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nLab > Latest Changes: a priori knowledge physics

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  1.  
    • CommentRowNumber1.
    • CommentAuthoraleksr
    • CommentTimeMar 23rd 2019

    A theory of everything must be one which is based on a priori knowledge physics.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019
    • (edited Mar 23rd 2019)

    Wait a moment.

    Let’s see. The page you just created with the title “a priori knowledge physics” is just a copy of an earlier page that you created a few days ago, called “hyperanalytic function”.

    Why did you do that (make a copy with that new title)? At the face of it, this seems an extremely crackpotterish move to do.

    I suggest you pause your a priori ambitions for a while and let us have a look at your entry on hyperanalytic functions first…

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019
    • (edited Mar 23rd 2019)

    I also take issue with your claim. This is a deep problem of physics and philosophy, and certainly not obviously true.

    I take the position (and I think most of the regulars here would too) that if you wish to contribute to the nLab, you should start by improving what is here first, by expanding entries around existing material, becoming familiar with the nLab style to some extent, interacting with the community via the nForum. If you have more speculative ideas, then you should find out if and where they fit. The nLab is an open notebook, yes, but not open for random speculation about deep problems by any passers-by. It certainly doesn’t help that we don’t know anything about you or your background.

    Drive-by edits to fix typos, add references, or add material from the literature are definitely welcome here. Dropping in and claiming new a theory of everything on your first or second edit is not ok.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019

    Actually, now that I look at a priori knowledge physics, I see the link http://www.gaussianfunction.com/. This leads me to a page that, shall I say, reads like material that should not be on the nLab. Sorry, but that’s how it goes: the nLab is not a platform to promote such personal theories.

    • CommentRowNumber5.
    • CommentAuthoraleksr
    • CommentTimeMar 23rd 2019

    This page is under construction. Please, wait a week.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019

    No, it’s not good enough. If you are porting stuff from http://www.gaussianfunction.com/, then I can already say it shouldn’t be here. The merits of the page should be clear from the get go, a week of slow reveal of material that seems like random unsupported claims is not the type of editing or contributions we want here.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMar 23rd 2019

    Yes, I agree with David; this page should be removed now.

    • CommentRowNumber8.
    • CommentAuthoraleksr
    • CommentTimeMar 23rd 2019

    Do You know that all intensities of all known and unknown interactions are equal to powers of the fine structure constant? Do You know that that there is only one type of functions that produces fine structure constant? I think You don’t know. Please, let me finish and then You will decide.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019

    I don’t think you know either. Please stop adding this material to the nLab. Final warning.

  10. I have deleted this page and ’hyperanalytic function’. These are the only pages which have been edited by the author except the Sandbox. I have the content of the pages logged in case it should be needed at some point (but the logs will be removed eventually). I have not blocked the author from making edits or from discussing here.

    I would just like to say that personally I feel we should always try to be as polite as possible in these situations, and in general with our communication with one another. A person may be misguided in something but a perfectly nice person nonetheless, who might be as worthy of respect as anyone else if we knew them better. We of course have to protect nLab content, and it is really great that so many take care about this and help out with it, but I at least always feel a bit uncomfortable when the tone may come across as a little commanding, or from ’on high’.

  11. Even in the recent announcements of the steering committee and in recent communication about security issues, I for example reacted slightly against the tone adopted at times. I am probably not alone in being much more likely to feel conducive to co-operation if someone says “Would it be a good idea to try …” rather than “Do this…” or “This is no good, you must do…”. The first is just more suggestive of respect and consideration.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019

    @Richard,

    pre-emptive action. We’ve spent months in the past trying to turn around peoples’ misguided attempts to contribute, that eventually led to a firm suggested to desist.

    Here are some relevant pages that Aleks should read https://ncatlab.org/nlab/show/writing+in+the+nLab#some_obvious_truisms and https://ncatlab.org/nlab/show/About#who_are_we.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2019

    @Richard again

    Also, I don’t say these things lightly. In the past I’ve been the one advocating for a light touch, trying to extract some interesting content out of what has been written. I’m surprised you deleted these pages so quickly. I was hoping to come to some sort of resolution here before figuring out what to do with the actual content.

    • CommentRowNumber14.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 24th 2019
    • (edited Mar 24th 2019)

    Re #13: My apologies if they should not have been removed. Here is the content. The pages can be put back if requested. The source compiles correctly in the actual nLab, the differences are due to the fact that only the old renderer is used on the nForum; I have lightly edited some syntax to make it display reasonably here.

    Hyperanalytic functions

    Idea

    An hyperanalytic function is a function that is locally given by a converging power series with a speed that corresponds to tetration.

