Dividing particles into 2 types in 3 ways

Tues. about 2:30 p.m. 3^rd day of June 2014.  Hoo boy.  It’s been 40 years since I enrolled in my first physics class.

Here’s a statement that I keep thinking about, mainly because I can’t remember it.  It’s from Freeman Dyson’s article in the NYRB of 6 March 2014:  “We can divide particles into two types in three different ways.”  This is a statement that I can’t forget, but also can’t precisely remember. In my mind, I start it as “We can divide two particles…,” so I’m immediately lost.  I guess the unhelpful choice of the words “divide particles” just throws me off the track.  Then, if I do remember to start with “We can divide particles” I can’t figure out how to bring the “two” in properly.  We can divide particles in two?  We can divide particles into two groups?  Or what?

One thing: “types” and “ways’ together—that bothers me somehow.  When I first read the sentence back in late February, I was impressed with how cleverly it covered that whole territory of charged vs. neutral, strong vs. weak interaction, and fermion vs. boson.  The three-fold way!  But where are the two types?  I’m sorry, that’s of course obvious.  But particles each have all three properties.  Once the properties are assigned, we have eight types, not two, or “eight possible combinations” as Dyson says later.

So can I say it better than Mr. Dicey Dyson does?  Not right now.  I need a nap here at 2:51 p.m.

4 pm. Nope, I can’t say it better.  Except “types” and “ways” still bothers me the way he used it.  Okay, so I’ll reverse it.  “We can divide particles into three different types in two different ways.”  I’ll think more about that.  Still bugs me.

3 types based on charge, interaction, spin.

2 ways:            yes/no, weak/strong, integer or not.

About 9 pm Friday, June 20^th.  Just a note to myself, a reminder of what books I’m currently keeping in mind as far as research is concerned (learning is concerned).  Lawrie’s Grand Tour, A&H’s Gauge Theories, Low’s Classical Field Theory, José & Saletan’s Contemporary Dynamics, J. Townsend’s Quantum Mechanics Intro, Holstein’s Weak Interactions in Nuclei, and some others that don’t come to mind right now.  Also I’m presently reading “Matrix Mechanics of the Infinite Square Well and the Equivalence Proofs of Schrödinger and von Neumann.”  A fine and richly referenced (rich with references of great interest) article in AJP.

7:15 a.m. June 21.  Summer Begins.

What bothers me about Dyson’s description is that by itself it makes no sense—at least not to me (see previous page).  In a way, it’s a parody of Dyson’s inclination to put numbers to his proclamations.  I may look up an example later but it’s something like saying there are two ways we could improve our understanding of so-and-so, or there are three reasons why we should improve such-and-such.  “Particles can be divided” is also a bad way to start off the sentence.  Nothing is being divided.

Electric charge, spin, and means of interaction are particle properties.  Actually, these are all “means of interaction,” a term I was using [as it is commonly used among theoretical physicists] only to describe the force—what type of force—the particle recognizes and responds to.  But charge or no-charge is a property that determines whether a particle responds to the electromagnetic force.  (Well, what about the neutron’s magnetic moment?)  (To be discussed later!)  Dyson, however, leaves out the electromagnetic force in his “divide into two types,” or maybe he includes it in the weak force without saying so.  His two types of force are strong and weak.  Spin is integer [boson] or half-integer [fermion].  Charge is either neutral or weak, oops, neutral or charged.

So I would say—as a first attempt to correct the indecipherable numerical shorthand of Dyson’s sentence—that elementary particles have three properties, each of which manifests itself in two ways.

But of course, this categorization [Dyson's categorization] leaves out the further property of plus or minus charge.  And I don’t know what all else (strangeness comes to mind, but I don’t recall its importance at the moment).

One thing I’ve managed to accomplish here is to again bring out the point that spin does have an influence very much like electric charge.  The exclusion principle causes Fermi-Dirac particles to interact in such a way as to prevent them [any two of them] from being in the same quantum state.  This is like particles of the same charge repelling each other.  The Bose-Einstein particles can all be in one quantum state, which gives them a property or means of interaction like neutral particles:  that is, no means of interaction!

