The Larson Research Center

In the previous post, I considered deriving a natural unit of motion, derived from the Rydberg frequency for hydrogen, as a basis for clarifying the meaning between dimensionless and dimensionful constants that John Baez has been discussing on his blog.  Of course, this is exactly what Larson did. Using his approach, we can define a “natural” unit of length and time in terms of a selected system of units, whether it is a system of scientific units, or a system of bananas.

However, it’s also clear that the dimensions of the space aspect of the natural unit of motion defined in this manner are not fixed at 1; that is, we can think of an increase of area, or of volume, over time, as motion, just as easily as we can think of an increase of length, as motion.

Logically, therefore, it’s just one more small step to formulate the equation of motion as the ratio of two, dimensionless scalars. However, while we can conceive of squared or cubed meters, we can’t conceive of squared or cubed bananas, even though we can square, or cube, the NUMBER of any quantity of bananas. So, in a sense, there is an isomorphism in the dimensions of space and the dimensions of , even though we are not accustomed to thinking of it that way.

The problem we run into is in dealing with unity as a number. While (2 meters)² and (2 bananas)² are both equal to 4, four square meters and four bananas mean different things, because the dimensions of one square meter are two, but the dimensions of one “square” banana are zero. Yet, if space is an “amount of substance” in , why are the dimensions of square space (in meters, yards, or furlongs) two and not zero? In other words, if we can’t conceive of square bananas, how can we conceive of square meters, if we are regarding space as an “amount of substance?”

The reason is reflected in one of John’s statements, posted in the discussion on his blog, explaining the concept of passive coordinate transformations:

We also use [passive coordinate transformations] when we switch from a system of units to a simplified system of units. For example, the SI system has seven units, which measure length, mass, time, current, temperature, amount of substance, and luminous intensity. So, any physical quantity gives a point in - the “dimensions” of this quantity in the SI system. But, we may choose to work with fewer units. For example, we may decide not to treat “amount of substance” as a dimensionful quantity. SI measures amount of substance in moles, but we can say a mole is just 6.0221415(10)x10²³ - a dimensionless number, Avogadro’s number. Our new system of units assigns to each physical quantity a point in , and we have a “change of units”

which is not an isomorphism. The ultimate extreme is to work in a system where all physical quantities are treated as dimensionless, so any physical quantity has dimensions living in . This is actually very popular in fundamental theoretical physics.

Therefore, even if it’s the ultimate extreme, if we regard the new natural unit of motion as a physical entity, John is saying that we can treat it with dimensions in , which is what we have also concluded, though not as formally and certainly not for the same reason!

But there is another “non-isomorphism” (sorry John, there’s just no way I can speak in the language of categories yet) that is found between the left and right side of the binomial expansion. On the left side, all the ones are in , but on the right side, all the ones are in , and when n = 3, the octonions, with dimensions , the binomial expansion is a map from a point to a volume, or from a zero-dimensional scalar to a three-dimensional pseudoscalar.

Indeed, whenever n > 0, is mapped to , and, I guess, when n = 0, is mapped to itself. Interestingly, however, the arrow in John’s passive tranformation goes in the other direction, from larger to smaller.

So, if we start with the larger, assuming n = 3, the dimensions of the natural unit of motion that we have defined would not be zero, but three, the dimensions of the pseudoscalar in the octonions. Thus, the equation of motion would be

as strange as this might seem to many, at first.

As John has explained in “This Week’s Finds” (I forget which one), the octonions are considered the “crazy uncles” of the math world and kept in the attic, because they are non-associative, yet they are mysteriously related to the Bott periodicity theorem, and some physicists are convinced that they play a key role in physics.

Well, if we define a natural unit of motion that has the dimensions of the octonions, and if we recall that the definition of motion is the heart and soul of physics, that should provide some motivation for taking the concept of a natural unit of motion seriously. However, it’s the mathematics that will have to be our guide, and here is the way we can think about it:

1) In general, the passive transformation that we are interested in is from to . Specifically, at n = 3, because BPT proves that there is no new phenomena beyond n = 3.

