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.”

 

Physics Today: Higgs as portal to the dark sector?

Abstract from a recent research paper:  We investigate dark matter (DM) in the context of the minimal supersymmetric extension of the standard model (MSSM). We scan through the MSSM parameter space and search for solutions that (a) are consistent with the Higgs discovery and other collider searches; (b) satisfy the flavor constraints from B physics; (c) give a DM candidate with the correct thermal relic density; and (d) are allowed by the DM direct detection experiments. For the surviving models with our parameter scan, we find the following features: (1) The DM candidate is largely a Bino-like neutralino with non-zero but less than 20% Wino and Higgsino fractions; (2) The relic density requirement clearly pins down the solutions from the Z and Higgs resonances (Z;h;H;A funnels) and co-annihilations; (3) Future direct search experiments will likely fully cover the Z;h funnel regions, and H;A funnel regions as well except for the "blind spots"; (4) Future indirect search experiments will be more sensitive to the CP-odd Higgs exchange due to its s-wave nature; (5) The branching fraction for the SM-like Higgs decay to DM can be as high as 10%, while those from heavier Higgs decays to neutralinos and charginos can be as high as 20%. We show that collider searches provide valuable information complementary to what may be obtained from direct detections and astroparticle observations. In particular, the Z - and h-funnels with a predicted low LSP mass should be accessible at future colliders. Overall, the Higgs bosons may play an essential role as the portal to the dark sector. 

(Let's hope physics tomorrow is somehow simpler and less nutty than physics today.  That's what I'm working on anyway.)