Why are the force laws the way they are?

From today's perspective, the crucial thing about electromagnetism is that it is a theory in which the dynamics (i.e. the behaviour of the forces) is intimately related to a symmetry principle.  In the everyday world, a symmetry operation is something that can be done to an object that leaves the object looking the same after the operation as before.  By extension, we may consider mathematical operations--or 'transformations'--applied to the objects in our theory such that the physical laws look the same after the operations as they did before.  Such transformations are usually called invariances of the laws.  Familiar examples are, for instance, the translation and rotation invariance of all fundamental laws:  Newton's laws of motion remain valid whether or not we translate or rotate a system of interacting particles.  But, of course--precisely because they do apply to all laws, classical or quantum--these two invariances have no special connection with any particular force law.  Instead, they constrain the the form of the allowed laws to a considerable extent, but by no means uniquely determine them.  Nevertheless, this line of argument leads one to speculate whether it might in fact be possible to impose further types of symmetry constraints so that the forms of the force laws are essentially determined.  This would then be one possible answer to the question:  why are the force laws the way they are?  (Ultimately of course this only replaces one question by another!)

       -- I. J. R. Aitchison and A. J. G. Hey, Gauge Theories of Particle Physics, 2nd ed., 1989, p. 42.

This is a good summation of the entirely pervasive idea that symmetry constraints (also called invariance principles) are behind the "behaviour of the forces" as we now understand them.  Exactly how you get a force field out of a symmetry principle is also discussed by A & H, but we'll have to come back to that later. Shades of Larry's visit with Rabi Nachtner here... questions that result in more questions.

I should add that there's the rather contradictory idea, one that I'm in agreement with, that gravity is not a force.  Yes, this most noticeable and inescapable presence is a different animal from, say, electromagnetism.  Objects responding only to gravity, like a satellite in orbit or (neglecting air's influence, which you really can't) a tossed basketball, are following their simplest paths in spacetime.  (Yes, spacetime, not just space.)  It takes a force of some kind to knock them into or onto some other path.  Or to hold them in one position, as you and I are being held in position now by electromagnetic forces keeping us from falling further toward the center of Earth. 

Llewyn Davis' undamaged face

Despite getting punched hard in the face during the alley-behind-the-Gaslight-Cafe scene in Inside Llewyn Davis, Llewyn's face is shown subsequently to be undamaged.  No bloody nose, no swelling or redness on the side of his face at the end of the movie. So was that sequence a dream?  Like many questions concerning the timeline and meaning of the scenes in the movie, it would be difficult to say where the reality and the dream separate, if there is a dream sequence in the movie.  But, especially on the morning AFTER getting clobbered in the face, we should be able to see swelling on Llewyn's face.  Once we see a repeat of the opening scenes in the last scenes  of the movie, we realize that the morning the cat gets out is not the morning after Llewyn's Gaslight gig.  Still, the closing scene should show Llewyn with some damage to his face from the fist of the big Arkansas fellow.

The unreality of that is like the unreality of Llewyn seeing the movie poster for The Incredible Journey, since the movie wasn't released until 1963 or '64.  Using the literal meaning of "incredible," or perhaps its original meaning, which is "not credible," one interpretation of the seeing of the movie poster is that it is Llewyn's journey that is not credible.  We know the seeing of the movie poster itself is not credible if we assume the year is still 1961 when he sees it.

Also we have the name  Llewyn to compare to the name Llewelyn of the character in No Country for Old Men.  I guess I'm like Larry in A Serious Man, who can't help feeling the questions.  Why do the Coens make us feel the questions if they aren't going to give us the answers?  To paraphrase Rabi Nachtner:  "They haven't told me."  That in itself, of course, is an example of a question without an answer. 

