fact, life would be killed if you changed the number by
> less than one per cent.
well, given the present trend, we have about 12 trillion years left
before life is killed off due to alpha getting "out of bounds".
> The fine structure constant can be seen at many places. For example, the
> (squared) speed of electrons in the hydrogen atom is roughly 1/137 of the
> (squared) speed of light. As a consequence of this, the spectrum of the
> hydrogen atoms have the famous lines with energies 1/n^2, but if you look
> at the lines with a better resolution, you find out that they are
> separated to several sublines; they form the so-called fine structure of
> the Hydrogen spectrum. The distance between the main lines of the spectrum
> is - if I simplify - 137 times bigger than the distance between the lines
> in the fine structure; therefore the name. If 137 was replaced by 10, the
> spectrum would look completely different, most known nuclei would decay
> radioactively (because proton repel each other electromagnetically and
> this force would be stronger than the extra "chromostatic" attraction
> between quarks - in our world, the electromagnetism is weaker and the
> attraction by gluons wins). Simply, it would be a different world.
that i understand. how much different would the world be if alpha
quickly changed by another 10 ppm? BTW, which way did it change in
the last 12 billion years? did it increase or decrease by 10 ppm?
> A
> couple of dimensionful numbers can be changed without changing the world
> (at most, their number can equal to the number of independent units); it
> just corresponds to redefining your units. However you cannot change
> dimensionless numbers.
*that* i understand! (at least that you cannot change dimensionless
numbers by very much without adverse consequences.)
thanks for the response, Lubos.
...
> Superstring/M-theory is the language in which God wrote the world.
some might say that instead "Superstring/M-theory is a language
construct of humankind to try to verbalize and understand what God
was/is doing when He wrote the world." kinda like reading
hieroglyphics. you never know, maybe in 200 years, they'll toss it on
the trash heap with Newton's Laws.
:-/
r b-j
Lubos Motl
> they were mad at me for saying that if the dependent variable of the
> dirac-delta function must be dimensionally in reciprocal units of the
> independent variable since the integral must be 1 (dimensionless).
And you were right. And if I have come here 3 years ago, there would be
two of us attacked by the people who don't know what is the dimension of
the delta function.
Delta(momentum) has units of 1/momentum because it can be written as the
derivative d stepfunction(momentum) / d momentum. Here, the
stepfunction is 0 or 1, so it is dimensionless, and therefore the only
dimension comes from the momentum in the denominator. Or just like you
say, the integral must be one. Delta is a distribution and this kind of
"function" has always the dimension of 1 / thedimension of its parameter.
> actually, i would put mu0 = 4*pi and epsilon0 = 1/(4*pi) so that the
> simple Coulomb force equation has a constant = 1. same for the
> gravitational force equation.
Right, I prefer to put epsilon0=1 as SI suggests but this difference is a
psychological one. Your convention with epsilon=1/4.pi corresponds to the
Gaussian units (CGS, centimeter-gram-second) in fact.
> which is one reason (caveat: i don't really know diddley about quarks)
> because of this e/3 charge thing that i didn't like about setting the
> unit charge to be e.
Quarks were known much later than the charge of the electron was called
"-e". ;-) Some string theory models admit even more exotic fractions such
as e/11 etc. (see the Chapter 9 of The Elegant Universe) but "e" is the
minimal unit of something that can exist freely and does not require too
huge energies. There would be a lot of mess if we suddenly decided that
the sign "e" should be replaced by "3e" in all the textbooks.
> of course not. it's just a matter about where one wishes to see it.
Exactly.
> we would wish so, but i hadn't really had any hope about it (how about
> 1/(exp(0.5*pi^2) - 2) ? - off by 70 ppm). anyway, my curiousity comes
Great formula. Much better than other people suggested even in their
papers submitted to xxx.lanl.gov. Unfortunately, your formula is most
likely wrong. :-)
> from (being a layman) just hearing that alpha has been measured to
> have changed by about 10 ppm (out of 4 ppbillion uncertainty) in the
> 12 billion years it took for the big-bang background radiation to
> befall us. and i'm wondering (if there were a much larger change in
> alpha) how that would be noticed. as a change in e? or a change in
> epsilon0 and z0? or something else? or is it moot?
Physically, you cannot note the change of the value of letter unless you
precisely define what they mean. Physically you can however measure
frequencies of the spectral lines of Hydrogen (the rainbow coming from the
Hydrogen contains some "lines", discrete strips of color). And the
distance between two lines in the fine structure is say 137 times smaller
than the big distance between two specific lines. So this ratio would
change. There would be very many things that would change. If it was more
than 1 ppm or so, you would certainly notice.
> well, i tried to say it as such. however our height not only depends
> on the number of atoms but their size and the Rydberg constant (or
> more precisely, its reciprocal), which depends on e, seems to have
> something to say about that.
It depends what constants you want to start with. If you start with the
radius of the atom, maybe you can use biological arguments to say that
beings as smart as we are :-) should better be 10 billion atoms tall.
Therefore a natural unit that they want to choose to measure things will
be 10 billion Angstroms i.e. 1 meter. And because of some relation between
the size of the atom and Planck length, you can also say that we are led
to use units 10^35 Planck lengths (we called this one "one meter",
approximately).
> charge of the electron. so, it seems to me that we cannot really
> perceive a change in the speed of light because our sense of length
> and time is relative to that.
Exactly! This is the correct viewpoint that I was just explaining to
someone else. In fact, the SI units directly reflect this approach. 1
meter is currently defined as 1/299 792 458 light seconds. So if you keep
your definition, the Universe can change in any way but the speed of light
is always fixed. A change of the speed of light is just a change of our
definitions and it is useful to keep it fixed because it implies a
relation between space and time which is so important, because of
relativity.
> that's why i've always thought that
> those "thought experiments" asking "what if the speed of light was 30
> miles per hour? what would life be like?" are similar to asking how
> many angels dance on the head of a pin.
Yes, this question physically means just "how it would look like if we
could moved by speeds c/5 or so".
> anyway, if our perception of reality *is* in terms of c, G, and hBar,
> then our perception of length, time, and mass must be in terms of the
> Planck Units which is a natural reason to use them for theoretical
> thinking.
Exactly. However a necessary amount of blood for a hospital is at least a
gallon and therefore we don't use Planck volumes to measure volume, for
instance. Maybe we will use them sometimes...
> E = m (not m * c^2 since we're normalizing c = 1)
> E = omega (not hBar*omega for the same reason)
Right.
