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This approach follows Veltman and van der Bij , who noticed that by using the homogeneity equation we have\n\s[\stag{homogDE}\n I(x,y,z)=\sfrac{1}{1-2\seps}(x\spd_x+y\spd_y+z\spd_z)I(x,y,z)\n\s]\nand that $\spd_xI(x,y,z):=I'(x,y,z)$ has a Feynman parametrisation in which all UV diveregence lies in the momentum integral. Thus we can find $I(x,y,z)$ by elementary methods. The advantage of this method is that it is manifestly symmetric in $x$, $y$ and $z$, and that the final result is free of branch cut ambiguities. Now\n\sbegin{align} \snon\n I'(x,y,z)&=-\sfrac1{x^{2\seps}}\sint_{k,p}\sfrac1{(k^2+1)^2}\sint_0^1\s!\s!\srmd\sa\n\sleft(p^2+\sa(1-\sa)(k^2+w)\sright)^{-2}~,\squad w=\sfrac{y/x}\sa+\sfrac{z/x}{1-\sa}\s\s\n &=\sfrac{-1}{(4\sp)^4}\sleft(\sfrac{4\sp\sm^2}x\sright)^{2\seps}\n \sfrac{\sG(\seps)}{\sG(2-\seps)}\n \sint_0^1\s!\s!\sfrac{\srmd\sa}{\sa^\seps(1-\sa)^\seps}\sint_0^\sinfty\s!\s!\srmd k^2\n \sfrac{k^{2(1-\seps)}}{(k^2+1)^2} \sfrac1{(k^2+w)^\seps} \snon\s\s\n &=\sfrac{-1}{(4\sp)^4}\sleft(\sfrac{4\sp\sm^2}x\sright)^{2\seps}\sfrac{\sG(2\seps)}{\seps}\n F\sleft(\sfrac yx,\sfrac zx ; \seps\sright)~,\n\send{align}\nwhere $F\sleft(\sfrac yx,\sfrac zx\sright)$ is manifestly $y$, $z$ symmetric and is defined by\n\s[ \slabel{F.homog.1} F\sleft(\sfrac yx,\sfrac zx\sright)=\seps\n\sint_0^1\s!\s!\srmd\sa\srmd\sb\sfrac{\sb(1-\sb)^{\seps-1}}{\sa^\seps(1-\sa)^\seps}\n \sleft(\sb+(1-\sb)w\sright)^{-2\seps}~.\n\s]\nSo we have\n\s[ \slabel{homog2}\n I(x,y,z)=\sfrac{(4\sp\sm^2)^{2\seps}}{(4\sp)^4}\sfrac{\sG(2\seps-1)}\seps\n \sleft(x^{1-2\seps}F\sleft(\sfrac yx,\sfrac zx\sright)+\scycl\sright)~.\n\s]\nThe case of $I(x,0,0)$ is easily found and reproduces \seqn{Ix00}, but unfortunately the Feynman parametrisation we have still contains divergences, so the other cases (and the general case) are not so easy. The result of [VB] is \n\s[ F\sleft(\sfrac yx,\sfrac zx\sright)=\n\sint_0^1\s!\s!\srmd\sa\srmd\sb\sfrac{\sb(1-\sb)^{\seps}}{\sa^\seps(1-\sa)^\seps}\n \sleft(\sfrac{2w\seps}{\sleft(\sb+(1-\sb)w\sright)^{1+2\seps}}\n +\sfrac{(2-\seps)}{\sleft(\sb+(1-\sb)w\sright)^{2\seps}}\sright)\n\s]\nBy a single integration by parts of \seqn{F.homog.1} I can obtain \n\s[ F\sleft(\sfrac yx,\sfrac zx\sright)=\n\sint_0^1\s!\s!\srmd\sa\srmd\sb\sfrac{\sb(1-\sb)^{\seps}}{\sa^\seps(1-\sa)^\seps}\n \sleft(\sfrac{2w\seps}{\sleft(\sb+(1-\sb)w\sright)^{1+2\seps}}\n +\sfrac{\sb+(1-\sb)w-2\seps\sb}{\sb\sleft(\sb+(1-\sb)w\sright)^{2\seps+1}}\sright)\n\s]\nthe difference of the integrands is $\spd_\sb\sbig(\sb(1-\sb)^{1+\seps}(\sb+(1-\sb))^{-2\seps}\sbig)$, a total derivative.\n\nNow let's examine the epsilon expansion, first we have\n\s[ \slabel{homog.eps1}\n \sleft(\sfrac{(4\sp\sm^2)}{x}\sright)^{2\seps}\sfrac{\sG(2\seps-1)}\seps=\n -\sfrac1{2\seps^2}-\sfrac1\seps\sleft(1-\slog\sfrac x{\sbar\sm^2}\sright)-\sleft(2+\sz(2)\n-2\slog\sfrac x{\sbar\sm^2}+\slog^2\sfrac x{\sbar\sm^2}\sright)+\sord(\seps)\n\s]\nand\n\sbegin{align} \slabel{homog.eps2}\n F(a,b|\seps)&=\sint_0^1\s!\s!\srmd\sa\srmd\sb \sb\n\sleft(\sfrac{(1-\sb)}{\sa(1-\sa)\sleft(\sb+(1-\sb)w\sright)^2}\sright)^\seps\n \sleft(\sfrac{2w\seps}{\sleft(\sb+(1-\sb)w\sright)}-\seps+2\sright) \snon\s\s\n &=1+\seps+\seps^2\sleft(2f(a,b)+1-\sz(2)\sright)+\sord(\seps)^3 \s\s\n\smbox{where}\squad f(a,b)&=\sint_0^1\s!\s!\srmd\sa\sleft(\sLi_2(1-w)\n+\sfrac{w\slog w}{w-1}\sright)~.\n\send{align}\nMultiplying the above expansions yields\n\sbegin{align} F(a,b|\seps)&=\n-\sfrac1{2\seps^2}+\sfrac1\seps\slog\sfrac{\shat\sm^2}x-\shalf\sleft(\sfrac52+\sz(2)\n+2\slog^2\sfrac{\shat\sm^2}x+f(a,b)\sright)+\sord(\seps)\n\send{align}\nthen combining with \seqn{homog2} yields our result\n\sbegin{align} \slabel{homog.eps.tot}\n (4\sp)^4I(x,y,z)&=-\sfrac{c}{2\seps^2}+\sfrac{\shat L_1}{\seps}\n-\shalf\sBigg(c\sleft(\sz(2)+\sfrac52\sright)+2\shat{L}_2+\stilde\sx(x,y,z)\sBigg)\s\s\n \stilde\sx(x,y,z)&=2\sleft[x f(y/x,z/x)+\scycl \sright],\n\send{align}\nThis is compatible with the JFJ result because we have the useful (and simplifying) relation\n\s[ \sleft[(x-y-z)\slog\sfrac{y}{\shat{\sm}^2}\slog\sfrac{z}{\shat{\sm}^2}+\scycl\sright]\n = \shat L_2-\sleft[x\slog(\sfrac yx)\slog(\sfrac zx)+\scycl\sright]~.\n\s]\nThis implies that \n\s[ \stilde\sx(x,y,z)=\sx(x,y,z)-\sleft[x\slog(y/x)\slog(z/x)+\scycl\sright]~,\s]\nA relation that has been sucessfully tested numerically using $\sx$ from [[JFJ|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/0111190]], [[CCLR|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/9805118]] and [[DT|http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B397,123]].
----\nJust for variety (with respect to the literature) we'll work with the proper-time expressions. The fundamental equation that we work with is\n\s[\stag{\stextcircled{s}}\sbegin{split} \nJ(y)J(z)&=-\sfrac{\sm^{4\seps}}{(4\sp)^d}\sintstu\sfrac\srmd{\srmd s}\n\sfrac{\srme^{-sx-ty-uz}}{(st+tu+us)^{2-\seps}} \s\s \n&=xI(x,y,z)+(2-\seps)\sfrac{\sm^{4\seps}}{(4\sp)^d}\n\sintstu\sfrac{(t+u)\srme^{-sx-ty-uz}}{(st+tu+us)^{3-\seps}}~,\sqquad \s{s\s}\n\send{split}\s] \nwhere $J$ is the [[one loop vacuum integral|One loop vacuum]]. All other equations just come from swapping $s$ with $t$ or $u$, taking mass derivatives and linear combinations.\ne.g. look at $\spd_x\stcs$,\n\s[\stag{$\spd_x$\stcs} 0=(1+x\spd_x)I(x,y,z)-(2-\seps)\sfrac{\sm^{4\seps}}{(4\sp)^d}\n\sintstu\sfrac{s(t+u)\srme^{-sx-ty-uz}}{(st+tu+us)^{3-\seps}}~,\n\s] \nby cycling $s\sto t\sto u$ and adding to get $\spd_x\stcs+\spd_y\stct+\spd_z\stcu$, the combined numerator in the final term cancels a power in the denominator, making the integral proportional to $I(x,y,z)$. We then see that we've reproduced the homogeneity equation, which is the starting point for this [[method|2LVacHomog]].\n\nNext we look at $\spd_y\stcs$\n\s[\stag{$\spd_y$\stcs} J'(y)J(z)\n= x\spd_y I(x,y,z)-(2-\seps)\sfrac{\sm^{4\seps}}{(4\sp)^d}\n\sintstu\sfrac{t(t+u)\srme^{-sx-ty-uz}}{(st+tu+us)^{3-\seps}}~.\n\s] \nto get rid of the $t^2$ term we subtract $\spd_y\stcu$ to get\n\s[ J'(y)(J(z)-J(x))\n=(x-z)\spd_y I(x,y,z)-(2-\seps)\sfrac{\sm^{4\seps}}{(4\sp)^d}\n\sintstu\sfrac{t(u-s)\srme^{-sx-ty-uz}}{(st+tu+us)^{3-\seps}}~,\n\s]\ncycle and add to cancel out the last term, leaving\n\s[J'(x)(J(y)-J(z))+\scycl=(z-y)\spd_x I(x,y,z)+\scycl \s] \nthe equation which forms the starting point to finding $I(x,y,z)$ via the [[method of characteristics|2LVacMOC]].\n\nFinally we look for a combination that forms an ordinary DE, say in z. We note\nthat we can also remove the integral terms by the combination\n$\spd_x(\stcs+\stct-\stcu)+\spd_z(\stcs-\stct-\stcu)$, yielding\n\s[ J'(x)\sleft(J(z)-J(y)\sright)+J'(z)\sleft(J(y)-J(x)\sright)\n =\sleft((x+y-z)\spd_x+(x-y-z)\spd_z\sright)I(x,y,z)\n\s]\nThis is an equation for $\spd_xI(x,y,z)$ in terms of $\spd_zI(x,y,z)$. \nWe substitute this, and the similar equation for $\spd_y I$, into the homogeneity\nequation to get\n\sbegin{equation}\sbegin{split} \sD\spd_zI(x,y,z)&= (1-2\seps)(x+y-z)I(x,y,z)\s\s\n &+xJ'(x)\sleft(J(y)-J(z)\sright)+yJ'(y)\sleft(J(x)-J(z)\sright)\n +(x-y)J'(z)\sleft(J(x)-J(y)\sright) \slabel{ODE}\n\send{split}\send{equation}\nwhere\n\s[ \sD(x,y,z)=2(xy+yz+zx)-x^2-y^2-z^2~. \s]\n----
