Equality under transformation

The idea of symmetry in physics and math can be described as equality under some sort of transformation.  

Our usual idea of mathematical equality is an equation, which is a test for numerical equality and involves finding the values of the variable that make the equality true, if such values can be found.

Or, in higher math, where symbols and not just numbers are the currency of the realm, when you "solve an equation" you have a new equation for the dependent variable (the strength of an electric field, or the temperature distribution on the surface of a frying pan, for instance) in terms of the independents (time and place, for instance), and an equation is a test of whether you can accomplish that or not. Boundary conditions and/or initial conditions are also relevant to solving an actual physical "problem".

In contrast to numerical equality, a "transformation" can be considered as a test for geometric equality: what value of the variable (the rotation angle, for instance) produces an arrangement that is identical or equal to the original arrangement?  A simple example is the rotation of a square about its center. There are four angles of rotation where exact symmetry or geometric equality is observed:  90, 180, 270 and 360 degrees.

The word used more commonly than "equality" when it comes to a geometric figure is "invariance."  In physics, symmetry and invariance are really synonyms. What sort of symmetry or invariance is involved is the next question that naturally arises, as in 4-fold symmetry (for example the square), 8-fold symmetry (octagon), and continuous symmetry (circle).  And that's just looking at a few cases in only two dimensions--there are other geometric figures besides the square that have 4-fold rotational symmetry (a plus sign is one of them--you can try to think of others).  In general, the type of symmetry or invariance is labeled or categorized by the symmetry group or transformation group, to be talked about later.

Cosmic inflation as supercooling; a 1979 memory

[In an earlier post that I recently deleted, I criticized Weinberg's recent writing and Steely Dan's Gaucho album for their cool mechanical/technical precision and lack of any hint that a human being was responsible for their creation.]

Let me get down off my high-as-the-cat’s-back horse and say I’m not really equipped to read Steven Weinberg’s Cosmology textbook.  Also, I need to listen to Steely Dan’s Gaucho album again before I comment on its crystalline production quality as a negative aspect.  There, I just put the album on.  I’ll write while I’m listening to it.  Here come those Santa Ana winds again. 

As regards Weinberg's book: I’ve told students before that there are three reasons why a person would have trouble reading and understanding something.  

One reason is that the vocabulary is beyond what you know, meaning the material is too advanced for your present state of knowledge.  I think this is my problem with Weinberg’s book.  It is, after all, a graduate-level textbook in cosmology and gravitation. I need to read and work some exercises at an advanced undergraduate level before I can tackle it.  Or else work harder at understanding it, and not expect it to be an easy read.  I've got a copy of J. V. Narlikar's Introduction to Cosmology (2nd ed, 1993) that uses general relativity at the advanced undergraduate level and also seems quite readable, with reasonable exercises at the end of each chapter.

Another reason you might not understand what you’re reading is that you are not in a tranquil enough emotional state to be able to concentrate.  Okay, maybe tranquil isn’t the right word.  You can possibly be jazzed up on caffeine and comprehend what you’re reading.  Usually my mind is going all over the place after drinking coffee—not an ideal state for reading.  The coffee buzz is better for writing than reading (I’ve had a cup and a half so far this morning). So let’s say one reason you might have trouble reading and comprehending something is that your mind is not in the best possible state for reading. 

The third reason is that the writing is not very good.  That’s a matter of taste, partly.  I guess many people enjoy Brian Greene’s books, but I’m not one of those people.  You shouldn’t have to work at reading a book.  It should be enjoyable, no matter what the subject.  I’ve tried to read The Elegant Universe and The Fabric of the Cosmos, but to no avail.  Too much work.

Of course, in the realm of textbooks the expectation for enjoyment is pretty low, but the book should be well written nevertheless, with similarly well-written exercises that aren’t overly obtuse and don’t require great cleverness to solve.  I’m kind of getting off the subject now, back into the complaint mode, so I’ll move on.

On Monday, walking into the first floor entrance of the Pine Bluff library, I stopped to look at some of the library books being offered for sale.  Some books had been added to the sale since the last time I stopped and looked, and Lo and Behold, the first book I set my eyes on was The Inflationary Universe, by Alan Guth!  It was among a bunch of physics and astronomy books that had been put in the sale since last time I passed by the tables.

Actually the sale is more of a giveaway.  No one is monitoring the books by sitting there to tell what the prices are or take money.  It’s the honor system.  A note taped to the wall says the prices are ten cents to fifty cents, which I guess means paperbacks are a dime and hardbacks are fifty cents.  In order to pay, you have to go up in the elevator to the circulation desk , since the first floor is just the entrance to the children’s library and also where the library's one elevator is located. I usually don't use the elevator and instead go up the civic center steps outside and use the door on the main floor near the circulation desk.

