The Larson Research Center

Peter Woit gave a talk at the University of Central Florida (UCF) recently, entitled “The Challenge of Unifying Particle Physics.” He does a real good job of explaining what the Standard Model (SM) is, and then, in the last slide, he makes the point that is the most interesting of all: the challenge of unifying particle physics, he insists, is really the challenge of unifying the mathematics of the SM. Specifically, he makes three important observations:

1. The mathematics of the SM is poorly understood in many ways.
2. The representation theory of gauge groups is not understood.
3. The unification of physics may require the unification of mathematics.

But in the midst of these he puts another bullet that states:

• One indication of the problem with string theory: [It is] not formulated in terms of a fundamental symmetry principle. What is the group?

In other words, the trouble with the SM is that it works very well, but we don’t know why, and the trouble with string theory is that it doesn’t work very well, and this may be because it doesn’t work the way the SM works. The answer to the question, “What is the group?” is, of course, that there isn’t a group that corresponds to the vibrations of a string, even though there are two ends of the string, one positive and the other negative, which is the basic form of a group.

Thus, it seems that theoretical physics is caught on the horns of a mathematical dilemma. On reading this, it’s only natural to ask what the new mathematics, the reciprocal system of mathematics (RSM), has to say, if anything, about this predicament? Well, there are so many aspects to the problem that the most difficult challenge in answering it is the decision of where to begin. After contemplating it for some time, I think the best thing is to just start over, at the very beginning, with the so-called fixed coordinate system of three dimensions.

After all, the fixed, 3D, coordinate system is our basic framework of numbers and magnitudes. Indeed, it’s the framework upon which we have built our modern technology. Like women in a worldwide sewing circle, constructing various quilts by means of quilting frames, mathematicians and physicists have worked together to sew the intricate patterns of modern physical concepts into frames of the 3D fixed reference system.

Interestingly enough, is the fact that, for the scientists with their functions and variables, as well as for the ladies with their needles and threads, the most important and central principle of their work is the principle of symmetry. So, here is Peter Woit, a passionate member of the inner circle of theorists, standing up at the edge of the frame to complain about the lack of a clear symmetry in the latest work of the group. “I don’t get it,” we can almost hear him saying, “This new pattern sucks!” We can imagine him walking around the frame, pointing out the how the intricate needle work, gorgeous and elaborate as it is, in places, is just not symmetrical overall, as a whole!

“But,” we can imagine someone like David Gross protesting, “the overall symmetry doesn’t matter, because you can’t see the damn thing as a whole anyway. The only thing that really matters is the local symmetry, which everyone must admit is breathtaking.” Of course, Peter, and the many others who agree with him, are not convinced at all, by this argument. Yet, what can they do? To go back now and start over is unthinkable, but, at the same time, to find a principle which would serve to unify all the independent patterns into one seems impossible, at this advanced stage of the work.

However, this is not the case for amateur investigators like Larson. As he so eloquently pointed out, the amateur investigator has no vested interest in the current fabric of mainstream theory and is free to start from the beginning. Certainly, Larson is not the only one to recognize this. In fact, so many amateur have tried their hand at coming up with their own alternative that their ranks have become legion, but they are usually summarily and derisively dismissed by the professional craftsmen, as naive, with a hopeless lack of proficiency in the essential skills to work on the elaborate quilt, and Larson is no exception.

Nevertheless, and notwithstanding the lack of interest of the professionals in amateur suggestions, Larson’s approach should not be lightly dismissed, and for one reason: because his suggestion is completely and totally different than all others. In effect, he suggests that the sought for global symmetry, which can accommodate the desired local symmetry, can be found, if we modify the fixed, 3D, reference frame first. Since, historically, modifying this frame itself, in appropriate ways, has always led to the greatest advances in physical theory, this alone ought to get the attention of the pros.

Unfortunately, however, Larson never clearly understood how his modification of the frame worked mathematically. Indeed, he always insisted that his new frame was consonant with existing mathematics, but, as we have discovered, here at the LRC, this is far from the case. The truth is, his new, fixed, 3D, frame of reference constitutes a revolutionary new approach to the mathematics of the frame that completely transcends the transformations of the past.

The fundamental reason that this is so is that Larson’s recognition of the reciprocity of space and time, wherein he redefines space and time, as nothing more or less than the reciprocal aspects of changing magnitudes in the equation of motion, applies equally well to rational numbers in general: What we call scalar quantities, or zero-dimensional numbers, are best understood as reciprocal aspects of positive and negative magnitudes. In other words, the observed properties of physical magnitudes, quantity, duality, and dimension, are also the properties of abstract numbers, when they are understood, as the ancients understood them, as two, reciprocal, aspects of magnitude.

With this exceeding simple change in our most fundamental abstract concept, so obvious, but so revolutionary, we can construct a new, totally different, fixed, 3D, frame of reference upon which to build beautifully symmetric patterns that contain all the local variations that enchant us so, but we can do so without sacrificing the overall symmetry of the quilt.

