THE INTER-REGIONAL RATIO
The inter-regional 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 inter-regional ratio we are inquiring as to how many possibilities are there for a motion unit to existthe 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 Inter-regional Ratio.
Rotational Degrees of Freedom in Three-dimensional Time (or Space):
Let us examine rotation in space in order to draw conclusions that are equally applicable to rotation in time. One-dimensional rotation means that one magnitude (or parameter) is required to fully specify the rotation. A one-dimensional rotation occupies two-dimensional space. Similarly, a two-dimensional rotation requires two magnitudes for its full specification and occupies three-dimensional space. Now a unit of one-dimensional rotation has two possible directions, +1 and -1, within the framework of three-dimensional space, as shown in fig. 1.
Fig. 1. The Two Possibilities of a One-dimensional Rotation
As a result, the total number of possibilitiesthe degrees of freedom, as we will call themin three-dimensional 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 one-dimensional rotation has eight degrees of freedom (that is, intrinsic existential possibilities) in three-dimensional 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 one-dimensional spaces instead of one three-dimensional space. If the three dimensions are independent, then the total possibilities are
In fact, this is what we have in the case of space-time dimensionsthe dimensions of scalar motionin distinction to the dimension of space (or time)which we have called the vector dimensions. Since the three space-time 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 32 = 9 possibilities rather than 23 = 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 two-dimensional motion, and not otherwise.
As the degree of complexity of the motion increases, the existential states possible to it decrease. The two-dimensional rotation, it is also remarked, requires two magnitudes to specify it fully. So the possible degrees of freedom for a two-dimensional rotation in three-dimensional 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 Two-dimensional Rotation
The two-dimensional rotation is a coupled rotation of two one-dimensional 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 two-dimensional rotation is that if the directions of both of the one-dimensional rotations are reversed, as in fig. 2(b), the net effect is to leave the sense of the two-dimensional 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 two-dimensional 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 two-dimensional (magnetic) and one one-dimensional (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 space-time (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 space-time dimensions, then motion in only one of these space-time dimensions can be represented in either three-dimensional space (or time)(2). This is depicted in fig. 3.
Gravitation (atomic rotation) is three space-time dimensional. The two space-time dimensions which cannot be represented in three-diomensional 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 Three-dimensional Time:
While a one-dimensional rotation has two possibilities (clockwise and counter-clockwise, as shown in fig. 1), a one-dimensional vibration has only one possibility, since both the directions (forward and backward) in any dimension constitute one oscillation. This is true of both one-dimensional linear and rotational vibrations. In view of this, the possible number of degrees of freedom of a one-dimensional vibration in three-dimensional 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 space-time dimensions, only one of which is occupied by the single unit of photon vibrational motion. This leaves the remaining two space-time dimensions vacant (unlike in the case of atomic rotation). Consequently the one unit of vibrational motion has three possible choices as far as the space-time dimensions are concerned. Notationally we can list these possibilities as
Thus the number of degrees of freedom of the one-dimensional 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 one-dimensional vibration on its own right, but rather in the additional degrees of freedom, if any, that this one-dimensional vibration makes available to the rotational unit that is built on it. Since the atomic rotation is a time-displacement while the basic photon vibration is is a space-displacement, both belong to different regions. As a result, by applying probability laws, we see that N degrees of freedom of the space-displacement of the photon is equivalent to 1/N degrees of freedom from the point of view of the time-displacement of the rotation.
The three degrees of freedom calculated by eq.(12) are specifically applicable to the case of a one-dimensional rotation founded on a one-dimensional vibration, giving the rotational unit an additional 1/3 degree of freedom. But the rotation basic to the atomic or subatomic structure is two-dimensional and not one-dimensional.(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 two-dimensional rotation as
This means that for every rotational degree of freedom in three-dimensional time there is an additional 1/9 degree of freedom due to the underlying vibration. However, since that the atomic structure consists of two two-dimensional rotational systemsthis is what distinguishes the atom from subatomic particles, the latter having only one two-dimensional rotational system in its structurethe 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 inter-regional 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.