Guth’s book, published in 1997, looks like a good one from what I’ve read of it so far. Another well-written book I’m reading  is Sean Carroll’s From Eternity to Here.   He has a chapter on cosmic inflation, and he quotes what Guth wrote in a notebook in December 1979 about the possibility of an inflationary period in the early part of the life of the universe. (That the universe "started" sometime in the past, and especially that it started with a big bang, just gives me the epistemological and ontological willies.  Hard to fathom.) 

Guth didn't use the word "inflation" when he wrote down his "spectacular realization" as he called it.  He instead used the word "supercooled," and that makes more sense to me than the word inflation.  Fluids such as water can be supercooled and superheated, and suddenly freeze or boil, respectively, because of a slight disturbance.  Our most common experience in normal life is a liquid that is superheated after being in a microwave oven.  It is at the boiling point, but doesn't boil until you reach in to get it out. Moving the container is the disturbance that causes it to suddenly boil.

Supercooling can happen to steam or to liquid water.  In the case of the liquid state, the temperature can be reduced to below water's freezing point, maybe very rapidly reduced, and the water doesn't freeze until there is a slight disturbance, then it all freezes at once.  Similarly, steam that is supercooled can suddenly condense into liquid water all at once.  These changes of state between gas, liquid, and solid, no matter how they occur, are known as phase transitions.

So in the Big Bang, the very hot, early  universe suddenly for some reason supercooled by expanding exponentially for a short time period.  Here is what Guth wrote in his notebook in December 1979, the same time I was finishing my work on a bachelor's degree in physics at Hendrix College:

SPECTACULAR REALIZATION:  this kind of supercooling can explain why the universe today is so incredibly flat--and therefore resolve the fine-tuning paradox pointed out by Bob Dicke in his Einstein Day lectures.

March 14, 1979, by the way, was the hundredth anniversary of Einstein's birth.  Nineteen-seventy-nine was also the year Steven Weinberg, Abdus Salam, and Sheldon Glashow won the Nobel prize for their theoretical work uniting the weak and the electromagnetic forces.  So, while Alan Guth was having his December revelation on inflation in the early universe, Glashow, Salam, and Weinberg were receiving their Nobels and I was creating a new student lab at Hendrix that involved using a light-emitting diode to find Planck's constant.

Now that I'm remembering it, I have to tell about my discovery when I was working on that lab.  LEDs were still something of a novelty in 1979.  Almost all of them emitted red light, but some green ones were starting to come into use.  The one I used for my lab was red, but the voltage I measured across it was not the right voltage I needed if my lab was going to predict an accurate value of Planck's constant.  Actually it was not the right voltage to correspond to an LED that emitted red light.  My lab project was looking pretty doomed at that point, in late November. 

Then I decided to look more closely at the LED itself while it was on.  I used a little cylindrical magnifier that I'd bought earlier that year at Radio Shack, made for inspecting a turntable's needle (stylus) for wear.  Under magnification, the LED could be seen to emit green as well as the dominant red light!  Green light takes more energy and therefore more voltage to produce.  The voltage across the LED was just right for producing green light, and based on that, my idea of using an LED was a new way to measure Planck's constant (with a plus or minus 10% error) in a student lab.  I tried other LEDs, both green ones and red ones, and found they could also be used reliably in the experiment.

Now I'll comment again on Gaucho, which I listened to while writing this. Throughout 1979 Steely Dan was working on this album, which was to be their last as a band.  The songs just aren't very good compared to Steely Dan's previous albums. They sound more like they were produced by a programmed song-writing machine than by people. So all this effort--two years' worth--was put into producing a great-sounding album (Mark Knoppler plays on it, for instance), but the "content" was lacking.  

Aja, Steely Dan's 1978 album that preceded Gaucho, is one of my favorite albums.  The sound quality on it is also superb, as is Steve Gadd's drumming, but mainly the songs are good, especially Black Cow, Deacon Blues, and Aja:  Up on the hill / they think I'm okay / or so they say. 

Discovery of evidence for cosmic inflation

First of all, I don't really understand it.  Haven't had time to read much about it yet, and was never that interested in the idea of cosmic inflation during the early part of the universe's existence, anyway.  But not understanding and not being that interested have never been good reasons for not discussing something.  If you understood something perfectly, I would think it would be the least interesting thing for you to discuss.  Which is how physics professors--with important exceptions that I have been fortunate to know--usually teach physics, as if they understand it perfectly.

