994 views

Skip to first unread message

Sep 6, 2001, 7:35:12 PM9/6/01

to

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

--

Sep 9, 2001, 11:06:23 PM9/9/01

to

Dear Robert,

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.

Sep 10, 2001, 8:17:02 PM9/10/01

to

Lubos Motl <mo...@physics.rutgers.edu> wrote in message news:<Pine.SOL.4.10.101090...@physsun3.rutgers.edu>...

> 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

Sep 16, 2001, 1:13:55 PM9/16/01

to

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.

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

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Sep 17, 2001, 3:38:08 AM9/17/01

to

Lubos Motl <mo...@physics.rutgers.edu> wrote:

> 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

Sep 16, 2001, 5:00:28 PM9/16/01

to

Lubos Motl <mo...@physics.rutgers.edu> wrote:

> 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

Sep 23, 2001, 9:53:10 PM9/23/01

to

Lubos Motl <mo...@physics.rutgers.edu <mailto:mo...@physics.rutgers.edu>> wrote

in message <news:Pine.SOL.4.10.101091...@physsun9.rutgers.edu>...

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)

Sep 23, 2001, 9:54:48 PM9/23/01

to

robert bristow-johnson wrote in part:

>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

Sep 24, 2001, 2:03:34 PM9/24/01

to

Thomas Larsson wrote in part:

>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

Sep 24, 2001, 3:50:44 AM9/24/01

to

Toby Bartels <to...@math.ucr.edu> wrote:

> 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

Sep 25, 2001, 3:36:20 AM9/25/01

to

J. J. Lodder wrote in part:

>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

Oct 3, 2001, 12:16:37 AM10/3/01

to

In article <1ezuonc.1cv...@de-ster.demon.nl>,

J. J. Lodder <j...@de-ster.demon.nl> wrote:

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.

Oct 3, 2001, 11:19:20 PM10/3/01

to

Paul Arendt <par...@black.nmt.edu> wrote:

> 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

Oct 4, 2001, 9:19:21 PM10/4/01

to

Toby Bartels <to...@math.ucr.edu> wrote in message

news:<9onshm$bl2$1...@glue.ucr.edu>...

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

------------

Oct 10, 2001, 10:44:48 PM10/10/01

to

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.

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

Oct 12, 2001, 9:02:14 PM10/12/01

to

In article <1f0p07l.poz...@de-ster.demon.nl>,

>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?

>

>> 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.

Oct 14, 2001, 4:47:46 PM10/14/01

to

Hi,

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]

Oct 14, 2001, 4:48:34 PM10/14/01

to

Toby Bartels <to...@math.ucr.edu> 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.

[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)

Oct 14, 2001, 8:08:16 PM10/14/01

to

J. J. Lodder wrote:

>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

Oct 15, 2001, 4:10:51 AM10/15/01

to

Eric Alan Forgy <fo...@uiuc.edu> wrote:

> 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)

Oct 15, 2001, 4:09:28 AM10/15/01

to

Paul Arendt <par...@black.nmt.edu> wrote:

> 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

Oct 16, 2001, 3:55:28 AM10/16/01

to

Hi,

"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

Oct 17, 2001, 3:37:32 PM10/17/01

to

"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.

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

Oct 17, 2001, 3:38:10 PM10/17/01

to

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.

Gerard

Oct 17, 2001, 3:44:01 PM10/17/01

to

Toby Bartels <to...@math.ucr.edu> wrote:

> 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

Oct 17, 2001, 8:51:54 PM10/17/01

to

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.

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

Oct 17, 2001, 8:53:24 PM10/17/01

to

Gerard Westendorp wrote:

>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

Oct 17, 2001, 8:54:00 PM10/17/01

to

Gerard Westendorp wrote:

>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

Oct 17, 2001, 8:56:34 PM10/17/01

to

Hi,

"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

Oct 18, 2001, 4:31:46 PM10/18/01

to

Hi,

"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 :)

Oct 18, 2001, 4:32:24 PM10/18/01

to to...@math.ucr.edu

Toby Bartels <to...@math.ucr.edu> writes:

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;'

Oct 18, 2001, 4:38:58 PM10/18/01

to

Toby Bartels wrote:

[..]

