Jan 29, 2013

A Remark on High Dimension Propagators

In a previous post, I calculated the Coulomb Law (gravity) in $d$-dimension. It is in fact also the free field propagator in $d$-dimension.

The equation of motion of a Klein-Gordon field is, \[ \left( \partial^2 - m^2\right) D_F(x-y) = \delta(x-y) \] where the Feynman propagator $D_F(x-y) = \left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right>$ is also called Feynman propagator. Because of this equation, it is also called Green's function.
The equation of motion can be solved exactly in free space boundary condition. Fourier transform shows, \[
D_F(x-y) = - \int \small{\frac{\mathrm{d}^d k}{(2\pi)^d}} \frac{ e^{i k\cdot (x-y)} }{k^2+m^2 - i\epsilon}
\] Observation: $D_F(x-y)$ is a Lorentz invariant. Therefore, it must be only a function of the Lorentz invariant $(x-y)^2$. Do the contour integral of $k^0$ component first,
D_F(x-y) = i\int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i \mathbf{k}\cdot \mathbf{(x-y)} - i k^0 \cdot |x^0 - y^0|},
= i\theta(x^0-y^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (x-y)} + i\theta(y^0-x^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (y-x)},
\] where $k^0 = \sqrt{\mathbf{k}^2 + m^2}$. Note that, $k \cdot x = -k^0 x^0 + \mathbf{k \cdot x}$.

Following the method in the previous post,  \[
D_F(x-y) = i \frac{1}{2(2\pi)^\frac{d-1}{2} r^{d-2}} \int_0^\infty \mathrm{d}\xi \frac{ \xi^\frac{d-1}{2} }{\sqrt{\xi^2 + m^2 r^2}} J_{\frac{d-3}{2}}(\xi) e^{-i\sqrt{\xi^2+m^2r^2}\frac{|x^0-y^0|}{r}},
\] where $r = |\mathbf{x - y}|$, $J_\nu(x)$ is the Bessel function of the first kind.

It seems this integral does not have a closed form representation. Nevertheless, the closed-form expression can be obtained for several special cases,

 1. $x^0 - y^0 = 0$

\left. D_F(x-y) \right|_{x^0=y^0} = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{(m r)^{\frac{d}{2}-1} K_{\frac{d}{2}-1}(m r)}{r^{d-2}}
Note that this is the same as Coulomb potential. Bessel functions of the second kind $K_\alpha(m r)$ are exponential decaying functions. At large $m r$, \[K_\alpha( m r) \sim \sqrt{\frac{\pi}{2 m r}} e^{- m r}.\]
Generally, power law is a sign of massless excitation (here from vacuum), exponential decay is a sign of excitation overcoming energy gap. In this instance, the energy gap is the particle mass (called mass gap).

(Corollary) $x^0 = y^0, m = 0$ 

\[ \left. D_F(x-y) \right|_{x^0=y^0} = \frac{i \Gamma\left(\frac{d}{2}-1\right)}{4\pi^{\frac{d}{2}}} \frac{1}{r^{d-2}}, \quad (m=0) \]

(Corollary) $(x-y)^2 > 0$, space-like

As we said above, $D_F(x-y)$ is only a function of $(x-y)^2$. If $(x-y)^2 = (\mathbf{x}-\mathbf{y})^2 - (x^0-y^0)^2 > 0 $, we can Lorentz-transform to a frame in which $x'^0 = y'^0$, $ r'^2 = \left| \mathbf{x'} - \mathbf{y'}\right|^2 = (x-y)^2$. Following 1,
D_F(x-y) = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{m^{\frac{d}{2}-1} K_{\frac{d}{2}-1}\left(m \sqrt{(x-y)^2}\right)}{(x-y)^{d/2-1}}

2. $\mathbf{x} - \mathbf{y} = 0$

D_F(x-y) = \frac{i}{(2\pi)^\frac{d-1}{2}\Gamma\left(\frac{d-1}{2}\right)} \int_0^\infty  \frac{\mathrm{d} k}{2\sqrt{k^2+m^2}} k^{d-2} e^{ - i \sqrt{k^2+m^2} |x^0 - y^0|},

3. $ m = 0 $

D_F(x-y) = \frac{i \Gamma\left( \frac{d}{2}-1 \right)}{4\pi^\frac{d}{2}}\frac{1}{ \left((x-y)^2\right)^{\frac{d}{2}-1}}
where $(x-y)^2 = -(x^0-y^0)^2 + (\mathbf{x-y})^2$.

Figure 1. The image of the free scalar field propagator in $d=4$D. $x$-axes is $\mathbf{x-y}$, $y$-axes is $x^0-y^0$.

Appendix: The Feynman Propagator

The following derivation for scalar field (Klein-Gordon field) is standard-textbook.