Flow equation preserves both $\sD$ and $c$, in terms of the [[mass simplex|2LVac_Geom]]
\sbegin{align} \nI(x,y,z) &=\sm^{4\seps}\sint\sfrac{\srmd^dk\srmd^dp\srmd^d\sk}{(2\sp)^{3d}}\n \sfrac{\sd^d(p+k+\sk)}{(k^2+x)(p^2+y)(\sk^2+z)} \s\s\n &=\sm^{4\seps}\sint\s!\srmd^d\sr\sint\sfrac{\srmd^dk\srmd^dp\srmd^d\sk}{(2\sp)^{3d}}\n \sfrac{\srme^{\srmi\sr\scdot(p+k+\sk)}}{(k^2+x)(p^2+y)(\sk^2+z)}\n\send{align}\nWe will examine the generalised object\n\sbegin{align} I_{abc}&=\sint\sfrac{\srmd^dk\srmd^dp\srmd^d\sk}{(2\sp)^{3d}}\n \sfrac{\sm^{4\seps}\sd^d(p+k+\sk)}{(k^2+x)^{a}(p^2+y)^{b}(\sk^2+z)^{c}}\s\s\snon\n &=\sfrac{\spd_{-x}^{\s,a-1}}{(a-1)!}\sfrac{\spd_{-y}^{\s,b-1}}{(b-1)!}\n \sfrac{\spd_{-z}^{\s,c-1}}{(c-1)!}I(x,y,z)~,\squad{\srm for~~} a,b,c > 0\n\send{align}\nWe can introduce Schwinger parameters by\n\s[\sfrac{n!}{\sleft(k^2+x\sright)^{n+1}}\n=\sint_0^\sinfty s^n\srme^{-s(k^2+x)}\srmd s~, \n\s]\nso\n\s[I(x,y,z)=\sm^{4\seps}\sint\s!\srmd^d\sr\sint\sfrac{\srmd^dk\srmd^dp\srmd^d\sk}{(2\sp)^{3d}}\n \sintstu\s,\srme^{\srmi\sr\scdot(p+k+\sk)-s(k^2+x)-t(p^2+y)-u(\sk^2+z)}\n\s]\nThen since\n\s[\sint\s!\sfrac{\srmd^d k}{(2\sp)^d}\srme^{\srmi\sr\scdot k-s k^2}\n =\sfrac{s^{\seps-2}}{(4\sp)^{2-\seps}}\srme^{-\sfrac{\sr^2}{4s}}\n \squad \smbox{and} \squad \sint\s!\srmd^d\sr\s,\n \srme^{-\sfrac{\sr^2}{4\ss}}=\sleft(\sfrac{4\sp}{\ss}\sright)^{2-\seps}\n\s]\nwe have\n\sbegin{align} I(x,y,z)&=\n \sfrac{\sm^{4\seps}}{(4\sp)^{3(2-\seps)}}\sint\s!\srmd^d\sr\n \sint_0^\sinfty\s!\s!\sfrac{\srmd{s}\s,\srmd{t}\s,\srmd{u}}{(stu)^{2-\seps}}~\n \srme^{-\sfrac{\sr^2}{4}\sleft(\sfrac1s+\sfrac1t+\sfrac1u\sright)-sx-ty-uz} \snon\s\s\n &=\sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}\sint_0^\sinfty\s!\s!\n \sfrac{\srmd{s}~\srmd{t}~\srmd{u}}{(st+tu+us)^{2-\seps}}~\srme^{-sx-ty-uz}~.\n\send{align}\n\nWe can do one of the proper-time integrals, choosing the $u$-integral we get\n\s[I(x,y,z)=\sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}z^{1-\seps}\n \sint_0^\sinfty\s!\s!\sfrac{\srmd{s}~\srmd{t}}{(s+t)^{2-\seps}}\n \sG(\seps-1,\sfrac{st}{s+t}z)\s,\srme^{\sfrac{st}{s+t}z-sx-ty}~,\n\s]\nwhere the incomplete gamma function is defined by\n\s[\sG(a,z)=\sint_{z}^{\sinfty} t^{a-1}\srme^{-t}\srmd{t}~, \n \squad \sG(a)=\sG(a,0)~.\n\s]\n\nFor $a,b,c > 0$ we can simply read off\n\sbegin{align} I_{abc}\n &=\sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}\sint_0^\sinfty\s!\s!\n\sfrac{\srmd{s}\s,\srmd{t}\s,\srmd{u}\s,s^{a-1}t^{b-1}u^{c-1}\srme^{-sx-ty-uz}}{(a-1)!(b-1)!(c-1)!(st+tu+us)^{2-\seps}},\n\send{align}\nwhile for and $a,b$ or $c=0$ the corresponding momentum integral yields a \n$\sd$-function, and the correct formula is found by removing the corresponding\nproper-time(s) from the above equation. The integral then factorises and\ncan be done in closed form, e.g. for $b,c\sgeq 1$\n\sbegin{align} I_{0bc}\n &=\sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}\sint_0^\sinfty\s!\s!\n \sfrac{\srmd{t}\s,\srmd{u}\s,t^{b-1}u^{c-1}\srme^{-ty-uz}}{(b-1)!(c-1)!(tu)^{2-\seps}}=J_b(y) J_c(z)~,\n\send{align}\nwhere $J(y)=J_1(y)$ is a well known [[one-loop integral|One loop vacuum]], and\n\s[J_b(y)=\sfrac{\spd_{-y}^{b-1}J(y)}{(b-1)!}\n:=\sfrac{\sm^{2\seps}}{(4\sp)^{2-\seps}}\sint_0^\sinfty\s!\s!\n\sfrac{\srmd{t}\s,t^{b-1}\srme^{-ty}}{(b-1)!t^{2-\seps}}\n=\sfrac{\sm^{2\seps}}{(4\sp)^{2-\seps}}\sfrac{\sG(b+\seps-2)}{\sG(b)}y^{2-\seps-b}~.\n\s]
<html>\n<iframe src="images/fig/2LVac_Geom_ang_unicode.svg" style="FLOAT: left" width="290" height="470" type="image/svg" frameborder=0 />\n</html>\nMasses and side (edge) lengths:\n\sbegin{align} \nm_1^2&=Y^2+Z^2~,&~ 2X^2&=-m_1^2+m_2^2+m_3^2 \s\s\nm_2^2&=Z^2+X^2~,&~ 2Y^2&=m_1^2-m_2^2+m_3^2 \s\s\nm_3^2&=X^2+Y^2~,&~ 2Z^2&=m_1^2+m_2^2-m_3^2\n\send{align}\nFaces (Defined to be opposite the points $P_w$):\n\sbegin{align} \nF_0&=\slc \sfrac xX+\sfrac yY+\sfrac zZ=1~ |~x\sgeq0,~y\sgeq0,~z\sgeq0\src \s\s\nF_X&=\slc x=0,~\sfrac yY+\sfrac zZ\sleq1~ |~y\sgeq0,~z\sgeq0\src \squad \smbox{and cyclic}\n\send{align}\nPerimeters of the faces:\n\sbegin{align} \nL_0&=m_1+m_2+m_3:=c:=2s \s\s&=\ssqrt{Y^2+Z^2}+\ssqrt{Z^2+X^2}+\ssqrt{X^2+Y^2} \s\s\nL_X&=m_1+Y+Z~,\squad L_Y=X+m_2+Z~,\squad L_Y=X+Y+m_3\n\send{align}\nAreas of the faces ($A_0$ found using [[Heron's formula|http://en.wikipedia.org/wiki/Heron's_formula]]):\n\sbegin{align} \nA_0&=\ssqrt{s(s-m_1)(s-m_2)(s-m_3)}\squad\smbox{Heron's formula} \s\s\n &=\squart\ssqrt{2(m_1^2m_2^2+m_2^2m_3^2+m_3^2m_1^2)-m_1^4-m_2^4-m_3^4}:=\squart\ssqrt\sD \s\s\nA_X&=\shalf YZ~,\squad A_Y=\shalf ZX~,\squad A_Y=\shalf XY\n\send{align}\nIt is straight forward to check that the areas satisfy [[de Gua's theorem|http://en.wikipedia.org/wiki/De_Gua%27s_theorem]] (3D Pythagoras theorem)\n\s[ A_X^2+A_Y^2+A_Z^2=A_0^2 \s]\nVolume:\n\s[ V={\ssmall\sfrac16}XYZ\n={\ssmall\sfrac1{48}}(-m_1^2+m_2^2+m_3^2)(m_1^2-m_2^2+m_3^2)(m_1^2+m_2^2-m_3^2)\n\s]\n\nThe normals to the faces are\n\sbegin{align}\nN_0&=(YZ,ZX,XY)=2(A_X,A_Y,A_Z)=(X,-Y,0)\stimes (X,0,-Z) \s\s\nN_X&=(X,0,0)~, \squad N_Y=(0,Y,0)~,\squad N_Z=(0,0,Z)~.\n\send{align}\nSo the the norm of $V_0=2A_0$ is twice its area. The dihedral angles are then calculated using the dot product to get\n\s[ \sQ_{0X}=\sarccos(A_X/A_0)=\sarccos(YZ/\ssqrt{\sD/4}) ~~\smbox{and sim for } X\sto Y,Z~.\s]\n\nThe angles that appear in the $\sx$ function found in the [[2 loop vacuum graph|Two-loop vacuum sunset diagram]] are\n\s[ \sq_x=\sarctan\sleft(\sfrac{-m_1^2+m_2^2+m_3^2}{\ssqrt\sD}\sright) ~~\smbox{and cyclic} \s]\ncan be rewritten as \n\s[ \sq_x=\sarcsin\sleft(\sfrac{-m_1^2+m_2^2+m_3^2}{2m_2m_3}\sright) ~~\smbox{and cyclic} \s]\nwhich are the complimentary angles in the front face, $F_0$, of the simplex, \n\s[ \sq_1:=\sfrac\sp2-\sq_x=\sarccos\sleft(\sfrac{-m_1^2+m_2^2+m_3^2}{2m_2m_3}\sright)~~\smbox{and cyclic}. \s]\n\nFinally, up to now we've assumed that the simplex 'exists', ie the $\sD>0$, but what happens if the masses are such that $\sD<0$? This implies that the area, $A_0$ is pure imaginary, and that the triangle can not close since for some $(i,j,k)={\srm perm}(1,2,3)$ we must have $m_i > m_j + m_k$. We choose $m_1>m_2+m_3>\ssqrt{m_2^2+m_2^2}>0$. \n\nAs a quick example lets examine the case of $m_2=0$ then $F_0$ becomes a line and $\sD=0$. From $m_2^2=X^2+Z^2$ we get $Z^2=-X^2>0$, the final inequality holds since $m_1>m_3$. \n\nThis holds in general,\n\s[ m_1^2>m_2^2+m_2^2 ~\sRightarrow~X^2<0~\sRightarrow~X \smbox{ is pure imaginary!} \s]\nEquivalently, our dot product must now have the signature $(-1,1,1)$.
