In the previous post below, I laid out what amounts to a 3D “coordinate” system of operationally interpreted (OI), rational, numbers. Of course, in turning our attention to the quantitative development of our RST-based theory, these numbers will not be referred to spatial locations in a vectorial reference system, where zero is at the origin of three, orthogonal, axes, as we are accustomed to using it in the legacy system of physics (LST).
Rather, these numbers will be referred to the net speed-displacements of the constituent units of S|T triplets (see “The Big Bet” and “Preon Models” and subsequent posts). Recall that these OI numbers are developed from the increasing degrees of freedom in the quantity, duality, and dimensional properties of OI numbers. To develop the physical magnitudes, which correspond to these numbers, requires the application of a set of physical principles of motion, just as to develop physical magnitudes that correspond to the numbers of the zero reference system also requires the application of a set of physical principles of motion.
In the latter case, however, continuous magnitudes of velocity, energy and momentum are developed from principles of mass, length and time, in what is referred to as classical physics. Beyond this, in the so-called quantum physics of the LST, discrete magnitudes of velocity, momentum and energy are developed from modified principles of mass, length and time, which include principles of uncertainty, of exclusion, and of complementarity, of non-commutativity, etc. These are principles that are not needed in the classical version of the mass, length and time system, but are indispensable in its quantum version.
Consequently, the continuous principles of physical magnitude are not replaced by the additional discrete principles of physical magnitude, in the LST, but the continuous principles are seen to arise out of the discrete principles, which apply only to the properties of particles existing below a certain scale. Nevertheless. because these particles can only be studied while moving relative to one another at very high speed, both sets of principles are required. In other words, in LST physics, both continuous and discrete magnitudes are needed to understand experimental observations correctly, and this necessitates the application of two separate sets of physical principles.
Of course, this approach has worked, to a certain extent, but it is ultimately problematic, because continuous and discrete magnitudes are ultimately different animals. Successfully modifying the physical principles of continuous magnitudes, in order to “quantize” them, making it possible to adapt these principles to serve our purpose in deriving the observed discrete magnitudes of nature, is admittedly ingenious, but, ultimately, it is doomed to failure, as we are discussing in the The Trouble with Physics blog.
However, it is important to recognize that basing the continuous and the discrete physical magnitudes of nature on one type of motion, what we call the M2 type, or the vectorial type, of continuous motion, is no longer necessary, since the discovery of a new set of physical principles, one that applies strictly to discrete magnitudes of motion, now portends a way out of the current continuous-discrete quagmire. In particular, we should now understand that the discrete principles, which the LST has employed in its quantum system, the rotational form of M2 motion, where magnitude, expressed as a continuously changing direction in one location, is substituted for a continuously changing location in one direction, is a contrived approach, used to generate discrete magnitudes, and that it is no longer necessary.
This changes everything. Now it should only be necessary to distinguish between the M2 type of motion, whether of the linear or rotational form, and the M4 type, or the scalar type, of motion, in calculating the discrete physical magnitudes of nature from the principles of discrete space and time only. The new principles should be completely capable of explaining how the continuous type of magnitudes arise from the discrete type of physical magnitudes, without the need to modify the continuous principles one iota. At least it seems clear to us, here at the LRC, that, in order to properly derive the discrete physical magnitudes of nature, it behooves us to seek to understand and to employ the new set of discrete physical principles that Larson has discovered, rather than to continue to rely on a modification of the continuous physical principles of the Newtonian system.
This means, among other things, that we cannot entertain a model of the nuclear atom, with a nucleus consisting of a tightly bound mix of quarks, surrounded by a cloud of orbiting electrons, occupying various levels of discrete energy, derived from complex treatments of angular momentum, the nature of which is mysterious. Hence, in the new system, Planck’s energy constant is not regarded as a unit of angular momentum, generated by the wavefunction of the massive electron. Indeed, the whole idea of kinetic and potential energy, in its Hamiltonian and/or Lagrangian form, as operators on the wavefunction, is rendered completely irrelevant, by our transition to an entirely new system of discrete physical principles, used to develop the discrete physical magnitudes of the new theory.
In short, we have developed a new set of n-dimensional, OI, rational numbers, based on 1, not 0, which we seek to use to express discrete physical magnitudes, and these magnitudes must be developed using a new set of physical principles of discrete units of motion, which does not include the idea of particles and atoms that consist of moving subparticles, bound together by autonomous, fundamental, forces. Instead, the new approach embraces the idea of combining discrete units of space|time, or scalar motion, from which the properties of mass, energy, and radiation emerge in such a way as to produce the observed forces of interaction.