Gauge (quantum field phase) symmetries

About 9:45 pm now [13 Jan 2013].  Wigner, whom I once heard give a talk at UALR (1981, with P. J. Cheek), makes a point of charge conservation being different somehow in its derivation than the other conservation laws--momentum, energy, angular momentum, for instance.  That was his point of view in 1949--I don't know how it changed later.  But, his discussion starting on p. 10 of "S & R" was very abstract anyway.  So I'm just noting it here, and also noting that he mentions . . .

                                                     charge quantization!

as something not explained by theory.  (Oh yeh, if only we had Dirac magnetic monopoles, but that seems forgettable now).

So, with this pain in my right side bugging me tonight as I sit in bed writing this, I will turn to Crease & Mann's page 92 discussion of, if not quantization or conservation exactly, at least the reason for the electron's charge to be a physical constant:

"As Fritz London showed, the reason the phase can shift and not affect the charge is because of the gauge symmetry, which compensates for such changes by creating virtual photons--and hence an electromagnetic field--whose actions ensure the conservation of electric charge."

Okay, so it is about the idea of charge conservation. I'm looking for a similar statement in 't Hooft's 1980 Sci Am article but can't find it at the moment.  But here's something else from the great Gerard, p. 8, 3rd column:  "It is the symmetries of this kind, where the phase of a quantum field can be adjusted at will, that are called gauge symmetries."

10:20, time to quit.  Actually, at the top of the last page [blogwise, at the bottom of my June 20th post] I quoted the sentences that follow the one of  't Hooft's I just quoted.  So I've come full circle somewhat since this morning just before noon, or just after.

Symmetries and Reflections brief excerpt

Okay, I said I'd look at Wigner, page 10, so here is a relevant quote.  The subject matter is electric charge conservation, the idea that the total amount of charge in the universe doesn't change.  Charge can be created and destroyed, but only in pairs of plus and minus charges.  Nowadays, there is a popular idea for why charge is conserved.  It's related to the phase of the electron's "matter wave."  The dissonance still exists, if you ask me--to be discussed later!  Here's Eugene:

"My account of the role of invariance in quantum mechanics would remain grossly incomplete if I did not mention a dissonant sound in the harmony of quantum mechanics and the older theorems of invariance.  This is the conservation law for the electrical charge.  While the conservation laws for all other quantities, such as energy or angular momentum, follow in a natural way from the principles of invariance, the conservation law for electric charge so far has defied all attempts to place it on an equally general basis.  The situation was, of course, the same in classical mechanics but the simplicity of the connection between invariance and the ordinary conservation laws makes the situation even more conspicuous in quantum mechanics."

From Symmetries and Reflections: Scientific Essays of Eugene P. Wigner, Indiana University Press, Bloomington and London, 1967.  This essay is called “Invariance in Physical Theory.” It was first presented as a talk at “the celebration honoring Professor Albert Einstein on March 19, 1949, in Princeton.” Einstein had turned 70 years old on March 14 of that year, thus the celebration.

Abolish virtual particlism!

What is gauge invariance, by the way?  Well, here on Sunday night, January 6^th of 2013, let’s not precisely answer that question just yet.  Instead, I would like to extract some quotes from “A Dictionary of Physics,” published by good ol’ Oxford University Press in 1996.

There is no entry for gauge invariance, but there is one for gauge boson, and one for gauge theory.

Last paragraph of the gauge theory entry—well, the 2^nd paragraph of two—is:

“In gauge theories the interactions between particles can be explained by the exchange of particles (intermediate vector bosons, or gauge bosons), such as gluons, photons, and W and Z bosons.”

Which leads us back to the entry for gauge bosons:  “A spin-one vector boson that mediates interactions governed by gauge theories.  Examples of gauge bosons are photons, in q.e.d., gluons in q.c.d. and W and Z bosons that mediate the interactions in the Weinberg-Salam model (see electro-weak theory) unifying electromagnetic and weak interactions.  If the gauge symmetry of the theory is unbroken, the gauge boson is massless.  Examples of massless gauge bosons include the photon and gluon.

