NIST lists alpha= 7.297352568 10E-3

[nightcleaner]

(this post was moved from the Using the Force Constant in Equations thread as it seems to be a branching question.)

I just got off work and I usually like to have something to think about while going about my chores. Cleaning is not very demanding, mentally, and radio is just an irritating noise to me these days. So I find that I like listening to what is going on in my head best.

Well, it seems natural to wonder why 1/137 and not some other number. Why one charge square per 137 separations square? (NIST lists alpha= 7.297352568 10E-3, which makes it 1/137.035999 by my calculator).

NIST also has an explanation under the heading "current advances" which gives the dimensional derivation of alpha as charge squared divided by (h-bar c 4 pi permitivity).

$$\alpha=\frac{e^2}{4 \hbar c \pi \epsilon_0}$$

NIST goes on to say that alpha is proportional to e^2, so it is the square of an effective charge "screened by vacuum polarization and seen from an infinite distance." Since alpha depends on the energy at which it is measured, NIST says, it is considered an "effective", or running coupling constant.

Wikipedia sent me the long way around the barn, as usual, but it was an interesting tour. I found little directly on the fine structure constant, but then found the hyperfine structure, which I learned is a small perturbation in Bohr energy levels due to proton-electron dipole moment. Hyperfine structure is much smaller than Lamb shift, which Wiki says is the same as fine structure.

Hyperfine structure is very interesting in itself, which gives the 21 cm line of hydrogen in intersteller medium, a useful tool for radio astronomers. It is also used to define the second (a unit of time, as in 60 seconds per minute) as 9,192,631,770 cycles of the hyperfine structure transition of caesium-133 atoms.

The Lamb shift, which Wiki identified with alpha, is a radio frequency transition between 2s 1/2 and 2p 1/2. The energy change was found to be about 1060 MHz of 2s 1/2 above 2p 1/2. I remember a little about s and p levels from chemistry. S is a spherical shell, p is a double-lobed shape, like a dumb bell, they say, although I have never seen a dumb bell shaped just like that. Two tear drops joined at the narrow end, if you have never seen a picture of it. The Wiki author then goes on to explain that this is a one-loop effect in QED, interpreted as the influence of virtual photons.

Whatis.com said alpha is equal to the ratio of the velocity of an electron in a hydrogen atom to the speed of light. This interested me because I once asked my chemistry prof. about the speed of electrons in orbitals and he assured me that it was a meaningless question, since electrons in orbitals are more like charge density clouds than like particles which can have velocities.

Answers.com said "for any length s, the fine structure constant is the ratio of two energies: (i) the energy needed to bring two electrons from infinity to a distance of s against their electrostatic repulsion, and (ii) the energy of a single photon of wavelength s/2pi. Then they noted that it is one of the 20-odd "external parameters" that have to be added manually from experimental data to the standard model of particle physics, and that it cannot be predicted by QED. All interesting stuff, but not really helpful in trying to imagine what alpha actually is.

Then there were a couple of numberology approaches, not very meaningful to me, and finally, Eric Weisstein at Wolfram.com noted that there is a "curious approximation" to alpha in the fact that 44pi - cos^-1 (e^-1)=137.036007939.

So out of all this, my best fix was from NIST, where I got the idea that the fine structure constant is due to shielding of the charges by virtual patricles. In this idea, the charge causes virtual particles near it to rotate, which in turn cause slightly more distant virtual charges to rotate, and this creates a sphere of positive virtual charges around the negative electron charge, and these charges then shield the negative charge. In this way local effects radiate outward to become actions at a distance.

In other news, my friend has decided to buy a feeder pig to raise up on restaurant scraps which I am to supply. I plan to call it "8 piG." If I do not grow too fond of it, it will eventually transition to "ate piG." I am sure this is only a coincidence.

Be well,

Richard

 

[marcus]

for this thread to "realize its potential" you should hopefully get comments and information from several people

there are a lot of interesting quantities here and alpha enters into them and sort of ties them together

and then that business about the electron's orbit "speed" is right if you imagine a primitive circa 1900 idea of atom with electron circling like a little planet
then the speed really would be 1/137 of the speed of light!
(to make the energy come out right)
but that model doesn't actually work so your teacher waved it off
tho I think it is an interesting handle on alpha----there are a lot of handles

 

[nightcleaner]

$$\alpha=\frac{e^2}{4 \hbar c \pi \epsilon_0}=1/137$$

So if $$e^2 =1$$, and following the units in the Force Constant thread, $$\hbar=1, c=1$$, then $$4 \pi \epsilon_0 = 137$$ so $$\epsilon_0=\frac{137}{4 \pi}$$.

By my calculator, using NIST value for alpha, and pi rounded to match the error in alpha, $$\epsilon_0=43.6199132$$ or about 44. Yikes!