> F = m1*m2 / r^2 (not G*m1*m2 / r^2)
> F = q1*q2 / r^2 (not k*q1*q2 / r^2)
Here I would put the usual 4.pi into the denominator as in SI units. The
reason is that 4.pi.r^2 is the surface of sphere - and the electric field
is kind of uniformly distributed over the sphere around your charge. If
you accept my SI conventions with 4.pi, the Maxwell equations (which are
more fundamental, I think) do not have any 4.pi's in them - with your
convention you need to put some 4.pi's into Maxwell's equations. In the
previous case of gravity, we should do the same (with those 4.pi), but in
fact we still use Newton's convention for his constant. A better
denominator could be perhaps 8.pi here. Sometimes a "gravitational
constant" differs from the "Newton's constant" by a factor of 8.pi. All of
this is just a convention.
> yes. and the unit charge is not e but would be e/sqrt(alpha),
> correct?
Correct - up to this convention of 4.pi - I would probably prefer to say
that the unit charge is e/sqrt(4.pi.alpha). Today we say that your
conventions with 4.pi are "not rationalized". :-)
> you do that definition first, *then* you do some kind of Miliken
> experiment and observe that the charge of an electron appears to be
> sqrt(alpha) times your unit charge.
Right, I would say that this is precisely how Millikan did it except for
the various powers of hbar and c that he used everywhere around (and two
of us set them equal to one). His result was something like sqrt(alpha)
times some powers of hbar and c, even without those 4.pi because he was
using your, old conventions.
> well, given the present trend, we have about 12 trillion years left
> before life is killed off due to alpha getting "out of bounds".
Well, and you did not know that we had exactly 1 day before something like
the World War III starts.
I hope that all of you and your families and friends are doing fine and
that the attacks on Tuesday (the day when I defended my thesis) will make
us stronger, not weaker.
God bless you
Lubos
P.S. I am not sure whether the experiments "showing" changing value of
alpha are reliable enough (they contradict some estimates derived from
successes of our theory of the primordial nucleosynthesis) and I don't
know which direction it goes. Sorry that I did not reply to everything.
______________________________________________________________________________
E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
J. J. Lodder
> Right, I prefer to put epsilon0=1 as SI suggests but this difference is a
> psychological one. Your convention with epsilon=1/4.pi corresponds to the
> Gaussian units (CGS, centimeter-gram-second) in fact.
With rationalised units it is a matter of convention
whether or not to take the 4pi as part of the equation,
or to incorporate it in epsilon_0 or mu_0.
MKSA chooses the second option.
In Heaviside-Lorentz or rationalized natural units
it seems best to write Coulomb's and Ampere's law
with an explicit 4pi in it
-and- say that you have units with epsilon_0 = 1,
if you would be crazy enough to worry about
what epsilon_0 should be in such a unit system.
To be discouraged, IMHO:
Saying that eps_0 equals 4pi in such unit systems.
The eps_0 and mu_0 are artefacts of the MKSA system,
without any physical meaning or interpretation,
and nobody would even think about introducing them
if a more sensible unit system had been chosen, long ago,
without them.
But indeed, conventions only,
Jan
J. J. Lodder
> On 10 Sep 2001, 1 day before the attacks, robert bristow-johnson wrote:
> > they were mad at me for saying that if the dependent variable of the
> > dirac-delta function must be dimensionally in reciprocal units of the
> > independent variable since the integral must be 1 (dimensionless).
>
> And you were right. And if I have come here 3 years ago, there would be
> two of us attacked by the people who don't know what is the dimension of
> the delta function.
The simplest way to see that is to note that delta(x) must be
homogeneous of degree -1 under scale transformations:
delta(ax) = 1/a delta(x),
since integrals involving a delta function should be scale invariant.
Best,
Jan
Thomas Larsson
in message <news:Pine.SOL.4.10.101091...@physsun9.rutgers.edu>...
> On 10 Sep 2001, 1 day before the attacks, robert bristow-johnson wrote:
> > we would wish so, but i hadn't really had any hope about it (how about
> > 1/(exp(0.5*pi^2) - 2) ? - off by 70 ppm). anyway, my curiousity comes
>
> Great formula. Much better than other people suggested even in their
> papers submitted to xxx.lanl.gov. Unfortunately, your formula is most
> likely wrong. :-)
Hello Lubos, and congratulations to your Ph.D.
Robert's formula for the inverse fine structure constant gives
exp(0.5*pi^2) - 2 = 137.0456367
The best formula I have seen on the Arxiv is
4 pi^3 + pi^2 + pi = 137.0363038
The author claimed that the three terms should have something to do with
the groups SU(3), SU(2) and U(1), but I didn't really understand how.
The best experimental data that I found (from a 20 year old PPDB) is
137.03604(11)
Toby Bartels
>actually, i would put mu0 = 4*pi and epsilon0 = 1/(4*pi) so that the
>simple Coulomb force equation has a constant = 1. same for the
>gravitational force equation.
This convention is called "unrationalised";
I (and apparently Lubos) prefer "rationalised".
The reason is that I find the Maxwell equations
more fundamental than the Coulomb equation.
The debate between rationalised and unrationalised never ends.
I am even more radical than most proponents of rationalisation
in that I also rationalise the constant of gravitation.
Since 8 pi G, rather than G itself, appears in
the Einstein equations of general relativity,
I like to set 8 pi G to 1 rather than G,
which makes my Planck units off from others'
by a factor of about 5.
>we sorta do that with Newton's 2nd law: we
>don't say that Force is proportional to mass times acceleration
>(although it is for |v| << c), we choose our unit of force so that
>force *is* mass times acceleration.
Exactly. This example is a good one to use
when trying to explain Planck units to people.
If you want a good *historical* example,
use our modern measurement of heat with energy units.
Once upon a time, people measured heat in different units,
so we had an extra fundamental constant of nature, 4.184 J/cal.
This reminds me that there is another quantity
that can be measured in energy units but usually isn't:
temperature. The fundamental constant of nature here
is Boltzmann's constant k_B, about 1.381e(-23) J/K.
If you set Boltzmann's constant to 1,
then entropy becomes dimensionless
and you can see that it can measure
the dimensionless quantity of information.
In fact, let's be as systematic as possible about this.
In the International System of units (SI),
there are 7 allegedly physical units,
so we need to set 7 fundamental constants to 1
in order to make everything dimensionless.
Actually, 1 unit, the candela, is not a physical unit at all
but a physiological unit for apparent brightness to a human eye.
So the 6 physical units are the metre, the second,
the kilogramme, the ampere, the kelvin, and the mole.
The 6 constants are c, hbar, G (I prefer 8 pi G),
epsilon_0 (you prefer 4 pi epsilon_0), k_B, and N_A.
(N_A is Avogadro's number, about 6.022e23/mol.
Setting it to 1 just says that a mole is about 6.022e23,
which comes as naturally as saying that a dozen is 12.)
Now everything is unitless.
>>Superstring/M-theory is the language in which God wrote the world.
>some might say that instead "Superstring/M-theory is a language
>construct of humankind to try to verbalize and understand what God
>was/is doing when He wrote the world." kinda like reading
>hieroglyphics. you never know, maybe in 200 years, they'll toss it on
>the trash heap with Newton's Laws.