> >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

Oct 18, 2001, 4:40:30 PM10/18/01

to

Gerard Westendorp <wes...@xs4all.nl> wrote:

> "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 :-)

Oct 18, 2001, 4:41:17 PM10/18/01

to

Gerard Westendorp <wes...@xs4all.nl> wrote:

> 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

Oct 19, 2001, 1:19:26 AM10/19/01

to

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

Oct 19, 2001, 1:20:27 AM10/19/01

to

Toby Bartels <to...@math.ucr.edu> wrote:

> 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

Oct 19, 2001, 1:20:57 AM10/19/01

to

Eric Forgy <fo...@uiuc.edu> wrote:

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

Oct 19, 2001, 1:22:30 AM10/19/01

to

Hi,

"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).

Oct 19, 2001, 4:32:29 PM10/19/01

to

On 18 Oct 2001 00:56:34 GMT, "Eric Forgy" <fo...@uiuc.edu> wrote:

>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

Oct 19, 2001, 11:54:27 PM10/19/01

to

J. J. Lodder wrote:

>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

Oct 19, 2001, 11:55:06 PM10/19/01

to

Gordon D. Pusch wrote:

>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

Oct 19, 2001, 11:55:19 PM10/19/01

to

Eric Forgy wrote at last:

>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

Oct 19, 2001, 11:55:38 PM10/19/01

to

J. J. Lodder wrote:

>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.

Right, this is why we should see immediately that,

even though we may be in a context where

it's appropriate to distinguish E from D

and even to make D a 2form while E is a 1form,

the underlying physical reality dictates that

D must actually be a pseudo2form,

since this underlying physical reality

provides only a metric and not an orientation.

-- Toby

to...@math.ucr.edu

Oct 20, 2001, 12:00:50 AM10/20/01

to physics-...@ncar.ucar.edu

ark <a...@cassiopaea.org> writes:

> 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.

>

Yes.

"Duality and conformal structure"

Tevian Dray, Ravi Kulkarni and Joseph Samuel

J. Math. Phys. 30, 1306-1309 (1989).

-charlie

Oct 20, 2001, 5:02:24 PM10/20/01

to

Hi,

Ok, thanks for expoundng a bit. Now, we are getting some where.

"J. J. Lodder" <nos...@de-ster.demon.nl> wrote:

> 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.

It seems you are saying that D is some field that only has meaning as a

macroscopic description of some underlying atomistic phenomenon. I would

charge that even at the microscopic scale that D is just as fundamental of a

field as E, while still being distinct.

Now, of the four fields E,B,H, and D, the fields E and B are actually more

closely related than E and D. Likewise H and D are more closely related than

H and B. E and B are simply rather arbitrary components of a single

geometrical object F. While H and D are simply rather arbitrary components

of the same geometrical object G.

====Entering math zone====

Stepping up the maths for a second, the action for Maxwell's equations can

then be written as:

S = int_M F/\G

Note that this looks a lot like the action for BF-theory, which is no

coincidence. BF-theory with some constraints leads to general relativity.

The above action with some constraints leads to electromagnetic theory.

A more standard, which doesn't mean more correct, way to look at

electromagnetic theory is to start right off declaring that G = #F (# =

Hodge star), then

S = int_M F/\#F.

This is particularly nice, because that is just the global inner product of

forms, i.e.

S = [F,F] := int_M F/\#F = int_M (F,F) vol,

where (F,F) is the local inner product of forms. Varying the action with

respect to A leads to

del F = 0,

where del is the adjoint of d, i.e.

[F,F] = [dA,F] = [A,del F].

Maxwell's equations without sources can then be stated simply as

dF = 0 (which is almost a tautology because F = dA)

and

del F = 0

(with sources, the second equation becomes del F = j).

There are several ways to interpret all of this. Here are two that come to

mind:

1.) You can begin with S = int_M F/\G with constraints (akin to BF-theory)

in which case you will find that

dF = 0

and

dG = 0,

with the constitutive relation G = #F, (with sources, you have dG = J, where

J is a source 3-form). Note that in this theory, # may be degenerate. In

this case, saying that E and D are the same amounts to saying F and G are

the same. However, this is like saying B and F are the same in BF-theory,

which I wouldn't do.