The key to understanding how this works is found in understanding what we call the operational interpretation (OI) of the rational number, where the operation is the reciprocal relation of rational numbers just described. Given the OI number, we can form both negative and positive numbers, and zero, without the use of imaginary numbers. One of the most important things that this then does for us, is to give us a better handle on the abstraction of infinity and infinitude, which promises to help us to eventually extricate ourselves from the horns of the continuous-discrete dilemma, which has so exacerbated our difficulties in constructing consistent physical theory.

We start with the infinite OI number,

∞|∞ = 0,

where “|” is the operator symbol used to indicate the reciprocal relation of the OI rational number. Clearly,

∞|∞ = 0,

because of the perfect symmetry of the unit reciprocal - nothing is perfect. In other words, the value of the OI rational number is the relation between the reciprocal quantities, and the unit relation means that there is zero difference between the two, reciprocal, quantities. On the other hand, the only other possibility, the non-unit relation, is where

n|m < 0 and m|n > 0,

when m > n. This means that there is a non-zero difference between two, reciprocal, unequal quantities. It follows from this that if ∞|∞ is a group, then m|m, regardless of the value of m, is an identity element of the group. Thus, we conclude that both

∞|∞ = 1|1, and m|m = 1|1, a subset of ∞|∞,

are valid, but different, equations of OI numbers. The number of elements in the group, ∞|∞, is infinite, but a subset of these elements, s, is finite and determined by the selected identity element, m|m, where

s = 2(m-1).

Hence, in a 3D system, combining three, orthogonal, subsets of these OI rational numbers, x, y, and z, we have

1. s_x = 2(m-1)
2. s_y = 2(m’-1)
3. s_z = 2(m”-1)

and, when m = m’ = m”, we have a finite set of 3D OI numbers = (m|m)³, which is a subset of the infinite group of 3D OI numbers, (∞|∞)³, which is perfectly symmetrical.

For example, when m = m’ = m” = 1, there are 0 elements in the 3D set; When they are all equal to 2, there are 8 elements in the set, and so on:

1. (1|1)³ = (2(1-1))_x * (2(1-1))_y * (2(1-1))_z = 0³ = 0
2. (2|2)³ = (2(2-1))_x * (2(2-1))_y * (2(2-1))_z = 2³ = 8
3. (3|3)³ = (2(3-1))_x * (2(3-1))_y * (2(3-1))_z = 4³ = 64
4. (4|4)³ = (2(4-1))_x * (2(4-1))_y * (2(4-1))_z = 6³ = 216
5. (5|5)³ = (2(5-1))_x * (2(5-1))_y * (2(5-1))_z = 8³ = 512
6. …

Now, the question is, what can we do with these symmetrical sets of positive and negative numbers? The answer is, of course, we can make a 3D, fixed, “coordinate” system out of them, but where the “coordinates” represent potential discrete units of magnitude, rather than potential discrete points of space. For example, If we combine the eight elements of the (2|2)³ = 2x2x2 = 8 set, as follows:

1. Q1 = (2|1)_x + ( 2|1)_y + ( 1|2)_z = (5|4) (right, top, front )
2. Q2 = (1|2)_x + ( 2|1)_y + ( 1|2)_z = (4|5) (left, top, front)
3. Q3 = (1|2)_x + ( 1|2)_y + ( 1|2)_z = (3|6) (left, bottom, front)
4. Q4 = (2|1)_x + ( 1|2)_y + ( 1|2)_z = (4|5) (right, bottom, front)
5. Q5 = (2|1)_x + ( 2|1)_y + ( 2|1)_z = (6|3) (right, top, back)
6. Q6 = (1|2)_x + ( 2|1)_y + ( 2|1)_z = (5|4) (left, top, back)
7. Q7 = (1|2)_x + ( 1|2)_y + ( 2|1)_z = (4|5) (left, bottom, back)
8. Q8 = (2|1)_x + ( 1|2)_y +( 2|1)_z = (5|4) (right, bottom, back)

then we can form four OI sets along the diagonals, as follows:

1. Q5|Q3 (right, top, back) | (left, bottom, front)
2. Q6|Q4 (left, top, back) | (right. bottom, front)
3. Q2|Q8 (left, top, front) | (right, bottom, back)
4. Q1|Q7 (right, top, front) | (left, bottom, back)

Clearly, the three elements in the diagonally opposed sets are qualitative inverses of each other, as can be seen by comparing their textual descriptions; that is each element in the front, top, left and right is the inverse of an element in the back, bottom, right and left, but to be a mathematical group, the numbers have to conform to the rest of the rules of the group, which means that there also has to be an identity element in each diagonal set, and the binary operation (addition in this case) on the elements of the group has to be closed and associative.

Well, we know that there must be an identity element in each diagonal set, because the OI numbers from which they are constructed, being elements of the ∞|∞ group, each have an identity element, in this case the number is (2|2) = (1|1), at the intersection of x, y and z. Consequently, adding the identity element to any element of the set doesn’t change the value of the element. This condition of the mathematical group is obviously met in the set, since

(n|m) + (m|m) = (n+m)|(m+m)

doesn’t effect the relative value of the two, reciprocal, quantities, even though it changes the quantities themselves; that is, in the equation

4|5 + 1|1 = 5|6,

the difference between 4 and 5 is the same as the difference between 5 and 6.