Eric Weisstein at Wolfram.com noted that there is a "curious approximation" to alpha in the fact that 44pi - cos^-1 (e^-1)=137.036007939."

If we jump to conclusions and go ahead and try to insert our calculation of the permitivity of vacuum into the curious approximation we have

$$\frac{137}{4 \pi}\pi -cos^(-1)(e^-1)$$

and we have already set e^2=1 so e=1 so the formula becomes


$$34.25-cos^(-1)(1)$$

and cos^-1 (1) is zero. So have we discovered that $$\epsilon_0$$is equal to about 34.25? And what, if anything, would that mean?

 

[marcus]

$$\alpha=\frac{e^2}{4 \hbar c \pi \epsilon_0}=1/137$$

So if $$e^2 =1$$, and following the units in the Force Constant thread, $$\hbar=1, c=1$$, then $$4 \pi \epsilon_0 = 137$$ so $$\epsilon_0=\frac{137}{4 \pi}$$.

By my calculator, using NIST value for alpha, and pi rounded to match the error in alpha, $$\epsilon_0=43.6199132$$ or about 44. Yikes!
...

you were going along good and then forgot to divide by the 4. you only divided 137.036..by pi. So you got 43.6... instead of 10.905...
but the basic reasoning is good

Personally i am skeptical about "permittivity of free space". Some CGS systems of units that had a good trackrecord from before the current version of metric SI simply do not bother with epsilon-naught.
They could sometimes have instead something called "coulomb's constant". there is a fair amount of variety in humanity's conventions about electrical units.

but if you think of epsilon-naught as a real physical quantity that should have a value in the system of units then I'd imagine the value would be 10.905 and the units would be the reciprocal or topsyturvy of whatever the units for the "coulomb constant" are

coulombconstant = 1/137 force x length²/charge²

that is so you can multiply it by two charges, and divide by a square distance between them, and be left with a force (which is the force betw. the charges). This BTW seems intuitive and straightforward to me.
It is like the gravitation constant---it tells the force betw. two things sep by a certain distance.

epsilon-naught is just another name for 1/(4pi coulomb)
so its units are going to be the upside down of the previous:
charge²/(length² x force)

and in a natural units system where coulomb constant's value is 1/137.036.. then epsilon-naught is going to be
10.905 charge²/(length² x force)
but in some other system where coulomb const is something else then it is
going to be something else.

Personally it does not seem intuitive or straightforward to me when you already have a perfectly good quantity called Coulomb constant to go and
define another thing epsilon-naught which is simply

1/(4pi coulomb)

but these things are to a certain extent matters of taste
==========

your post has a lot of other stuff I hope gets discussed, I am too sleepy now so we get off to bed and return to this alpha discussion in morning

 

[marcus]

Hi Richard,
what I hope happens in this thread is that several people contribute what their take on alpha is. like it is the elephant and one describes the trunk the tail the ears the warty wrinkled knees etc. Do elephants have belly buttons? they must!

in QED calculations alpha is what you expand in powers of
Like you want to make a theoretical calculation of the magnetic moment of the electron.
So you start with a good guess and you add a correction term (something times alpha) and then a tinier correction term (something times alpha²) and then an even tiner "third order" correction term
(something times alpha³)

It is like a power series A + Bx + Cx² + Dx³ + Ex⁴ +...

where the "somethings" or coefficients are calculated from feynmandiagrams by theory

and then they brag that their QED theory agrees with experimental measurement out to seven or eight or ten digits or whatever.

One thing to notice is that alpha is like 1/100 so POWRS of alpha get small fast. alpha-square is like 1/10000. So in the power series you don't have to go many terms before the corrections are tiny-ass little-bitty no-count fur.

OK so that is the trunk of the elephant: alpha is the "x" of the powerserieses that you expand in terms of alpha in QED. alpha is the basic number input in QED from which other numbers are calculated
(well with the help of the mass of the electron and Planck constant and suchlike basic quantitities)

Now we go around to the back end of the elephant.
that is where I am
what fascinates me is the TAIL of the elephant. which is the fact that when the COULOMB CONSTANT is expressed in QUANTUM GRAVITY units then it turns out to be alpha.

this is a very obvious thing, much too obvious to interest theoretical physicists (about a million times too obvious to be interesting)
but it speaks to me as someone fascinated with symbol systems and human language in general.
what it says (to me personally) is that the coulomb constant is kind of a good choice of constant. it is a good constant worth keeping in mind

I say don't forget the Coulomb constant because it is the tail on the elephant! It just happens that when you express it in QG natural units that its numerical value in those units is alpha.

the coulomb const is what tells you the force between two point charges which are a specified distance apart. You multiply the charges together and divide by the square sep-----and then to finish off you multiply by coulombconst and bingo there's the force.

of course it "runs". 1/137.036... is the macroscopic limit when the two charges are some sensible distance apart, if you push them so close together they start climbing into each other's underwear then the number gets bigger

but for practical purposes, like atoms and magentic moments and Xray machines and MRI scans at the hospital and superconductivity and the machines chemists use to analyse chemicals and that stuff, AFAIK the largescale limit alpha that we know and love is OK,

an atom is large enough so you can use 1/137.o36 and forget about
"running"

these are, so to speak, my cherished prejudices that I am telling you.
maybe selfAdj or others can lend a different perspective.