“If the gauge symmetry of the theory is a broken symmetry, the gauge boson has a non-zero mass, examples being the W and Z bosons.  Treating gravity, as described by the general theory of relativity, as a gauge theory, the gauge boson is the massless spin-two graviton.”

I don’t want to treat gravity as a gauge theory!  So how to do it?  Goodnight.

Well, first I want to write down the dictionary’s definitions of broken symmetry, electroweak theory, q.e.d. and q.c.d.  The initials are my abbreviations, not the dictionary’s, on previous page as well as on this one.  Now maybe I can say goodnight and mean it!  9:45 p.m.

5 a.m. almost, Thurs. Jan. 10^th 2013.  So, if the e.m. field could be “turned off,” isotopic spin symmetry would be exact.  This is stated in ‘t Hooft’s Sci. Am. article of June 1980, first page.

If you are wanting to have unified description of forces, it would be best to have the isotopic spin be exact when the e.m. field is turned ON, I would think.

“In order to make a theory invariant with respect to a local transformation, something new must be added:  a force.”  P. 5 of ‘t Hooft article.  Need to look at Holstein book to compare how he discusses this.

We’ve got:  force = field = particle (virtual).  Not good!  How to make it better?

See also Icke’s description in his Force of Symmetry book of how uncertainty prinziple governs virtual particle so-called interaction.  Now, Icke’s book is pretty Ickey, but I do want to keep his virtual particle description in mind.  Gerard ‘t Hooft writes well.  Can’t say same for his fellow Dutchman, Vincent Icke.

Later.

All right, 8:45 pm Jan 10^th Thurs., in bed now, with Jessie and Icke.  And a pen, this journal, and a desire to poke holes in the modern description of symmetry.

Which is why I’ve returned to Icke’s book.  Chapter 7, “Symmetries,” to be precise.  “There must be at least some things that are more allowed than others.” = end of 1^st paragraph.

2^nd paragraph talks about many things being forbidden, “and it is the forbidding rules that give structure to the world.”  Who boy, that’s a compelling thought!  But summarizing the “thou shalt not” rules is overwhelming.

So we want to “lump together a large (possibly infinite) number of rules in a single all-encompassing one.  Thus you might write in the Constitution of your universe, “Do unto others as thou wouldst others do unto thee.”  Such an overall rule, Icke says, “that summarizes a multitude of individual possibilities, is called a symmetry.”

Redundancy is what it’s coming from, however.  Think about that, and realize you don’t want redundancies to rule the basic theory of … everything!

Sleepy now.  Went to Lake Village today, for SEAEDD meeting.  Goodnight.

10:30 a.m. Sunday January the 13^th 2013.

How would I explain “gauge invariance” if asked?  (Dvorak Symphonies 5, 7, 9 playing on CD changer.)  Well, it would be difficult.  On page 112 herein, I note that there was no entry in the Oxford Dictionary of Physics for gauge invariance, only one for “gauge theories,” and one for “gauge bosons”.

So let me start trying to think of how to explain gauge invariance by enumerating other terms in physics where “gauge” is present:  gauge theory, gauge boson, gauge transformation, gauge condition, pressure gauge – ha ha ha got you there!

I need to go back to Holstein’s book.  I’ve quoted it in here, I’m sure, but don’t recall when, and so will turn pages backwards and see what I said about what he said.  Pardon me while I turn back the pages.

Okay, pages 92 thru 96.  Symmetry group, or transformation group, is the key idea or at least the primary tool used in gauge theories.  After all, “invariance” means invariance with respect to some transformation.

So “gauge invariance” means invariance with respect to a gauge transformation.

What is a gauge transformation?  (9^th Dvorak Symphony, “New World,” 1^st movement, playing.)

