A few 2001 journal entries: ref frames, forces

Later on, Sonday . . . . no, ain’t drunk or drinkin’ or gone crazy either.  11:57 ~ midnight.  April 23  (doin’ assignment in Stat. Mechanics).  So the thing einstein did, really, is to make a reference frame physically meaningless? (combination of question/exclamation).  The reason I mention this is that which reference frame is chosen is arbitrary.  If it’s arbitrary, in physics, it doesn’t have a physical significance.  Back to van der Waals stuff, friends.

12:55 a.m.  May 1 Tuesday/Belgium

Taking a break from doing HW—from trying to get started on some HW, really—I’d like to mention the possibility that in looking for a unified field theory, we are looking for the wrong thing.  Wanting to unify the forces of nature under one theoretical roof, or hold them up with one theoretical foundation, sounds very admirable.  Could we also consider the possibility that the forces are a mere façade, however?  Then where do we want to go with our theories of gravity, weak/em, strong forces?

Well, we want a causative description of motion, for one thing.  We observe attraction and repulsion, and want to explain them.  Unifying thus means describing the attraction-repulsion of the known forces with one theory.  “One theory” then seems to imply that there would then be one force, with different disguises.

I’m suggesting that “force” itself is a disguise for something else.  What else?  Einstein found curvature of spacetime gave the description of gravitational force.  So that is how the disguise works in that case.  To construct a “field” out of particles seems like a big step away from getting behind the disguise.  It seems like putting another level of disguise on the phenomena.  A better solution would be to not have the need for the particles, like general relativity has no need for a “force” once spacetime curvature is accepted.

Related to getting rid of the force, Luke, is getting rid of the force carrier.  What then is left?  Somehow we need to look beyond attraction/repulsion in order to answer that question.

The idea of inertia is a good starting point for thinking about a better way to describe observed motions—better than attraction/repulsion mindset.

So, how about starting with the idea I wrote down on the previous page—which reference frame is chosen is physically meaningless (a la Einsteiner), so first we just abolish the idea of reference frames.  Banish reference frames from the “force” discussion, see if forces can then be banished, too!

8:15 p.m. Sat. May 5, 2001   [page 45 in blue journal]

I’m about to catch up with the page number equal to my age.

After I wrote the previous entry, I realized that in Jeff Wilson’s class, when he mentioned one of us asking what the force was that kept fermions apart (I was the one who’d asked about it), that this case of “exclusion principle repulsion,” as Tipler (Modern Physics p. 289) puts it, is a case of a force acting without force carriers.  Either that or I misunderstood it.  Jeff Wilson was trying to explain why it was other forces or other sources of energy [other than the standard four] that caused the exclusion principle repulsion, but whatever he said didn’t convince me.  I plan to bring up the subject with him later.

Can I do formal physics for someone besides myself?  I have trouble latching onto the formal requirements of homework and tests, because I’m not really attracted to the sort of career that would result from being successful in the formal sense.  I’ve just been reading Walker Percy’s book The Second Coming and wondering if I could say, okay Walker Percy, I’ll do this for you.  Then I thought of Daddy, a Walker Percy fan himself.  Could I say to him I’ll do this for you and Walker Percy?

Saturday May 12

One of the conclusions I came to a few years ago was that, in the Einstein non-simultaneity experiment, the events are actually determined to by simultaneous by each observer, but are not simultaneous when each observer transforms [his measurement] to the other’s rest frame.  I was looking at this same issue on pages 12 and 13, and previous pages, in here.

The simultaneity issue is closely related to that other relativity issue I’ve been thinking about lately, how all reference frames must have the same standards but those quantities that aren’t Lorentz invariant don’t transform as “standards.”

Which brings up another question:  e and c are Lorentz invariants, but is G?  Oh, yeh, and is h?  They would seem to be, as they are taken as the fundamental constants, along with alpha, a measure of the strength of the EM force [equal to e²/ħc].

And from last Saturday, what are the force carriers that supposedly carry the “exchange force” causing the exclusion principle repulsion?  Haven’t seen J.W. at a good time to discuss the question again.