2.) You can begin with S = [F,F] := int_M F/\#F in which case you will find

that

dF = 0

and

del F = 0,

where del is the adjoint of d, (with sources, you have del F = j, where j is

a source 1-form). In this case, there IS NO G. This amounts to not even

having D or H in the model at all, so the question of whether E is the same

as D is moot :)

====End of math zone====

Sorry about that :)

I can appreciate the atomistic viewpoint of D. In fact, you can probably say

that most, if not all, EM treatise present D like that. However, I am also

of the opinion that physics is not a democracy, so I tend to give very

little weight to what the majority says (i.e. counting the number of

publications on a certain subject as an indication of the physical validity

of that subject is nonsense) and give the weight to the physics and

mathematics :)

Mathematics has a voice, and I think we should listen to it. If the voice of

differential geometry/differential forms has any relevance to physics,

electromagnetic theory in particular, then that voice says that E and D are

different mathematical objects (even at the microscopic level) that are

related through the Hodge star.

Here is just a thought (that will reveal the exotic things that run through

my mind). Magnetic charges are somewhat contraversial entities, but if they

were to exist, they appear as purely topological beasts, i.e. as points

removed from the topological space.

Electrons (fermions) are much less contraversial entities, but it may be

that even these guys are also topological beasts. For instance, this paper

Fermions and Topology

Authors: Lee Smolin

http://xxx.lanl.gov/abs/gr-qc/9404010

(and check out the "refers to" and "cited by" as well) suggests that

fermions are actually exotic topological objects, i.e. wormholes. To say it

in less sci-fi terms, it is possible that fermions result from the

identification of distinct points in space-time, where the electron is one

end of the wormhole and the positron is the other.

Thus, even the atomistic viewpoint may involve complicated topological

spaces and quantum gravity *shudder*. Hence, any book that claims to

understand the distinction (or non-distinction) between E and D is

automatically suspect. No matter what turns out to be the "true" explanation

for all this stuff, I am fairly certain that differential forms *in one way

or another* will survive (the emphasis is there because it may be that

space-time is not even a smooth manifold, but even then you will probably be

able to construct some kind of algebraic version of differential forms that

survives).

I think your post answered my question. We just think of things differently

and that is fine. I was afraid I had made some blunder :) I may have an

unpopular view of things, but that is nothing new to me, so I'll keep on

going the way I have been :)

Thanks,

Eric

Oct 20, 2001, 5:03:45 PM10/20/01

to

Also earlier: "Electromagnetic permeability of the vacuum and the

light cone structure"

A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94.

Online at http://www.cassiopaea.org/quantum_future/emp.htm with extra

info.

ark

Oct 20, 2001, 5:04:51 PM10/20/01

to

Toby Bartels <to...@math.ucr.edu> wrote:

> J. J. Lodder wrote:

>

> >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!

No, the relative ones, water at eps_r = 80 etc,

stay the same,

Jan

Oct 20, 2001, 5:05:16 PM10/20/01

to

Gerard Westendorp <wes...@xs4all.nl> wrote:

But there is:

In the laws of physics as we know them,

whatever choice for eps_0 you make,

it is only the combination e^2 / eps_0 that occurs.

(And eps_0 E^2, of course, leaving force eE , eps_0 free)

You apparently want to see some meaning for eps_0 by itself,

without a corresponding e^2.

Therefore it is up to you to elucidate

which law of physics you want changed,

and what the -physical- (that is observable)

consequences will be.

As long as you don't you have said nothing,

physicaly speaking,

except perhaps that you prefer other units,

Jan

Oct 20, 2001, 5:23:29 PM10/20/01

to to...@math.ucr.edu

Toby Bartels <to...@math.ucr.edu> writes:

> Gordon D. Pusch wrote:

>

>> 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).