Well, of course we have the global and the local transformations of “gauge.” So “gauge group” seems to be the most relevant of the terms involving the word “gauge.”  In fact, saying the more meaningfully, we would say “gauge transformation group.”

We are talking about “potentials” also.  Strange word, but quite relevant!  The fields, say E & B, are invariant with respect to CERTAIN changes in the potentials, A_μ , in electromagnetism.  Now a new term comes in: “gauge field.”  For instance, Holstein says, “the price for achieving local U(1) invariance is the introduction of a new gauge field, A_μ .

So then this means we’re calling the potential(s) A_μ a field? 

And don’t forget about the most excellent term “Lagrangian,” used as a noun. Holstein: “… by demanding the mathematical requirement of local U(1) invariance, we have produced a Lagrangian that is known to have physical significance.”

To say it another way: By requiring our Lagrangian to be invariant with respect to a certain gauge transformation group, U(1), we find that it can be used to generate the Maxwell equations for the electromagnetic sources and fields.

Of  course, saying something is a “member” of a group is more proper, so leaving out the word “group” above is more proper, too. But never mind PROPER!  P-ROPER!  We gotta get beyond proper, elegant, and sophisticated.  “We” meaning I.  Abolish virtual particlism!

Now it’s about 11:45 a.m. and I’d like to finish this up by noon and get out to the BB Trail with knee boots on, to take Jessie and myself for a walk.  (Steely Dan’s “Black Cow” is playing now.)

The thing to be discussed is electric charge conservation.  Gerard ‘t Hooft notes that although the absolute value of the phase of the electron wave in quantum electrodynamics is irrelevant for the outcome of experiments, “in constructing a theory of electrons it is still necessary to specify the phase.  The choice of a particular value is called a gauge convention.”

Well now it’s 12:10.  Time for a break.  Wigner’s Symmetries and Reflections book, p. 10 on the subject of charge conservation, will be up next.

The cat in Inside Llewyn Davis

Time to take a break from math and physics and discuss a movie.  Well, it's related to the subject of this blog, in both the Serious Man and Schrödinger's Cat senses.  Because it's related to the Coen Bros messing with our minds (again) again.

The timeline of Inside Llewyn Davis is a lovely Coen Brothers' thought experiment.  The last scenes in the movie are the same as the first scenes. There's not a one-to-one correspondence in dialogue, but that's because some sentences are left out either in the first or the last version. The setting is the Gaslight coffeehouse.  Llewyn does his time on the stage, one song in the first version and two songs in the last version, but it's the same night in each case. Then he gets whupped up on by the man in the alley, with the last version of this incident showing more of it while also leaving out some dialogue that was in the first version.  In the last version, Llewyn says to the disappearing taxi with Elizabeth Hobby's husband--the Arkie who hit and kicked him--in it:  "Au revoir."  These are the last words of the movie. They literally mean "to the seeing again."

So the end is also the beginning, and it's a Groundhog Day continuous loop of a movie--apparently!  There's something else going on with the cat, however.  Possibly. The cat first appears in a segue with the man in the alley walking away.  The cat, also seen from behind, is walking down the hallway in the Gorfein's apartment on Riverside Drive, where Llewyn is asleep on the couch.  The cat wakes up Llewyn by sitting on his chest.  We are led to think this is the morning after his gig at the Gaslight and his alley confrontation.  But the movie's superposition of the end of the alley incident and the cat walking down the hall--the fading out of the man in the alley as the cat in the hallway fades in--is a clue that something weird may be going on.  And it's not until the final scenes that we see what it is.

So timewise the movie really starts with the cat in  the hallway.  The preview of Llewyn's Gaslight gig and alley scene is just that, a preview.  We think we know it's the next morning when the cat wakes Llewyn up (the first time), but  further events in the movie point to it being the morning of his gig, not the morning after That's the Coen brothers' time trick number one. The cat then gets out of the apartment door in spite of Llewyn's trying to block it with his foot.