Nope. String theory is the one final theory of physics that explains
every possible phenomenon -- or at least will once we've finished it.
If we ever find something that contradicts string theory,
then that will only prove that what we've found does not exist.
^_^ ^_^ ^_^
(This is just me teasing Lubos; you can ignore it, robert.)
-- Toby
to...@math.ucr.edu
Toby Bartels
>The best experimental data that I found (from a 20 year old PPDB) is
>137.03604(11)
20 years old? I can beat that!
137.0359895+-61
This is from Brookhaven Natl Lab's Nuclear Wallet Cards, 1995 Jul.
But really it's not that good, since they reveal in the fine print
that their table of fundamental constants is simply swiped from
the 1986 Nov CODATA Bulletin, published 9 long years earlier.
So surely somebody else here can do better!!!
-- Toby
to...@math.ucr.edu
J. J. Lodder
> This convention is called "unrationalised";
> I (and apparently Lubos) prefer "rationalised".
> The reason is that I find the Maxwell equations
> more fundamental than the Coulomb equation.
> The debate between rationalised and unrationalised never ends.
>
> I am even more radical than most proponents of rationalisation
> in that I also rationalise the constant of gravitation.
> Since 8 pi G, rather than G itself, appears in
> the Einstein equations of general relativity,
> I like to set 8 pi G to 1 rather than G,
> which makes my Planck units off from others'
> by a factor of about 5.
No need to do that: Planck units are defined up to a proportionality
constant anyway, (it is only dimensional analysis)
so you can mess up the Einstein equation
without changing the Planck units.
There is no end to the confusion you can produce,
once you start meddling.
(I asked John, sometime ago, on precisely this point,
whether he wanted to mess up Planck also, or only Einstein.
No answer, if I remember correctly)
My preference: don't change the established -numerical- values
of the Planck length/time/mass/etc/, change the definitions,
if you feel you must change at all.
Best,
Jan
Toby Bartels
>Toby Bartels wrote:
>>I like to set 8 pi G to 1 rather than G,
>>which makes my Planck units off from others'
>>by a factor of about 5.
>My preference: don't change the established -numerical- values
>of the Planck length/time/mass/etc/, change the definitions,
>if you feel you must change at all.
Fair enough. Say this:
<<I like to set 8 pi G to 1 rather than G,
<<which makes my natural units off from the Planck units
<<by a factor of about 5.
In real life, I wouldn't confuse anybody
by introducing my natural units as "Planck units"
unless we were only talking about order of magnitude.
-- Toby
to...@math.ucr.edu
Paul Arendt
J. J. Lodder <j...@de-ster.demon.nl> wrote:
>The eps_0 and mu_0 are artefacts of the MKSA system,
>without any physical meaning or interpretation,
>and nobody would even think about introducing them
>if a more sensible unit system had been chosen, long ago,
>without them.
Well, I agree that these things can be set to "1" if you
choose your units of charges (and fields) appropriately. But
I don't agree that there is no physical meaning or
interpretation to them!
For instance, eps_0 is more than a pure constant. It's the
relationship between D and E in Maxwell's equations, in free
space. A convention may be chosen so that D = E in free
space, but D and E are still very different entities!
The difference between them is that E is used to find the
force on a charged nonmoving test particle, while D is used
to integrate over a region's boundary to find the total charge
contained within that region (Gauss' law). In other words,
E measures the effect of the field on charged particles, while D
measures the effect of charged particles (as sources of the field).
E tells what charges will do, while D tells where the charges are!
One might argue that these should equal each other by some sort
of action=reaction Newtonian argument (conservation of momentum).
But the fact remains that geometrically, they represent different
entities! This is easiest to understand if one represents E and D
by differential forms. (Eric Forgy just got extremely interested,
didn't he? :-) )
As a differential form in 3-space, E is a one-form. The physical
interpretation is that it represents the differential contribution
to a charged particle's energy (per unit charge) when the particle
moves across the surfaces of the one-form. (For electrostatic
configurations, this one-form field is integrable, so that you
may define E = - grad Phi everywhere.) When the one-form E is
integrated over a line in 3-space, the result is the change in
energy (per unit charge) on a charged particle which has moved
along that line.
As a differential form in 3-space, D is a two-form! The
interpretation is that the integral of D over a closed region
equals the charge contained in that region, which is Gauss' law.
So, as forms, D and E are related by the Hodge star operator, even
when the units of the vectors #(*D) and #E are chosen to be equal to
one another. (Notation: #(one-form) is the vector obtained by
applying the inverse metric tensor to the one-form, and * is the
Hodge star.)
Similar remarks apply when D and E are forms in (3+1)-D spacetime:
they are both 2-forms, but E is a space-time 2-form, while D is a
space-space 2-form.
J. J. Lodder
> In article <1ezuonc.1cv...@de-ster.demon.nl>,
> J. J. Lodder <j...@de-ster.demon.nl> wrote:
>
> >The eps_0 and mu_0 are artefacts of the MKSA system,
> >without any physical meaning or interpretation,
> >and nobody would even think about introducing them
> >if a more sensible unit system had been chosen, long ago,
> >without them.
>
> Well, I agree that these things can be set to "1" if you
> choose your units of charges (and fields) appropriately. But
> I don't agree that there is no physical meaning or
> interpretation to them!
>
> For instance, eps_0 is more than a pure constant. It's the
> relationship between D and E in Maxwell's equations, in free
> space. A convention may be chosen so that D = E in free
> space, but D and E are still very different entities!
What -physical- experiment would you propose to demonstrate
a physical (as opposed to conceptual) difference between E and D,
in vacuum?
> The difference between them is that E is used to find the
> force on a charged nonmoving test particle, while D is used
> to integrate over a region's boundary to find the total charge
> contained within that region (Gauss' law). In other words,
> E measures the effect of the field on charged particles, while D
> measures the effect of charged particles (as sources of the field).
> E tells what charges will do, while D tells where the charges are!
Indeed, this is a -conceptual- distinction only:
in vacuum there is only one field,
which you may -call- either E or D.
> One might argue that these should equal each other by some sort
> of action=reaction Newtonian argument (conservation of momentum).
> But the fact remains that geometrically, they represent different
> entities! This is easiest to understand if one represents E and D
> by differential forms. (Eric Forgy just got extremely interested,
> didn't he? :-) )
I would not argue anything of the kind.
Instead I would say that there is only one field E,
and that only the Maxwell egns in vacuum are fundamental.
The Maxwell eqns in matter, with D in them, are approximate eqns,
to be derived from the fundamental eqns in vacuum
by appropriate statistical mechanics.
snip forms, not that they aren't nice, but nice formalism
cannot substitute for physical desciption.
Best,
Jan
zirkus
news:<9onshm$bl2$1...@glue.ucr.edu>...
> 20 years old? I can beat that!