BTW alpha gets into so many things including the ionization energy of the hydrogen atom.

if you take the massenergy of an electron, the energy of the electrons existence, in whatever units, and multiply by alpha-squared, then you get twice the energy needed to remove the electron from the H atom

so then divide by two, and that is the ionization energy (or potential energy of the ground state, however you like to say it)


alpha gets into the formula for the 21 centimeter line too, but I have forgotten the formula. it is a player in a lot of things that one can calculate,

oh, the magnetic moment of the electron is a famous case of alpha
appearing in the formula

 

[Kea]

Hi

I have been selfishly enjoying nightcleaner's eloquence and Marcus' industriousness without contributing much.

I just had a nice argument with a bright young postgrad here, who is much more reasonable than me, which I think is worthwhile recording. There's clearly something really elementary that I just don't get, and I was hoping that someone could help me understand. Let's call my colleague Bob, and let's call me Alice. I apologise to Bob for possibly misrepresenting him.

Alice and Bob go and get a cup of coffee in the tea room.

Bob: Why do you think it is possible to calculate alpha, when we know it changes with time?

Alice: Whose time? You mean the time of a classical human observer. I'm interested in the local value of alpha.

Bob: I mean an observer 12 billion years ago!

Alice: 12 billion years? Oh, according to our classical observations. But you cannot infer anything about a local observation of alpha out there somewhere 12 billion years ago.

Bob: What? But if we see alpha was different, it was different.

Alice: You're imposing the conditions of a classical human observer on your spacetime.

Bob: But if we were on a planet in a galaxy back then, which we can see, alpha would have been different.

Alice: How can you possibly claim to know how things looked back then? How can you say there could have been human type observers?

Bob: There were planets and atoms and stars and alpha is alpha and we could measure it.

Alice: All of that is a part of our reality. You're making some wild extrapolations which I do not accept are valid.

Bob: All right. When did things change? How far back can we go? 100 years?

Alice: There were similar observers then, sure.

Bob: 200 years?

Alice: Sure.

Bob: 300 years?

Alice: Sure.

Bob: I can keep going. 5000 years?

Alice: Again, similar observers, using classical cosmology. Look, we simply disagree about the conditions under which the theory breaks down.

Bob: Are you seriously trying to tell me that, suddenly, a few thousand years ago the nature of physics changed and that all the stars and everything aren't really there? We have a classical reality and we can extrapolate all the way back to $10^{-13}$ seconds, or whatever, after the Big Bang.

Alice: No. You're thinking of a universal human type observer. I don't accept such a thing. Quantum Gravity doesn't work that way. I'm not disagreeing that standard cosmology is quite correct within its domain of applicability. But Quantum Gravity does say some radical things about cosmology.

Bob: What? Quantum Gravity applies to small systems. Standard cosmology must be correct. We have astronomers and people out there who measure stuff, and it agrees with the theory.

Alice: I'll say again, I agree that the standard cosmology must be correct from the point of view of a classical observer here. Of course the new theory must reproduce the standard cosmology as an approximation.

Bob: All the way back to the Big Bang!

Alice: According to a human observer. But when, assuming, we measure alpha to be different back then, we're not talking about a hypothetical alien from 12 billion years ago. We're talking about a human observer now.

Cycle through most of the above 3 or 4 times.

Bob: So if we went to alpha centauri in a spaceship we wouldn't find an ordinary star?

Alice: If we naively tried to go to alpha centauri in an ordinary spaceship built according to current technology we might be in for a few surprises, and it would take an awfully long time to get there.

Bob: Let's say we built a spaceship that traveled at close to the speed of light.

Alice: Really? Well, you need to think about what sort of resources that would take.

Bob: We might need a bit more fuel, that's all.

Alice: No. I disagree. The resources are on a sufficient scale that it is outside the realms of any experiment we have done yet. We don't know how to build such a spaceship.

Bob: Let's say there is an observer B over there somewhere 12 billion years ago. Assume that he can measure alpha and it is 1/137. He holds up a placard saying "I measure alpha to be 1/137". If we were to measure alpha over there to be different from what observer B says it is, we would have a contradiction. I think the two measurements should agree. That means alpha really is whatever we measure it to be here, and he will agree with us.

Alice: We haven't done that experiment. We haven't seen any aliens. Things over there need not be what we think they are. There is no objective reality.