The subsequent scenes in the movie lead up to his getting the gig at the Gaslight, and to his staying at the Gorfeins the night before the gig.  The next morning he wakes up again with the cat on his chest--the details are different, but it could be thought of as identical to the Gorfein-apartment scene  near the beginning of the movie.  However, this time Llewyn is able to block the cat with his foot as he's leaving.

So we have a Schrödinger's Cat superposition situation:  the cat gets out and the cat doesn't get out.  Apparently it's not the same day, but I've gotta look at the movie a little more before I could say that.

Now that I've looked at those scenes again, I have to say:  apparently, it is the same day!  In the beginning of the movie, we see the note Llewyn is writing to the Gorfeins as he's leaving.  He is apologizing for being "a mess" the previous night.  In the end-of-movie scenes of Llewyn leaving the apartment we see him writing the note, but aren't able to see what he's writing.  However, we know he was a mess the previous night because we see him spitting on the floor at the Gaslight and then heckling Elizabeth Hobby--apparently because he's pissed at Pappi, the owner of the Gaslight, for having slept with Jean. Apparently Jean got Llewyn the gig by having sex with the rather undesirable Pappi.

Here's my best guess on the apparently-the-same-day scenes:  this is trick number two.  All we are shown is that the experimental set-up is the same, meaning the cat-gets-out morning looks enough like the cat-doesn't get out morning that we can't really say it's not the same morning.  The Coen brothers start the two morning sequences with a different cinematic effect, however.  Near the beginning of the movie we have the segue of alley scene into cat scene.  Near the end, as Llewyn is going sleep, there is a fade out, or fade-to-black. So that we are given a very different cinematic effect of the next mornings' opening scene.  It fades in, rather quickly, from the  fade out of the previous night.  Then we see the "experimental set-up" as it was in the Gorfein-apartment-morning-scenes near the beginning of the movie.  Not identical, but a close approximation.

Then, the cat doesn't get out.  But on his way to the Gaslight, Llewyn sees a poster of the Walt Disney movie The Incredible Journey.  According to the Wikipedia article of the same name, this book was published in 1961 but the movie didn't come out until 1963.  Aha!  Or actually, not "aha!"  Just another Coen bros conundrum. (If  we consider that it could be February 1964 that the movie was playing in NYC, then we have another "three years" anachronism, similar to the Santana Abraxis one in A Serious Man.) Anyway, we can imagine using the many-worlds interpretation of quantum mechanics to say the cat superposition of not-getting-out and simultaneously getting-out divides upon a measurement--a viewing of the cat--into two separate universes.  Have fun thinking about that!  And also you might think about how Larry in A Serious Man and Llewyn are both in one very real sense serious men:  we never see them laugh, except for a little chuckle from Llewyn when Jean tells him he'll have to share the money from his Gaslight gig donations with another musician who's performing.  In regards to laughter, the Coens themselves are just the opposite, even to the point of responding to recognition of human vulnerability with laughter.  I'm like that, too.

Speaking of vulnerability, we can't forget Mikey's suicide, a specter that haunts the movie, although oddly enough, if you zoom in on the album notes under Mike Timlin's photo (Gorfein morning # 1), the name Van Ronk is shown in the little bit of text that can be seen.  However, it's Llewyn himself who is  supposed to be a simulacrum of Dave Van Ronk.  Also, the fat and nasty jazz musician Roland Turner (John Goodman) claims nobody used the George Washington Bridge for suicide--it's supposed to be the Brooklyn Bridge, he says.  But the only suicide I've ever heard of taking place from either bridge (yeh, I'm not really up on that subject) is the one in James Baldwin's novel Another Country.  Rufus Scott is a despondent jazz drummer who commits suicide by jumping off the George Washington bridge.  The novel was published in 1962.  (Hmm.)  I read it in 1977 or '78.  The only thing I really remember from reading it is the description of one of Rufus Scott's shoes coming off because of the air rapidly rushing by as he is falling