Hi all, and bless.(adsbygoogle = window.adsbygoogle || []).push({});

Sorry about the mystic greeting but I feel like I need all the grisgris I can get these days. A frustrating week.

However, i was last working on Pascal triangles, tetrahedra, and higher dimensional simplexes. N-choose-k was the math of the day. Now I have recently acquired, in the usual cheap paperback version, a copy of Princeton Guide to Advanced Physics, Alan C. Tribble, 1996. I find low and behold, N-choose-K formula for statistical physics of Bose-Einstein gas (bosons) and Fermi-Dirac gas (Fermions). Most of it is Greek and gives me headaches, but look and see, there is the N-choose-k formulations!

But what does this mean to a poor country lad?

I suppose it has something to do with how many paths there are to integrate among interactions between the boson and fermion crowd. Twist all the paths together into a bundle, eh? Is it something to do with that? And octonions, are they something to do with the octahedral planes in the cubeoctahedron?

Well anyway a discrete spacetime structure of minimal actions makes things a lot easier. In the lattice there are really only a few ways to get from one vertice to another. But what do we mean when we talk about going? A movement involves time as well as space. Each member of the lattice is one space and one time unit, both at once. And there is a spacetime ratio, which is velocity. And velocity is relitive. The spacetime ratio changes as an observer moves through the lattice. What is changing? We know the changes are very small, almost continuous. And at the Planck limit, really, nothing changes much at all.

How does an observer move through the lattice? By occupying portions of it in sequence. How does this occupation take place? In two ways. In one way, the observer advances positions at some spacetime rate. In the other way, information about the surrounding lumosphere infalls and meets the observer even as the observer advances. This information changes the state of the observer.

But again, at the Planck scale, there is really very little change to worry about. Huge regions of Planck space (well, according to scale they are huge, as the scale is very small), that is, very large numbers of Planck units, are all nearly in identical relationship to each other. Locally, change is very slow. It is so slow that you can almost examine it frame by frame as if there were no time, and the universe in all its dimensions were no more lively than a still photograph.

If this is still unclear, this vision of the frozen river of time, the unalterable record of all time histories, then only remember that we in our ordinary human lives are moving pretty slow compared to light. If the spacetime ratio is one space to one time, the velocity is that of light. At our daily raceing pace, we must have a velocity ratio that has gadzillions of spaces to every time. Since the time element is required to register a change, All those gadzillion spaces are identical, and only one will register a tick of time.

As in Pascals triangle, and I guess by the useages of Occam's razor, when all the units are the same unchangeing, we can just call them out as one and not actually have to count. Look at the Pascal triangle. Then look at the subset of the Pascal triangle that calls out the numbers of simplexes in any dimension. The subset excludes a layer made all of ones. As in the Einstein convention, if it is always going to be a one, you don't really need to talk about it. The only interesting stuff in the formulation is that which changes.

Each way there is to change in a Planck unit is both a distance and a time. Each way there is to change is a dimension. At the Planck scale, how many dimensions are there for bosons and fermions?

Well, some things we want to observe about statistical behavior of particles can be examined in terms of scale, with no dimension at all, just a point. And some other things can be examined in terms of a line, like a spacetime unit member of the lattice. And some things that can be examined are two dimensional some three, and so on. Some things about the behavior of particles which we wish to examine are four or more dimensional.

Well, that's my best thinking about dimensionality. For the Time Being. But I still have to find out what, if anything, all this has to do with those funny Greek letters.

Be well,

Richard

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# Best thinking about dimensionality

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