    Definitions

    Let VV and WW be complete Hausdorff topological vector spaces, let WW be locally convex, let cc be an element of VV, and let (a 0,a 1,a 2,…)(a_0,a_1,a_2,\ldots) be an infinite sequence of homogeneous operators from VV to WW with each a ka_k of degree kk.

    Given an element cc of VV, consider the infinite series

    ∑ ka k(x−c) k \sum_k a_k(x - c)^k

    (a power series). Let UU be the interior of the set of xx such that this series converges in WW; we call UU the domain of convergence of the power series. This series defines a function from UU to WW; we are really interested in the case where UU is inhabited, in which case it is a balanced neighbourhood of cc in VV (which is Proposition 5.3 of Bochnak–Siciak).

    Let DD be any subset of VV and ff any continuous function from DD to WW. This function ff is hyperanalytic if, for every c∈Dc \in D, there is a power series as above with inhabited domain of convergence UU such that

    f(x)=∑ ka k(x−c) k f(x) = \sum_k a_k(x - c)^k

    for every xx in both DD and UU and just rare a ka_k are not equal zero. (That ff is continuous follows automatically in many cases, including of course the finite-dimensional case.)

    Examples

    {#Examples}

    It is known that there is a fundamental connection between analyticity of the function and the convergence of its Fourier coefficients. The better the function, the faster its coefficients tend to zero, and vice versa. The power decrease of Fourier coefficients is inherent in functions of the C kC^{k} class while exponential to analytical functions. Here there is a possibility of existence of the hyperanalytic functions, for which the decrease of the Fourier coefficients corresponds to tetration.

    Natural hyperanalytic function occurs when considering reticulum with a step L, in which nodes there are not defined yet objects. The distribution of center’s objects can be described using the reticulum functions (RF). The definition of a one-dimensional RF is based on the following identity:

    1σ2π∫ −∞ ∞e −12(xσ) 2dx=1σ2π∫ −L2 L2∑ n=−∞ ∞e −12(x−nLσ) 2dx=1. \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x}{\sigma})^{2}}dx=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\frac{L}{2}}^{\frac{L}{2}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}}dx=1.
    Definition

    From here RF is

    ℝ(x)=1σ2π∑ n=−∞ ∞e −12(x−nLσ) 2. \mathbb{R}(x)=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}}.

    It is obvious that the RF can not be laid out in the Fourier series because it does not have antiderivative that can be expressed as elementary functions. By virtue of this RF cannot be decomposed into even and odd functions, while an arbitrary analytic function ff can be only presented in the form of sum of odd and even functions in the interval [a,b][a,b]:

    f(x)=g(x)+h(x), f\left(x\right)=g\left(x\right)+h\left(x\right),

    where

    g(x)=f(x−a)−f(b−x)2, g\left(x\right)=\frac{f\left(x-a\right)-f\left(b-x\right)}{2}, h(x)=f(x−a)+f(b−x)2. h\left(x\right)=\frac{f\left(x-a\right)+f\left(b-x\right)}{2}.

    Due to this the RF can be laid out in an endless row of two primitive hyperanalytic functions by sequential attempts to decompose on even and odd functions. Thus, the RF can be decomposed by the simplest way, but such a series is not one like the orthonormal basis of Fourier series.

    Decomposition of RF

    The mean value of RF

    As it follows from (1.1) the mean value of RT is 1. However as will be seen from the further, it is expedient to choose the greater value of the decomposition’s constant member. Introduce the following definitions:

    Definition

    ℝ(0)\mathbb{R}\left(0\right) is

    ℝ(0)=ℝ max=1σ2π∑ n=−∞ ∞e −12(−nσ) 2, \mathbb{R}\left(0\right)=\mathbb{R}_{max}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{-n}{\sigma}\right)^{2}},

    ℝ(1/2)\mathbb{R}\left(1/2\right) is

    ℝ(1/2)=ℝ min=1σ2π∑ n=−∞ ∞e −12(1/2−nσ) 2. \mathbb{R}\left(1/2\right)=\mathbb{R}_{min}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/2-n}{\sigma}\right)^{2}}.

    Then A 0A_{0} is the mean value of RF:

    A 0=ℝ max+ℝ min2. A_{0}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}.

    First difference

    One can approximate first difference by the following way:

    A 1(x)=ℝ max−ℝ min2cos(2πx). A_{1}\left(x\right)=\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{2}cos\left(2\pi x\right).
    Definition

    Let introduce parameter of the fine structure α\alpha as function of σ\sigma:

    α(σ)=12ℝ max−ℝ minℝ max+ℝ min. \alpha\left(\sigma\right)=\frac{1}{2}\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{\mathbb{R}_{max}+\mathbb{R}_{min}}.

    Now A 1(x)A_{1}\left(x\right) can be expressed:

    A 1(x)=ℝ max+ℝ min2(2α(σ)cos(2πx)). A_{1}\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right).

