In exploiting the gauge principle, we explained in the previous post that the LST physicists use the properties of complex numbers, which were invented by virtue of the imaginary number, ‘i’, to invent a set of infinite numbers, corresponding to the infinite radii, contained in the unit circle of the complex plane. This infinite set of complex numbers constitutes a group under multiplication, wherein elements of the group correspond to the set of infinite points along the circumference of the unit circle, which is interpreted as an interval from 0 to 1, in terms of rotation. Figure 1, below, illustrates the idea:
Figure 1. The Unit Circle in the Complex Plane
This ingenious invention effectively transforms the natural set of numbers between 0 and 1, and 0 and -1, into a corresponding set of rotations and counter-rotations, but it was accomplished unawares, because the natural set of numbers in this set, the continuous set of operationally interpreted reciprocal numbers (RNs), was not, and still isn’t, recognized in the LST community.
Nevertheless, now that we understand how the symmetry of reciprocity works to define the discrete units of integers, and the continuous units of non-zero rationals, in the form of the dual interpretations of RNs (see here for more), there’s no going back. Obviously, physicists can’t continue to use an ad hoc invention to formulate our most fundamental theory of physics, when a purely inductive set of numbers has been found that is free of such free inventions of the human mind.
Clearly, however, not many of this generation will be convinced that the operational interpretation of RNs is so remarkable anytime soon, so we won’t hold our breath, waiting for the rush of graduate students and post docs to apply the new numbers in building a new physical theory, based on the new system of theory and the new system of mathematics.
Nonetheless, researchers at the LRC are committed to do so. We have the daunting task to begin afresh in developing a new, RST-based, physical theory that promises to surpass what the geniuses like Bohr, Heisenberg, Pauli, Dirac, Schrodinger, and Einstein managed to accomplish in the 20th Century, which today is regarded as the greatest intellectual achievement of that century. Just articulating such a thing is enough to make anyone want to run home, jump into bed, and assume a fetal position.
Yet, we can’t shrink from the responsibility to press on. We must press on, regardless of the intimidating prospect of what lies ahead and simply trust that we are on solid ground here and that great things are brought to pass by means that seem small to men. So let’s cut to the chase: The dual interpretations of the operationally interpreted RNs, forming two sets of numbers, one of which constitutes an infinite group under addition, the other of which constitutes an infinite set of finite groups under multiplication, allow us to do two things:
- Add (subtract) discrete units to form combinations of units.
- Divide (multiply) the “size one,” or unit element of the group in 1 above into sub-units of arbitrary size.
The ability to do these two things is very relevant to the theoretical universe of motion, because the only existents in this universe are units of discrete motion, combinations of units of discrete motion, and relations between these. Thus, to develop a unified theory of the structure of the physical universe, we must use these two groups, one discrete, but infinite, and the other continuous, but finite.
What’s surprising is that the elements of the infinite group (integers) are discrete (“size one”), while the elements of the finite group (non-zero rationals) are continuous (variable size). Of course, in the LST view, there are gaps in the set of non-zero rationals, which must be filled in with irrationals, but, as we have discussed before (see here), this too is an ad hoc invention, which in the new inductive science of the LRC, must be excluded.
Obviously, Larson’s unit speed-displacement, which is a discrete unit of scalar motion that takes two, reciprocal, forms, one as the space|time progression ratio of ds|dt = 1|2, and the other as the space|time progression ratio of ds|dt = 2|1, corresponds to the positive and negative elements of the discrete group. We label these two ratios as the space unit-displacement and the time unit-displacement, respectively, or SUDR and TUDR.
Since the SUDRs and TUDRs are elements of the discrete group, as an infinite number of positive and negative elements in the group, they can be combined under addition, forming new units of positive and negative elements of the group, and, since for every combination of positive units, there is a corresponding combination of negative units (satisfying the inverse requirement of a group), the combinations of these inverse elements forms the identity element of the group. Moreover, adding the identity element of the group to any element of the group, including the combination units, doesn’t change the element, and adding positive or negative elements of the group, to the identity element of the group, creates new elements of the group (i.e. subtraction works too).