Assuming *I* understand Burke's terms correctly, then I believe you do... :-/

Insofar as I can figure it out from his prose and figures, Burke's

``twisting'' of geometric objects appear to associate ``circulations''

with them instead of ``directedness;'' hence, a ``twisted'' vector or

1-form has a ``screw-sense'' instead of an ``arrowhead,'' a ``twisted''

bi-vector or 2-form has a ``circulation'' around its plaquette or

``egg-crate'' cell, etc. I can't visualize what sort of ``circulation''

a ``twisted'' tri-vector or 3-form would have --- and apparently neither

can Burke,because he doesn't have an illustration of one in his book... :-/

Oct 22, 2001, 12:49:34 AM10/22/01

to

J. J. Lodder wrote:

>Toby Bartels wrote:

>>J. J. Lodder wrote:

>>>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!

>No, the relative ones, water at eps_r = 80 etc,

>stay the same,

So you have just wasted your time defining eps_0 to be 37,

switching to relative epsilons, and getting back where you started.

Certainly one *can* do this, but it's silly!

And now esp_r = 1 is a physical property of the vacuum.

-- Toby

to...@math.ucr.edu

Oct 22, 2001, 12:58:41 AM10/22/01

to physics-...@ncar.ucar.edu

ark <a...@cassiopaea.org> writes:

> Also earlier: "Electromagnetic permeability of the vacuum and the

> light cone structure"

> A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94.

> Online at http://www.cassiopaea.org/quantum_future/emp.htm with extra

> info.

>

> Also earlier: "Electromagnetic permeability of the vacuum and the

> light cone structure"

> A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94.

> Online at http://www.cassiopaea.org/quantum_future/emp.htm with extra

> info.

>

Nice. I will keep your result showing how Hodge * on 2-forms determines

conformal structure in mind next time the issue comes up, which it

does from time to time.

As long as I am writing, I might mention (changing the subject entirely)

that your series of papers with Coquereaux and the corresponding book (all

from long ago) on Kaluza-Klein reductions have been very useful to us here

at USU since we are working on various aspects of symmetry reduction in

gravitational theories. I would highly recommend them to anyone who is

interested in such things.

-charlie

Oct 22, 2001, 11:34:09 PM10/22/01

to

Eric Alan Forgy <fo...@uiuc.edu> wrote:

snip agreements

> I think your post answered my question. We just think of things differently

> and that is fine. I was afraid I had made some blunder :) I may have an

> unpopular view of things, but that is nothing new to me, so I'll keep on

> going the way I have been :)

We appear to agree that (as far as is known now)

all of the physics (so far) can be done with E only,

D being an averaged macroscopic E-field.

We also agree that you can build mathematical structures

in which E and D appear differently.

Indeed, these mathematical structures -may- point to new

(as yet undiscovered) aspects of reality.

Reformulating theories in as many way as possible

is indeed a way to see new possible openings.

But, until you manage to find some physical content,

the added formalism remains extra overhead.

(and with Ockham against it :-)

The D=pseudo? in this thread is a case in point:

trivially wrong in the standard view,

arguments (with possibility for errors) with the extra formalism.

But, of course, some new physics may come out of it eventually.

Good luck with your quest,

Jan

Oct 23, 2001, 2:25:59 PM10/23/01

to

In article <gizo6mx...@pusch.xnet.com>,

Gordon D. Pusch <gdp...@NO.xnet.SPAM.com> wrote:

Gordon D. Pusch <gdp...@NO.xnet.SPAM.com> wrote:

>Toby Bartels <to...@math.ucr.edu> writes:

>> 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).

>Assuming *I* understand Burke's terms correctly, then I believe you do... :-/

>Insofar as I can figure it out from his prose and figures, Burke's

>``twisting'' of geometric objects appear to associate ``circulations''

>with them instead of ``directedness;''

Hmm, and I thought we were talking about "twisting" in the usual

mathematical sense of "tensoring with a line bundle": we are

taking the bundle whose sections are p-forms and tensoring it

with the line bundle whose sections are pseudoscalars, to get

a new bundle whose sections are "twisted p-forms". People

also call this sort of trick "twisting by a line bundle".

Twisting by a line bundle is a handy way to get new bundles from

old ones. In physics we do this a lot using the line bundle whose

sections are "densities". Twisting by one of this bundle is

also called "densitizing". So you'll also see people, especially

in general relativity, running around talking about "densitized

vector fields", "doubly densitized 2-forms", and the like.