Define the partition function, \[
Z[J] = Z[0] \int \mathcal{D}\varphi \exp\{i S[\varphi] + \int \mathrm{d}^d x J(x) \varphi(x)\}.
\] The two-point time-ordering correlation function, \[
\left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right> = \left. Z^{-1}[0] \frac{\delta}{\delta J(x)}\frac{\delta}{\delta J(y)} Z[J] \right|_{J \to 0}.
\] The equation of motion (the first Dyson-Schwinger equation) reads, \[
\left< 0 | \right. \mathcal{T}\varphi(x) \frac{\delta S}{\delta \varphi(y) } \left. | 0 \right> = i \delta(x-y). \] Now, Dyson-Schwinger Equations (DSEs) are normally a series of tower. Fortunately for free field theory, the equation of motion is self-contained. In free field theory, \[
S_0 = \frac{1}{2}\int \mathrm{d}^d x \varphi(x) \left( \partial^2 - m^2 \right) \varphi(x).
\] The partition function of the free field theory can be expressed in terms of Feynman propagator, \[
Z[J] = Z[0] \exp\{-\frac{i}{2}\int \mathrm{d}^d x \mathrm{d}^d y J(x) D_F(x-y) J(y) \}.
\] To prove this, just substitute $\chi(x) = \varphi(x) + \int \mathrm{d}^d y J(y) D_F(x-y) $. That's why propagator (classical Green's function) is important in free field theory (subsequently perturbation theory).

Jan 16, 2013

How to display source codes in Blogger

0. Post your codes directly in RTF editor!

for example:
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Westerner_on_a_camel.jpg/450px-Westerner_on_a_camel.jpg/" />

1. Use <textarea> tag in html editor

for example,
<textarea cols="60" rows="6">
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Westerner_on_a_camel.jpg/450px-Westerner_on_a_camel.jpg/" />
will produces,

2. Use advanced source code brushes/JavaScript syntax highlignters

There are several brushes available. I use SyntaxHighlighter here. To use SyntaxHighligher for example, go to layerout -> add a Gadget -> HTML/JavaScript. Post the following lines and save.

Use SyntaxHighlighter as following
<pre class="brush:language" >

some source code ...

</pre >
Available language depends on the installed script above normally including cpp, html, plain, text, csharp, java, jscript, python, ruby etc.

For a comprehensive list of supported languages with public domain JavaScripts, see this post by ABEL BRAAKSMA.

The syntax highlight effect of these js codes themselves is,

<link href='http://alexgorbatchev.com/pub/sh/current/styles/shCore.css' rel='stylesheet' type='text/css'/>

<link href='http://alexgorbatchev.com/pub/sh/current/styles/shThemeDefault.css' rel='stylesheet' type='text/css'/>

<script language='javascript'>

SyntaxHighlighter.config.bloggerMode = true;

SyntaxHighlighter.config.clipboardSwf = 'http://alexgorbatchev.com/pub/sh/current/scripts/clipboard.swf';



A piece of C code,
#include <stdio.h>
/* hello world in C */

int main(int argc, char* argv[]){
     printf("Hello, world!\n");
     return 0;

// Hello World in Microsoft C# ("C-Sharp").

using System;

class HelloWorld
    public static int Main(String[] args)
        Console.WriteLine("Hello, World!");
        return 0;

Mathematica code,
Print["Hello, world!"];

$\LaTeX$ Tips

I use $\LaTeX$ a lot. In this post, I summarize the tips I believe useful.

Tools and general references:


The Com­pre­hen­sive $\TeX$ Archive Net­work (CTAN) is the cen­tral place for all kinds of ma­te­rial around $\TeX$. CTAN has cur­rently 4431 pack­ages. They have been con­tributed by 2078 au­thors. Most of the pack­ages are free and can be down­loaded and used im­me­di­ately.
url: http://www.ctan.org/

1. Detexify
A $\LaTeX$ symbol classifier

2. The Comprehensive $\LaTeX$ Symbol List
The Comprehensive $\LaTeX$ Symbol List is an organized list of over 5900 symbols commonly available to LaTeX users.

3. $\LaTeX$, a tutorial Wikibook
$\LaTeX$ is a guide to the LaTeX markup language. It is intended that this can serve as a useful resource for everyone from new users who wish to learn, to old hands who need a quick reference.

4. CodeCogs
CodeCogs is a online LaTeX equation editor. BTW: CodeCogs is also useful if you want to hide your email and/or telephone number in a picture. For example,

5. Art of Problem Solving
This page provides a short list of commonly used LaTeX symbols.

6. LaTeX Lab
LaTeX Lab is an online LaTeX document processing app based on Google Document.

7. Kile
Kile is a user-friendly TeX/LaTeX editor for the KDE desktop environment. KDE is available for many architectures and operating systems such as PC, Mac, and BSD, including Linux and Microsoft Windows.

8. VIM-LaTeX
Vim-LaTeX attempts to provide a comprehensive set of tools to view, edit and compile LaTeX documents without needing to ever quit Vim. Together, they provide tools starting from macros to speed up editing LaTeX documents to compiling tex files to forward searching .dvi documents.