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/***\nname: AllTagsExceptPlugin\nauthor: Garrett\nversion: 0.1.0\nThis is a revision of Clint Checketts' allTagsExcept plugin, which lists all tags except those listed.\n\n<<option chkDisableExcept>> show hidden system tags\n\n!!Usage\n{{{\n<<AllTagsExcept tag1 tag2 ...>>\n}}}\n!!!Code\n***/\n/*{{{*/\nversion.extensions.AllTagsExcept = {major: 0, minor: 1, revision: 0};\n\nif (!config.options.chkDisableExcept) config.options.chkDisableExcept=false; // default to standard action\n\nconfig.macros.AllTagsExcept = {tooltip: "Show notes tagged with '%0'",noTags: "There are no tags to display"};\n\nconfig.macros.AllTagsExcept.handler = function(place,macroName,params)\n{\n var tags = store.getTags();\n var theDateList = createTiddlyElement(place,"ul");\n if(tags.length == 0)\n createTiddlyElement(theDateList,"li",null,"listTitle",this.noTags);\n for(var t=0; t<tags.length; t++)\n {\n var includeTag = true;\n for (var p=0;p<params.length; p++) if ((tags[t][0] == params[p])&&(!config.options.chkDisableExcept)) includeTag = false;\n if (includeTag)\n {\n var theListItem =createTiddlyElement(theDateList,"li");\n var theTag = createTiddlyButton(theListItem,tags[t][0] + " (" + tags[t][1] + ")",this.tooltip.format([tags[t][0]]),onClickTag);\n theTag.setAttribute("tag",tags[t][0]);\n }\n }\n}\n/*}}}*/\n
${\scal N}=4$ SYM written in terms of ${\scal N}=1$ superfields\n$$S[W,\sPhi_i]=\sfrac{1}{g^2}\sint{\srm d}^6z\s,{\srm tr} W^\salpha W_\salpha + \sint{\srm d}^8z\s,{\srm tr} \sPhi_i^\sdagger \sPhi_i+\sleft\s{g\sint{\srm d}^6z\s,{\srm tr}\s,\svarepsilon_{ijk}\sPhi_i\sPhi_j\sPhi_k + {\srm c.c.} \sright\s}~,\squad i=1,2,3$$\n$\sbeta$-deformation yields (with $q=\sexp({\srm i}\sbeta)$)\n\sbegin{eqnarray}\nS[W,\sPhi_i]&=&\sfrac{1}{g^2}\sint{\srm d}^6z\s,{\srm tr} W^\salpha W_\salpha + \sint{\srm d}^8z\s,{\srm tr} \sPhi_i^\sdagger \sPhi_i \n%\s\s&&\n+\sleft\s{h\sint{\srm d}^6z\s,{\srm tr} \sbig( q\sPhi_1\sPhi_2\sPhi_3 -q^{-1}\sPhi_1\sPhi_3\sPhi_2 \sbig) + {\srm c.c.} \sright\s}~\n%,\squad i=1,2,3\n\send{eqnarray}\n\nExplain why is good,\ndetails of cacl.\nreferences
Proof of Bianchi identity for super-commutators: $[a,b\s}:=ab-(-1)^{\seps_a\seps_b}ba$\nWant to show that\n$(-1)^{\seps_a\seps_c}[a,[b,c\s}\s}+(-1)^{\seps_b\seps_a}[b,[c,a\s}\s}+(-1)^{\seps_c\seps_b}[c,[a,b\s}\s}=0$\n\nA mathematica notation:\n${\srm ac}[a,b]:= [a,b\s}$ where $e[a]:=\seps_a\sin\s{0,1\s}$.\nThe following code tells [[mathematica]] how do deal with everything:\n{{{\nClearAll[e, ac, CenterDot, MNumericQ]\nac[a_, b_] := a·b - (-1)^(e[a] e[b]) b·a\ne /: e[a_]^n_Integer := e[a] (* ie e[a]==1 or 0 *)\ne /: n_Integer?(# > 1 &) e[a_] := Mod[n, 2] e[a] (* 1+1=2~0 *)\ne /: e[a_ + b_] := e[a] (*need to idiot proof *)\ne /: e[a_·b_] := e[a] + e[b]\ne /: e[a_ b_] := e[a] + e[ b]\ne /: e[a_?MNumericQ] := 0\nMNumericQ[expr_?(Not[# === Blank[]] &)] := NumericQ[Evaluate[expr /. e[_] -> 1]]\nCenterDot /: a___·(k__?MNumericQ b_)·c___ := k (a·b·c)\nCenterDot /: a___·(b_·c__)·d___ := a·b·c·d\nCenterDot /: a___·(b_ + c_)·d___ := a·b·d + a·c·d\nCenterDot /: a_·(b_ + c_) := a·b + a·c\nCenterDot[a___, b_?MNumericQ, c___] := b (a·c)\n}}}\nand this next line should evaluate to zero!\n{{{\n(-1)^(e[a] e[c]) ac[a, ac[b, c]] + (-1)^(e[b] e[a]) \n ac[b, ac[c, a]] + (-1)^(e[c] e[b]) ac[c, ac[a, b]] // ExpandAll\n}}}
Creating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n{{{\nCreating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n}}}
/***\nAuthors: Eric Shulman & Bradley Meck\nversion: 2007.30.03\nsource: http://www.tiddlytools.com/\n***/\n/*{{{*/\nconfig.commands.collapseNote = {\ntext: "-",\ntooltip: "Collapse this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nif(e.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(e.getAttribute("template") != t ){\ne.setAttribute("oldTemplate",e.getAttribute("template"));\nstory.displayNote(null,title,t);\n}\n}\n}\n}\n\nconfig.commands.expandNote = {\ntext: " | ",\ntooltip: "Expand this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.displayNote(null,title,e.getAttribute("oldTemplate"));\n}\n}\n\nconfig.macros.collapseAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"-","Collapse all notes",function(){\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE])\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nstory.displayNote(null,title,t);\n})})\n}\n}\n\nconfig.macros.expandAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"expand all","Expand all notes",function(){\nstory.forEachNote(function(title,note){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(note.getAttribute("template") == t) story.displayNote(null,title,note.getAttribute("oldTemplate"));\n})})\n}\n}\n\nconfig.commands.collapseOthers = {\ntext: "Ø",\ntooltip: "Expand this note and collapse all others",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif (e==note) t=e.getAttribute("oldTemplate");\n//////////\n// ELS 2006.02.22 - removed this line. if t==null, then the *current* view template, not the default "ViewTemplate", will be used.\n// if (!t||!t.length) t=!readOnly?"ViewTemplate":"WebViewTemplate";\n//////////\nstory.displayNote(null,title,t);\n}\n})\n}\n}\n/*}}}*/
<div>\n<span class='toolbar' macro='toolbar +editNote expandNote collapseOthers closeOthers -closeNote'></span>\n<span class='title' macro='view title'></span>\n</div>
[[Mathematica|mathematica]] will simplify complex expressions easier if you use the TargetFunction option, e.g.\nComplexExpand[expr, TargetFunction -> {Re, Im}]
This site is powered by [[TiddlyWiki|http://www.tiddlywiki.com]] <<version>>\n!I installed these plugins (need t(T)iddler -> n(N)ote find and replace):\n*[[InlineJavascriptPlugin]]\n**used for the [[DisplayControl]]\n**and for [[HideTags]]\n*[[TextAreaPlugin]] -- disabled for now\n**deselect the edit contents, and adds ctr-f,ctrl-g,cmd-v search/replace to editing.\n*[[jsMathPlugin]]\n**this processes the [[LaTeX]]. The AJAX part had problems, so I put the jsmath load into the source directly.\n**inserted custom LaTeX/jsmath command abbreviations into plugin.\n*[[CollapsePlugin]]\n**[[CollapsedTemplate]]\n*[[RearrangeNotesPlugin]]\n*[[ListTaggedPlugin]]\n**used for folder/tag listings\n*[[AllTagsExceptPlugin]]\n**advanced checkbox to see system tags\n*[[CopyNotePlugin]]\n*[[DisableWikiLinksPlugin]]\n**remove checkbox so it's always on\n**this is very tricky in combination with [[jsMathPlugin]] and \sss pytw problem\n*[[FaviconPlugin]]\n*[[ReferencesPlugin]]\n*[[RecentPlugin]]\n**set to show last 2\ncheck to make sure I didn't install any<<tag plugin>>and forget to list it here. Try using the [[PluginManager]].\n\n!I changed these notes to configure operation and appearance:\n*These control the content of several boxes:\n**[[SiteTitle]]\n**[[SiteSubtitle]]\n**[[SiteUrl]]\n**[[DefaultNotes]]\n**[[MainMenu]]\n**[[SideBarOptions]]\n**[[OptionsPanel]]\n***[[SideBarOptionsText]]\n**[[AdvancedOptions]]\n**[[SideBarTabs]]\n***[[TabContents]]\n***[[TabTimeline]]\n***[[TabAll]] - nope, for some reason this has the text built in. :(\n***[[TabTags]]\n**[[DisplayControl]]\n*These are css layout templates:\n**[[PageTemplate]]\n**[[ViewTemplate]]\n**[[EditTemplate]]\n**[[CollapsedTemplate]]\n*And these change the system and css options:\n**[[SystemConfig]]\n**[[StyleSheet]]\n***Trouble with [[MyColors]] conflicting with [[ColorPalette]]\n**[[StyleSheetPrint]]\nThe default config files are invisible and listed as [[ShadowNotes]]. These:\n*[[StyleSheetLayout]]\n*[[StyleSheetColors]]\nare augmented and overriden by the [[StyleSheet]]. If they change in the future, with updates, the old version content will likely have to be added to the new [[StyleSheet]]. \n\n!Evil raw html/javascript TW source code tweakage\n*edit cookie options, since setting them in [[SystemConfig]] overrides user cookies\n*maybe add ctrl-w accessKey -- just fooling around\n*comment out a couple of displayMessage s\n*switch line order in {{{config.macros.search.handler}}} for search button after search field\n*comment out tag prompt line in {{{config.macros.tags.handler}}}\n*Insert this just after body. (This starts jsmath)\n**{{{<scriipt src="jsMath/jsMath.js"></scriipt>}}}\n*add B's logging script call\n**make index executable so log script will run\n\n!edited jsMath.js\n*reduced vertical margins, {{{margin-top: 0.5em; margin-bottom: 0.5em;}}}\n*changed cookie default font scaling to {{{scale: 110}}} and {{{warn: 0}}}\n*remove doubleclick show by commenting out a few lines in {{{CheckDblClick}}}\n\n!And finally\nI did a global find/replace of "t(T)iddler" -> "n(N)ote" in the base file or directory via editor or the noteify shell script. Make sure to do this when more plugins are installed, so they'll work.\n\nThen, save a bare copy, without folders or editing tips, and a minimal copy, with them. Then try to [[ImportNotes]].