Bob: But we could do the experiment. You need to be internally consistent. If I see a star over there 12 billion light years away, and I send a man over there to check that it's really a star and not a carrot, and he goes over there, he'll find a star. And his spaceship will be a spaceship. It won't turn into a banana!

Alice: If we observe it from here within our classical observational framework, I agree. When we can see it, it will look like a spaceship.

Bob: So, what? The spaceship might actually turn into a banana?

Alice: Sure, why not? You're talking about a completely different class of observer.

Bob: But all observers see stars as stars.

Alice: You're making a very unreasonable extrapolation! You're saying that if a person wanted to travel 12 billion light years through our local classical reality that things should appear to them to be roughly as one would expect according to that local classical reality. I think that's unreasonable. I don't believe it's possible to travel 12 billion years without taking some pretty radical quantum gravitational effects into account.

Bob: But the stars over there exist. There is only one classical reality, and it's like a patchwork of local realities that we must be able to piece together.

Alice: According to us that reality is valid, and the new theory agrees completely with it. But we haven't tried to travel 12 billion light years! We haven't gone further than the edge of the solar system! And it even looks like that might need to take QG into account.

Bob: But, you agree that the solar system is there. The sun is there?

Alice: According to the local reality, yes.

Bob: Then a star is a star and if I travel to it, it should be a star and not a banana.

Alice: From here it will look like a star, much as we expect. But you can't extrapolate that picture to a qualitatively different type of observer, such as one far away from us in our classical spacetime. There is no objective reality based on your classical reality. That's what the maths tells us. We have to do away with the objective reality.

Bob: But stars are made of ordinary matter. Atoms and stuff. When Rutherford did his experiments he inferred the existence of nuclei. But those things have an objective reality. A galaxy 12 billion light years away is made of the same stuff.

Alice: I'm not disagreeing that from our local point of view the theory is perfectly reasonable. And we can extrapolate all the way back to the early universe and infer the existence of ordinary matter back then.

Bob: You're contradicting yourself! How can you have both ordinary matter and bananas? You're insane!

Argument degenerates into shouting match. Alice and Bob return to their desks.

I would like to thank Bob for the time and effort he has put into indulging my insanity.

 

[marcus]

there is not need to raise the issue of sanity, I think, since
any quantum theory tends to be about describing what information one system has about another
in a quantum setup doesn't one give up the idea of objective reality for good, and just talk about what some observer is measuring?

well, that wasnt what I wanted to say (and it may well be wrong)
what I wanted to say was that I like the image Bob evoked of
looking thru a telescope and seeing some guy in a distant galaxy and the
guy is waving a placard at us saying "I measured alpha here and it is 1/137.036..."

so he is voluntarily taking part in an experiment to check something. He has figured out that the information on the placard might do somebody else (like us) some good. that was very nice of him.

because we can measure alpha in his galaxy on our terms by analyzing the light coming from his galaxy, but that is with our instruments here and we are the observers----it is slightly different (I think) if HE is the observer and then sends us the result of his measurment

a widely scattered NETWORK of observers who cooperate (or simply act altruistically like the little alien with the placard) seems able to get a different kind of grip on things from just one isolated observer.

Or at least one could do thought-experiments to see if a network of observers can learn things which might escape one loner. come to think of it this has probably been investigated plenty already (I just don't read so widely and havnt encountered it)

 

[marcus]

BTW how do other readers like the dialog? anybody?
I like it. maybe people who are able to write dialog-type posts
should try it more. notice that it is balanced: it is not just a set up
to make Bob viewpoint look foolish, there are two reasonable and
tenable points of view, which makes it more interesting than a
onesided pitch for just one. not everybody can write dialogs
because some people are too singleminded and serious for it.
Ive never tried. Galileo was a good science dialog writer
and maybe has not ever been topped.

 

[Kea]

A widely scattered NETWORK of observers who cooperate (or simply act altruistically like the little alien with the placard) seems able to get a different kind of grip on things from just one isolated observer.

Hello Marcus

Yes, lots of people now study networks of observers. One of the earlier papers seems to be the short paper

Multi-Player Quantum Games
Simon C. Benjamin, Patrick M. Hayden
http://arxiv.org/abs/quant-ph/0007038

But as far as I know most of this work uses standard quantum theory and does not address the issue of no objective spacetime. So it cannot be resorted to directly as an argument against Bob's line of thinking. On the other hand, there is a strong connection between the mathematics of quantum computation and informational cosmology.

 

[marcus]

"where is setAI when we need him"
quantum computation and informational cosmology is outside my ken
so someone else should respond
even "multiplayer quantum games" intriguing as it sounds is outside my present reach

but one thing you say rings a big alarmbell for me and that is this:
if there would be some quantum theory of spacetime able to replace general relativity, then in that theory there would be no
objective spacetime (oh horrors)
there would only be the universe as you and I see it
and the universe as the alien in the distant galaxy sees it (if there is such an alien) and no guaranteed objective common spacetime.