9. MathJaX
Powered by MathJax;
MathJax is an open source JavaScript display engine for mathematics that works in all modern browsers.

I've introduced MathJax for Blogger in a previous post based on (mainly copying) this post.

Exotic symbols

1. "d-bar" notation,
Used in integration measures, as

2. Roman numbers

3. Feynman slash notations

4. Inline Feynman diagrams


1. Tabular raw spacing
from Everything You Forget About LaTeX. Use

instead of \hline to add extra spaces above and below the horizontal line \hline.

2. Less margins

3. Equation justification

argument fleqn means,
fleqn % justify all equations to the left

other arguments are also available:
leqno (reqno) % put all equation numbers on the left (right)

4. Combining images

5. Multiline subscripts and superscripts
reference: http://www.nada.kth.se/~carsten/latex/amsldoc/node51.html

Long subscripts could be ugly:

\[ \sum_{0\le i\le m, 0 < j < n} P(i, j) \] Use \substack:
\[ \sum_{\substack{0\le i\le m\\ 0 < j < n}} P(i, j) \] Use \subarray:
\sum_{\begin{subarray}{l} 0\le i\le m \\ 0 < j < n \end{subarray}} P(i,j)

Jan 5, 2013

on the size of leaves

Written to Phys. Rev. Lett., Kaare H. Jensen and Maciej A. Zwieniecki try explain the physical limits to leaf size by modeling the fluidity of the botanic vascular system. Their seemingly over-simplified model established the upper and lower limits to leaf size with respect to tree height and surprisingly shows good agreement with a large set of botanical data.

Fig. 1
In their modeling, a plant is treated as three phloem tubes: the leaf, the stem and the root. Each part is assigned hydraulic resistance, \[ \mathcal{R}_\text{leaf} = 1/(2 \pi r l L_p), \quad \mathcal{R}_\text{stem} = 8\eta h/(\pi r^4), \quad \mathcal{R}_\text{root} = 1/(2\pi r s L_p), \] where $l, h, s$ are the length (height) of the leaf, stem and root respectively, $\eta \simeq 5 \text{ mPa}$ the viscosity of the sap, $L_p$ the hydraulic permeability of the membrane, $r$ the phloem radius. Note the difference on resistance between leaf/root and stem: the longer is the leaf/root, the smaller is the resistance, reflecting the role of leaf/root as source/sink; however, the longer is the stem, the larger is the resistance, reflecting the role of stem is to transport. Therefore, the sugar transport speed is, \[
u = \frac{1}{\pi r^2} \frac{\Delta p}{\mathcal{R}} \simeq \frac{2r^2 L_p l}{r^3 + 16 \eta L_p l h}\Delta p, \] where $\Delta p \simeq 1 \text{ MPa}$ is the osmotic pressure difference. The fact that the root is much longer than leaves $s \gg l$, is used for approximation. The transport speed is directly proportional to energy flux $E = k c u$.

First of all, there is a maximum energy flux for fixed $h$, that is achieved when the leaf size $l$ is sufficiently large, \[ E_\max = k c \Delta p \frac{r^2}{8\eta h}. \] Jensen and Zwieniecki argue that, $l$ cannot be unbounded. As the leaf size $l$ increases, the energy flux gain becomes small from increasing the leaf size. Other factors may step in and stop the leaf size from increasing. They model all other factors by setting a energy flux threthold, $E_t = \mu E_\max (\mu \lesssim 1)$. Once the energy flux has reach $E_t$, the increase in leaf size will stop. This implies a upper limit to the leaf size, \[l_\max = \frac{r^3}{16(1-\mu)L_p\eta h} \]
There also exists a lower limit on leaf size for fixed $h$, as if the energy flux is lower than some value $E_\min$, the plant will not survive,
\[ l_\min = \frac{r^3}{16 L_p\eta (h_\max - h)} \] where $h_\max = k c \Delta p r^2/(8\eta E_\min)$ is the constraint on tree height $h$. $h$ large than that, $u_\max < u_\min$ implies $E < E_\min$.

The authors compared their seemingly over-simplified model with a set of botanical data covering 1925 species from 327 genera and 93 families. Surprisingly, their model shows a good agreement with the upper and lower limits of leaf size. With the fitting, they also predicts $h_\max = 104 \pm 6 \text{ m } \quad (95\%), \mu = 0.916 \pm 0.004 \quad (95\%)$.

Fig. 2: Gray triangles show the reported range of leaf sizes for particular species as the longest and shortest leaf lamina length plotted as a function of tree height h. Circles show the five longest (red, dark gray) and five shortest (green, light gray) leaves in each 20 m height bin for trees taller than h = 20 m. Solid lines are fits to theoretical predictions with parameters corresponding to a minimum flow speed of $u_\min = 100 \mu \mathrm{ m/s}$ and energy output efficiency of 90%. Dashed lines indicate 95% confidence intervals.


Kaare H. Jensen and Maciej A. Zwieniecki, Physical Limits to Leaf Size in Tall Trees, PRL 110, 018104 (2013)