Groups intro

Here's some discussion of the mathematical idea of a group, from a college-level textbook titled simply Algebra, by Michael Artin. [From chapter 2 (Groups), © 1991 Prentice-Hall.]  Notice that the "set of invertible n x n matrices" and matrix multiplication are one type of group and one example of a law of composition, respectively:

"A group is a set on which a law of composition is defined, such that all elements have inverses.  …  For example, the set of nonzero real numbers forms a group R^x under multiplication, and the set of all real numbers forms a group R⁺ under addition.  The set of invertible n x n matrices, called the general linear group, is a very important example in which the law of composition is matrix multiplication.  We will see many more examples as we go along.

By a law of composition on a set S, we mean a rule for combining pairs a,b of elements S to get another element, say p, of S.  The original models for this notion are addition and multiplication of real numbers.  Formally, a law of composition is a function of two variables on S, with values in S, or [in other words] it is a map

S X S → S

a, b [squiggly arrow pointing right] p .

Here, S X S denotes, as always, the product set of pairs (a, b) of elements of S.

Functional notation p = f(a,b) isn’t very convenient for laws of composition.  Instead, the element obtained by applying the law to a pair (a,b) is usually denoted using a notation resembling those used for multiplication or addition:

P = ab, a x b, a ° b, a + b,         and so on,

a choice being made for the particular law in question.  We call the element p the product or the sum of a and b, depending on the notation chosen.

Our first example of a law of composition, and one of the two main examples, is matrix multiplication on the set S of n x n matrices.

We will use the product notation ab most frequently.  Anything we prove with product notation can be rewritten using another notation, such as addition.  It will continue to be valid, because the rewriting is just a change of notation.

It is important to note that the symbol ab is a notation for a certain element of S.  Namely, it is the element obtained by applying the given law of composition to the elements called a and b.  Thus if the law is multiplication of matrices and if

     1     3               1      0                                               7        3

a=           and  b=             , then ab denotes the matrix   

     0     2               2      1                                               4        2  .

Once the product ab has been evaluated, the elements a and b cannot be recovered from it.

Let us consider a law of composition written multiplicatively as ab.  It will be called associative if the rule

(ab)c  =  a(bc)    (associative law)

holds for all a,b in S, and commutative if

ab  =  ba        (commutative law)

holds for all a,b in S.  Our example of matrix multiplication is associative but not commutative.

When discussing groups in general, we will use multiplicative notation.  It is customary to reserve additive notation a + b for commutative laws of composition, that is, when a + b = b + a for all a,b.  Multiplication carries no implication either way concerning commutativity.

In additive notation the associative law is (a + b) + c = a + (b + c), and in functional notation it is

f(f(a,b), c) = f(a, f(b,c)).

This ugly formula illustrates the fact that functional notation isn’t convenient for algebraic manipulation.

The associative law is more fundamental than the commutative law; one reason for this is that composition of functions, our second example of a law of composition, is associative."  (end of Artin quotation)

After showing how composition of functions is described mathematically and giving examples, Artin notes “Composition of functions is not commutative.”   

That’s enough group discussion for now.  Oh, wait.  Three pages later, Artin gives the definition of a group, so I should write that down:

Definition.  A group is a set G together with a law of composition which is associative and has an identity element, and such that every element of G has an inverse.

Matrices intro

Remember the commutative property that we all learned in Algebra I?  It's

                                       ab = ba

where (in Algebra I at least) a and b are real numbers.  That property bugged me when I started taking algebra.  Maybe the associative and distributive properties (of real numbers) also bothered me, but according to my memory of it, I was especially perplexed and put out with the commutative idea because I'd already learned that numbers could be multiplied in forward or backward order. Why did the textbook make such a big deal out of it?  The answer has to do with the emphasis on set theory in the New Math of the sixties and early seventies. 

What I didn't know is there are things besides plain old real numbers that are noncommutative under multiplication. I wish someone had told me that way back then.  (Yeh, don't we all wish someone had told us something way back when?)

For instance, two ARRAYS of real numbers multiplied together don't in general have the commutative or reversal-of-order property.  An array of numbers is written as a rectangular or square table called a matrix.  Actually, an array of any sort of mathematical animal such as variables, operators, or functions is a matrix.  