Accordingly, in the previous post below, we show how the new “coordinate” system of n-dimensional OI numbers, with varying degrees of freedom, are derived. Notice how, in the 2D case, the pattern of numbers that we developed follows that of the periodic table of elements, as developed by Larson; that is, it follows the 4n2 pattern:
- (2|2)2 = 4*12 = 4
- (3|3)2 = 4*22 = 16
- (4|4)2 = 4*32 = 36
- (5|5)2 = 4*42 = 64
Please note that, if the normal, quantitative, interpretation (QI) of rational number is used, the above equations are incorrect: That is to say, in the QI case, all four of these numbers are equal to 1 raised to the second power, but when the new, operational interpretation, of rational number is used, as shown in figures 1 - 4 of the previous post, then they are all valid equations, reflecting the total, discrete, number combinations that are possible in each case (except that (1|1)2 = 0 was shown in the graphic instead of (5|5)2 = 64.)
However, due to our natural inclination to think of orthogonality in terms of multiplication, we need to strive to understand that, in the discrete case, the mathematical meaning of dimension signifies the number of independent terms involved, not necessarily the geometric dimensions involving directions differing by 90 degrees of angle. Thus, while we represent the independence of the dimensional degrees of freedom in figures 1-4, as orthogonal lines, separated by 90 degree angles of direction, this does not imply a geometric relationship between the dimensions. We can just as easily represent them as independent, parallel, lines, for example.
We are simply saying that with a value of 1 unit of speed-displacement, the second power of this number, (2|2)2, has two, dual, values, in each of the two dimensions, or a positive and negative value in each dimension, if you will. Thus, there are four possible magnitudes in the 2D number 2:
- D1 +1
- D1 -1
- D2 +1
- D2 -1
Similarly, the 2D number 3, (3|3)2, as show in figure 3 of the previous post, has 4 values in each of 4 quadrants, or 16 numbers, because now we have two positive and two negative values in each of the two dimensions:
- D1 +1, +2, -1, -2
- D2 +1, +2, -1, -2
So, combining the 4 numbers of dimension 1, with the four numbers of dimension 2, will give us 4*4 = 16 unique values, but these have nothing to do with orthogonal geometric directions. They are just numbers. We can add them together, as weights, multiply them together, as areas, or use them, as binary inputs, in a logic circuit, but the numbers are always independent of the applications that employ them. They are only abstract symbols with a logical, 4n2, pattern.
Therefore, to develop multi-dimensional physical magnitudes that we can express in terms of these multi-dimensional numbers, we simply need three, independent, physical variables, such as the speed-displacement of the three constituent S|T units of the S|T triplet combination of scalar motion that we’ve been discussing (see previous posts below). These units of discrete motion are independent in the sense that they constitute three adjacent physical locations, partially merged together as one, as illustrated in figure 1 below:
Figure 1. The Physical Representation of the S|T Triplet
In the figure above, we see a graphical illustration of three, independent, S|T units merging together to form a combination unit. Notice that, while the geometry of the combination is necessarily two-dimensional, because the center points of the three spheres must lie on a plane, the independent space|time magnitudes of displacement (represented by the three different colors) are three-dimensional and each dimension has the dual “directions” of the OI numbers; that is, the net magnitude of the space|time displacement of each sphere can be green, red or blue, corresponding to magnitudes that are balanced (neutral green), or unbalanced in one of two “directions,” red (negative) or blue (positive).
A balanced (green) S|T unit can be any value that has the same number of SUDRs as TUDRs, but the minimum number it can have is, of course, one of each. If we denote the constituent S|T units in the triplet as A,B,C, then the equation for the minimum green unit in the triplet is
S|TA = (1|2) + (1|1) + (2|1) = 4|4 nm,
where nm is the symbol used for natural units of motion. The equation for the minimum blue unit is then
S|TB = (1|2) + (1|2) + (4|2) = 6|6 nm,
and the equation for the minimum red unit is
S|TC = (2|4) + (2|1) + (2|1) = 6|6 nm
The total number of natural units of motion is the sum of these, or
(4|4) + (6|6) + (6|6) = 16|16 nm
However, in terms of independent variables, we have 0 (green), +1 (blue), and -1 (red), and we can see that, whenever one of the S|T triplet’s three constituent units is green, the 22 = 4 possibilities of magnitude that the triplet can take are those of two, orthogonal, dimensions:
- blue + red = +, -
- blue + blue = +, +
- red + blue = -, +
- red + red = -,-
But now an important question arises: What is the difference between the (green + red + blue) combination, and the (green + blue + red) combination? Numerically, (0 + (-1) + 1) = 0 and (0 + 1 + (-1)) = 0, because quantitatively interpreted (QI) numbers are always zero-dimensional numbers (scalars). However, the numbers of the S|T units are comprised of three terms, not one, and while these three terms represent one space|time location, which is separate and distinct from adjacent space|time locations, the values of the space and time displacements, in a given location, vary.