To meet the closure condition, if group elements a and b are combined, then the combined element (a + b) must also be an element of the group. In this case, there are eight numbers in the set of (2|2)³, and combining any two of them clearly does not result in another member of the set. Nevertheless, the sum is an OI number in the infinite group, which seems to fulfill the requirement, even though it is not part of the finite set. If we combine the two inverses of the diagonals, however, the result is the identity element, the only other member of the set in this case, so, in that sense, the set is closed, I think. For example,

Q1 + Q2 = [(2|1)_x + ( 2|1)_y + ( 1|2)_z] + [(1|2)_x + ( 1|2)_y + ( 2|1)_z] = (3|3 + 3|3 + 3|3) = 9|9 = 1|1 = 0.

Consequently, we kill two birds with one stone, so-to-speak, showing that the inverse and the closure condition are both met in the diagonal sets; that is, an element plus its inverse equals the identity element (inverse condition), and the sum of two elements in the group is also in the group (closure), even though it’s, again, the identity element:

1. Q5|Q3 = 6|3 + 3|6 = 9|9 = 1|1
2. Q6|Q4 = 5|4 + 4|5 = 9|9 = 1|1
3. Q2|Q8 = 4|5 + 5|4 = 9|9 = 1|1
4. Q1|Q7 = 3|6 + 6|3 = 9|9 = 1|1

As m increases, however, the two tests can be performed separately, even though the result will always include one case where the sum is the identity element, regardless of the value of m. For example, in the case of (3|3)³, there are two non-identity numbers and their inverses, 2|3 and 1|3. The sum of these,

(1|3) + (2|3) = 3|6,

is the inverse of the sum on the other side,

(3|1) + (3|2) = 6|3,

which, combined together, is

(3|6) + (6|3) = 9|9 = 1|1.

However, remember these numbers are nothing but the potential magnitudes, or well-defined sets of static magnitudes, that we can form out of the number, our metric, we might say. It’s the same with the LST fixed coordinate system. The numbers of coordinate pairs, or triplets, and the patterns they form, are a framework to which the numbers of physical magnitude relate, not equate. Hence, there may not be a reason that they should constitute a group.

Fortunately, when we consider the units of this metric, or the interval between the numbers, the set of OI numbers does form a group. This is the set of numbers that conforms to actual physical magnitudes. To illustrate, if we assume only one and two are fundamental quantities, we can form all other magnitudes from these two quantities with OI numbers, just as all integers can be formed with the three fundamental QI numbers, the integers

-1, 0, 1.

The corresponding OI numbers,

1|2, 1|1, 2|1,

are the three fundamental OI numbers of the RSM; that is, all other OI numbers can be formed from them. The operation of the group that this set of numbers forms is addition, and any element of the group is a sum of multiple instances of either 1|2 or 2|1. For example,

(1|2) + (1|2) + (1|2) + (1|2) = 4|8,

which is equivalent to

(-1) + (-1) + (-1) + (-1) = -4.

The inverse of this,

(2|1) + (2|1) + (2|1) + (2|1) = 8|4,

is equivalent to

1 + 1 + 1 + 1 = 4.

Hence, while subtraction, multiplication and division are operations that are not used in the definition of the group, they are nevertheless valid operations that can be defined on elements of the group. For example,

5 x (4|8) = (4|8) + (4|8) + (4|8) + (4|8) + (4|8) = 20|40,

which is equivalent to

5 x (-4) = -20,

is really only another way of specifying the number of elements to be summed. Likewise, the other operations can be seen as an alternate form of specifying the elements to be summed. The important thing is that this set qualifies as a group. In other words, it’s a numerical expression of symmetry, a symmetrical system of numbers.

Now, combining the “coordinate” system of OI numbers, as magnitudes of “distance,” with OI numbers, as multiple units of unit magnitude, we can assign magnitudes to the coordinate “positions” of a given identity element, but since the available “coordinate positions” are limited, by the magnitude of the coordinate identity element, according to the identity element selected, the unit magnitudes, specifying those “positions,” are limited. For example, in a 2x2x2 framework, only one “position,” on either side of unity, in four diagonals, are available. Thus, if we order these, according to some arbitrary scheme, as shown above, then

Q1 = (2|1)_x + ( 2|1)_y + ( 1|2)_z = (5|4),

is the “coordinate position” of one unit of 3D magnitude, the absolute value of which is one unit, but the sign of which is determined by its Q index, indicating its “coordinate position.” In other words, the OI number, designating the magnitude (number of discrete units), at that position, is independent of the OI number designating the position of the magnitude itself. The magnitude has a coordinate position, or a numerical characteristic, we might say, which has both “direction” and limit.

Since these magnitudes form a group, they possess the local symmetry of the finite group that they are part of, but, at the same time, they are also elements of the infinite group and its global symmetry.  Isn’t this exactly what is needed?