/***\nAuthors: Eric Shulman\nversion: 2.1.2\nsource: http://www.tiddlytools.com/\nadds a "copy" option to duplicate a note\n***/\n/*{{{*/\nversion.extensions.copyNote= {major: 2, minor: 1, revision: 2, date: new Date(2007,5,17)};\nconfig.commands.copyNote = {\n text: '\sxA9',\n hideReadOnly: true,\n tooltip: 'Make a copy of this note',\n prefix: "Copy of ",\n handler: function(event,src,title) {\n var text=store.getNoteText(title); // get text from note (or shadow)\n var tags=[]; var tid=store.getNote(title); if (tid) tags=tid.getTags();\n var textfield=story.getNoteField(title,"text");\n if (textfield&&textfield.getAttribute("edit")=="text") var text=textfield.value; // edit mode, use field value\n var tagsfield=story.getNoteField(title,"tags");\n if (tagsfield&&tagsfield.getAttribute("edit")=="tags") var tags=tagsfield.value; // edit mode, use field value\n var newTitle = this.prefix + title;\n story.displayNote(null,newTitle,DEFAULT_EDIT_TEMPLATE);\n story.getNoteField(newTitle,"text").value=text;\n story.getNoteField(newTitle,"tags").value=tags;\n story.focusNote(newTitle,"title");\n return false;\n }\n};\n/*}}}*/
On my Windows box I use CrimsonEditor mainly for writing [[LaTeX]] documents, with the occasional HTML and other stuff thrown in. In Dec06 the source code for CE was released, and a community was formed to update CE and write a new program called [[EmeraldEditor|http://about.emeraldeditor.com/]] in the vane of CE that will eventually replace it.\n\nHere's my list of commands for CrimsonEditor,\nnote the forward and inverse searching for [[Yap]].\n{{{\nName: LaTeX Context Help\nCmd: C:\sMiKTeX\sMain\sdoc\slatex\shelp\slatex2e.chm\nArg: $(CurrWord)\nDir: $(FileDir)\nKey: F1\n\nName: BibTex\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\sbibtex.exe\nArg: $(FileTitle)\nArg: $(FileDir)\nKey: F6\n\nName: Compile LaTeX Document\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\slatex.exe\nArg: --src-specials $(FileTitle)\nArg: $(FileDir)\nKey: F7\n\nName: DVI -> PS\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\sdvips.exe\nArg: "-t a4 $(FileTitle)"\nArg: $(FileDir)\nKey: Ctrl-F7\n\nName: PS -> PDF\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\sps2pdf.exe\nArg: "$(FileTitle).ps"\nArg: $(FileDir)\nKey: Ctrl-Alt-F7\n\nName: View dvi file with Yap\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\syap.exe\nArg: -1 -s$(LineNum)$(FileName) $(FileTitle).dvi\nArg: $(FileDir)\nKey: F8\n\nName: LaTeX -> PDF\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\spdflatex.exe\nArg: "$(FileName)"\nArg: $(FileDir)\nKey: F9\n\nName: LaTeX -> HTML\nCmd: C:\sMiKTeX\sMain\smiktex\sbin\shtlatex.bat\nArg: "$(FileName)"\nArg: $(FileDir)\nKey: Ctrl-F9\n\nName: View PDF with Foxit\nCmd: C:\sProgram Files\sFoxit Software\sFoxit Reader\sFoxit Reader.exe\nArg: $(FileTitle).pdf\nArg: $(FileDir)\nKey: F10\n\nName: FeynMP-mpostall\nCmd: C:\sWINDOWS\ssystem32\scmd.exe\nArg: /C "for %i in (*.mp) do mpost %~ni"\nArg: $(FileDir)\nKey: \n\nName: Open in Browser\nCmd: C:\sPROGRA~1\sMOZILL~1\sFIREFOX.EXE\nArg: -url "$(FileDir)\s$(FileName)"\nArg: $(FileDir)\nKey: \n}}}
For the math kernel to get the directory of the currently open file, it needs to interface with the frontend.\nHere is how it can be done....\n\nFor [[mathematica]] 5.x - on a Windows machine:\ncurDir := ToFileName@Rest@First["FileName" /. NotebookInformation@EvaluationNotebook[]];\n\nFor [[mathematica]] 6.x - should work on all machines:\ncurDir := NotebookDirectory[EvaluationNotebook[]];\n\nThen of course you can use this to save data in the current directory:\nSave[curDir <> "abcd.dat", "abcdList*"];\nand later you can 'get' the data back:\nGet[curDir <> "abcd.dat"];
Welcome
/***\nAuthors: Eric Shulman\nversion: 1.0.0\nsource: http://www.tiddlytools.com/\nThis plugin allows you to disable TiddlyWiki's automatic WikiWord linking behavior, so that WikiWords embedded in note content will be rendered as regular text, instead of being automatically converted to note links. To create a note link when automatic linking is disabled, you must enclose the link text within {{{[[}}} and {{{]]}}}.\n!!!!!Code\n***/\n//{{{\nversion.extensions.disableWikiLinks= {major: 1, minor: 0, revision: 0, date: new Date(2005,12,9)};\n\n// G changed to have this on, without checkbox\nconfig.options.chkDisableWikiLinks= true;\n\n// find the formatter for wikiLink and replace handler with 'pass-thru' rendering\nfor (var i=0; i<config.formatters.length && config.formatters[i].name!="wikiLink"; i++);\nconfig.formatters[i].coreHandler=config.formatters[i].handler;\nconfig.formatters[i].handler=function(w) {\n // if not enabled, just do standard WikiWord link formatting\n if (!config.options.chkDisableWikiLinks) return this.coreHandler(w);\n // supress any leading "~" (if present)\n var skip=(w.matchText.substr(0,1)==config.textPrimitives.unWikiLink)?1:0;\n w.outputText(w.output,w.matchStart+skip,w.nextMatch)\n}\n//}}}
&nbsp;<script label="O" title="toggle sidebar">\n var sb=document.getElementById('sidebar');\n var da=document.getElementById('displayArea');\n if (sb.style.display == 'none') {\n da.style.marginLeft = '18.5em';\n sb.style.display = 'block';}\n else {\n da.style.marginLeft = '0em';\n sb.style.display = 'none';}\n</script>&nbsp;<script label="O" title="toggle title">\n var h=document.getElementById('head');\n if (h.style.height == '1em') {\n h.style.height = '5.8em';}\n else {\n h.style.height = '1em';}\n</script>&nbsp;\n<<saveChanges>>\n[[cheat|cheat.pdf]]
<div class='toolbar' macro='toolbar +saveNote copyNote deleteNote closeOthers -cancelNote'></div>\n<div class='title' macro='view title'></div>\n<div class='editor' macro='edit title'></div>\n<div class='editor' macro='edit text'></div>\n<div class='toolbar' macro='toolbar +saveNote copyNote deleteNote closeOthers -cancelNote'></div>\n<div class='editorFooter'><span macro='tagChooser'></span><span macro='message views.editor.tagPrompt'></span></div>\n<div><span class='editor' macro='edit tags'></div>
!!Guidelines\n*Keep the notes short &mdash; if it gets long, split it. Somewhere between a paragraph and a page is about right.\n*Don't use objects or methods without linking references (once) for the reader.\n*The first time you reference another object in a note, link to its eponymous note.\n**If you reference an object defined in a non-eponymous note, link to it again by [ [ object | note ] ]. \n*If you define some object, make the object name ''bold''.\n**If you include a pseudonym for a defined object, make it //''bold italic''//.\n*Include all steps of calculations, with the manipulations established from referenced notes.\n*Link to [[Wikipedia|http://en.wikipedia.org/]] for standard stuff, or to arxiv papers and other things as needed.\n*Include introduced symbols in the [[Symbols]] table.\n*Include introduced tags in [[Tags]] and<<tag folder>>. //Don't do this often//\n*Examples are OK, even if they run a little long.\n*It is good to start a note as "A ''this thing'' is a kind of [[link to more general case]] which //more detailed properties//..."\n*In general, don't link "down" to more specific instances -- that's what the "referenced by" button is for. Unless it seems pedagogically useful.\n*Lateral (mutual) linking is OK, such as in the [[differential form]] note &mdash; for similar objects, special generalized cases, or extended discussions.\n*Attempt to duplicate the same structure of links for similar structures of mathematical objects and instantiations.\n*Links that establish a hierarchical relationship should probably be treated differently then links to define used objects -- but for now they're the same.\n*If you link to a note that doesn't exist, create that note and tag it with<<tag 0>>if you leave it empty.\n*Tag notes that desperately need editing with<<tag 0>>.\n*Put editorial comments (//like this one//) in parens and italics.\n!!Markup\nThere are many markup formating commands and features that can be used when <<tag editing>> notes:\n<<ListTagged editing>>\nYou can do other neat stuff with more <<tag plugin>>s.\n
Images can be included by their filename or full URL. It's good practice to include a title to be shown as a tooltip, and when the image isn't available. An image can also link to another note or or a URL\n[img[Romanesque broccoli|images/fractalveg.jpg][http://www.flickr.com/photos/jermy/10134618/]]\n{{{\n[img[Romanesque broccoli|images/fractalveg.jpg]\n [http://www.flickr.com/photos/jermy/10134618/]]\n[img[title|filename]]\n[img[filename]]\n[img[title|filename][link]]\n[img[filename][link]]\n}}}\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]][>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]You can also float images to the left or right: the forest is left aligned with {{{[<img[}}}, and the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):You can use CSS to clear the floats@@\n{{{\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]]\n[>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]\nYou can also float images to the left or right:\n the forest is left aligned with {{{[<img[}}},\nand the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):\nYou can use CSS to clear the floats@@\n}}}