Richard has been toying with the idea of switching from physical speculation to icehockey because, as he points out, icehockey at least is unambiguous. Maybe one should not dwell overmuch on these matters.

Actually the little guy in Andromeda is a friend of mind and I was the one that suggested to him that he write his alpha = 1/137 on a piece of cardboard and hold it up for Bob to see thru the telescope. He and I share the same dreams. So I am totally confident of a single objective universe. Trust me, it's real.

a wink smiley is obligatory here

 

[nightcleaner]

Hi Marcus and Kea

I havn't gone over to ice hockey yet, altho I found it advisable to take a break from abstract reasoning. Ok, the truth is my abstract-reasoning-thing was flummoxed and not really producing any new doorways. I wish I could say I feel much better now, but the fact is I just don't have the certitude required in a good sports fan, or any other kind of fanatic at all I suppose. Actually I don't like sports in general and ice hockey in particular, but I don't try to force that opinion on anyone. I rather envy the exhileration one must feel when "HE SCORES!" has some importance beyond the empty hollow meaningless shell of ephemeral and oft-repeated victory. He scored yesterday, he will score tomorrow, or someone else will. What has been accomplished? It is only noise and confusion, occasionally mixed with blood.

Well, Alice and Bob were having an interesting and probably unresolvable discussion, weren't they? If I had been there I might have tried to tie the two ends of the argument together by suggesting that what is going on twelve or thirteen billion lightyears away and what is going on in the extremely local realm of quantum gravity might just be the same thing. If a spaceship can be a banana, why not?

Anyway things had to evolve somehow after the big bang. Somehow localities becomes events at a distance. Perhaps the two processes share some common features. But the argument (aside from the weird human courtship ritual thing) seems to be about the second sense of identity: It is.

To illustrate, consider a cup of coffee between Alice and Bob. There it is on the table, by all means a verifiable object. Bob looks at it from his point of view. Alice looks at it from her point of view. They can agree on some terms. It contains coffee. It is a cup. Which side is the handle on? You ask them and they each write it down on a scrap of paper. You compare the scraps. Bob says the handle is on the right side. Alice says the handle is on the left. Can they both be right when their answers obveously conflict?

Two observers look from different points of view at what they both agree is one object. By virtue of their separation, each of them inhabits a different universe from the other. Bob's universe contains things that are not in Alice's universe, and in Alice's universe are things not contained in Bob's universe. But there is commonality between the universes, a set of things common to both. If the two points of view are removed from each other spatially to the extreme, what remains in common between them? Can the two points of view be removed from each other entirely so that they have nothing in common?

In spacetime terms, the act of separation, no matter how extreme, implies that separation cannot ever be complete. They were together once, and in space-time equivalence terms, that event cannot be erased or neglected. The act of separation implies that there was, and still is in some sense, unity. So universes have this quality, that if they have ever or will ever share a single point, they must always have at least one point, a spacetime point, in common. Further, if by the act of separation we mean that there is a direction in time in which the two universes share fewer and fewer commonalities, then there is a sense in which all of the commonalities continue to exist as spacetime events, so their degree of unity in that sense can never be reduced.

Note that in close intimacy (it was Marcus who mentioned underwear) two pointlike particles must experience a running value of alpha. Note that close to the "big bang" singularity, alpha was much different than it is now. Note that a hypothetical spacetime ship that could take us close to the big bang or close to point zero might as well be a banana for all the chance we have of building one. So in what way can we say with any certainty that Alpha Centauri is not King Kong just waiting for another nice banana to come into view?

"AND HE SCORES!" Ok, now I do feel much better. Thanks, Marcus, Kea, selfAdjoint, Shoshanna, and all, for the encouragement. I am very pleased to have something to think about while cleaning behind the line, other than ice hockey.

Be well,

Richard

 

[nightcleaner]

but one thing you say rings a big alarmbell for me and that is this:
if there would be some quantum theory of spacetime able to replace general relativity, then in that theory there would be no
objective spacetime (oh horrors)
there would only be the universe as you and I see it
and the universe as the alien in the distant galaxy sees it (if there is such an alien) and no guaranteed objective common spacetime.

Hi Marcus, Kea, and others who happen by (why not stop a moment and say hello? Even if only to let us know you have been here?)

First I regret that crack about human courtship rituals, and that other blatant opinion about ice hocky. I was stretching for some humor to fill the emptyness in my creative thought process, which temporarily (I hope) took a little vacation. I am sorry.

