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Best thinking about dimensionality

  1. Jan 12, 2005 #1
    Hi all, and bless.

    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,

    Last edited by a moderator: Jan 12, 2005
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  3. Jan 12, 2005 #2


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    Nightcleaner, that Princeton guide is for people who are taking, or have taken, graduate physics courses. It contains all the main equations and formulas, with their math derivations, but has none of the real intuitive physics sense behind them. The students are supposed to pick that up in class.

    The reason n-take-k formulas turn up in thermodynamics and such is that you are interested in sets of molecules and their energy. In how many ways can you get k molecules if you start with n of them? An example of this idea, which goes back to Laplace, is pulling balls out of an urn. Suppose there are n balls in it and you reach in blindly and pull out k. How many ways can you do this? That is how many different combinations of balls are there? The answer is n-take-k, your Pascal number. This is the start of probability theory, and it is the probability aspect which brings these numbers into statistical physics.

    Octonions are a kind of higher arithmetic. (the technical name is "Clifford Algebra") after William Kingdom Clifford who devised them) They have nothing (special) to do with octahedra. The basic idea is this;

    Start with the real numbers (symbolized by the letter R in bold or open face type). They have one generator, the number 1.

    Construct the complex numbers a + bi, where a and b are real numbers and i is a new generator which turns out to have the property that i2 = -1. Then complex numbers form a complete arithmetic (add, subtract, multiply, divide) and are symbolized by the letter C.

    William Rowan Hamilton discovered how to build the quaternions a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are new generators. The real generator 1 and the three new ones make four, which is where the 'quater' prefix comes from. The quaternions are symbolized by H for Hamilton, and have all the arithmetic operations, but multiplication is not commutative q*r = - r*q.

    The octonions have eight generators a + bi + cj + dk + el + fm + gn + ho. They are weaker than a full arithmetic because they aren't associative a* (b*c) isn't equal to (a*b)*c.

    There is a theorem that you can get all possible Clifford algebras by taking matrices with numbers of these four kinds as entries.
    Last edited: Jan 12, 2005
  4. Jan 12, 2005 #3
    Thank you selfAdjoint this is very helpful. I don't have access to any graduate level physics classes, but it amuses me to look at the pretty squiggles and try to prise out just exactly what it is I do not know.

    For example I notice that the Bose statistics are different, a little, from the Fermi statistics. It is the Fermions that behave in the n-choose-k manner, while the Bosons do something more complex. The book says that the formula gives the number of ways to choose g objects such that the sum is n objects.

    You are right of course about the book not giving any hint of intuitive meaning. It is a little like looking at a Russian dictionary when you haven't yet learned the Cryllic alphabet. You not only do not know what the words mean, you do not even have any way to know what they sound like.

    I know the large font Greek "E" (it is a sigma, actually, isn't it?) is the summation sign, and I sort of know what that does. The large font Greek "Pi" is still more of a puzzle for me. I have read that it is a product sign, and does to the formula which it precedes the same as the Greek "E", only with multiplication instead of addition. I get lost about there.

    Well I've been shoveling snow and am a bit tired so will go have a rest and come back for more of this later. I hope you are well,

  5. Jan 12, 2005 #4


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    You're right about the [tex]\Pi[/tex] symbol being used for multiplication. Suppose you had two numbers, call 'em x1 and x2 and you wanted to multiply them. You could just write x1x2, or you could write [tex]\prod_{i=1}^2 x_i [/tex]; they mean exactly the same thing. Of course the utility of the product notation comes when there are lot of factors to multiply, even an infinite number; [tex]\prod_{i=1}^{\infty} x_i[/tex]. Of course you'd darn well better be sure all those x's converge when they are multiplied together!
  6. Jan 13, 2005 #5
    By convergence I imagine you mean that the numbers do not multiply to infinity? So there would be some distribution about unity, some of the numbers tending to reduce the total while others increase it. There would be a PI operator notation for the factorial series, something like PI from i=1 to n of n_i

    [tex]\prod_{i=1}^{n} n_i[/tex] ?

    Now about all those ijklmnop's in the octonion. Each of them is a variable within some limits, as in the above where i= 1 to n. Each variable in the simplest case is a dimension. So I imagine an octonion mathematics would be useful when observing some system in eight dimensions.

    Most of us do not have an intuitive idea of what kind of observable would inhabit eight dimensions. Actually it is difficult but not impossible to imagine a system that would need eight dimensions. You might want to look at a sports team and analyse the value of the players by eight different measures. But it is harder to imagine what the simplex for an eight dimensional system is. What is the basic shape of the data, the least complex view that displays all the variables? One might be interested in such a view if one were trying to determine which player to cut or add to the team.

    Anyway I have read somewhere that the Hamiltonian is used to determine the total energy of a system. The Hamiltonian uses quarternions? That would be a set of {i,j,k,l} variables? Or I guess we might wish to use xyzt, so that the Hamiltonian of a system in xyzt would describe its movements in three dimensional space, the idea of movement accounting for the dimension t. Darn lucky that t occurs in the denominator, so that it keeps it all from blowing up to infinity, eh?

    Well I have to go do chores. Thanks for the help.

    Be well,

    Last edited by a moderator: Jan 13, 2005
  7. Jan 13, 2005 #6


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    The formula for the factorial would be

    [tex]n! = \prod_{i=1}^n i[/tex]

    The actual numbers that are in the index are multiplied together. This is the same as n! = 1*2*3*...*n.

    The Hamiltonian uses whatever math system desribes the physics, be it real numbers, complex numbers, or vectors of some higher space.
    Last edited: Jan 13, 2005
  8. Jan 14, 2005 #7
    But energy has natural units which give it a specific dimensionality which means it behaves in a way described by a limited set of math systems, not just any system. Have you looked at the terms of energy, force and power under spacetime equivalence? Space exponents cancel time exponents.
  9. Jan 14, 2005 #8


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    Quite so, but althout the output, energy, is described as a real number, that doesn't mean the input math has to be. The physics is described by "canonical" coordinates - versions of the position and momentum coodinates as specialized for the given physical environment. And when you get into LQG or string theory those environments can get hairy.

    I think you mean the time components have opposite sign to the space components, due to the metric of Minkowski space (and locally in GR). Perfectly true, although they don't generally exactly cancel out.
    Last edited: Jan 14, 2005
  10. Jan 14, 2005 #9
    Well that's my theory and i'm sticking to it!

    I did some graphing of a spacetime equivalence analysis and posted it at SIP, but it didn't come out very pretty on the board. I could send it to you email. I know this has been a point of difference between our views. Still, I think it may mean something. Graphing the exponents as a unit basis, a log scale space, and the derived units make an interesting picture. I think they don't generally cancel out in our observations because the point of observation is always moving, giving it a self-preferred reference frame. The preferrence is not absolute but is a part of the observer system. There is no real absolute preferred frame, but that does not stop us from having to choose one. Every time we make an observation, we have to begin by choosing a frame, is it not so? Then we go on to normalize everything else to the chosen locality?

    I have to go to work in the kitchen to night but will try to think on it some more.

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