Cynosure对《Fractional Calculus》的笔记(1)

第1页 Chapter 26 On the Origin of Space

Fractional Calculus_ An Introduction for Physicists
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Chapter 26 On the Origin of Space  （把空间视为一种quantum phenomenon，研究分数阶的量子简谐振动方程，对“空间为什么是三维的，以及空间的起源和演化”给出了一个解释。）

所用到的假设：
- space is a quantum phenomenon， –--> use the simple model of the fractional quantum harmonic oscillator
- (complete adiabaticity) the system has always time enough to occupy the equilibrium state ---->treat the time t as a parameter and the problem is reduced to the solution of a one dimensional stationary
Schrodinger equation in x, where t parametrizes the solutions
- the dimensionality of a given space element is directly related to the multiplicity of the occupation probability p$^{α(t)}_n$ for a given state ψ$_n$(x, t)
- propose an evolution of the space dimension as a function of time:
α(t) = 1/2+ 3/2*arctan(t)     0 ≤ t ≤ ∞
- The threshold dimension dc may be considered as： dc = 1/2 − perturbation




In this chapter we want to introduce a further facet, which emerges naturally
from a new point of view based on fractional calculus with variable order.
不仅是导数的阶α是实数，而且导数的阶α是自变量

分数阶微分方程使得time and/or space 依赖于α

基于分数阶微分方程，本章讨论了一个新的方法来描述空间的起源和演化。
。
we consider the genesis of space as a quantum phenomenon
。
26.1 The interplay between matter and space
各种理论对space and time的看法：
- Newtonian physics:
the concept of matter was established as an independent quality in an absolute space and time, which left themselves reduced to passive containers of a dynamically developing material world.
- quantum theory:
space … serves more and more as a waste deposit for dispensable virtual matter of all kind
- general relativity:
space, time and matter are connected via the Einstein field equations. Hence in Einstein’s field theory besides curvature there is no other property of space affected. The number of space and time dimensions is invariant (3+1) under any Lorentz-transformation, which results in a very static picture of space and time at the end. The Kaluza-Klein theory realized 5-dimensional theory of relativity, which allowed for a simultaneous treatment of gravity and electro-magnetism
- string theory:
a higher dimensional realization introducing extra dimensions avoids divergences in a perturbation expansion(指的是研究(3+1)D的各种其他理论)
- generalized Einstein’s field equations
这些理论都没有解释为什么空间是三维的，以及三维空间是怎么开始的。（the classical big bang model: the number of space dimensions is fixed to a value, which remains unchanged till today）
相对于以上的那些理论，fractional calculus能够给出一些解释。

two different strategies to apply fractional calculus methods in cosmology：
- the Einstein field equations with the  fractional derivatives
- derives the fractional extension of the field equations and geodesic equations starting with the least action principle and replacing the usual integral with a fractional integral

我们这里的方法与上面两个都不一样，我们的方法：
Considering space as a quantum phenomenon, it evolves smoothly from a point-like vacuum fluctuation with d = 0 up to it’s final form d = 3, where d is a real number.(It  may be compared to gas bubbles in boiling water). And space dimension is considered a dynamic quality d = d(t), which clearly emphasizes the evolutionary aspect of space genesis. We will model this behavior by investigating the properties of a fractional wave equation with time dependent α = α(t)
。
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26.2 Fractional calculus with time dependent α in the adiabatic limit

Since we consider the genesis of space as a quantum phenomenon, we describe this process with the simple model of the fractional quantum harmonic oscillator:




It has also been demonstrated, that the transition from vibrational (α ≈ 1) to rotational type (α ≈ 2) of spectra may be interpreted within the context of the fractional quantum harmonic oscillator as
a transition from one- d ≈ 1 to three-dimensional d ≈ 3 space, as long as the corresponding multiplicities are correctly implemented

so, α = α(t) –-> d = d(t)
.
类比In α：
the infrared spectra of diatomic molecules may be repro-
duced based on the level spectrum of the fractional quantum harmonic os-
cillator by defining the relative intensities I n α as a Boltzmann distribution
weighted with the degeneracy of the n-th energy level，where β = 1/(k B T ):
得出In vib， In rot
。
the multiplicity of a given state n:  m = (d(t) − 1)n + 1  = 2n(α(t) - 1) +1
d(t) = 2α(t) − 1
the requirement of normalizability of the solutions of the fractional quantum harmonic oscillator
---> α ≤ 2 ---> d<=3

m>0 –--> n < 1/( 2(1-α(t)) ), 同时由于是讨论space，所以n是>=0的整数 ---> 1/2<=α<1 --->0<=d<1
所以最小值是α=1/2，d=0
当α增大时，由Fig. 26.3，第一个分岔点c1 (α= 0.7463952, d= 0.4927904)～occupation probability～dimensionality
Below this threshold (指的是c1(α= 0.7463952, d= 0.4927904)) this space element is dark(有物质也看不到—暗物质)
同时，由于
The threshold dimension dc may be considered as： dc = 1/2 − perturbation（为什么？）
由分数阶泰勒级数展开得：Dc = ½ – perturbation =  1/(1+αs) -1/2  ≈ 0.492756
这和c1点的 α值很接近。所以认为得到了验证。


Space dominated regions are considered as matter free, shown as white areas, 
matter dominated areas may be considered as boundaries and are presented in dark color.
The white regions are therefore permanent sources of space generation and cause a major contribution to the expansion of the universe.


electromagnetic interaction is rather a property of space than of matter.
Furthermore, e.g. the constancy of the speed of light in the vacuum
may be understood immediately, assuming it a property of space,
instead of matter.
.
Dark energy was first introduced as a necessary attribution for the explanation for the accelerated expansion of the universe, deduced from the observed redshift of distant galaxies.
.
Dark matter has been considered to explain the rotation velocity of stars in galaxies.
.
In the above presented fractional model darkness in the early stage of space creation is a natural outcome of a threshold space dimension, above which the minimum requirement for visibility, a single additional
.
regions of space genesis ～～ high concentrations of dark energy Ed ~~ space volume
.
matter dominated regions  ～～ a manifestation of a boundary effect with a compressive contribution  influenced by  dark matter Md . ~~ space boundary area
.
The fractional ansatz therefore connects the concepts of dark energy and dark matter as a unified concept of space generation. In a foamy large scale structure of the universe, the empty areas are considered as sources of expanding space and may be responsible for a concentration of matter
at the boundaries of those space bubbles, which simulates the influence of an imaginary matter contribution.


P392 In this chapter the creation.....是本章的总结
- CH26对“空间为什么是3D的，空间的起源和演化“给出了一种解释。
- 当空间的维度小于0.49时，空间不能吸收和发射光子(暗物质/暗能量)。
- “光速不变“是空间自身的性质，所以光速与光源速度无关、与观测者速度无关。
- 空间的演化联系着暗能量和暗物质：真空空间与暗能量相关，真空空间和物质的边界与暗物质有关。