>
> 137.0359895+-61
>
> This is from Brookhaven Natl Lab's Nuclear Wallet Cards, 1995 Jul.
>
> But really it's not that good, since they reveal in the fine print
> that their table of fundamental constants is simply swiped from
> the 1986 Nov CODATA Bulletin, published 9 long years earlier.
>
> So surely somebody else here can do better!!!
There is an astronomy paper [1] which shows that the fine
structure constant used to be about one part in 10^5 less than it is
now! Hopefully, some team
can verify or refute this evidence because this surprising result has
importance for e.g. variable-
speed-of-light (VSL) cosmology.
[1] http://arxiv.org/abs/astro-ph/0012539
------------
Toby Bartels
>Instead I would say that there is only one field E,
>and that only the Maxwell egns in vacuum are fundamental.
>The Maxwell eqns in matter, with D in them, are approximate eqns,
>to be derived from the fundamental eqns in vacuum
>by appropriate statistical mechanics.
Well, I would argue that the vacuum equations aren't fundamental either.
They are merely a classical approximation to QED.
And even that is merely an approximation to the GWS electroweak theory.
And even that is merely an approximation;
even if there is no grand unification of electroweak and strong forces,
still the appearance of the metric tensor in the GWS theory
must be modified by a theory of quantum gravity.
The value of a physical theory isn't determined by its fundamentalness.
The Maxwell equations in matter, where E and D are different,
are quite useful and accurate across a broad range of phenomena.
You need a relationship between E and D given by the type of matter;
for many types, this can be given by a single constant epsilon.
The value of epsilon in vacuum is epsilon_0 = 1, in appropriate units.
Good, it should be!
-- Toby
to...@math.ucr.edu
Paul Arendt
>
>> For instance, eps_0 is more than a pure constant. It's the
>> relationship between D and E in Maxwell's equations, in free
>> space. A convention may be chosen so that D = E in free
>> space, but D and E are still very different entities!
>
>What -physical- experiment would you propose to demonstrate
>a physical (as opposed to conceptual) difference between E and D,
>in vacuum?
Well, from what I wrote below:
>> The difference between them is that E is used to find the
>> force on a charged nonmoving test particle, while D is used
>> to integrate over a region's boundary to find the total charge
>> contained within that region (Gauss' law). In other words,
>> E measures the effect of the field on charged particles, while D
>> measures the effect of charged particles (as sources of the field).
>> E tells what charges will do, while D tells where the charges are!
...you can imagine the following experiment: measure the force on
small charged particles (whose charge is as small as possible),
and divide by the charge of the particle. This gives you the
electric field E at a point. Continue this over the entire (2-D)
boundary of a (3-D) volume, and you have defined E everywhere on
the boundary.
Now, take small coiled loops of paramagnetic material, and measure
(somehow!) the total induction H integrated along the loops as you
rotate the loops (at constant angular velocity) around at the
same points where you measured E. This gives you the flux of
D through the loops. Repeat this over the same boundary that
was done for the E field, and you will have the flux of D through
the boundary.
Gauss' Law now says that this total flux of D equals the charge
contained within the volume (whose boundary was the region D and
E were measured over).
You may choose a unit system in which D = E in vacuum in Euclidean
space. Suppose that we do this. If the experiment above has
been performed in Euclidean space, then the total flux of E through
the boundary will also give the charge contained in the volume.
But in a curved space, the flux of E though the surface will *not*
generally give the charge enclosed, unless it happens to be 0.
So, we can conclude that E cannot equal D at every point on that
surface.
You may instead try to adjust the *electric charge* so that the "E-charge"
(giving the force on a particle) is different from the "D-charge" (to be
used in Gauss' Law) in curved spaces, but D and E are defined to be
the same things. However, to be fair, you should also start having
the "D-charge" change in dielectric media too, if that is the route
taken.
Eric Alan Forgy
It's nice to see people pointing out that E and D are two different
things :) I saw a comment in some other thread where it was written E
= D, and I was tempted to pipe in, but now I can't resist :)
I know you know this, and what you said was precisely that E is a
1-form and D is a 2-form. The relation between them involves the
space(time) metric. I personally think it is misguided (and
misleading) to write E = D ever! But I'm probably more passionate
about this than most people because I work in applied electromagnetics
:)
Eric
"Paul Arendt" <par...@black.nmt.edu> wrote:
> J. J. Lodder <j...@de-ster.demon.nl> wrote:
> >Paul Arendt <par...@black.nmt.edu> wrote:
[snip]
J. J. Lodder
> J. J. Lodder wrote:
>
> >Instead I would say that there is only one field E,
> >and that only the Maxwell egns in vacuum are fundamental.
> >The Maxwell eqns in matter, with D in them, are approximate eqns,
> >to be derived from the fundamental eqns in vacuum
> >by appropriate statistical mechanics.
>
> Well, I would argue that the vacuum equations aren't fundamental either.
> They are merely a classical approximation to QED.
[snip more irrelevantia]
Sure, but why drag in these irrelevantia?
The physical meaning of eps_0 and mu_0, if any,
is an issue on the pre-1900 level.
It could, and should, have been settled then, once and for all,
by following Heaviside and Lorentz' proposals
for a sensible EM unit system.
If that had been done then you now would not even have known
that it is actually possible to introduce these notions,
unless you had happened to study history of science.
Best,
Jan
--
"The electrical intensity is given in square root psi" (Thomson)
Toby Bartels
>Toby Bartels wrote:
>>J. J. Lodder wrote:
>>>Instead I would say that there is only one field E,
>>>and that only the Maxwell egns in vacuum are fundamental.
>>>The Maxwell eqns in matter, with D in them, are approximate eqns,
>>>to be derived from the fundamental eqns in vacuum
>>>by appropriate statistical mechanics.
>>Well, I would argue that the vacuum equations aren't fundamental either.
>>They are merely a classical approximation to QED.
>Sure, but why drag in these irrelevantia?
>The physical meaning of eps_0 and mu_0, if any,
>is an issue on the pre-1900 level.
In the sense that this physical meaning could be understood before 1900, yes.
But the physical meaning remains, albeit a very simple meaning.
>It could, and should, have been settled then, once and for all,
>by following Heaviside and Lorentz' proposals
>for a sensible EM unit system.
I quite agree.
>If that had been done then you now would not even have known
>that it is actually possible to introduce these notions,
>unless you had happened to study history of science.
But now I disagree. Using Heaviside Lorentz units,
I would still have studied Maxwell's equations for dielectric media
and been introduced to the concept of epsilon and mu.
(These quantities would be dimensionless, of course.)
Then I would learn that epsilon and mu for the vacuum are both exactly 1.
How nice!
That the dielectric constant of the vacuum is 1
has as much physical meaning as that the speed of light there is 1.
The speed of light in vacuum may be a very trivial quantity in good units,
but it retains its physical meaning -- that is the speed that light travels.
