THE INTERREGIONAL RATIO Introduction The interregional ratio is an important concept discovered in the development of the Reciprocal System of theory. The works of Larson, notably Nothing But Motion and The Structure of the Physical Universe are to be referred to for an explanation of the origin and significance of this ratio. This paper only attempts to clarify the factors involved in its calculation, as applied to the basic properties of matter. At the outset, I feel that the word “orientation” that we have been using in this context does not seem appropriate because of its strong connotation of direction in space. The word “possibility” might seem preferable, since in evaluating the interregional ratio we are inquiring as to how many possibilities are there for a motion unit to exist—the intrinsic existential possibilities, we might say. Another word that comes to mind is “eigenstate.” But “degrees of freedom” seems very much suitable, provided we refrain from smuggling in some of its spatial connotations. The Reciprocal System shows that there are several types of regions or domains in the structure of the physical universe and that there are interactions across the regional boundaries. During the interactions it is not always the case that the effect of a unit of motion transmitted across the boundary is also one unit. For example, if there are f number of equipossible alternatives within the region for a unit of motion, then by probability laws we know that there is 1/fth chance of the unit effect being transmitted, or what is tantamount, only 1/fth part of the unit motion gets transmitted. The number of possibilities or degrees of freedom, f, is called the Interregional Ratio. Rotational Degrees of Freedom in Threedimensional Time (or Space): Let us examine rotation in space in order to draw conclusions that are equally applicable to rotation in time. “Onedimensional” rotation means that one magnitude (or parameter) is required to fully specify the rotation. A onedimensional rotation occupies twodimensional space. Similarly, a twodimensional rotation requires two magnitudes for its full specification and occupies threedimensional space. Now a unit of onedimensional rotation has two possible directions, +1 and 1, within the framework of threedimensional space, as shown in fig. 1. Fig. 1. The Two Possibilities of a Onedimensional Rotation As a result, the total number of possibilities—the degrees of freedom, as we will call them—in threedimensional space with two possibilities in each dimension is 2*2*2 = 8. Notationally we can express the eight possibilities as
As such, a unit of onedimensional rotation has eight degrees of freedom (that is, intrinsic existential possibilities) in threedimensional space (or time). The question is sometimes raised as to whether the two possibilities in each of the three dimensions do not make up a total of six rather than of eight. This would indeed be true if we were considering three onedimensional spaces instead of one threedimensional space. If the three dimensions are independent, then the total possibilities are
In fact, this is what we have in the case of spacetime dimensions—the dimensions of scalar motion—in distinction to the dimension of space (or time)—which we have called the vector dimensions. Since the three spacetime dimensions, being scalar, are independent, the possible number of degrees of freedom is six.^{(1)} So if n is the number of scalar dimensions and p the number of possibilities per dimension, we can write down the formula for the number of degrees of freedom available in the scalar dimensions as
On the other hand, if the three dimensions are interrelated, the total number of degrees of freedom, as given by eq.(2) is
Another question that is sometimes raised is why two possibilities per dimension and three dimensions do not imply 3^{2} = 9 possibilities rather than 2^{3} = 8. But it is not difficult to see that this would be the case only if we had three possibilities in each of the dimensions of a twodimensional motion, and not otherwise. As the degree of complexity of the motion increases, the existential states possible to it decrease. The twodimensional rotation, it is also remarked, requires two magnitudes to specify it fully. So the possible degrees of freedom for a twodimensional rotation in threedimensional space (or time) are 8/2 = 4. This can easily be understood with the help of the diagrams shown in fig. 2. Fig. 2 The Degeneracy of a Twodimensional Rotation The twodimensional rotation is a coupled rotation of two onedimensional rotations. This coupling causes a “degeneracy.” In fig. 2(a), the directions of the two component rotations are indicated by two plus signs. The characteristic of the twodimensional rotation is that if the directions of both of the onedimensional rotations are reversed, as in fig. 2(b), the net effect is to leave the sense of the twodimensional rotation unchanged, in view of the fact that
Due to this feature, the eight possibilities listed in statement (1) above reduce to four, for the case of the twodimensional rotation, because each of the possibilities listed in the upper line of statement (1) turns out to be the same as the one listed immediately below it, in the second line. For example, for the coupled rotation
Therefore if d is the vector dimensionality of the motion, then eq.(2) is modified to give f, the number of degrees of freedom available in vector space (or time) as
We finally arrive at the total number of degrees of freedom available for a unit of motion in the atom which comprises two twodimensional (magnetic) and one onedimensional (electric) rotations, as
There is another point of relevance that needs to be mentioned at this juncture before turning attention to the inquiry of the vibrational degrees of freedom. We have already distinguished between the dimensions of spacetime (the scalar dimensions) and the dimensions of space (or time) (the vector dimensions). If we have an instance of motion existing in two or three spacetime dimensions, then motion in only one of these spacetime dimensions can be represented in either threedimensional space (or time)^{(2)}. This is depicted in fig. 3. Gravitation (atomic rotation) is three spacetime dimensional. The two spacetime dimensions which cannot be represented in threediomensional time (or space) are fully occupied by scalar motion and therefore leave no more degrees of freedom than calculated by eq. (9).
Vibrational Degrees of Freedom in Threedimensional Time: While a onedimensional rotation has two possibilities (clockwise and counterclockwise, as shown in fig. 1), a onedimensional vibration has only one possibility, since both the directions (forward and backward) in any dimension constitute one oscillation. This is true of both onedimensional linear and rotational vibrations. In view of this, the possible number of degrees of freedom of a onedimensional vibration in threedimensional time (or space), as calculated by eq.(2), with p =1 and n=3, is
However, this number is increased by an additional factor, the freedom available in the three spacetime dimensions, only one of which is occupied by the single unit of photon vibrational motion. This leaves the remaining two spacetime dimensions vacant (unlike in the case of atomic rotation). Consequently the one unit of vibrational motion has three possible choices as far as the spacetime dimensions are concerned. Notationally we can list these possibilities as
Thus the number of degrees of freedom of the onedimensional vibrational unit becomes, by eq.(3) or (4)
At this juncture we recall that that we are not so much interested in the degrees of freedom available to the onedimensional vibration on its own right, but rather in the additional degrees of freedom, if any, that this onedimensional vibration makes available to the rotational unit that is built on it. Since the atomic rotation is a timedisplacement while the basic photon vibration is is a spacedisplacement, both belong to different “regions.” As a result, by applying probability laws, we see that N degrees of freedom of the spacedisplacement of the photon is equivalent to 1/N degrees of freedom from the point of view of the timedisplacement of the rotation. The three degrees of freedom calculated by eq.(12) are specifically applicable to the case of a onedimensional rotation founded on a onedimensional vibration, giving the rotational unit an additional 1/3 degree of freedom. But the rotation basic to the atomic or subatomic structure is twodimensional and not onedimensional.^{(3)} Therefore, with p = 3 and n = 2, by eq.(2), we obtain the total vibrational degrees of freedom from the point of view of the twodimensional rotation as
This means that for every rotational degree of freedom in threedimensional time there is an additional 1/9 degree of freedom due to the underlying vibration. However, since that the atomic structure consists of two twodimensional rotational systems—this is what distinguishes the atom from subatomic particles, the latter having only one twodimensional rotational system in its structure—the additional degree of freedom due to the vibrational contribution is 2/9 (being 1/9 for each of the rotational systems) in the case of atoms, whereas it is only 1/9 in the case of the subatoms. The interregional ratio, which is simply the number of total degrees of freedom, is
in the case of the atomic rotation, and is
in the case of the subatomic rotation. Summary
References