I don't know Burke or his book. If Burke is a fancy-ass mathematical

guy, he was probably talking about twisting by a line bundle. If

he is a down-to-earth physical guy, he was probably alluding to

what you're talking about - the fact that pseudovectors describe

"circulation" rather than "direction".

If you cheat somewhat and draw a 2-form as a little "oriented

plaquette", you should draw one of these twisted 2-forms as an

"unoriented plaquette".

("Cheat somewhat" because it's really bivectors that look like

oriented plaquettes, and even then, it's only the decomposable

ones that really look like that - i.e. those of the form v ^ w.)

Oct 23, 2001, 3:29:18 PM10/23/01

to

ba...@galaxy.ucr.edu (John Baez) writes:

> In article <gizo6mx...@pusch.xnet.com>,

> Gordon D. Pusch <gdp...@NO.xnet.SPAM.com> wrote:

>> Insofar as I can figure it out from his prose and figures, Burke's

>> ``twisting'' of geometric objects appear to associate ``circulations''

>> with them instead of ``directedness;''

> Hmm, and I thought we were talking about "twisting" in the usual

> mathematical sense of "tensoring with a line bundle": we are

> taking the bundle whose sections are p-forms and tensoring it

> with the line bundle whose sections are pseudoscalars, to get

> a new bundle whose sections are "twisted p-forms". People

> also call this sort of trick "twisting by a line bundle".

>

> Twisting by a line bundle is a handy way to get new bundles from

> old ones. In physics we do this a lot using the line bundle whose

> sections are "densities". Twisting by one of this bundle is

> also called "densitizing". So you'll also see people, especially

> in general relativity, running around talking about "densitized

> vector fields", "doubly densitized 2-forms", and the like.

Yes, Burke explicitly states that ``twisted'' forms are ``densities.''

> I don't know Burke or his book. If Burke is a fancy-ass mathematical

> guy, he was probably talking about twisting by a line bundle. If

> he is a down-to-earth physical guy, he was probably alluding to

> what you're talking about - the fact that pseudovectors describe

> "circulation" rather than "direction".

Burke appears to be a ``fancy-ass mathematical guy'' who is trying to

write to the ``down-to-earth physical guys'' and ``convert'' them to the

``fancy-ass mathematics'' way of doing things, e.g., differential forms,

line bundles, and all that. (The dedication of his book reads: ``To all

those who, like me, have wondered how in hell you can change $\dot{q}$

without changing $q$.'')

Oct 25, 2001, 9:28:26 PM10/25/01

to

In article <1f17ur9.1wi...@de-ster.demon.nl>,

J. J. Lodder <j...@de-ster.demon.nl> wrote:

>Paul Arendt <par...@black.nmt.edu> wrote:

J. J. Lodder <j...@de-ster.demon.nl> wrote:

>Paul Arendt <par...@black.nmt.edu> wrote:

(snip descriptions of how to separately measure E and flux of D)

>> 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.

On the contrary -- they are essential to the point I was trying to

make! If we restrict the situation to Euclidean spaces, then I

would have to agree with your original statement: that if sensible

units where D = E were chosen long ago, there would have never been

any reason to introduce the constant epsilon_0. I disagree with

this strongly: epsilon_0 will still show up in some guise or

another when electromagnetic experiments are performed in situations

where the curvature of space changes with position.

There are even easier experiments to measure D and E than the ones I

proposed, in Bamberg and Sternberg's "A Course in Mathematics for

Students of Physics." Measure the kinetic energy change imparted to

various charged particles when they have moved in various directions.

In the limit of small charges and short distances, the ratio of this

energy change to the product of the distance traveled and the charge

is the (component of the) electric field E (in the direction traveled).

Now, take two very thin conducting sheets of metal of equal area, touch

them together, and bring them back apart. Measure the charge on

each plate, and divide by the area of the plate. In the limit as this

area becomes very small, this number is the component of D (oriented

with the plates' orientation).

My point is that: if units are chosen such that the magnitudes of

D and E are equal in a flat space, then they will NOT be equal,

using the exact same procedures, in certain locations in curved

spaces. And if they are found to be equal at some point in a space of

varying curvature, then they will not generally be equal at another

point in the same space.