-- Toby
to...@math.ucr.edu
J. J. Lodder
> It's nice to see people pointing out that E and D are two different
> things :) I saw a comment in some other thread where it was written E
> = D, and I was tempted to pipe in, but now I can't resist :)
>
> I know you know this, and what you said was precisely that E is a
> 1-form and D is a 2-form. The relation between them involves the
> space(time) metric. I personally think it is misguided (and
> misleading) to write E = D ever! But I'm probably more passionate
> about this than most people because I work in applied electromagnetics
Things which hold in one particular representation
of a theory may be helpful to some,
but they cannot have -physical- content.
They are different descriptions of the same thing.
Likewise, you would not claim that a particle
actually has two positions, x_\mu and x^\nu,
because a covariant vector is something entirely different
than a covariant one, mathematically speaking.
Best,
Jan
--
"Mathematicians are like Frenchmen:
They translate everything you say to them
immediately into their own language,
after which it is something entirely different" (Goethe)
J. J. Lodder
> In article <1f0p07l.poz...@de-ster.demon.nl>,
> J. J. Lodder <j...@de-ster.demon.nl> wrote:
> >Paul Arendt <par...@black.nmt.edu> wrote:
> >> For instance, eps_0 is more than a pure constant. It's the
> >> relationship between D and E in Maxwell's equations, in free
> >> space. A convention may be chosen so that D = E in free
> >> space, but D and E are still very different entities!
> >What -physical- experiment would you propose to demonstrate
> >a physical (as opposed to conceptual) difference between E and D,
> >in vacuum?
> ...you can imagine the following experiment: measure the force on
> small charged particles (whose charge is as small as possible),
> and divide by the charge of the particle. This gives you the
> electric field E at a point. Continue this over the entire (2-D)
> boundary of a (3-D) volume, and you have defined E everywhere on
> the boundary.
Sure, gives you E(r) for all r, in principle.
And that is all there is to know,
in an electrostatic situation.
> Now, take small coiled loops of paramagnetic material, and measure
> (somehow!) the total induction H integrated along the loops as you
> rotate the loops (at constant angular velocity) around at the
> same points where you measured E. This gives you the flux of
> D through the loops. Repeat this over the same boundary that
> was done for the E field, and you will have the flux of D through
> the boundary.
No need to introduce paramagnetic matter: a flip coil will do.
And: this measurement will not tell you anything new:
The results of any further experiments can be predicted
from E(r) measured above.
> Gauss' Law now says that this total flux of D equals the charge
> contained within the volume (whose boundary was the region D and
> E were measured over).
>
> You may choose a unit system in which D = E in vacuum in Euclidean
> space. Suppose that we do this. If the experiment above has
> been performed in Euclidean space, then the total flux of E through
> the boundary will also give the charge contained in the volume.
>
> But in a curved space, the flux of E though the surface will *not*
> generally give the charge enclosed, unless it happens to be 0.
> So, we can conclude that E cannot equal D at every point on that
> surface.
Let's not drag curved spaces into this discussion.
The emptiness of the argument can also be seen in Euclidean space,
by using a coordinate system with metric tensor not the identity.
Indeed, there are two ways then to calculate the charge in a given
volume: a correct and an incorrect one.
> You may instead try to adjust the *electric charge* so that the "E-charge"
> (giving the force on a particle) is different from the "D-charge" (to be
> used in Gauss' Law) in curved spaces, but D and E are defined to be
> the same things. However, to be fair, you should also start having
> the "D-charge" change in dielectric media too, if that is the route
> taken.
The 'cure', (two charges) would be worse
than the (non-existent :-) disease,
in my opinion.
Jan
Eric Alan Forgy
"J. J. Lodder" <nos...@de-ster.demon.nl> wrote:
> Eric Alan Forgy <fo...@uiuc.edu> wrote:
> >
> > I know you know this, and what you said was precisely that E is a
> > 1-form and D is a 2-form. The relation between them involves the
> > space(time) metric. I personally think it is misguided (and
> > misleading) to write E = D ever! But I'm probably more passionate
> > about this than most people because I work in applied electromagnetics
>
> Things which hold in one particular representation
> of a theory may be helpful to some,
> but they cannot have -physical- content.
> They are different descriptions of the same thing.
I agree 100%
> Likewise, you would not claim that a particle
> actually has two positions, x_\mu and x^\nu,
> because a covariant vector is something entirely different
> than a covariant one, mathematically speaking.
I agree 100%
Hmm... if I agree 100% with what you said, then why does it seem like
you were disagreeing with what I said? :) If you are really
disagreeing with what I said, would you mind spelling out a bit more
clearly in what way you disagree? I'd like to know. I'm supposed to be
an expert in EM, so if I am missing something basic, I'd like to know
about it :)
Maybe I'll explain what I mean more precisely so that if there is any
hole in my logic, it will be easier to spot. I'd say that given E, a
metric, and some information about the material properties, you can
find D. Conversely, given D, a metric, and some information about the
material properties, you can find E. So, essentially, given a metric
and some information about the material properies, then perhaps, in
some sense of the word, you can say E and D are the "same".
But then what about this scenario? Say we have a time-harmonic system
so that
d/dt -> i*w,
and we know J and H. Then we can find D simply by
D = [curl H - J]/(i*w).
With this D, a metric, and information about the material properties,
we can then find E. Conversely, if we know E, then we can similarly
find B by
B = [-curl E]/(i*w).
With this B, a metric, and information about the material properties,
we can then find H. So, under the prescribed scenario, given E we can
find H, and given H we can find E. Would you then say that E and H are
the "same"?
One of the reasons I am such a stickler about saying E and D are not
the same comes from experience with numerical solutions to Maxwell's
equations. E, being a 1-form, is naturally associated to the edges of
some mesh. D, being a 2-form, is naturally associated to the faces of
some mesh. To me, saying E and D are the same is like saying edges are
the same as faces :) In 3d, there is a nice way to associate edges and
faces. That is by constructing a dual mesh. For instance, if the mesh
is a simplicial complex, then you can construct the dual mesh in many
ways, e.g. a Poincare dual or a barycentric dual. Then, for every
p-simplex of the primary mesh, you have an (n-p)-cell of the dual
mesh, which is not simplicial. Still, I'd hesitate to say E and D were
the same because that would be like saying an edge is the same as a
dual cell. Sure, they are related, but I wouldn't call them the same.
The last paragraph was based on lattice arguments, but those arguments
do manifest themselves when you go to the continuum limit (if you
desired to do so... I'm personally of the opinion you should do away
with the continuum model of space-time altogether, but that is a
different story). I think it is a subtle, yet important, distinction
between E and D, but I think it should be made. (By the way, those
arguments are of relevance for spin foam models as well.)
Is there really a disagreement here, or is it a semantic issue about
the meaning of the word "same"? If it is the latter, then there is no
need to argue over it. You say tomato, I say tomato... c'est la vie :)
Cheers,
Eric
Gerard Westendorp
[..]
