The issue of spin effects has arisen in my discussion of Dirac's equation and it brings me back to a comment made by Qfwfq a couple of years ago:
Quote:
Originally Posted by Qfwfq
Well, I still don't get how the combination of that vector notation with
could include all the cases I have proposed.
|
There is an issue I think has to be made clear here. Complex numbers are a very valuable concept as they bring ordinary addition and multiplication to a two dimensional vector picture which is applicable to many rather complex physical problems. But, in another sense, they can be seen as a short term solution to a very important complex problem embedded in any internally self consistent explanation of anything.
The original problem here was to obtain "a way of getting the probability of finding a specific set of numbers" which would include all possibilities. The issue initially brought up was shift symmetry and it was shown that any function which was consistent with shift symmetry (“a” added to each and every member of that set of specific numbers can result in no changes) led to

. The second point was the fact that

(since it by definition, defines a process from getting from one specific set of numbers to another) can represent any conceivable mathematical function. The squared magnitude of that function summed over all possible arguments is a number N (we can worry about infinity if you are concerned). Thus it is that

can always be seen as a probability. Or rather that absolutely any conceivable mathematical function can be seen as generating a probability via a very specific process. The problem then degenerates into finding the function which yields the desired probability; essentially a search problem.
This is where the vector nature of

becomes significant. As you have said many times, all that

does is generate a phase relationship which has no consequence in

. What we are really talking about here is the fact that shift symmetry requires that

cannot depend upon those coordinates: i.e., P must equal a constant. If

is a real scalar function (i.e., not a vector function) then the requirement of shift symmetry also yields

equals a constant, a rather useless paradigm. On the other hand, since we want to include "ALL" possibilities, it could be a vector function. It follows that there must be another constraint which will guarantee that the consequences of it not being a constant will not carry over to P. That constraint is the simple fact that the various components of that inner product (the process of obtaining the squared magnitude) must be phase related such that the required sum of that inner product will exactly cancel out all the variations in

depending upon specific values of the coordinates.
Clearly if the abstract vector space of

is two dimensional, the vector can be seen as possessing two components ninety degrees out of phase with one another. Rotation in that in that two dimensional space can easily provide exactly the required complementary terms. In other words the two vector components of

must be intimately related to one another. In the two dimensional circumstance, that relationship is essentially handled by the fact that [imath]1^2+i^2=1-1=0[imath]: i.e., we need only allow

to be complex and define the other component to be exactly the same except for a change in the sign of the imaginary components (the standard meaning of

).
Thus it is that allowing

to be complex (which is totally equivalent to a two dimensional abstract vector space) totally alleviates the required constraint that the requirements of shift symmetry demand that P equal a constant without requiring

to be a constant. At the same time, the idea that

can be an n-dimensional abstract vector opens up a complexity with profound consequences. The opening example would be
Pauli's spin matrices. These matrices (in the abstract vector space of

) specificly designed to fill the purpose of the required anti-commuting

operators used in my deduction of my fundamental equation, allows the possibility of new complex constructs of

which would otherwise be difficult to assign meaning to. These things are called “spin” matrices for a very simple reason: their commutation properties are exactly the same as the commutation properties of ordinary angular momentum as conceived of in classical mechanics. The “isospin” interpretation of charge quantization is no more than an extension of the same phenomena into the tau dimension.
As I have said many times, I am not a mathematician and I leave mathematics to those more qualified than myself. I only use mathematics as a language able to express complex internally self consistent relationships which are fundamentally beyond expression in ordinary verbal logic. I leave it to the professional mathematicians to guarantee that their constructs are indeed logically consistent. All I am doing here is pointing out that the vector nature of

(in an abstract space) opens up many possible internal relationships consistent with the symmetries defining the constraints on the probabilities of interest to us.
It is my opinion that the physics community has chosen to see those various possibilities embedded in that abstract space as real characteristics of the ontological elements underlying their explanation. My interpretation, on the other hand, is that they are no more than additional possibilities, available within the definition of

(which I have defined to be “the collection of "ALL Possible" mathematical functions”), consistent with the symmetries required by the fact that the ontological elements can not be defined.
As one moves to higher dimensional abstract space, the possibilities have to include additional quantized rotational phenomena. I do not have this extension worked out in detail but I strongly suspect the requirements will be pretty well in alignment with the experimental results as clearly the additional symmetries, being fundamentally rotations in that abstract space, are going to yield quantized results analogous to those up/down, charm/strange, and top/bottom designations taken as fundamental ontological qualities desired by the physics community.
At one point Qfwfq asked me if these symmetries were represented by Lie algebra and I sort of pulled away from commenting on the issue. The reason was my ignorance of Lie algebra. I have looked into the subject a bit and now strongly suspect that there is an intimate connection there. The standard model, which is apparently the consequence of considering SU3 symmetries was introduced in the early seventies which is a number of years after I left the physics community so I really have little familiarity with it.
But seriously folks, if what I have presented is correct, is it really necessary for me to work out the whole of physics before my paradigm is even considered to be worth looking at? I would really like to see a reason for such an extreme attitude.
Have fun -- Dick