Being on lunch break, however, makes it something I will discuss only briefly.  There are two important things involved in this discovery announced yesterday.  One is that it gives very good evidence for the idea of cosmic inflation, and the other is that it seems to be the first confirmation of the existence of gravitational waves.  Well, actually there's a third reason it's a great discovery, if it's not a "trick of light" deluding our radio antennas.  The inflation theory of Alan Guth and Andre Linde is based on quantum mechanics, so these gravitational wave signatures that have been detected were produced by a quantum process, implying a heretofore unknown but much sought connection between  quantum physics and the classical physics of general relativity .

See the New York Times article by Dennis Overbye for a more detailed discussion.

This discovery involves detecting and interpreting the polarization of very faint radio waves.  To get an idea of the meaning of polarization as it applies to electromagnetic waves, see my 11 June 2011 post.

Note added later: The next year, this discovery announcement was retracted. But, non-primordial gravitational waves were discovered that year (2015), coincident with the 100th anniversary of Einstein's publication of his general relativity theory of gravity. Gravitational wave detection has occurred several times since then.

The influence of symmetry in physics

The symmetery…

Is symmetry a force, like gravity or magnetism?  One of the books I quoted in my previous post is titled The Force of Symmetry.  A literal reading of the title implies that symmetry is a force, like the force of gravity.  But the title should be read impressionistically rather than literally, maybe as The Influence of Symmetry in Physics, for instance. The influence of symmetry in physics nowadays is huge, because symmetry in its mathematically most abstract form is considered to be the explanation for the existence of forces.

What is a force? That’s easy to answer. A force is a push or a pull.  Your body is being pulled toward the center of Earth by gravity right now, and the chair you’re sitting in is pushing up on you against the pull of gravity.  In terms of Newton’s third law, these aren’t the action-reaction forces, however.  The reaction force of the Earth’s gravity pulling on your body is your body’s gravity pulling back on Earth.  The action-reaction forces in the case of the chair you’re sitting in are the “contact forces” of the chair pushing up on you and your butt pushing back on it.  Well, partly your thighs pushing back also.

The contact forces are electromagnetic forces in the atoms of your body and of the chair.  The action-reaction pair in the case of you pulling up on Earth’s mass and Earth pulling down on your mass is due to the gravitational force.

What causes forces?  Physics has a pretty easy answer for that, too.  Forces are caused by fields. Even the contact forces of two objects in apparent direct contact with each other are caused by the interaction of the electromagnetic fields of the objects. But when you ask “what is a field?” well, now you’re getting into the land of abstraction. Now you’re getting closer to seeing the influence of symmetry in physics.  

Before getting into that, however, let’s not forget the simple symmetries we see all around us.  We see, for instance, numerous examples of squares, rectangles, circles and other such geometric figures.   We see the human body’s bilateral symmetry.  In all these examples, if you perform some transformation or operation such as rotating the object around a certain axis by a certain angle, you find that in some cases the object is unchanged by the transformation.

Draw a square on a piece of paper and then rotate the paper by 90°. The square looks just the same as it did before you rotated it.  Of course, there are things you can do to prevent the square from looking the same after you rotate it by 90°.  You can draw each side of the square in a different color.  Or if you use a rectangular sheet of paper, you will have an external reference that tells you your square has been rotated by 90° (but once you rotate it by 180° the external reference itself looks the same as it originally did). 

 

The square can be said to have a four-fold symmetry.  If you start rotating it around an axis through its center, you find there are four angles at which symmetry clicks in and the square looks like it originally did, no matter what the original position of the square was.  A hexagon has a six-fold symmetry, as its name implies.  Snowflakes also have sixfold symmetry.

How many “folds” of symmetry does a circle have?  We’re getting into the physicist’s idealized notion of symmetry here.  The circle has “perfect” two-dimensional symmetry, or an infinite number of choices of rotations about its center that leave it unchanged.  Squares, hexagons, etc. have discrete or countable degrees of symmetry.  The circle has continuous 2-dimensional symmetry.   

Likewise, a sphere has continuous or complete 3-dimensional symmetry.  Another way of putting this is that you can crawl around on the surface of a perfect sphere as much as you want, and no place on it will appear different from any other place. On a cube or any other polyhedron, you can crawl around and find its sides and corners. You won't be able to identify any corner or any side as being different from the others, but you can observe that corners are different from sides, and both of these are different from the "vertices" where more than two sides meet. 

Abstract symmetry: Low and Icke

Francis E. Low, writing in his 1967 book Symmetries and Elementary Particles, says:

A symmetry principle can usually be formulated as a statement about the impossibility of knowing something. Thus translation invariance states that all points in space-time are equivalent—there is no way of knowing where you are. Any inhomogeneity observed so far has always been accounted for by identifying a field which produces it. Thus translational invariance applies to an isolated system.