> Sure, but why drag in these irrelevantia?
> The physical meaning of eps_0 and mu_0, if any,
> is an issue on the pre-1900 level.
As you may remember from previous threads, I disagree with this.
Because eps_0, together with e, h and c fixes the fine-structure
constant, any change in the fine structure constant (I think it
changes at very high energies, like just after the big bang)
will impact either eps_0, e, h or c.
I would choose eps_0 to change, leaving the others as
fundamental constants that can be set to unity. But you could
also change e. (Although that would be on the pre-2000 level)
Gerard
Gerard Westendorp
[..]
> One of the reasons I am such a stickler about saying E and D are not
> the same comes from experience with numerical solutions to Maxwell's
> equations. E, being a 1-form, is naturally associated to the edges of
> some mesh. D, being a 2-form, is naturally associated to the faces of
> some mesh.
Doe that mean that D transforms as a pseudo-vector?
E is the force on a charge, so it is a vector (Newton/Coulomb).
D is the displacement of (imaginary) charges through a surface.
(Coulomb/m2). It could be a pseudo vector.
Gerard
J. J. Lodder
> J. J. Lodder wrote:
snip agree
> >If that had been done then you now would not even have known
> >that it is actually possible to introduce these notions,
> >unless you had happened to study history of science.
>
> But now I disagree. Using Heaviside Lorentz units,
> I would still have studied Maxwell's equations for dielectric media
> and been introduced to the concept of epsilon and mu.
> (These quantities would be dimensionless, of course.)
> Then I would learn that epsilon and mu for the vacuum are both exactly 1.
> How nice!
Indeed :-) But, being propery educated in this way
you would be less likely to make the mistake
of thinking of eps = 1
is a physical property of the vacuum.
> That the dielectric constant of the vacuum is 1
> has as much physical meaning as that the speed of light there is 1.
> The speed of light in vacuum may be a very trivial quantity in good units,
> but it retains its physical meaning -- that is the speed that light travels.
I guess this is the old confusion between
'fundamental speed in our universe' and 'speed of light' again.
The first can be taken to be 1,
and then it cannot be measured.
For light, it is of course necessary to establish
that it is actually massless,
which can in principle be done by verifying experimentally
that it travels one nanosecond in a nanosecond.
Best,
Jan
Toby Bartels
>Toby Bartels wrote:
>>But now I disagree. Using Heaviside Lorentz units,
>>I would still have studied Maxwell's equations for dielectric media
>>and been introduced to the concept of epsilon and mu.
>>(These quantities would be dimensionless, of course.)
>>Then I would learn that epsilon and mu for the vacuum are both exactly 1.
>>How nice!
>Indeed :-) But, being properly educated in this way
>you would be less likely to make the mistake
>of thinking of eps = 1 is a physical property of the vacuum.
Well, this *is* how I was educated, more or less --
I was originally taught in SI units, but I already knew how to
reduce the number of units by setting constants to 1,
so I immediately set eps_0 to 1 and thought about things in that way --
and I *do* think that eps = 1 is a physical property of the vacuum.
A property of a rather vacuous (ha! an unintentional pun!) sort,
but a property nonetheless.
Like saying cardinality = 0 is a mathematical property of the empty set.
Well, this analogy probably won't be so clear to people here,
but it really makes it click for me!
>>That the dielectric constant of the vacuum is 1
>>has as much physical meaning as that the speed of light there is 1.
>>The speed of light in vacuum may be a very trivial quantity in good units,
>>but it retains its physical meaning -- that is the speed that light travels.
>I guess this is the old confusion between
>'fundamental speed in our universe' and 'speed of light' again.
>The first can be taken to be 1,
>and then it cannot be measured.
When I say "speed of light", I mean the speed that light travels.
For the fundamental speed in the universe, I say "one".
>For light, it is of course necessary to establish
>that it is actually massless,
>which can in principle be done by verifying experimentally
>that it travels one nanosecond in a nanosecond.
Yes, and this is a physical fact.
Thus it is a physical property of the vacuum that
light travels there at the speed of 1,
just as it's a physical property of other materials that
light travels at certain speeds in them.
The speed of light in certain materials, as you know,
can be calculated as c = 1/sqrt(eps mu)[*].
Thus, c = 1/sqrt(1*1) = 1 in the vacuum.
[*]Or something like that.
-- Toby
to...@math.ucr.edu
Toby Bartels
>Because eps_0, together with e, h and c fixes the fine-structure
>constant, any change in the fine structure constant (I think it
>changes at very high energies, like just after the big bang)
>will impact either eps_0, e, h or c.
Yeah, specifically e ^_^.
>I would choose eps_0 to change, leaving the others as
>fundamental constants that can be set to unity. But you could
>also change e. (Although that would be on the pre-2000 level)
Should we reopen the discussion about which to change?
It seems patently obvious to me that you would change e,
having set eps_0 and c to 1, and h to 2 pi.
Planck, as we know, originally set e to 1 (and h to 1),
but Planck was not perfect.
-- Toby
to...@math.ucr.edu
Toby Bartels
>Eric Alan Forgy wrote:
Yeah, given a spacial metric, pseudovector and 2forms are equivalent.
Most people turn 2forms into vectors of course,
but this requires an orientation in addition to the metric.
I do find it more fundamental not even to assume the metric
and just to deal with the 2form itself -- in certain contexts.
-- Toby
to...@math.ucr.edu
Eric Forgy
"Gerard Westendorp" <wes...@xs4all.nl> wrote:
>
> Doe that mean that D transforms as a pseudo-vector?
> E is the force on a charge, so it is a vector (Newton/Coulomb).
> D is the displacement of (imaginary) charges through a surface.
> (Coulomb/m2). It could be a pseudo vector.
That is a really good question :) I am not an expert on pseudo vectors
because I usually think of them as artifacts of misinterpreting 2-forms (or
bivectors) as vectors. But if that is really true then it would be tempting
to think of D as a pseudo vector also, so there must be something else to
it. I don't think D is a pseudo vector because I have never heard of that
and I probably should have by now if it were (not a very scientific reason,
eh? :)). So let's try see why not (if not).
Let A be a 1-form in 4d space-time and let
F = dA.
This is a 2-form in space-time. As such, it is (4!)/(2!2!) = 6 dimensional
with three space-space dimensions and three space-time dimensions. That is
just big enough to accomodate 2 3d vectors. So, if you choose a reference
frame, which amount to choosing a time axis, you can decompose F into two
parts:
F = B + E/\dt.
(Note: This decomposition is quite arbitrary because dt is arbitrary.)
If you write out F in all its gory details it becomes:
F = (B_23 dx^23 + B_31 dx^31 + B_12 dx^12)
+ (E_1 dx^1 + E_2 dx^2 + E_3 dx^3)/\dt
where dx^ij = dx^i /\ dx^j. Under a parity transformation dx^i -> -dx^i, the
components of E change sign whereas the components of B do not change sign.