''Bold''\n{{{''Bold''}}}\n==Strikethrough==\n{{{==Strikethrough==}}}\n__Underline__ \n{{{__Underline__}}}\n//Italic// \n{{{//Italic//}}}\n2^^3^^=8 \n{{{2^^3^^=8}}}\na~~ij~~ = -a~~ji~~ \n{{{a~~ij~~ = -a~~ji~~}}}\n@@highlight@@ \n{{{@@highlight@@}}}\n\n//The highlight can also accept CSS syntax to directly style the text://\n@@color:green;green coloured@@\n{{{@@color:green;green coloured@@}}}\n@@background-color:#ff0000;color:#ffffff;red coloured@@\n{{{@@background-color:#ff0000;color:#ffffff;red coloured@@}}}\n@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@\n{{{@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@}}}\n@@display:block;text-align:center;centered text or image@@\n{{{@@display:block;text-align:center;centered text or image@@}}}\n\n//For backwards compatibility, the following highlight syntax is also accepted://\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n{{{\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n}}}
var n = document.createElement("link"); \nn.rel = "shortcut icon"; \nn.href = "favicon.ico"; \ndocument.getElementsByTagName("head")[0].appendChild(n);
I will add more here, but for now, here is some notes on the Feynman and Schwinger parametrisations, by Joe Shapiro [[schwingertrick.pdf|http://www.physics.rutgers.edu/grad/615/lects/schwingertrick.pdf]]\n\nWant notes on\n[[Feynman Parameters]]\n[[Schwinger Parameters]]\n[[Dimensional Regularisation]]\n[[Epsilon Expansion]]\n[[Differental Equation Method|FeynIntDE]]\n[[Mellin-Barnes Evaluation|FeynIntMB]]\n[[Two Loop]]
The observation of Feynman was based on the partial fraction decomposition\n\s[ \sfrac1{AB}=\sfrac1{B-A}\slp\sfrac1A-\sfrac1B\srp ~.\s]\nThis may be written in integral form as\n\s[ \sfrac1{AB}=\sfrac1{B-A}\sint_A^B\sfrac{\srmd x}{x^2}=\sint_0^1\sfrac{\srmd x}{\slp xA+(1-x)B\srp^2}\n=\sint\s!\s!\sint_0^1\sfrac{\srmd x\srmd y\sd(x+y-1)}{\slp Ax+By\srp^2} \n\s]\nDifferentiating wrt $A$ and $B$ yields\n\s[ \sfrac1{A^nB^m}=\sfrac{\sG(n+m)}{\sG(n)\sG(m)}\sint\s!\s!\sint_0^1\s!\s!\srmd x\srmd y\sfrac{x^{n-1}y^{m-1}\sd(x+y-1)}{\slp Ax+By\srp^{n+m}} \s]\nAlso repeated application gives\n\sbegin{eqnarray} \sfrac1{ABC}&=&\sfrac{1}A\sint_0^1\s!\s!\sfrac{\srmd y}{\slp By+C(1-y)\srp^2}\n=2\sint_0^1\s!\sint_0^1\sfrac{\srmd x\srmd y(1-x)}{\slp Ax+By(1-x)+C(1-x)(1-y)\srp^3}\s\s\n&=&2\sint_0^1\s!\sint_0^{1-x}\sfrac{\srmd x\srmd y}{\slp Ax+By+C(1-x-y)\srp^3}\n=2\sint_0^1\s!\s!\srmd^3x\sfrac{\sd(x_1+x_2+x_3-1)}{\slp Ax_1+Bx_2+Cx_3\srp^3}\n\send{eqnarray}\nwhere we went to the second line using the change of variables $y'=y(1-x)$.\n\nIt is useful to realise that\n\sbegin{eqnarray} \n\sint_{\ssum{x_i}\sleq 1}\s!\s!\s!\s!\s!\s!\s!\s!\s!\srmd^{n-1}x\s,f(x_1,\sldots, x_{n-1},1-\ssum_{i=1}^{n-1}{x_i})\n&=&\sint_0^1\s!\s!\s!\srmd{x_1}\sint_0^{1-x_1}\s!\s!\s!\s!\s!\srmd{x_2}\scdots\sint_0^{1-\ssum_{i=1}^{n-2}{x_i}}\s!\s!\s!\s!\s!\s!\s!\s!\srmd{x_{n-1}}\s,\n f(x_1,\sldots, x_{n-1},1-\ssum_{i=1}^{n-1}{x_i})\s\s\n&=&\sint_0^1\s!\s!\s!\srmd{x_1}\scdots\sint_0^1\s!\s!\s!\srmd{x_n}\s,f(x_1,\sldots, x_n)\sd(\ssum_{i=1}^{n}{x_i}-1) \n\send{eqnarray}\n\nThe same tricks lead to\n\s[ \sfrac1{A_1\scdots A_n}=\sG(n)\sint_0^1\s!\s!\srmd^nx\sfrac{\sd(\ssum{x_i}-1)}{\slp \ssum{A_ix_i}\srp^n} \s]\nRepeated differentiation leads to the general formula for combining denominators,\n\sbegin{equation}\n\sfrac1{A_1^{a_1}}\scdots\sfrac1{A_n^{a_n}}=\sfrac{\sG(\ssum_{i=1}^na_i)}{\sprod_{i=1}^n\sG(a_i)}\n\sint_0^1\s!\s! {\srm d}^nx \sfrac{\sdelta(\ssum_{i=1}^nx_i-1)\sprod_{i=1}^nx_i^{a_i-1}}{\slp\ssum_{i=1}^nx_iA_i\srp^{\ssum_{i=1}^na_i}}\n\send{equation}\n\nFeynman's parameterisation is related to [[Schwinger's|Schwinger Parameters]] via a fairly simple change of variables. First write the $\sG$ function in integral form\n\s[ \sG(\ssum_{i=1}^na_i)=\sint_0^\sinfty\s!\s!\srmd\sl \s,\sl^{\ssum_{i=1}^na_i-1}\srme^{-\sl} \s]\nthen rescale $\sl\sto\sl\ssum{A_ix_i}$ to get\n\sbegin{equation}\n\sfrac1{A_1^{a_1}}\scdots\sfrac1{A_n^{a_n}}=\sfrac{1}{\sprod \sG(a_i)}\n\sint_0^\sinfty\s!\s!\srmd\sl \s,\sint_0^1\s!\s! {\srm d}^nx \sl^{\ssum a_i-1}\srme^{-\sl\ssum{A_ix_i}} \n\sdelta(\ssum x_i-1)\sprod x_i^{a_i-1}\n\send{equation}\nnow write $x_i=s_i/\sl$ and use $\sd(ax)=\sfrac1{|a|}\sd(x)$ to get\n\sbegin{equation}\n\sfrac1{A_1^{a_1}}\scdots\sfrac1{A_n^{a_n}}=\sfrac{1}{\sprod \sG(a_i)}\n\sint_0^\sinfty\s!\s!\srmd\sl \s,\sint_0^\sl\s!\s! {\srm d}^ns \srme^{-\ssum{A_ix_i}} \n\sdelta(\ssum s_i-\sl)\sprod s_i^{a_i-1}\n\send{equation}\nthe $\sd$ function now makes the $\sl$ integral trivial. We recover the Schwinger parametrisation \n\sbegin{equation}\n\sfrac1{A_1^{a_1}}\scdots\sfrac1{A_n^{a_n}}=\sfrac{1}{\sprod \sG(a_i)}\n\sint_0^\sinfty\s!\s! {\srm d}^ns \srme^{-\ssum{A_ix_i}} \sprod s_i^{a_i-1}\n\send{equation}\n\ngive simple examples\n
(Most of this note is inspired by the exercises on pp419-420 of Siegel's [[Fields|http://insti.physics.sunysb.edu/~siegel/errata.html]] .)\n\nThe (Euler) Gamma function is an analytic continuation of the factorial, $\sG(n+1)=n!$. \nIt has an integral representation\n\s[ \sG(z)=\sint_0^\sinfty\s!\s!\srmd\sl \s,\sl^{z-1}\srme^{-\sl} \squad \sRe(z)>1\s]\nfrom which we recover, after an integration by part, the expected functional relation\n\s[ \sG(z+1)=z\sG(z)~\s]\nThis relation allows us to extend the domain of definition by using\n\s[ \sG(z)=\sfrac1{z(z+1)\scdots(z+k-1)}\sG(z+k)=\sfrac1{(z)_k}\sG(z+k) \squad \sRe(z+k)>1\s]\nwhich can be considered as the definition of the [[Pochhammer symbol|http://mathworld.wolfram.com/PochhammerSymbol.html]], $(z)_\sz$.\nSome special, non-integer, values of $\sG(z)$ that can be exactly computed are\n\s[ \sG(\shalf)=\sint_0^\sinfty\s!\s!\srmd\sl \s,\sl^{-\shalf}\srme^{-\sl}=2\sint_0^\sinfty\s!\s!\srmd x \s,\srme^{-x^2}=\ssqrt\sp\n\squad\sRightarrow\squad\n \sG(n+\shalf)=(n-\shalf)(n-\ssmall\sfrac32)\scdots\shalf\sG(\shalf)=\sfrac{(2n)!\ssqrt\sp}{4^nn!}\n\s]\n\nA related function, and one that often occurs in [[Feynman Integrals]] is the [[Beta function|http://mathworld.wolfram.com/BetaFunction.html]],\n\s[ B(x,y)=\sfrac{\sG(x)\sG(y)}{\sG(x+y)}~.\s]\nThe easiest way to get an integral representation for it is to use the [[Feynman parametrisation|Feynman Parameters]] for a product of two denominators,\n\s[ \sfrac1{A^nB^m}=\sfrac{\sG(n+m)}{\sG(n)\sG(m)}\sint\s!\s!\sint_0^1\s!\s!\srmd x\srmd y\sfrac{x^{n-1}y^{m-1}\sd(x+y-1)}{\slp Ax+By\srp^{n+m}}\n\s]\nand set $A=B=1$ to get\n\s[ B(x,y)=\sint_0^1\srmd\sa(1-\sa)^{x-1}\sa^{y-1}\squad\sRe(x)>1,~\sRe(y)>1~.\s]\nWe could also have started from the [[Schwinger parametrisation|Schwinger Parameters]]\n\s[\sG(n)\sG(m)=\sint_0^\sinfty\srmd x\srmd y\srme^{-(x+y)}x^{n-1}y^{m-1} \s]\nand factorised the integral using the handy change of variables\n\s[ \sint_0^\sinfty\srmd x\srmd y f(x,y)=\sint_0^\sinfty\st\srmd\st\sint_0^1\srmd \sa f(\st(1-\sa),\st\sa)~. \s]\nBy making the change of variables $4\sb^2=\sa$ and symmetrising the resulting integral, it's easy to see that\n\s[B(\shalf,x)=2^{2x-1}B(x,x)~,\s]\nthis implies that $\sG(x+\shalf)=\sfrac{\sG(2x)\ssqrt\sp}{2^{2x-1}\sG(x)}$, as seen above.\nAnother useful parameterisation of $B(x,y)$ is obtained by substitution $\sa=1/(1+z)$ to get\n\s[ B(x,y)=\sint_0^\sinfty\srmd z\sfrac{z^{x-1}}{(z+1)^{x+y}}~,\s]\nthis is used, eg, in the direct proof of the last equation in [[Schwinger Parameters]].\nNow let us examine \n\s[ B(z,1-z)=\sG(z)\sG(1-z)=\sint_0^\sinfty\srmd t\sfrac{t^{z-1}}{1+t}\n=\sint_{-\sinfty}^{\sinfty}\srmd u \sfrac{\srme^{zu}}{1+\srme^u}~,\squad t=\srme^u\s]\nSince $Re(z)>1$ we close the contour in the bottom half plane. The poles of the integrand are at $u_n=(2n+1)\srmi\sp$, where $\sexp{u_n}=-1$, and have residue\n\s[ {\srm Res}_{u=u_n}\slp\sfrac{\srme^{zu}}{1+\srme^u}\srp=\slim_{u\sto u_n}\slb (u-u_n)\sfrac{\srme^{u_nz}\n\slp1+z(u-u_n)+\shalf z^2(u-u_n)^2+\sldots \srp}{1-\slp 1+(u-u_n)+\shalf(u-u_n)^2+\sldots \srp}\srb \n=-\srme^{zu_n}\n\s]\nso that, by the Cauchy residue theorem, (note the minus sign from the clockwise contour)\n\s[ B(z,1-z)=\sG(z)\sG(1-z)=2\sp\srmi\ssum_{n=0}^{\sinfty}\srme^{-(2n+1)\srmi\sp z}\n=\sfrac{2\sp\srmi\srme^{-\srmi\sp z}}{1-\srme^{-2\srmi\sp z}}=\sp\scsc(\sp z)~.\n\s]\n\nA formula that is often needed is the Laurent expansion of $\sG(n+\seps)$ around a non-positive integer $n$.\n.....
Really useful integrals!!\n\n....