Kea, I agree with you that we have to acknowlege the possibility that observables are transformed by questions of distance, scale, and age in such a way that a hypothetical human observer, translated via wormhole or quantum entanglement or whatever, into a locality which we on Earth today can only see as a boundary condition, could plausibly find herself unable to interpret local events at that boundary using skills and tools brought wholesale along from home. Telescopes might not work, and stars might look like bananas, altho the banana image is probably too high a correlation. More likely in such a case a star or other object would not look like anything we could recognise or recall from home.

Marcus, I understand your argument, and that of the young post-doc, for a gauranteed objective common spacetime, but I think it is mostly a negative emotional response to the "insanity" of universes which might share a common boundary condition with our own, but have no other points of correspondance. We all like the rule that the laws of physics don't change from one locality to another. That doesn't make it the truth, but it is a premise we can work with. The alternative idea, even if it turned out to have some real basis, just doesn't work for us.

However, I wonder how you can assert "a gauranteed objective common spacetime" and in the same context uphold the idea that a good working scientific theory has to be background independant. Isn't an objective spacetime a background?

I myself prefer the middle view that the boundary conditions themselves are illusary. It is an horizon problem. If you project your co-ordinate grids to some extreme of any scalar field, such as that which is very far away, or very long ago, or very large or very small, or very intensely radient or very intensely gravitic, you will find, by my premise, that the projective grid co-ordinates do not, any longer, match local conditions. This is an unavoidable consequence of the premise that the universe is curved at infinity in every dimension.

In any location it seems pretty flat. In any other location it still seems pretty flat. But a comparison of the extended grid lines from the two locations does not have to agree. If the locations are sufficiently separate, there will be no agreement at all. The spectrum of any observable locally will not match or even be recogniseable as the spectrum of any observable taken at infinity.

I personally prefer to believe that all extreme boundary conditions are horizons, which means that if you could somehow approach the position of the boundary, as it is measured from home, you would not find any indication of a boundary when you got to that distant location. There is no barber pole indicating a sharp edge below at the north or south pole. The boundary condition is a clearly visible and definable object for any location, but if a being at a location tries to approach the boundary, she finds that there isn't any "there", there. No fence. No brick wall. No dotted line or other condition that indicates an absolute edge.

My position is that you carry your local conditions along with you when you travel, just as you might carry a tool box. When you get there, using your tool box, you measure things locally and find that the observables are still the same as they were at home. If you could travel to a quasar, you might not find an intensely bright object, but instead conditions much like those we have here in our own part of the universe. The boundary conditions make it seem like there is something unique and radically different at the location where we imagine there is a quasar, but if we actually could go there/then, we would find the universe looks very much like it does from where we are now. And, if we used the telescopes we brought with us, we would still look at the "edge" of the universe, and we would still find quasars there.

Thanks,

nc

 

[arivero]

The extended name for alfa is Sommerfeld's fine structure constant. Its first aparition is not in the Lamb Shift (where it appears, as Marcus tells, in a successive power series of alpha^n), but in the fine structure of atomic spectra. This is "easily" got from solving Dirac Equation in the atom of hydrogen.

There is a detail some people calls the "Sommerfeld puzzle". How did Sommerfeld get the precise formula for the fine structure, without using neither spin nor modern quantum mechanics? It is usually adscribed to a fortunate cancellation of both "errors", sort of mirroring Erathostenes fortune in calculating the circunference of Earth meridians.

 

[nightcleaner]

The extended name for alfa is Sommerfeld's fine structure constant. Its first aparition is not in the Lamb Shift (where it appears, as Marcus tells, in a successive power series of alpha^n), but in the fine structure of atomic spectra. This is "easily" got from solving Dirac Equation in the atom of hydrogen.

There is a detail some people calls the "Sommerfeld puzzle". How did Sommerfeld get the precise formula for the fine structure, without using neither spin nor modern quantum mechanics? It is usually adscribed to a fortunate cancellation of both "errors", sort of mirroring Erathostenes fortune in calculating the circunference of Earth meridians.

Hi arivero. Thanks for posting here.

I would like to know more about how to solve the Dirac Equation in the atom of hydrogen.

My quick review of Alpha gave me the idea that the spherical 1s shell and the dumbell shaped 1p shell are very close together in energy, and so it is "easy" (i.e. requires very little energy) for an electron to transition between the two states. Each transition is accompanied by the capture or release of a low energy photon. My guess is that this capture/release energy is somehow related to the fine structure constant. I am going to post this, and return with the Dirac equation in an attempt to open this for discussion.

Thanks,

nc

Dirac equation:
$$1+ \frac{E}{m_0 c^2}=\frac{1}{\sqrt{(1+\frac{\alpha^2}{|p+\sqrt{(j+\frac{1}{2})^2 -\alpha^2})|}}}$$

well, I am amazed that turned out on the first try. Only the phrase in the denominator under alpha^2 should be in absolute value brackets, which I don't know how to make in latex. (I made some touchups to latex next day...found abs. brkts on keyboard, and extended root sign to cover terms...rth)

This is my translation into latex from equation 3.67 in

http://www.autodynamicsuk.org/ApplicationBohr6.htm

 

[Hans de Vries]

Hi arivero. Thanks for posting here.