    The choice of the name and symbol of this parameter is due to the fact that

    α(0.4992619105929628)=α=e 24πε 0ℏc \alpha\left(0.4992619105929628\right)=\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}

    is the value known in physics as a fine structure constant.

    Even differences

    Even differences are a primitive hyperanalytic function 𝕍¯(2i×2πx)\overline{\mathbb{V}}(2i\times2\pi x), which is quasisymmetric relative to the point x=0.25\text{x=0.25}.

    Its symmetrical part approximated in the following way:

    A 2i(x)=c 2i(cos(2i×2πx)−1) A_{2i}\left(x\right)=c_{2i}\left(cos\left(2i\times2\pi x\right)-1\right)

    and

    ∑ i=1 ∞c 2i=ℝ max+ℝ min2−1=2*∑ k=1 ∞α 4 k. \sum_{i=1}^{\infty}c_{2i}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-1=2 * \sum_{k=1}^{\infty} \alpha^{4^{k}}.

    Using the value

    ℝ(1/4)=ℝ 1/4=1σ2π∑ n=−∞ ∞e −12(1/4−nσ) 2 \mathbb{R}\left(1/4\right)=\mathbb{R}_{1/4}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/4-n}{\sigma}\right)^{2}}

    define the amplitude for c 2c_{2}: 12(ℝ max+ℝ min2−ℝ 1/4)=2α 4\frac{1}{2}\left(\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-\mathbb{R}_{1/4}\right)=2\alpha^{4}. This definition allows to select approximation A(x)A\left(x\right) in the form:

    A(x)=ℝ max+ℝ min2(1+2α(σ)cos(2πx))+2α 4(cos(2×2πx)−1). A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(1+2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right)+2\alpha^{4}\left(cos\left(2\times2\pi x\right)-1\right).

    Odd differences

    Odd differences are a primitive hyperanalytic function 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right), which is quasiantisymmetric relative to the point x=0.25\text{x=0.25}.

    Quasiantisymmetry of 𝕎(2πx)\mathbb{W}\left(2\pi x\right) follows from the fact that the integral of A(x)A\left(x\right) differs from 1:

    ∫ −1/2 1/2A(x)dx−1=14(ℝ max+ℝ min)+12ℝ 1/4−1≃1.02E−34. \int_{-1/2}^{1/2}\text{A}\left(x\right)\text{dx}-1=\frac{1}{4}\left(\mathbb{R}_{max}+\mathbb{R}_{min}\right)+\frac{1}{2}\mathbb{R}_{1/4}-1\simeq1.02E-34.

    Thus function 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right) should be decomposed in the even and odd function. Its even part is:

    𝕎 qs((2i−1)×2πx)=𝕎((2i−1)×2πx)+𝕎((2i−1)×2π(0.5−x))2=𝕍¯(2(i+1)×2πx). \mathbb{W}^{\text{qs}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)+\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}=\overline{\mathbb{V}}(2(i+1)\times2\pi x).

    However, as shown above, 𝕍¯(2i×2πx)\overline{\mathbb{V}}(2i\times2\pi x) is not an even function.

    The odd part of 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right) is no longer a hyperanalytic function and is equal to:

    W qa((2i−1)×2πx)=𝕎((2i−1)×2πx)−𝕎((2i−1)×2π(0.5−x))2. W^{\text{qa}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)-\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}.

    It can be approximated with any degree of accuracy following way:

    A(W qa((2i−1)×2πx))=β(cos(3(2i−1)×2πx)−cos((2i−1)×2πx)), A(W^{\text{qa}}\left((2i-1)\times2\pi x\right))=\beta(cos\left(3(2i-1)\times2\pi x\right)-cos\left((2i-1)\times2\pi x\right)),

    where β\beta is a normalizing multiplier.

    Thus, the approximation of ℝ(x)\mathbb{R}(x) is:

    A(x)=ℝ max+ℝ min2(1+2αcos(2πx))+2∑ i=1 ∞α 4 i(cos(2i×2πx)−1)+2𝕎 max∑ i=1 ∞α 9i 2(cos(3×(2i−1)×2πx)−cos((2i−1)×2πx)), A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}(1+2\alpha cos\left(2\pi x\right)) +2\sum_{i=1}^{\infty}\alpha^{4^{i}}\left(cos\left(2i\times 2\pi x\right)-1\right)+\frac{2}{\mathbb{W}_{max}}\sum_{i=1}^{\infty}\alpha^{9{i}^2}\left(cos\left(3 \times (2i-1)\times 2\pi x\right)-cos\left((2i-1) \times 2\pi x\right)\right),

    where 𝕎 max\mathbb{W}_{max} is a normalizing multiplier.