The eight [[trace]]less, Hermitian, ''Gell-Mann matrices'', $\sl_A$, are\n$$\n\sbegin{array}{cccc}\n\sl_0 = \sl_8 = \sfrac{1}{\ssqrt{3}} \sleft[\sbegin{array}{ccc}\n1 & 0 & 0\s\s\n0 & 1 & 0\s\s\n0 & 0 & -2\n\send{array}\sright]\n&\n\sl_1 = \sleft[\sbegin{array}{ccc}\n0 & 1 & 0\s\s\n1 & 0 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n&\n\sl_2 = \sleft[\sbegin{array}{ccc}\n0 & -i & 0\s\s\ni & 0 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n&\n\sl_3 = \sleft[\sbegin{array}{ccc}\n1 & 0 & 0\s\s\n0 & -1 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n\s\s\n\sl_4 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 1\s\s\n0 & 0 & 0\s\s\n1 & 0 & 0\n\send{array}\sright]\n&\n\sl_5 = \sleft[\sbegin{array}{ccc}\n0 & 0 & -i\s\s\n0 & 0 & 0\s\s\ni & 0 & 0\n\send{array}\sright]\n&\n\sl_6 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 0\s\s\n0 & 0 & 1\s\s\n0 & 1 & 0\n\send{array}\sright]\n&\n\sl_7 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 0\s\s\n0 & 0 & -i\s\s\n0 & i & 0\n\send{array}\sright]\n\send{array}\n$$\n
/%\n|Name|HideTags|\n|Source|http://www.TiddlyTools.com/#HideTiddlerTags|\n|Version|0.0.0|\n|Author|Eric Shulman - ELS Design Studios (edited by Garrett)|\n|License|http://www.TiddlyTools.com/#LegalStatements <<br>>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|\n|~CoreVersion|2.1|\n|Type|script|\n|Requires|InlineJavascriptPlugin|\n|Description|hide a note's tagged/tagging/references display elements|\n\nUsage: <<note HideTags>>\n\n%/<script>\n var t=story.findContainingNote(place);\n if (t && t.id!="noteHideTags")\n for (var i=0; i<t.childNodes.length; i++)\n {if (hasClass(t.childNodes[i],"tagging")||hasClass(t.childNodes[i],"tagged"))\n t.childNodes[i].style.display="none";\n if (hasClass(t.childNodes[i],"references"))\n t.childNodes[i].style.display="none";}\n</script>
A horizontal dividing line.\n----\n{{{----}}}
Entities in HTML documents allow characters to be entered that can't easily be typed on an ordinary keyboard. They take the form of an ampersand (&), an identifying string, and a terminating semi-colon (;). There's a complete reference [[here|http://www.htmlhelp.com/reference/html40/entities/]]; some of the more common and useful ones are shown below. Also see [[Paul's Notepad|http://thepettersons.org/PaulsNotepad.html#GreekHtmlEntities%20HtmlEntitiesList%20LatinHtmlEntities%20MathHtmlEntities]] for a more complete list.\n\n|>|>|>|>|>|>| !HTML Entities |\n| &amp;nbsp; | &nbsp; | no-break space | &nbsp;&nbsp; | &amp;apos; | &apos; | single quote, apostrophe |\n| &amp;ndash; | &ndash; | en dash |~| &amp;quot; | " | quotation mark |\n| &amp;mdash; | &mdash; | em dash |~| &amp;prime; | &prime; | prime; minutes; feet |\n| &amp;hellip; | &hellip; | horizontal ellipsis |~| &amp;Prime; | &Prime; | double prime; seconds; inches |\n| &amp;copy; | &copy; | Copyright symbol |~| &amp;lsquo; | &lsquo; | left single quote |\n| &amp;reg; | &reg; | Registered symbol |~| &amp;rsquo; | &rsquo; | right single quote |\n| &amp;trade; | &trade; | Trademark symbol |~| &amp;ldquo; | &ldquo; | left double quote |\n| &amp;dagger; | &dagger; | dagger |~| &amp;rdquo; | &rdquo; | right double quote |\n| &amp;Dagger; | &Dagger; | double dagger |~| &amp;laquo; | &laquo; | left angle quote |\n| &amp;para; | &para; | paragraph sign |~| &amp;raquo; | &raquo; | right angle quote |\n| &amp;sect; | &sect; | section sign |~| &amp;times; | &times; | multiplication symbol |\n| &amp;uarr; | &uarr; | up arrow |~| &amp;darr; | &darr; | down arrow |\n| &amp;larr; | &larr; | left arrow |~| &amp;rarr; | &rarr; | right arrow |\n| &amp;lArr; | &lArr; | double left arrow |~| &amp;rArr; | &rArr; | double right arrow |\n| &amp;harr; | &harr; | left right arrow |~| &amp;hArr; | &hArr; | double left right arrow |\n\nThe table below shows how accented characters can be built up by subsituting a base character into the various accent entities in place of the underscore ('_'):\n\n|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>| !Accented Characters |\n| grave accent | &amp;_grave; | &Agrave; | &agrave; | &Egrave; | &egrave; | &Igrave; | &igrave; | &Ograve; | &ograve; | &Ugrave; | &ugrave; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; |\n| acute accent | &amp;_acute; | &Aacute; | &aacute; | &Eacute; | &eacute; | &Iacute; | &iacute; | &Oacute; | &oacute; | &Uacute; | &uacute; | &nbsp; | &nbsp; | &Yacute; | &yacute; | &nbsp; | &nbsp; |\n| circumflex accent | &amp;_circ; | &Acirc; | &acirc; | &Ecirc; | &ecirc; | &Icirc; | &icirc; | &Ocirc; | &ocirc; | &Ucirc; | &ucirc; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; |\n| umlaut mark | &amp;_uml; | &Auml; | &auml; | &Euml; | &euml; | &Iuml; | &iuml; | &Ouml; | &ouml; | &Uuml; | &uuml; | &nbsp; | &nbsp; | &Yuml; | &yuml; | &nbsp; | &nbsp; |\n| tilde | &amp;_tilde; | &Atilde; | &atilde; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &Otilde; | &otilde; | &nbsp; | &nbsp; | &Ntilde; | &ntilde; | &nbsp; | &nbsp; | &nbsp; | &nbsp; |\n| ring | &amp;_ring; | &Aring; | &aring; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; |\n| slash | &amp;_slash; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &Oslash; | &oslash; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; |\n| cedilla | &amp;_cedil; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &nbsp; | &Ccedil; | &ccedil; |
<<<\nA whole block\nof text to be quoted.\n<<<\nor\n>>>Multiple levels of indented quotes.\n>>Just like [[Bullet Points]].\n>yep\n>>or like [[Numbered Lists]]\nThat's what they said.\n{{{\n<<<\nA whole block\nof text to be quoted.\n<<<\nor\n>>>Multiple levels of indented quotes.\n>>Just like [[Bullet Points]].\n>yep\n>>or like [[Numbered Lists]]\nThat's what they said.\n}}}
TiddlyWiki lets you write ordinary HTML by enclosing it in {{{<html>}}} and {{{</html>}}}:\n<html>\n<a href="javascript:;" onclick="onClickNoteLink(event);" \ntiddlyLink="Welcome"\nstyle="background-color: yellow;">\nLink to Welcome constructed in HTML</a>\n</html>\n{{{\n<html>\n<a href="javascript:;" onclick="onClickNoteLink(event);" \ntiddlyLink="Welcome"\nstyle="background-color: yellow;">\nLink to Welcome constructed in HTML</a>\n</html>\n}}}\nHTML can enable some exotic new features (like [[embedding GMail and Outlook|http://groups.google.com/group/TiddlyWiki/browse_thread/thread/d363303aff5868d0/056269d8409d121f?lnk=st&q=embedding+gmail&rnum=1#056269d8409d121f]] in a TiddlyWiki). But, care needs to be taken with including things like JavaScript code.
/***\n''InlineJavascriptPlugin''\n^^version: 1.6.0\nauthor: Eric Shulman - ELS Design Studios\nsource: http://www.tiddlytools.com/\nlicense: [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]^^\nsee source link (above) for usage.\n!!!!!Code\n***/\n/*{{{*/\nversion.extensions.inlineJavascript= {major: 1, minor: 6, revision: 0, date: new Date(2007,2,19)};\n\nconfig.formatters.push( {\n name: "inlineJavascript",\n match: "\s\s<script",\n lookahead: "\s\s<script(?: src=\s\s\s"((?:.|\s\sn)*?)\s\s\s")?(?: label=\s\s\s"((?:.|\s\sn)*?)\s\s\s")?(?: title=\s\s\s"((?:.|\s\sn)*?)\s\s\s")?( show)?\s\s>((?:.|\s\sn)*?)\s\s</script\s\s>",\n\n handler: function(w) {\n var lookaheadRegExp = new RegExp(this.lookahead,"mg");\n lookaheadRegExp.lastIndex = w.matchStart;\n var lookaheadMatch = lookaheadRegExp.exec(w.source)\n if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {\n if (lookaheadMatch[1]) { // load a script library\n // make script tag, set src, add to body to execute, then remove for cleanup\n var script = document.createElement("script"); script.src = lookaheadMatch[1];\n document.body.appendChild(script); document.body.removeChild(script);\n }\n if (lookaheadMatch[5]) { // there is script code\n if (lookaheadMatch[4]) // show inline script code in note output\n wikify("{{{\sn"+lookaheadMatch[0]+"\sn}}}\sn",w.output);\n if (lookaheadMatch[2]) { // create a link to an 'onclick' script\n // add a link, define click handler, save code in link (pass 'place'), set link attributes\n var link=createTiddlyElement(w.output,"a",null,"tiddlyLinkExisting",lookaheadMatch[2]);\n link.onclick=function(){try{return(eval(this.code))}catch(e){alert(e.description?e.description:e.toString())}}\n link.code="function _out(place){"+lookaheadMatch[5]+"\sn};_out(this);"\n link.setAttribute("title",lookaheadMatch[3]?lookaheadMatch[3]:"");\n link.setAttribute("href","javascript:;");\n link.style.cursor="pointer";\n }\n else { // run inline script code\n var code="function _out(place){"+lookaheadMatch[5]+"\sn};_out(w.output);"\n code=code.replace(/document.write\s(/gi,'place.innerHTML+=(');\n try { var out = eval(code); } catch(e) { out = e.description?e.description:e.toString(); }\n if (out && out.length) wikify(out,w.output,w.highlightRegExp,w.note);\n }\n }\n w.nextMatch = lookaheadMatch.index + lookaheadMatch[0].length;\n }\n }\n} )\n/*}}}*/
Mathematica 6 now supports vector derivatives: {{{D[f[x,y,...],{{x,y,...}}]}}}\nSo take a vector derivative of a list, eg\n{{{\nD[{f[x,y,z], g[x,y,z], h[x,y,z]}, {{x,y,z}}]\n}}}\n\nOld Mathematica 1 - 5\n(From the [[Mathworld|http://mathworld.wolfram.com/]])\n\nThe neatest way to find the Jacobian of a transformation in [[mathematica]] is to use:\n{{{\nJacobianMatrix[f_List?VectorQ, x_List] := Outer[D, f, x] \n /; Equal@@(Dimensions/@{f,x})\nJacobianDeterminant[f_List?VectorQ, x_List] := Det[JacobianMatrix[f, x]]\n /;Equal @@ (Dimensions /@ {f, x})\n}}}\nWhy this isn't a standard, built-in function I don't know!\n
Access keys are shortcuts to common functions accessed by typing a letter with either the 'alt' (PC) or 'control' (Mac) key:\n|!PC|!Mac|!Function|\n|Alt-F|Ctrl-F|Search|\n|Alt-J|Ctrl-J|NewJournal|\n|Alt-N|Ctrl-N|NewNote|\n|Alt-S|Ctrl-S|SaveChanges|\nThese access keys are provided by the associated internal [[Macros]] for the functions above. The macro needs to be used in an open note (or the [[MainMenu]] or SideBar) in order for the access keys to work.\n\nWhile editing a note:\n* ~Control-Enter or ~Control-Return accepts your changes and switches out of editing mode (use ~Shift-Control-Enter or ~Shift-Control-Return to stop the date and time being updated for MinorChanges)\n* Escape abandons your changes and reverts the note to its previous state\n\nIn the search box:\n* Escape clears the search term
Useful links:\n[[LaTeX project|http://www.latex-project.org/]]\n[[amsMath FAQ|http://www.ams.org/tex/amsmath-faq.html]]\n[[TeX Resources - TUG|http://www.tug.org/interest.html#doc]]\n[[some LaTeX tutorials|http://theoval.sys.uea.ac.uk/~nlct/latex/]]\n[[LaTeX and Xfig|http://www.xfig.org/userman/latex_and_xfig.html]]\n\nProgramming:\n[[Donald E. Knuth|http://www-cs-faculty.stanford.edu/~knuth/]]\n[[plain.tex|http://www-cs-faculty.stanford.edu/~knuth/plain.tex]] - Good to see how some command is defined\n[[TeX by Topic, A TeXnician's Reference|http://www.eijkhout.net/tbt/]] by Victor Eijkhout\n[[LaTeX2e for Class and Package writers|http://www.latex-project.org/guides/clsguide.pdf]]
*Q: My references to figures and/or tables are showing incorrect numbers. What is wrong[[?|http://www.michaelshell.org/tex/ieeetran/]]\n**Remember that \slabel must go after or within \scaption because \scaption changes the counter that \slabel references. Placing \slabel prior to \scaption within a figure or table will reference the section number rather than the figure/table number. This is one of the most frequently asked LaTeX questions of all time. \n\n*Q: How do I use ispell with latex[[?|http://web.mit.edu/answers/latex/latex_spelling.html]]\n** ispell -t filename.tex\n**This will ignore anything that comes after a "\s\s", until it reaches the next blank space.