I hope that the above experiments make the difference between D and

E very clear: E is associated with the direction "radial" to a

point charge, while D is associated with the two *transverse*

directions. (Going between the two is the role performed by

the Hodge star operator.)

If we never consider non-flat spaces, then I agree that D and E can

always be chosen to have the same magnitude in vacuum. But that's like

trying to argue that gauge fields can have no physical meaning -- by

restricting oneself to gauge fields that are "pure gauge" only! Not

fair.

In another article in this thread, J. J. Lodder wrote:

> 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.

The metric is certainly used to raise and lower indices on a

position vector. The metric can also be used to get D from E

in vacuum and vice-versa. So I think I can see your point here:

that knowledge of g allows us to do either.

But I think that the example may be somewhat misleading beyond

that, for two reasons. The first is that D and E are *not* simply

related to each other by raising and lowering indices! The Hodge

star operator is also involved (although g determines it by providing

a preferred way to measure volumes). The second reason is that

although I cannot think of a way to experimentally measure a

particle's covariant versus its contravariant position, the above

experiments are conceptually and operationally *different* ways

of getting numbers for D and E out.

And in another article:

>D is just a partial field,

>which arises because we find it convenient

>to split the total electric field -mentally- into parts.

Now, this I do not agree with at all! Maxwell's equations show

quite clearly that the way E can be derived from a four-potential

is *very* different from the way D can be, for instance.

Oct 26, 2001, 1:59:43 PM10/26/01

to

Hi,

"John Baez" <ba...@galaxy.ucr.edu> wrote:

> Twisting by a line bundle is a handy way to get new bundles from

> old ones. In physics we do this a lot using the line bundle whose

> sections are "densities". Twisting by one of this bundle is

> also called "densitizing". So you'll also see people, especially

> in general relativity, running around talking about "densitized

> vector fields", "doubly densitized 2-forms", and the like.

Since you mention it, I'll go ahead and ask a question that's been bugging

me for a while. In the "modern" canonical formulation of general relativity,

you have a Lie-algebra valued 1-form (connection) A and a densitized Lie

algebra valued (n-2)-form E.

One thing that has bothered me, and was re-enforced while discussing the

senselessness of "pseudo" and "twisted" differential p-forms was that both

of these are simply Hodge duals of regular p-forms. I don't care for the

additional adjectives "pseudo" and "twisted" because it seems to imply that

these are somehow something "different" that needs to be learned in addition

to regular forms. This is not true. Regular forms are all you need to learn,

and these other beasts are just their Hodge duals. The additional

"complication" that seems to be necessary when people start throwing around

terms like pseudo and twisted forms makes it seem like a burden rather than

a blessing to learn differential forms. For someone like me who is trying to

convince engineers to use differential forms, this is a big problem. Forms

are simple and beautiful!! Why mess them up?! Maybe it is a conspiracy by

physicists and mathematicians. They don't want engineers to know how simple

all this "fancy-ass mathematics" really is ;)

Anyway, my complaint against the use of "pseudo" and "twisted" adjectives to

describe Hodge duals of regular forms seems to apply to densitized forms as

well. But this is probably a more serious complaint because it goes at the

heart of canonical QG. Instead of using a 1-form connection A and densitized

(n-2)-form

E, why not just use a regular 2-form?! What is being called the densitized

(n-2)-form E, is just the Hodge dual of a regular 2-form. In fact, E seems

to be just the Hodge dual of the curvature F = DA itself! So there really

aren't two different fields E and A, there is one field F and the Hodge star

*. I don't know which is easier, to subsume the Hodge into a new field E =

*F and vary E and A independently, or to just vary A and * (or maybe vol).

For instance, the Langrangian for EF-theory (going with the suggestion in

gr-qc/9905087) is given by

L = tr(F/\E),

but why not just write this as

L = tr(F/\*F) = (F,F) vol

? Then you can vary A as usual, but then you could also vary either * or

vol. Is there really some good reason for dealing with "densitized pseudo

twisted differential forms" that I am just not aware of, or is this an

example of academic inertia, "Because that's the way it's always been around

here." (http://www.ccem.uiuc.edu/ericf/apes.html)

Eric

Oct 25, 2001, 5:35:59 PM10/25/01