So, if the three magnitudes were QI numbers, the sum of the (red + blue), or (blue + red), would be the same, in both cases, but the OI numbers do not work in the same way. There are three, adjacent, space|time locations in the S|T fermion triplet, each with its red (space) location, and each with its blue (time) location. However, because the configuration of this triplet is a triangle, the time location of S|T unit A is associated with the space location of S|T unit B, and its space location is associated with the time location of S|T unit C. The same arrangement holds for all three S|T units: Each is joined to the other two, through a space|time connection.
Thus, this situation seems to raise the same type of issue raised by multiplication of M2 vectors, a and b. In taking the outer product of two vectors, the order of the operation makes a difference; that is, a^b is not the same as b^a. Yet, we are not multiplying n-dimensional vectors here, but, rather, we are summing n-dimensional scalars. Regardless, however, we see that we must still distinguish the sum of ((+) + (-)) from the sum of ((-) + (+)), just as vector algebra must distinguish the bivector of a^b from the otherwise identical bivector b^a, by making it -b^a, indicating a directional property of the bivector that Hestenes calls “orientation.” The orientation of an outer product, in Hestenes’ GA, indicates the direction of its construction, if you will, because there are two ways you can raise the dimension of a number, represented by sweeping the vector a along the vector b, or the vector b along the vector a.
However, in the RSM, we don’t have n-dimensional vector magnitudes, only n-dimensional scalar magnitudes, which sounds like a contradiction in terms, without an understanding of the difference made by the OI rational number. Yet, as explained in the previous post below, the new OI numbers do, in reality, have the three properties, quantity, duality, and dimension, the same three properties observed in physical magnitudes. They are not just symbols of quantities alone. Therefore, the sum (-A) + (+B) is different from the sum (+A) + (-B), because all three properties of the OI numbers must be taken into account. The easiest way to understand this is to consider the four quadrants of a plane formed from two, independent, reciprocal numbers (i.e. two, three-term, OI numbers), as shown in figure 2 below.
Figure 2. Plane of Two Independent OI Numbers
In figure 2 above, the magnitudes of quadrants 1 and 3 (disregarding the middle terms for now), in terms of polarity, or color, are straightforward, but quadrants 2 and 4 are ambiguous:
- Q1 = +, + (blue + blue)
- Q2 = -, + (red + blue)
- Q3 = -, - (red + red)
- Q4 = +, - (blue + red)
Yet, summing the OI numbers in each quadrant, we can see that, actually, there is a big difference between Q2 and Q4:
- Q1 = +, + (blue + blue) = (4|2) + (2|1) = 6|3
- Q2 = -, + (red + blue) = (1|2) + (2|1) = 3|3
- Q3 = -, - (red + red) = (1|2) + (2|4) = 3|6
- Q4 = +, - (blue + red) = (4|2) + (2|4) = 6|6
It turns out that Q4 is twice the value of Q2. Notice that this is not the same thing as the non-commutative property of bivectors, because (-A) + (+B) = (+B) + (-A). Thus, OI numbers are commutative, but they can be misinterpreted, if all three properties of the numbers are not considered.
An interesting aspect of all this is the light it throws on the concept of non-commutative mathematics. It’s clearly the result of raising the dimensions of numbers through rotation via the ad hoc invention of the imaginary number, used to form complex numbers from real numbers. If we do it this way, we still have to somehow preserve the duality property of the number, and we do this in GA (unwittingly), by invoking the direction of rotation as a numerical property; that is, the representation of the two sides of the plane, the positive and negative sides we might say, is accomplished by changing the sign of the bivector, under multiplication, indicating that the direction of rotation (the “sweeping of the vectors”) changes with a change in the order of the multiplication operation.
However, when we use OI numbers, this expedient is no longer necessary, because the duality of the numbers is preserved through the summing operation, since it is built into the numbers themselves, in a natural way. Actually, what we are doing is replacing the artificial plane of rotation, the complex plane, with its zero origin, with the plane of duality and its origin of unity. Fortunately, doing so is perfectly logical, from a mathematical point of view, and now we see that it is well motivated from a theoretical point of view, as well. All that remains is to justify it, by deriving actual physicical magnitudes, based on the same, reciprocal, principles of discrete values that the “coordinate” numbers are based on, and see if we can identify them with the magnitudes of observed physical phenomena.
A tall order, to say the least, but what could be more fun?