I would like to know more about how to solve the Dirac Equation in the atom of hydrogen.

I liked this here:

http://zopyros.ccqc.uga.edu/~kellogg/docs/rltvt/node1.html

Specially because he writes out Dirac in the "full matrix" form which
they should do in all textbooks. It's the only way to get some deeper
insight. see 1.2 and 1.3


Regards, Hans

 

[nightcleaner]

Thank you, Hans.

I have started a serious reading. In section 1.2, eq 1.4, alpha and beta are introduced. I guess alpha is our friend, the FSC, but what is beta? First thought: beta has to be a non-dimensional constant like alpha.

 

[Hans de Vries]

Thank you, Hans.

I have started a serious reading. In section 1.2, eq 1.4, alpha and beta are introduced. I guess alpha is our friend, the FSC, but what is beta? First thought: beta has to be a non-dimensional constant like alpha.

Mmm. That's a lack of characters, $\alpha$ and $\beta$ are 4x4 matrices whereby $\alpha_0 = \beta$

$$\alpha_0 = \left( \begin{array}{cccc} +1 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \ \ \ \ \ \ \ \alpha_1 = \left( \begin{array}{cccc} 0 & 0 & 0 & +1 \\ 0 & 0 & +1 & 0 \\ 0 & +1 & 0 & 0 \\ +1 & 0 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \ \alpha_2 = \left( \begin{array}{cccc} 0 & 0 & 0 & +i \\ 0 & 0 & -i & 0 \\ 0 & +i & 0 & 0 \\ -i & 0 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \ \alpha_3 = \left( \begin{array}{cccc} 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & -1 \\ +1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \$$

Often the gamma representation is used with the values in the lower-left 2x2 matrix inverted.

$$\gamma^0 = \left( \begin{array}{cccc} +1 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \ \ \ \ \ \ \ \gamma^1 = \left( \begin{array}{cccc} 0 & 0 & 0 & +1 \\ 0 & 0 & +1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \ \gamma^2 = \left( \begin{array}{cccc} 0 & 0 & 0 & +i \\ 0 & 0 & -i & 0 \\ 0 & -i & 0 & 0 \\ +i & 0 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \ \gamma^3 = \left( \begin{array}{cccc} 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \end{array} \right) \ \ \ \ \ \ \$$

Our friend alpha appears on the next page in the solutions of the hydrogen atom where it determines the ratio between the
major and minor parts of the solutions. Whereby the minor part is represented by the "anti-particle" fields, row 3 and 4.
Yet another place where our friend shows its importance. See section 1.3, eq 1.45

http://zopyros.ccqc.uga.edu/~kellogg/docs/rltvt/node5.html



Regards, Hans

 

[arivero]

Dirac equation:
$$1+ \frac{E}{m_0 c^2}=\frac{1}{\sqrt{(1+\frac{\alpha^2}{|p+\sqrt{(j+\frac{1}{2})^2 -\alpha^2})|}}}$$

well, I am amazed that turned out on the first try.

TeX is your friend... Now, that webpage is not very rigurous in calling this equation "The Original Dirac Equation". It can be deduced, that is true, by expanding from the solution of Dirac equation. But Dirac Equation knows about spin, while this one knows only about total angular momentum. Such is the "Sommerfeld Puzzle" I told before: How gets the Relativistic Bohr-Sommerfeld equation (which is the accurate name for it) to target the right fine structure splitting?

The usual lore -I am not sure I buy it- is that the equation has two opposite failures. On one hand, using the Bohr-Sommerfeld approach, the vacuum has energy zero, instead of the usual wh/2. Ie the levels are proportional to n h instead to (n+1/2) h. And this "failure" exactly compensates the lack of spin. Or so they say. I do not like this explanation because when one goes up to quantum field theory, the first thing one does is to apply "normal-ordering", an standard procedure to remove the w h/2 from the low energy state. In some sense, Sommerfeld (Bethe's tutor, by the way) was doing it better that Heisenberg.

NOTE: the quoted webpage follows Sommerfeld work up to equations 3.41 and 3.42, then it "rederives" the result from the point of view of the author's interpretation of relativity. As it is mostly a point of view, one can not see fundamental changes. The problem with Sommerfeld equation was the intensity of the lines, ie how to determine the allowed and forbidden transitions; this become cleared thanks to Dirac spin.

NOTE2: Hmm the name of the author of the webpage sounds strange... Is he the son of some Spanish or Italian anarchist?