Vincent Icke, a Dutchman, says in his 1995 book The Force of Symmetry:

If you wanted to organize a universe, you could compile a whole book full of rules, a long list of entries that say ‘Thou shalt not this,’ ‘Thou shalt not that.’ But if your universe consisted of a possibly infinite multitude of inhabitants, which could do an infinite multitude of things at an infinite number of points in time and space, then such a rule book would surely be multiply infinite. . . .

Alternatively, you could 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’, or even more briefly, ‘Love thy neighbour as thyself’. Such an overall rule, that summarizes a multitude of individual possibilities, is called a symmetry. The symmetry just mentioned is between you and your neighbor. You are surely familiar with social symmetries: you and your neighbour have the same rights, even though you are certainly not the same, and may differ as to age, race, sex, creed, ethnic origin, and what not. It turns out that the symmetries that govern the quantum world are of a very special type, called a symmetry group by mathematicians.  . . .

Symmetry and redundancy in physics

A February 1992 journal entry of mine, the first entry in that particular journal:

 

Last 2013 journal entries: Order, disorder, symmetry

Dec. 19th, damn near 20th (11:55 p.m.).  Bed.

Long range order.  What is it?  Kitaigorodsky says on p. 47 of Order and Disorder in the World of Atoms, "This term can rightfully be applied to the arrangement of atoms in a crystal.” He also says, as the reason for this crystalline property, “along the directions of axes of the cell, similar atoms will be found at strictly equal distances hundreds and thousands of times.”  I put that comma in there myself.  But I do like K’s writing and would like to own a better—not falling apart—copy of the book.

I want to look at this idea in relation to the idea of symmetry.  Long range order implies a lack of symmetry.  This idea has been bugging me since I don’t know when.  Nineteen-ninety-four or a little earlier.  Long range disorder implies, as I understand it, perfect symmetry.  In other words, rotational and translational invariance.  Or, in other other words, isotropy and homogeneity.  I don’t like this idea of symmetry.  I like to think of symmetry as having to do with orderly structure, not just nothingness!  Well, one more thought: look into order, noncommutativity, and addition (summing).

6:15 p.m.  December 22.  Also, to continue the discussion above, I need to order another copy of Kitaigorodsky’s book.  Mine has fallen to pieces (it’s an old paperback, old meaning printed in 1980, although I have had it probably only since the mid-nineties.)

Thought number two:  I’ve really gotten habitual with writing the time of my journal entries.  I’m not sure I even like that.

Number three:  Oppenheimer had such great promise as a theoretical physicist, but he failed.  Why?  As Crease & Mann see it, he had no personal feeling in him for where he wanted physics to go.  He was not “his own man” is one way they describe Oppenheimer’s failure to do great things in physics.  He was clever, subtle, but in the end a physics failure.  (I’m using some applicable words from the classroom dream scene in A Serious Man.)

I view quantum field theory in exactly that way.  Gauge invariance, virtual particles, and . . . well, I’m not sure what all I’m thinking of that I don’t like, oh, yeh, the application of perturbation theory and of renormalization—these are all like Oppenheimer himself.  Cleverness to the n^th power, precocious, lacking in a clear philosophical view of what physics should be.  Now, like Oppenheimer in the 1930s, QFT and the Standard Model are having their field day.  In spite of the Higgs discovery—an amazing effort and accomplishment—I believe this field day of the Standard Model will also pass.  I think it will be replaced by something much simpler and better that puts it to shame by explaining all the current experimental high-energy data and also the electron, proton and neutron charges and masses.

1:55 a.m.  (I will keep writing down the time. Seems significant that I'm awake at 2 a.m., doesn't it?)  The day after Christmas.  "The Effervescence of the Vacuum" seems like it should be a term to describe some of the observed effects now attributed to virtual particles.

8:10 p.m.  Sunday Dec. 29.  The “law of conservation of electric charge, which is exact” (p. 194 Crease & Mann) is something I would like to now discuss.  The charge could change subtly in some situations, but may be unmeasurable in these situations.  This is in fact like symmetry:  the changes are unobservable.  E. g., changes in electromagnetic potentials are such that electromagnetic fields are not changed.

Can we also have, then, changes in electric charge of electron and proton such that the electromagnetic field is not observably different?  Huh.  Far-fetched, Frank.

Also, here’s an old thought scribbled on a piece of paper I’m now transcribing into this notebook:  “What was the thing about time I was thinking this morning?  About t = 0 in this diagram?  [Upside down T, with t = 0 at the intersection point.]   And how it should be used in the usual Einstein rest frame Lorentz transformation?  More to it!  I’ll remember it later.  Had some other inchoate thought before that.  Also:  shoot the monkey and the question of the relativity of simultaneity.”