    Three-dimensional RF

    Three-dimensional RF ℝ(x,y,z)\mathbb{R}\left(x,y,z\right) can be obtained from the definition (1.2)\left(1.2\right):

    ℝ(x,y,z)=ℝ max 2ℝ(x). \mathbb{R}\left(x,y,z\right)=\mathbb{R}_{max}^{2}\mathbb{R}\left(x\right).

    Thus, the approximation of the three-dimensional RF is also the series of the fine structure constant α\alpha along any axis of the reticulum three-dimensional space, and the constant itself is a function of the dimensionless parameter σ\sigma, which is equal to quotient of the “diameter” of some physical object, located in each cell, to the grid step L.

    Quantum derivative with respect to time

    To quantize the time the direct use of the lattice idea is too formal. It is therefore appropriate to use a definition of derivative with respect to time but without moving to the limit. Let ℝ(t)\mathbb{R}\left(t\right) is RF on a unit interval [−T/2,T/2]\left[-\text{T/2},\text{T/2}\right] and τ=σ\tau=\sigma и T=1\text{T}=1:

    ℝ(t)=1τ2π∑ i=−∞ ∞[exp(−12(t+T/4−iτ) 2)−exp(−12(t−T/4−iτ) 2)]. \mathbb{R}\left(t\right)=\frac{1}{\tau\sqrt{2\pi}}\sum_{i=-\infty}^{\infty}\left[\exp\left(-\frac{1}{2}\left(\frac{t+\text{T/4}-i}{\tau}\right)^{2}\right)-\exp\left(-\frac{1}{2}\left(\frac{t-\text{T/4}-i}{\tau}\right)^{2}\right)\right].

    By consistently subtracting sinuses, one can show that the approximation of the ℝ(t)\mathbb{R}\left(t\right) has the following form:

    A(t)=∑ k=0 ∞(−1) k+1a ksin(2π(2k+1)t). A\left(t\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).

    Let use k+1k+1 equations with different values of ll to determine the coefficient’s values a ka_{k}:

    ∑ i=0 k(−1) ia isin(2i+12l+12π4)=ℝ(14(2l+1)). \sum_{i=0}^{k}\left(-1\right)^{i}a_{i}sin\left(\frac{2i+1}{2l+1}\frac{2\pi}{4}\right)=\mathbb{R}\left(\frac{1}{4\left(2l+1\right)}\right).

    Given that A(1/4)A\left(1/4\right) is numerically equal to 2(ℝ max(τ)+ℝ min(τ))α(τ)2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)\alpha\left(\tau\right), equation can be written as follows:

    α eff(t,τ)=12(ℝ max(τ)+ℝ min(τ))∑ k=0 ∞(−1) k+1a ksin(2π(2k+1)t). \alpha_{eff}\left(t,\tau\right)=\frac{1}{2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)}\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).

    ℝ(t)\mathbb{R}\left(t\right) is also a hyperanalytic function, as the next approximation takes place:

    α eff(t,τ)=∑ k=0 ∞(−1) k+1α (2k+1) 2sin(2π(2k+1)t). \alpha_{eff}\left(t,\tau\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}\alpha^{(2k+1)^{2}}sin\left(2\pi\left(2k+1\right)t\right).

    Related concepts

    References

    The theory of hyperanalytic function was constructed to some extent by A. Rybnikov (2014) http://www.gaussianfunction.com/.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 24th 2019
    • (edited Mar 24th 2019)

    @Richard,

    I didn’t foresee a long-term future for the material, it’s just nice to arrive at a bit more of a consensus. Now at least others who didn’t get a chance to make a call on the page can at least see what we are discussing (without a copy of it, who’s to say we didn’t just scrub someone’s contributions because we didn’t like it?)

    I just noticed that the material implicitly gives a formula for the fine structure constant, but then substitutes a finite decimal approximation for it. This is a big red flag, given the unusual presentation.

    • CommentRowNumber16.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 24th 2019
    • (edited Mar 24th 2019)

    Indeed, I agree in this case it’s probably a good idea to have a copy available. With obvious spam (so not the case here) I tend to have a zero-tolerance policy, and will immediately delete and block the user; I am keen to delete properly rather than do things like rename to empty pages. As has been discussed before, in the long run it’d be great with better support in the software for this.

    As mentioned, every edit is now logged outside of the database, including updates which are invisible in the database due to being within the 30 minute window, so it is always possible (until the logs are removed, which will not be for some time) to recover a deleted edit.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2019

    Thanks everyone!

    Richard is right. Since we are all occupied otherwise, when in the thick of it and something disconcerting comes in here on top of it all, it can be hard to conjure the energy required to deal with it in Solomonian fashion. But good to try harder, sure.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeMar 24th 2019

    I agree we should of course save a copy. With that said, I also agree that our experience (and those of many other people) is that trying to have a productive conversation with crackpots is a waste of time.

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