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Aside:  In some dialects, matrix is pronounced “mah-trix,” as opposed to the English-speaking version, “may-trix.”  When I moved into a University-owned apartment with a fellow graduate physics student in Columbia, South Carolina, at the turn of the 21^st century, he asked me if I wanted to watch a movie he’d rented.  He was from Kazakhstan, and spoke decent English, but I could not understand what movie he was talking about.  The nearest I could guess, he was saying “The Mattress.” He must have finally gotten the box with the DVD (or VHS tape) in it and showed me: It was The Matrix.

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A matrix is usually designated with a capital letter rather than a lower case letter. So let's let A and B be 2 x 2 matrices of consecutive real numbers, like so: 

          1       2                         5        6 
 A =                     and    B =        

          3       4                         7        8   .

Multiplying A times B gives (row elements times column elements, then add the result, and repeat till finished)

1   2            5   6             1*5+2*7      1*6+2*8           

            x                  =                            

3   4            7   8             3*5+4*7      3*6+4*8 

                                           19                 22
                                =  
                                           43                 50  . 

(The multiplications shown here are done first, then the addition.)   And here’s the “backwards” multiplication of these two matrices:

5   6            1   2             5*1+6*3      5*2+6*4 

            x                  =                                     

7   8            3   4             7*1+8*3      7*2+8*4

                                             23                34
                                =
                                             31                46   .

So, we have AB ≠ BA.  A and B don’t commute, or, writing it differently,

                                        AB – BA ≠ 0.

We can give a name to the difference in AB and BA, and also get rid of the pesky “not-equal” sign.  We could just use a name or symbol such as C, but the difference in two matrices multiplied forwards and backwards has important properties in quantum mechanics, so we’ll use a different symbology, called “the commutator"

                                     AB – BA = [A, B].

In our consecutive numbers case [A, B] is equal to

19    22        23    34           -4    -12
               -                   =                     

43    50        31    46           12      4   .

This little example turned out to be more interesting than I thought it would be:  I discovered that the commutator of any consecutive numbers put into 2 X 2 matrices is equal to this -4, -12; 4, 12 matrix.  For instance, let

           57    58                              61    62 

A =                            and  B =    

           59    60                              63    64   .

Then 

                      -4    -12

[A, B]   =                            

                       12     4   .

This happens because the consecutive numbers are of the form n+1, n+2, …, n+8 where n can be any integer. The n’s can be put in their own matrix which is merely added to the basic 1, 2, 3, 4, etc, matrices, like so

n+1    n+2              n     n            1     2

                     =                     +

n+3    n+4              n     n            3     4    .

The “n” matrices cancel each other when the commutator is calculated, and so do cross-terms that have a factor of n in them, leaving only the commutator of the two matrices with 1, 2, …, 8 in them.  Very cool!

I’m writing about matrices because matrix multiplication of operators and wave functions (state vectors) is the bread and butter of quantum mechanics.  For instance, certain quantum mechanical matrices that commute, meaning AB = BA, or that “anti-commute,” meaning AB = -BA, are important for predicting the results of physical measurements.  Matrices that don’t commute are responsible for the theory behind the Heisenberg Uncertainty Principle--well, that's also important for physical measurements! I could just say "measurements" instead of "physical measurements." Or in general: experiments (as opposed to just thought experiments).

Matrices used in quantum mechanics can be transposed (rows and columns interchanged) and have their complex conjugate taken (replace i by -i in a complex number).  Sometimes they can be inverted (have their reciprocal taken). When a matrix is equal to its transpose, it’s called symmetric.  When it’s equal to the negative of its transpose, it’s called anti-symmetric or skew-symmetric.  The most important matrices for quantum mechanics are the ones that equal their own complex-conjugate transpose (Hermitian matrices), and ones that equal their own complex-conjugate inverse (unitary matrices).  

Question:  is the above "consecutive number commutator" matrix,

     -4      -12

      12       4  ,

symmetric or maybe anti-symmetric?  It has a common factor of 4 in all its elements that can be taken out, giving 

               -1      -3

4 times

                3       1  ,

which really looks (even more) symmetric.  But if you take its transpose (interchange the -3 with the 3) you don't get the same matrix.  To get the same matrix, you'd have to have both 3's be positive or both be negative.  It's also not anti-symmetric, which you can check for yourself (adding the transpose to the original would give the zero matrix if the original matrix were anti-symmetric).

Okay that’s enough self-directed math review for today.