Thus, you can conclude that E is a vector and B is a pseudo vector. This
follows simply because B is a 2-form.
Now, the Hodge star # acts on the space-space basis elements as
#dx^23 = O(123)*dx^1/\dt
#dx^31 = O(123)*dx^2/\dt
#dx^12 = O(123)*dx^3/\dt
where O(123) is +/-1 and keeps track of the orientation. The Hodge star acts
on the space-time basis elements as
#(dx^1/\dt) = -O(123)*dx^23
#(dx^2/\dt) = -O(123)*dx^31
#(dx^3/\dt) = -O(123)*dx^12
Therefore
#F
= #B + #(E/\dt)
= O(123)*(B_23 dx^1 + B_31 dx^2 + B_12 dx^3)/\dt
-O(123)*(E_1 dx^23 + E_2 dx^31 + E_3 dx^12)
= H/\dt - D
= H_1 dx^1 + H_2 dx^2 + H_3 dx^3)/\dt
- (D_23 dx^23 + D_31 dx^31 + D_12 dx^12)
so that
H_1 = O(123)*B_23
H_2 = O(123)*B_31
H_3 = O(123)*B_12
and
D_23 = O(123)*E_1
D_31 = O(123)*E_2
D_12 = O(123)*E_3.
Ok, now the trick is that under the parity transformation
O(123) -> -O(123)
so you pick up one "-" for the basis elements of H, but then you pick up
ANOTHER "-" from the Hodge star. Then the overall sign of H is not changed
under the parity transformation so that H is a pseudo vector as well.
On the other hand, the sign of the basis elements of D do NOT change sign
under the parity transformation, but the components DO pick up a "-", so
overall, D picks up a "-" under the parity transformation. Hence D is a
vector as well (as it
should be based on my earlier unscientific reasoning :))
After all that mess, the short answer to the question "Is D a pseudo
vector?" is apparent. Although D is a 2-form, that does not mean that D is a
pseudo vector because D is the Hodge dual of a 1-form E and the HODGE STAR
PICKS UP AN ADDITIONAL SIGN UNDER A PARITY TRANSFORMATION (in this case, but
not in general).
Thanks for a nice question! It made me think. I hope my answer makes sense.
Eric
Eric Forgy
"Toby Bartels" <to...@math.ucr.edu> wrote:
> Gerard Westendorp wrote:
>
> >Doe that mean that D transforms as a pseudo-vector?
> >E is the force on a charge, so it is a vector (Newton/Coulomb).
> >D is the displacement of (imaginary) charges through a surface.
> >(Coulomb/m2). It could be a pseudo vector.
>
> Yeah, given a spacial metric, pseudovector and 2forms are equivalent.
> Most people turn 2forms into vectors of course,
> but this requires an orientation in addition to the metric.
> I do find it more fundamental not even to assume the metric
> and just to deal with the 2form itself -- in certain contexts.
Are you saying D DOES correspond to a pseudo vector? I just wrote a
long post declaring that D does NOT correspond to a pseudo vector, but
rather to a vector. Unless I made an error, the Hodge star causes a
sign reversal (in the case I was considering) under a parity
transformation. This additional sign due to the Hodge star makes the 2
form D correspond to a vector while the 1 form H corresponds to a
pseudo vector. This seemed to make perfect sense while I was writing
it :)
Eric
PS: The moral to this story is that VECTORS ARE EVIL!! :) Everyone
should start using differential forms and all this "pseudo" nonsense
will disappear once and for all :)
Gordon D. Pusch
In ``Applied Differential Geometry,'' [1] William Burke claims
that D is not an ``ordinary'' 2-form, but a ``twisted'' 2-form;
``twisted'' 2-forms apparently transform oppositely to ordinary
2-forms under parity. So apparently, Burke does not think one can
dismiss the orientation so cavalierly... (In fact, he appears to
explicitly represent the orientation by introducing two different
Hodge-operators into his 3+1 decompositions of forms: One for 3-D
forms, and one for 4-D forms.)
[1] <http://www.amazon.com/exec/obidos/ASIN/0521269296>
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Gerard Westendorp
[..]
> >I would choose eps_0 to change, leaving the others as
> >fundamental constants that can be set to unity. But you could
> >also change e. (Although that would be on the pre-2000 level)
>
> Should we reopen the discussion about which to change?
Probably not, because there is no objective way to decide
which makes more sense.
Gerard
J. J. Lodder
> "J. J. Lodder" wrote:
>
> [..]
>
> > Sure, but why drag in these irrelevantia?
> > The physical meaning of eps_0 and mu_0, if any,
> > is an issue on the pre-1900 level.
>
> As you may remember from previous threads, I disagree with this.
Yes I remember. Somebody should write a FAQ on this sometime :-)
> Because eps_0, together with e, h and c fixes the fine-structure
> constant, any change in the fine structure constant (I think it
> changes at very high energies, like just after the big bang)
> will impact either eps_0, e, h or c.
Perhaps it does occur in -your- finestructure constant,
it doesn't occur in mine.
This by itself is sufficient to show that the occurence of eps_0
in alpha is a matter of human convention,
rather than an aspect of nature.
> I would choose eps_0 to change, leaving the others as
> fundamental constants that can be set to unity. But you could
> also change e. (Although that would be on the pre-2000 level)
Of course you can,
(no need even to change mu_0 accordingly,
who cares about yet another arbitrary constant :-)
but it would merely imply
that you choose to describe nature
with a time-dependent unit system.
Possible, but rather inconvenient,
Jan
PS If you really want to create a mess of this kind
you could also define c to be a function of time,
by decreeing that the meter shall be the distance
that light travels in f(t) seconds :-)
J. J. Lodder
> Eric Alan Forgy wrote:
>
> [..]
> > One of the reasons I am such a stickler about saying E and D are not
> > the same comes from experience with numerical solutions to Maxwell's
> > equations. E, being a 1-form, is naturally associated to the edges of
> > some mesh. D, being a 2-form, is naturally associated to the faces of
> > some mesh.
>
> Doe that mean that D transforms as a pseudo-vector?
> E is the force on a charge, so it is a vector (Newton/Coulomb).
> D is the displacement of (imaginary) charges through a surface.
> (Coulomb/m2). It could be a pseudo vector.
This is a very typical example of the misunderstandings
that may arise from erroneous understanding of unit systems.
D is just a partial field,
which arises because we find it convenient
to split the total electric field -mentally- into parts.
Since D is a partial electric field,
it must transform as an electric field.
Jan
J. J. Lodder
Matter is composed of particles,
so on a fundamental level there is just particles
and (microscopic) electric fiels.
Now one can average these to obtain a macroscopic field.
(You may recognise Lorentz' program here)
It is often convenient to split the total macroscopic field
-mentally- into the field that would have been there
without the particles, and the rest.