2) When we do this, we have to incorporate a view of unity that embraces the passive transformation; that is, the natural unit of motion takes center stage. This means that we replace the quantity one, with the ratio 1, and then the expression:

in the binomial expansion, changes meaning, and, consequently, the ones down the left side of the expansion (the scalars) are

,

and the ones down the right side (the pseudoscalars) are

which seems to be a mathematically meaningless distinction, but the point is, it is far from meaningless, as we will see below.

3) The displacement of direction (n-dimensional rotation) that normally forms the coordinate system of vectorial units, which we can map to “all possible coordinate charts” (i.e. higher, lower dimensional units), at a given value of n, is replaced by the displacement of “direction” (“n-dimensional” expansion) that forms a new coordinate system of higher, or lower, “n-dimensional” scalar units, which we want to map to all possible coordinate charts, at a given value of n.

How does this work?

At n = 0, there are no coordinate charts to work with (), but at n = 1 there are two (); At n = 2 there are four (), and at n = 3, there are eight ().

The binomial mapping is possible, because, by defining the unit value as a ratio of proportions, there are two “directions” in each dimension, n/1 and 1/n, which together, relative to 1/1, form a “bidirectional” field of scalar values.

So, applying the expansion of this scalar “bidirecton” at n = 1, instead of the rotation of a vectorial direction, which we normally use imaginary numbers to define, we have a new coordinate chart, the 1/n and n/1 scalars, which is isomorphic to the positive and negative integers.

However, n=1 also includes n=0, because . So we have two charts, the original scalar and the coordinate chart (signed integers). At n=2, we have four charts, the original scalar, two sets of signed integers, and … what? How do we describe the pseudoscalar at n = 2? For that matter, how do we talk about two sets of signed integers?

The key is in understanding the role of the scalar, which we have now placed in the middle of our 1D coordinate system. The scalar coordinate system of rational units makes no more sense without the reference point of unity, than the signed integers do without the reference point of zero.

So, the n=1 expansion, must include the unit scalar in the pseudoscalar; that is, it is isomorphic to a one-dimensional “line,” but not described from point to point, but from the center to one point and from the center to the opposite point. Numerically, the expression

becomes a “line” magnitude, where unity is “displaced” in two “directions,” which we can designate positive and negative, but together they form a composite scalar number similar to the way in which the reals and imaginaries are combined into a composite number that is a number of higher dimension than the reals alone.

Now, let me draw the obvious conclusion, and then I’ll end this: The square of the composite scalar number above, which is composed of and , is in , and its cube is in , which means that all three, , , and are contained in . Yet, it doesn’t matter as long as

It’s not like those nested groups and n-spheres that Baez and company talk about, because the dimensions of scalars are always zero. Yet, while that’s true in the ususal sense, it’s also obvious that the square of a scalar “line” magnitude, as defined in the composite number above, is isomorphic to an “area” magnitude, and such a “line” magnitude cubed is isomorphic to a “volume” magnitude, in this new sense. In other words, given

,

then,

,

which is now a value in , and if we cube it,

,

we get a value in , which contains copies of all the lower dimensional values, just as the octonions contain points, lines, and areas in the pseudoscalar. Only now, the algebra of these “multi-dimensional” psuedoscalars has to have the ordered, commutative, and associative properties of the zero dimensional scalars!

In other words, because we’ve defined the scalar value of 4/4 as the natural unit of motion, but in the form of a 1D magnitude (a “line” magnitude) the square of this unit is (4/4)² = 16/16 natrual units of motion, or four of the 1D magnitudes (4 “line” magnitudes),  which are the minimum required to form a square around a center point. Its cube is (4/4)³ = 64/64, or 16 of the 1D magnitudes (16  “line” magnitudes) of motion, which are the minimum required to form a cube with a center plane.

There is much more to say about this idea, but the point I wish to make here is that this is system of units that is like a system of coordinates, but it is not a system of coordinates. It’s a system that lets us describe the reciprocal aspects of the reality of motion, space and time, using numbers, but without depending on our arbitrary choices of units.

(more later)