Link to notes, such as [[Horizontal Rule]].\n{{{\nLink to notes, such as [[Horizontal Rule]].\n}}}\nLink to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,\nwithout the messiness of the full URL appearing.\n{{{\nLink to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,\nwithout the messiness of the full URL appearing.\n}}}\nOr just type out http://www.osmosoft.com and it will be automatically linkified.
/***\nThis is a revision of the listTags plugin &mdash; [[I|Garrett Lisi]] mashed this up with Udo's big plugin.\n!!Usage\n{{{\n<<ListTagged tag>>\n}}}\n!!!Code\n***/\n/*{{{*/\nversion.extensions.ListTagged = {major: 0, minor: 1, revision: 1};\n\nconfig.macros.ListTagged = {\ntext: "Hello"\n};\n\nconfig.macros.ListTagged.handler = function(place,macroName,params)\n{\nvar notes = store.getTaggedNotes(params[0]);\nvar list = document.createElement("ul");\n place.appendChild(list);\n for (var i = 0; i < notes.length; i++) {\n var note = notes[i];\n var listItem = document.createElement("li");\n list.appendChild(listItem);\n createTiddlyLink(listItem, note.title, true);\n }\n\n}\n/*}}}*/\n
Macros let you write notes containing more exotic objects than just text. Macros may be added as plugins. If so, they should be tagged<<tag plugin>>, and described in [[Configuration]].\n!These are some of the built-in macros:\nToday is <<today>>\n{{{\nToday is <<today>>\n}}}\nClick on <<tag editing>> to popup all notes tagged "editing".\n{{{\nClick on <<tag editing>> to popup all notes tagged "editing".\n}}}\nTransclude one note into another via\n<<note 'Horizontal Rule'>>\n{{{\nTransclude one note into another via\n<<note 'Horizontal Rule'>>\n}}}\n//There is no protection against inadvertently setting up endless loops. And this may have problems if the transcluded note isn't loaded.//\nSlider: <<slider chkTestSlider 'Horizontal Rule' 'press me»' "Click here to see the Horizontal Rule slide out">>\n{{{\nSlider: <<slider chkTestSlider 'Horizontal Rule' 'press me»' "Click here to see the Horizontal Rule slide out">>\n}}}\nThe slider parameters are:\n* cookie name to be used to save the state of the slider\n* name of the note to include in the slider\n* title text of the slider\n* tooltip text of the slider\n
&nbsp;[[About]]&nbsp;&nbsp;&nbsp;[[Tags]]&nbsp;&nbsp;&nbsp;[[Symbols]]&nbsp;&nbsp;&nbsp;[[To Do]]
[[David Park's mathematica page|http://home.comcast.net/~djmpark/Mathematica.html]]\n[[Example of a Feynman diagram calc|http://www.scientificarts.com/feynman/feynman.html]]\n[[Combinatorica homepage|http://www.combinatorica.com]]
Inline {{{monospaced text}}} with no editing commands executed inside the brackets.\nInline <html>{{{</html>monospaced text<html>}}}</html> with no editing commands executed inside the brackets.\n{{{\nmonospaced\n blocks\n(useful for source code)\n}}}\nBy putting "<html>{{{</html>" and "<html>}}}</html>" on their own lines.
!Colors Used\n*@@bgcolor(#8cf): #8cf - Background blue@@ -- popup\n*@@bgcolor(#18f): #18f - Top blue@@ -- header top\n*@@bgcolor(#04b): #04b - Mid blue@@ -- header \n*@@bgcolor(#014):color(#fff): #014 - Bottom blue@@ -- commands\n*@@bgcolor(#ffc): #ffc - Bright yellow@@ --\n*@@bgcolor(#fe8): #fe8 - Highlight yellow@@ --\n*@@bgcolor(#db4): #db4 - Background yellow@@ --\n*@@bgcolor(#841): #841 - Border yellow@@ --\n*@@bgcolor(#703):color(#fff): #703 - Title red@@ --\n*@@bgcolor(#866): #866 - Subtitle grey@@\n*@@bgcolor(#888): #888 - footer@@\n*@@bgcolor(#999): #999 - dark grey@@\n*@@bgcolor(#bbb): #bbb - tag2 grey@@\n*@@bgcolor(#ccc): #ccc - grey@@\n*@@bgcolor(#ddd): #ddd - tag1 grey@@\n*@@bgcolor(#eee): #eee - light grey@@\n*@@bgcolor(#ff0): #ff0 - error1@@\n*@@bgcolor(#f00): #f00 - error2@@\n*@@bgcolor(#eef): #ff0 - cascade1@@\n*@@bgcolor(#aac): #f00 - cascade2@@\n*@@bgcolor(#666): #666 - quote@@\n*@@bgcolor(#333): #333 - table@@\n*@@bgcolor(#996): #996 - thead@@\n*@@bgcolor(#fe8): #fe8 - pre1@@\n*@@bgcolor(#ffc): #ffc - pre2@@
To hide text within a note so that it is not displayed you can wrap it in {{{/%}}} and {{{%/}}}. It can be a useful trick for hiding drafts or annotating complex markup. Edit this note to see an example.\n/%This text is not displayed\nuntil you try to edit %/
Creating numbered lists is simple.\n# Just add a pounds sign\n# at the beginning of a line.\n## If you want to create sub-lists\n## start the line with two pounds\n### And if you want yet another level\n### use three pounds\n# You can also do [[Bullet Points]]\n{{{\nCreating numbered lists is simple.\n# Just add a pounds sign\n# at the beginning of a line.\n## If you want to create sub-lists\n## start the line with two pounds\n### And if you want yet another level\n### use three pounds\n# You can also do [[Bullet Points]]\n}}}
\s[J(x):=\sm^{2\seps}\sintk\sfrac1{k^2+x}=\n\sfrac{\sm^{2\seps}}{(4\sp)^{2-\seps}}\sint_0^\sinfty\s!\s!\n\sfrac{\srmd{s}\s,\srme^{-sx}}{t^{2-\seps}}\n=\sfrac{\sm^{2\seps}}{(4\sp)^{2-\seps}}\sG(\seps-1)x^{1-\seps}~.\n\s]
These options are saved in the browser:\n \n<<note SideBarOptionsText inputFix>>\n\n<<option chkRegExpSearch>> regular expression search\n<<option chkCaseSensitiveSearch>> case sensitive search\n<<option chkAnimate>> animation\n \n[[change advanced options|AdvancedOptions]]
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Format blocks of CSS definitions as:\n{{{\n/***\nDescription and comments go here, surrounded by comment brackets.\nFollowed by CSS code, which will be displayed in a code block.\n***/\n/*{{{*/\ndiv {color: #ff0000;}\n/*}}}*/\n}}}\nThat way the code will be run without the wikitext messing it up, and it will still be displayed nicely.
/***\n|''Name:''|RearrangeNotesPlugin|\n|''Source:''|http://www.TiddlyTools.com/#RearrangeNotesPlugin|\n|''Author:''|Joe Raii|\n|''License:''|[[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|\n|''~CoreVersion:''|2.1|\n***/\n//{{{\nStory.prototype.rearrangeNotesHijack_refreshNote = Story.prototype.refreshNote;\nStory.prototype.refreshNote = function(title,template,unused1,unused2,unused3,unused4,unused5)\n{\n this.rearrangeNotesHijack_refreshNote.apply(this,arguments);\n var theNote = document.getElementById(this.idPrefix + title); if (!theNote) return;\n var theHandle;\n var children=theNote.getElementsByTagName("*");\n for (var i=0; i<children.length; i++) if (hasClass(children[i],"title")) { theHandle=children[i]; break; }\n if (!theHandle) return theNote;\n\n Drag.init(theHandle, theNote, 0, 0, null, null);\n theHandle.style.cursor="move";\n theHandle.title="drag title to re-arrange notes"\n theNote.onDrag = function(x,y,myElem) {\n if (this.style.position!="relative")\n { this.savedstyle=this.style.position; this.style.position="relative"; }\n y = myElem.offsetTop;\n var next = myElem.nextSibling;\n var prev = myElem.previousSibling;\n if (next && y + myElem.offsetHeight > next.offsetTop + next.offsetHeight/2) { \n myElem.parentNode.removeChild(myElem);\n next.parentNode.insertBefore(myElem, next.nextSibling);//elems[pos+1]);\n myElem.style["top"] = -next.offsetHeight/2+"px";\n }\n if (prev && y < prev.offsetTop + prev.offsetHeight/2) { \n myElem.parentNode.removeChild(myElem);\n prev.parentNode.insertBefore(myElem, prev);\n myElem.style["top"] = prev.offsetHeight/2+"px";\n }\n };\n theNote.onDragEnd = function(x,y,myElem) {\n myElem.style["top"] = "0px";\n if (this.savedstyle!=undefined)\n this.style.position=this.savedstyle;\n }\n return theNote;\n}\n\n/**************************************************\n * dom-drag.js\n * 09.25.2001\n * www.youngpup.net\n **************************************************\n * 10.28.2001 - fixed minor bug where events\n * sometimes fired off the handle, not the root.\n **************************************************/\n\nvar Drag = {\n obj:null,\n\n init:\n function(o, oRoot, minX, maxX, minY, maxY) {\n o.onmousedown = Drag.start;\n o.root = oRoot && oRoot != null ? oRoot : o ;\n if (isNaN(parseInt(o.root.style.left))) o.root.style.left="0px";\n if (isNaN(parseInt(o.root.style.top))) o.root.style.top="0px";\n o.minX = typeof minX != 'undefined' ? minX : null;\n o.minY = typeof minY != 'undefined' ? minY : null;\n o.maxX = typeof maxX != 'undefined' ? maxX : null;\n o.maxY = typeof maxY != 'undefined' ? maxY : null;\n o.root.onDragStart = new Function();\n o.root.onDragEnd = new Function();\n o.root.onDrag = new Function();\n },\n\n start:\n function(e) {\n var o = Drag.obj = this;\n e = Drag.fixE(e);\n var y = parseInt(o.root.style.top);\n var x = parseInt(o.root.style.left);\n o.root.onDragStart(x, y, Drag.obj.root);\n o.lastMouseX = e.clientX;\n o.lastMouseY = e.clientY;\n if (o.minX != null) o.minMouseX = e.clientX - x + o.minX;\n if (o.maxX != null) o.maxMouseX = o.minMouseX + o.maxX - o.minX;\n if (o.minY != null) o.minMouseY = e.clientY - y + o.minY;\n if (o.maxY != null) o.maxMouseY = o.minMouseY + o.maxY - o.minY;\n document.onmousemove = Drag.drag;\n document.onmouseup = Drag.end;\n Drag.obj.root.style["z-index"] = "10";\n return false;\n },\n\n drag:\n function(e) {\n e = Drag.fixE(e);\n var o = Drag.obj;\n var ey = e.clientY;\n var ex = e.clientX;\n var y = parseInt(o.root.style.top);\n var x = parseInt(o.root.style.left);\n var nx, ny;\n if (o.minX != null) ex = Math.max(ex, o.minMouseX);\n if (o.maxX != null) ex = Math.min(ex, o.maxMouseX);\n if (o.minY != null) ey = Math.max(ey, o.minMouseY);\n if (o.maxY != null) ey = Math.min(ey, o.maxMouseY);\n nx = x + (ex - o.lastMouseX);\n ny = y + (ey - o.lastMouseY);\n Drag.obj.root.style["left"] = nx + "px";\n Drag.obj.root.style["top"] = ny + "px";\n Drag.obj.lastMouseX = ex;\n Drag.obj.lastMouseY = ey;\n Drag.obj.root.onDrag(nx, ny, Drag.obj.root);\n return false;\n },\n\n end:\n function() {\n document.onmousemove = null;\n document.onmouseup = null;\n Drag.obj.root.style["z-index"] = "0";\n Drag.obj.root.onDragEnd(parseInt(Drag.obj.root.style["left"]), parseInt(Drag.obj.root.style["top"]), Drag.obj.root);\n Drag.obj = null;\n },\n\n fixE:\n function(e) {\n if (typeof e == 'undefined') e = window.event;\n if (typeof e.layerX == 'undefined') e.layerX = e.offsetX;\n if (typeof e.layerY == 'undefined') e.layerY = e.offsetY;\n return e;\n }\n};\n//}}}\n