In short, all the operations of combining SUDRs and TUDRs into SUDR|TUDR combo units (S|T units), are justified by virtue of the fact that the set of these discrete units of motion constitutes a group under addition. However, since, physically, SUDRs are fixed spatial locations, relative to one another that are progressing in time, and TUDRs are fixed temporal locations, relative to one another that are progressing in space, combining them creates a physical unit that is progressing in both space and time, and therefore cannot remain fixed in space or time.
Thus, the motion of S|T units is unit motion, relative to the fixed spatial and temporal locations of their constituent SUDRs and TUDRs, which can be understood in the world line charts that we’ve discussed previously (see here).
We color the SUDRs red, and the TUDRs blue, and when they are combined together in an S|T unit, we change the color to green, indicating the S|T combo, which is the identity element, but where identity is understood as meaning that an equal number of SUDRs and inverse TUDRs constitutes the combo. Hence, all equal combinations form the identity element, but this means that 1|1, 2|2, 3|3, …n|n are all the identity element. In other words, the identity element of the group itself, 1, is infinite, but under the operational interpretation of the discrete group, it is also 0 at the same time.
This union of infinity with zero is very fortuitous and has many implications that are beyond our scope right now, but suffice it to say that the value of the same identity element, under the operational interpretation of the continuous group, is 1, not 0. The binary operation of the continuous group is multiplication (actually it’s division), but remember that the size of the unit elements of this group is not “size one,” as in the discrete group, but are the continuously variable units that can be defined inside one discrete unit, from -0 to 1, and from +0 to 1.
Thus, we see both the similarity and the disconnect between the RST groups viz-a-viz the LST groups. However, the major, and most significant, difference between the two groups of operationally interpreted RNs, which we are using to develop an RST-based physical theory, and the two (actually three) “unitary” groups of rotationally interpreted complex numbers, used in the LST development of the standard model, U(1) and SU(2) (and also SU(3)), is that the RST-based groups are inductively derived, while the LST-based groups are free, ad hoc, inventions.
The discrete group of RNs is an infinite group under addition (actually subtraction) and the continuous group of RNs is a finite group under multiplication (actually division), but the rotational groups of the complex numbers are infinite, continuous, groups that are equated to two infinite, continuous, groups of rotation, R(2) and R(3). What the heck is going on here?
It behooves us to understand why the LST community needs to invent an infinite, continuous, “complex space” of rotations that is equated to the infinite, continuous, “real space” of rotations, in order to get the infinite, continuous, group they need, in developing their quantum theories.
However, this is not just a question for us outsiders. They themselves do not have the answers, which is great clue that we are on to something, in taking a fresh look at their motivations and thinking on this. Indeed, Peter Woit has specifically pointed this out for us when he explained in a recent presentation that (see our discussion on this here):
One indication of the problem with string theory [is that it is] not formulated in terms of a fundamental symmetry principle. What is the group?
implying that there is an accepted consensus in the LST community, which recognizes that the closure, identity, inverse, associative, etc. properties of the mathematical group are essential to the descriptions of correct physical theory. What he only dimly understands is that, while string theorists don’t have the necessary group, the group that the particle theorists have, and that they have used as the basis for developing the standard model, is an ad hoc invention that mixes up the infinite with the finite, the discrete with the continuous, to form one unclean and confused concoction of numeric concepts. Is it any real wonder that they don’t understand it? Duh!!!
Of course, understanding how they use their invented group to get the success of the standard model will be an important part of this investigation and in our efforts to obtain the same level of success, and beyond, in our new theory. These guys are really, really smart. What they did can only go so far, as they are now finding out, but it’s just amazing to see how clever they are in inventing ad hoc concepts to keep their theories together.
Ironically, it turns out that we need one of their invented concepts, the concept called the gauge principle, but in a different context that transforms it from an invention to a geniune principle of scalar mathematics. Hopefully, we will be able to see why in the next post or two.