 

[arivero]

A related theme... alpha contains both c and h. So what about the limits h->0 and c->infty

 

[marcus]

...
NOTE2: Hmm the name of the author of the webpage sounds strange... Is he the son of some Spanish or Italian anarchist?

I think the author's name is Charles Brian Kellogg
and he was at Uni Georgia and he is now at NIST, in Maryland

but the URL has a strange name in it "zopyros"
is this the name you noticed?

 

[nightcleaner]

I am getting into my work mode, as tonight I will be degreasing the lines again. Business is picking up so there is more to do. Owner will probably ask me to work more hours soon, maybe even five or six nites a week.

About the matrices. Why are they always two dimensional? Wouldn't a three dimensional matrix be useful?

I have to eat and rest before tonight. I may get to check in here for a few minutes before I go to work. Plenty to think about. I may have to go over matrices again, and Dirac delta.

Be well,

thanks,

Richard

 

[arivero]

I think the author's name is Charles Brian Kellogg
and he was at Uni Georgia and he is now at NIST, in Maryland

but the URL has a strange name in it "zopyros"
is this the name you noticed?

No, no. I refer to "Ricardo Libertario Carezani", in the webpage about "autodynamics". Very amusing name.

 

[selfAdjoint]

About the matrices. Why are they always two dimensional? Wouldn't a three dimensional matrix be useful?

Matrices are always 2-D grids, but tensors can be multidimensional; a rank 2 tensor looks locally just like a matrix and behaves like one too, till you change coordinates. But tensors can be any rank (as well as covariant or contravariant, see the thread on that down in the differential geometry forum). A rank 3 tensor is a 3-D array and a rank 4 tensor like the famous Riemann-Christoffel curvature tensor would take four dimensions to display! Try that in LaTeX!

 

[nightcleaner]

Thanks, selfAdjoint.

I am working again tonight so only have a little time to go into this. I think I understand then that a rank three tensor can be completely displayed by three matrices, altho it has other ways in which it can be divided up. Think of a cubic stack of twentyseven boxes, each box containing an element in the array. It has a front, a side, and a top. You can see every box in the array, by looking at three matrices, the front nine boxes, then the nine immediately behind them, then the nine immediately behind them. Or, you could first look at the nine boxes in the top layer, then the nine under that, then the nine on the bottom. One further arrangement of three matrices is possible, starting from the side.

The boxes could be uniquely identified in the three dimensional array, or rank three tensor, by some notation involving three indices, such as a_(ijk) where i, j, and k can have values of 1, 2, or 3. The set of permutations of (ijk) in 1,2,3 would uniquely name each of the boxes in the set.

I remember reading some of this in Bohm, Quantum Theory, also in Byron and Fuller, Mathematics of Classical and Quantum Physics, also in Shadowitz, The Electromagnetic Field. I have these three books from the big box book store in the Dover editions. I have started to read each of them, gotten through the first few chapters of each before stalling out. But I am making progress and understand the ideas better now than two years ago.

Is the above essentially correct? Usually I start getting confused when they talk about the Dirac delta, but slowly I learn.

Thanks to all for helping

Richard

 

[selfAdjoint]

Thanks, selfAdjoint.

I am working again tonight so only have a little time to go into this. I think I understand then that a rank three tensor can be completely displayed by three matrices, altho it has other ways in which it can be divided up. Think of a cubic stack of twentyseven boxes, each box containing an element in the array. It has a front, a side, and a top. You can see every box in the array, by looking at three matrices, the front nine boxes, then the nine immediately behind them, then the nine immediately behind them. Or, you could first look at the nine boxes in the top layer, then the nine under that, then the nine on the bottom. One further arrangement of three matrices is possible, starting from the side.

The boxes could be uniquely identified in the three dimensional array, or rank three tensor, by some notation involving three indices, such as a_(ijk) where i, j, and k can have values of 1, 2, or 3. The set of permutations of (ijk) in 1,2,3 would uniquely name each of the boxes in the set.

I remember reading some of this in Bohm, Quantum Theory, also in Byron and Fuller, Mathematics of Classical and Quantum Physics, also in Shadowitz, The Electromagnetic Field. I have these three books from the big box book store in the Dover editions. I have started to read each of them, gotten through the first few chapters of each before stalling out. But I am making progress and understand the ideas better now than two years ago.

Is the above essentially correct? Usually I start getting confused when they talk about the Dirac delta, but slowly I learn.

Thanks to all for helping

Richard

Yes you have it right. That 27 is for a tensor in 3-space. For 4-D you'd have 4X4X4 = 64 cells; that's what the Riemann-Christoffel tensor (also called the curvature tensor) has. Generally a tensor of rank k in n-dimensional space has n^k components.

 

[nightcleaner]

I want to say that it should be 4x4x4x4, or a_(ijkl) where ijkl are permutations of 1,2,3,4. Is that wrong, then?

nc