Both components are electric fields,
even if you happen to prefer to measure one
in different units than the other.
Best,
Jan
J. J. Lodder
> J. J. Lodder wrote:
>
> >Toby Bartels wrote:
>
> >>But now I disagree. Using Heaviside Lorentz units,
> >>I would still have studied Maxwell's equations for dielectric media
> >>and been introduced to the concept of epsilon and mu.
> >>(These quantities would be dimensionless, of course.)
> >>Then I would learn that epsilon and mu for the vacuum are both exactly 1.
> >>How nice!
>
> >Indeed :-) But, being properly educated in this way
> >you would be less likely to make the mistake
> >of thinking of eps = 1 is a physical property of the vacuum.
>
> Well, this *is* how I was educated, more or less --
> I was originally taught in SI units, but I already knew how to
> reduce the number of units by setting constants to 1,
> so I immediately set eps_0 to 1 and thought about things in that way --
> and I *do* think that eps = 1 is a physical property of the vacuum.
> A property of a rather vacuous (ha! an unintentional pun!) sort,
> but a property nonetheless.
Guess we are down to semantics in the meantime.
You prefer to think of eps_0 = 1 as a property of the vacuum,
I cannot possibly see how a quantity which could just as well
have defined to be 37 can be a property of anything
but your particular definitions.
Best,
Jan
J. J. Lodder
snip
> After all that mess, the short answer to the question "Is D a pseudo
> vector?" is apparent. Although D is a 2-form, that does not mean that D is a
> pseudo vector because D is the Hodge dual of a 1-form E and the HODGE STAR
> PICKS UP AN ADDITIONAL SIGN UNDER A PARITY TRANSFORMATION (in this case, but
> not in general).
>
> Thanks for a nice question! It made me think. I hope my answer makes sense.
Of course!
It is an excessively beautiful way
to deduce an excessively trivial result.
Jan
Eric Forgy
"Gordon D. Pusch" <gdp...@NO.xnet.SPAM.com> wrote:
>
> In ``Applied Differential Geometry,'' [1] William Burke claims
> that D is not an ``ordinary'' 2-form, but a ``twisted'' 2-form;
> ``twisted'' 2-forms apparently transform oppositely to ordinary
> 2-forms under parity. So apparently, Burke does not think one can
> dismiss the orientation so cavalierly... (In fact, he appears to
> explicitly represent the orientation by introducing two different
> Hodge-operators into his 3+1 decompositions of forms: One for 3-D
> forms, and one for 4-D forms.)
>
> [1] <http://www.amazon.com/exec/obidos/ASIN/0521269296>
Good point. This concept of twisted forms is related to another long post I
wrote in this thread. I happen to have a distaste for "twisted" forms. They
are about as bad as "pseudo" vectors. Differential forms are beautiful
objects, why de-beautify them with such un-natural complications? A twisted
p-form is simply the Hodge dual of a regular (n-p)-form. The Hodge star
carries information about the orientation and hence may alter the behavior
under parity transformations. The way Burke presents them is as if they are
something "different" that needs to be learned in addition to regular forms.
I'll have to check to see what he actually says, but he probably could have
saved some ink by simply writing "twisted forms are Hodge duals of regular
forms." Apologies if he DOES say that :)
Eric
PS: ALL of this can be avoided if people just worked with forms properly and
stopped trying to write silly things like B = B_1 dx^23 + B_2 dx^31 + B_3
dx^12, when it should be B = B_23 dx^23 + B_31 dx^31 + B_12 dx^12. The
former is a consequence of the desire to work exclusively with vectors when
bivectors and 2-forms are the appropriate objects. This goes back to the
original subject of the thread, E and D are NOT the same physically or
mathematically (unless you relax the definition of "same" to mean
isomorphic, in which case E could be the same as H in time-harmonic cases,
but that is just semantics and not really worthy of a discussion).
ark
>After all that mess, the short answer to the question "Is D a pseudo
>vector?" is apparent. Although D is a 2-form, that does not mean that D is a
>pseudo vector because D is the Hodge dual of a 1-form E and the HODGE STAR
>PICKS UP AN ADDITIONAL SIGN UNDER A PARITY TRANSFORMATION (in this case, but
>not in general).
>
>Thanks for a nice question! It made me think. I hope my answer makes sense.
One observation that is not strictly related to the original equation,
but, somehow, is relevant:
The relation between D and E, as well as the relation between H and B,
in vacuum, involves Hodge operator on 2-forms. Hodge operators on
2-forms in 4d (as well as hodge operators on n/w forms in nd) is
conformally invariant. Thsi is an easy exercise. But it less known
that Hodge * operator on 2-forms in (-+++) signature , with the
properies *^2=-1 and Hermitian (with respect to the natural inner
product) is sufficient to define uniquely the light cone (=
conformal) structure of space time.
ark
Toby Bartels
>Toby Bartels wrote:
>>A property of a rather vacuous (ha! an unintentional pun!) sort,
>>but a property nonetheless.
>You prefer to think of eps_0 = 1 as a property of the vacuum,
>I cannot possibly see how a quantity which could just as well
>have defined to be 37 can be a property of anything
>but your particular definitions.
Well, if you redefine epsilon_0 to be 37,
then you redefine all of the other epsilons by a factor of 37 too.
So I guess that none of them are physical properties!
-- Toby
to...@math.ucr.edu
Toby Bartels
>Toby Bartels wrote:
>>Yeah, given a spacial metric, pseudovector and 2forms are equivalent.
>>Most people turn 2forms into vectors of course,
>>but this requires an orientation in addition to the metric.
>>I do find it more fundamental not even to assume the metric
>>and just to deal with the 2form itself -- in certain contexts.
>In ``Applied Differential Geometry,'' [1] William Burke claims
>that D is not an ``ordinary'' 2-form, but a ``twisted'' 2-form;
>``twisted'' 2-forms apparently transform oppositely to ordinary
>2-forms under parity.
Yes, you are correct. As Eric pointed out in another post,
D corresponds (given a spacial metric) to a spacial vector.
Thus it is, fundamentally, a *pseudo*2form
(or "twisted" 2form, if I understand that term correctly).
Given a metric, the following are equivalent:
p-vector = p-form = pseudo(n-p)vector = pseudo(n-p)form.
Given an orientation, the following are equivalent:
p-vector = pseudo-p-vector;
p-form = pseudo-p-form.
Given a volume form, the following are equivalent:
p-vector = (n-p)form;
pseudo-p-vector = pseudo(n-p)form.
Given a metric and an orientation (and hence a volume form),
then all of the above are equivalent.
-- Toby
to...@math.ucr.edu
Toby Bartels
>PS: ALL of this can be avoided if people just worked with forms properly
*if* you have a metric (or an orientation) around.
Otherwise you can't get rid of the pseudos.
-- Toby
to...@math.ucr.edu