//''Shows DefaultNotes + most recently modified notes as default when any TiddlyWiki or adaptation is first loaded.''//\n//To use, copy this note's contents to a new note on your site and tag it "systemConfig".//\n\n{{{\nvar num = 3;\nvar ignore_tags = ['systemConfig', 'systemNotes', 'plugin', 'system'];\n\nfunction in_array(item, arr){for(var i=0;i<arr.length;i++)if(item==arr[i])return true};\nfunction get_parent(note){while(note && in_array('comments', note.tags)) note=store.fetchNote(note.tags[0]);return note};\nfunction unique_list(list){var l=[];for(i=0;i<list.length;i++)if(!in_array(list[i], l))l.push(list[i]);return l};\nfunction get_recent_notes(){\n var notes = store.getNotes('modified');\n var names = store.getNoteText("DefaultNotes").readBracketedList();\n var ignore_notes = [];\n for(var i=0; i<ignore_tags.length; i++)\n ignore_notes=ignore_notes.concat(store.getTaggedNotes(ignore_tags[i]));\n for(var i=notes.length-1; i>=0; i--) {\n if(in_array('comments', notes[i].tags)) {\n var t = get_parent(notes[i]);\n if(t)names.push(t.title)\n }\n else if(!in_array(notes[i], ignore_notes))\n names.push(notes[i].title);\n }\n return unique_list(names).slice(0, num);\n}\nvar names = get_recent_notes();\n_restart = restart\nrestart = function() {\n if(window.location.hash) _restart();\n else story.displayNotes(null,names);\n}\n}}}
/***\n''Name:'' ReferencesPlugin\n''Author:'' Garrett Lisi\n''Description:'' Places a comma separated list of referring notes at the bottom of each note -- replacing the "references" command bar button.\n''Installation:'' Copy this note, change the [[StyleSheet]] to set the references class style, and add a line in the [[ViewTemplate]].\n\n''Code:''\n***/\n/*{{{*/\nconfig.macros.references = {};\nconfig.macros.references.handler = function(place,macroName,params,wikifier,paramString,note)\n{\n var references = store.getReferringNotes(note.title);\n if(references.length>0)\n {\n// createTiddlyText(place,"\sxAB "); \n createTiddlyLink(place,references[0].title,true);\n }\n for(var r=1; r<references.length; r++)\n if(references[r].title != note.title)\n {\n createTiddlyText(place,", ");\n createTiddlyLink(place,references[r].title,true);\n }\n}\n/*}}}*/
(AKA Propertime representation)\nThe basic observation of Schwinger was\n\s[ \sfrac{1}{A} = \sint_0^\sinfty\s!\s! \srmd s \srme^{-s A}=\srmi\sint_0^\sinfty\s!\s!\srmd s\srme^{-\srmi s(A-\srmi\s,0)} \s]\nBy taking derivatives this can then be extended for all $n\sin\sdsN$ (and then analytically continued) as\n\s[ \sfrac{\sG(n+1)}{A^{n+1}} = \sint_0^\sinfty\s!\s! \srmd s\s,s^n \srme^{-s A} =\smathcal{L}\s{s^n\s}\s]\nWhere we recognise the integral is just a [[Laplace transform|http://en.wikipedia.org/wiki/Laplace_transform]].\nThe general expression looks like\n\s[ \sfrac{1}{A_1^{\sa_1}} \scdots \sfrac{1}{A_n^{\sa_n}} \n =\sfrac{1}{\sprod\sG(\sa_i)} \sint_0^\sinfty\s!\s! \srmd^ns \s, \srme^{-\ssum s_i A_i}\sprod s_i^{\sa_i-1}\n\s]\n\nNote1: For terms like $\slog(A)$ the trick becomes\n\s[ \slog{\sfrac{A}{A_0}} =- \sint_0^\sinfty\s!\s! \sfrac{\srmd s}{s}\slp \srme^{-s A} -\srme^{-s A_0} \srp \n =-\sint_0^\sinfty\s!\s! \sfrac{\srmd s}{s}\slp \srme^{-\srmi s (A-\srmi\s, 0)} -\srme^{-\srmi s (A_0-\srmi\s, 0)} \srp\n\s]\nNote2: Numerators can be included by using the generating function approach.\n\nTo connect the Schwinger parametrisation to [[Feynman's|Feynman Parameters]] we introduce $1=\sint_0^\sinfty\srmd\sl\sd(\sl-\ssum s_i)$ and make the change of variable $s_i=\sl x_i$ to get\n\s[ \sfrac{1}{A_1^{\sa_1}} \scdots \sfrac{1}{A_n^{\sa_n}} \n=\sint_0^\sinfty\sfrac{\srmd\sl}{\sl^{1-\ssum a_i} } \sint_0^\sinfty\s!\s! \srmd^nx \s,\sd(1-\ssum x_i) \srme^{-\sl\ssum x_i A_i}\sprod \sfrac{x_i^{\sa_i-1}}{\sG(\sa_i)}\n\s]\nNow rescale $\sl$ and use the definition of the [[Gamma|Gamma and related functions]] function to perform the $\sl$ integral, \n\s[ \sfrac{1}{A_1^{\sa_1}} \scdots \sfrac{1}{A_n^{\sa_n}} \n=\sfrac{\sG(\ssum a_i)}{\sprod \sG(\sa_i)} \sint_0^1\s!\s! \srmd^nx \s,\sd(1-\ssum x_i) \sfrac{\sprod x_i^{\sa_i-1}}{(\ssum x_i A_i)^{\ssum a_i}}\n\s]\nwhere the $x$ integration is limited by the $\sd$ function. This is exactly the expression found in [[Feynman Parameters]].\n\nWe get get a representation that is half-half by restricting the $\sd$ function to a subset of the denominators, explicitly (ordering such that the unmodified integrals come last) we have\n\s[ \sfrac{1}{A_1^{\sa_1}} \scdots \sfrac{1}{A_n^{\sa_n}} \n=\sfrac{\sG(\ssum_{i=1}^n a_i)}{\sprod_{i=1}^n \sG(\sa_i)} \n\sint_0^\sinfty\s!\s! \sprod_{i=\sn+1}^n \s! \srmd x_i \sint_0^1\s!\s! \sprod_{i=1}^\sn \srmd x_i\n \s,\sd(1-\ssum_{i=1}^\sn x_i) \sfrac{\sprod_{i=1}^n x_i^{\sa_i-1}}{(\ssum_{i=1}^n x_i A_i)^{\ssum_{i=1}^n a_i}}\n\s]\n[[Smirnov|http://books.google.com/books?id=oB81LbnamTgC&pg=PA42&lpg=PA42&dq=%22cheng+wu%22+theorem&source=web&ots=1QQ9xcIrto&sig=bwpwg0LiBoZvcnfCNoc3DPs8d_8]] calls this a "folklore Cheng-Wu theorem".\n\nIf we now use the above in a simpler case we obtain the corrected version of [[Smirnov (3.36)|http://books.google.com/books?id=oB81LbnamTgC]],\n\s[ \sfrac1{A^n B^m}=\sfrac{\sG(n+m)}{\sG(n)\sG(m)}\sint_0^\sinfty\srmd x\sint_0^1\srmd y\sd(1-y)\n\sfrac{x^{n-1}y^{m-1}}{(xA+yB)^{n+m}}\n=\sfrac{\sG(n+m)}{\sG(n)\sG(m)}\sint_0^\sinfty\srmd x\sfrac{x^{n-1}}{(xA+B)^{n+m}}\n\s]\nthis can be directly integrated using the change of variables $y=xB/A$ and the second integral form of the beta function given [[here|Gamma and related functions]].\n\ngive examples
<<list shadowed>>
<<search>><<newNote>><<permaview>><<collapseAll>><<closeAll>><<slider chkSliderOptionsPanel OptionsPanel '»' 'Change options'>>
username : <<option txtUserName>>
<<tabs txtMainTab Contents 'Hierarchy of tags and content' TabContents Latest 'Recently modified notes' TabTimeline Tags 'List all tags' TabTags All 'List all notes' TabAll>>
I am best summarised by:\n<html> \n<a href="http://www.nerdtests.com/nt2ref.html">\n<img src="http://www.nerdtests.com/images/badge/nt2/d4016ebf59b33ffd.png" alt="NerdTests.com says I'm a Cool High Nerd. What are you? Click here!">\n</a>\n</html>\n\n\s:)
some notes on my research
Collection of useless thoughts
http://www.physics.uwa.edu.au/~styler/
/***\nThe StyleSheet holds the custom style modifications. It augments and overrides the hidden default pages.\n***/\n/*{{{*/\nbody {background:#ddd;\n position: static;\n font-size: .75em;\n font-family: arial,helvetica;}\n.header {height: 5.8em;}\n.headerShadow {\n padding: 2em 1em 1em 20em;\n left: 1px;\n top: 1px;}\n.headerForeground {\n padding: 2em 1em 1em 20em;\n color: #ddd}\n.siteTitle {font-size: 2.5em;}\n.siteSubtitle {font-size: 1em;}\n#displayControl {\n position:absolute;\n top:0;\n left:0;\n margin:0em;\n padding: 0em;\n background: #ddd;}\n#displayControl a{text-decoration: none; font-size: 1em; color: #04b;}\n#sidebar {\n font-size: 1em;\n left: 0em;\n width: 19em;\n overflow: hidden;}\n#mainMenu {\n position: relative;\n width: 17em;\n margin: .5em 0em 0em .5em;\n padding: 0.3em .5em 0.3em .5em;\n text-align: left;\n line-height: 1.4em;\n font-size: 1em;\n border-bottom:1px solid #bbb;\n border-right:1px solid #bbb;\n background:#eee;}\n#sidebarOptions {\n width: 17.5em;\n margin: .5em .5em 0em .5em;\n padding: 0.3em 0em 0.3em 0.4em;\n overflow: hidden;\n font-size: 1em;\n border-bottom:1px solid #bbb;\n border-right:1px solid #bbb;\n background:#eee;}\n#sidebarOptions a {\n margin: 0em 0em;\n padding: 0.15em .3em;\n font-size: 1.1em;\n display: inline;}\n#sidebarOptions input {\n margin: 0em 0.1em;\n width: 7.5em;\n font-size: 1em;\n padding: 0.2em 0.2em;\n border: 1px solid #888;}\n#sidebarOptions .button {border: 1px solid #eee;}\n#sidebarOptions .button:hover {\n color: #014;\n background: #fe8;\n border: 1px solid #db4;}\n#sidebarOptions .sliderPanel {\n margin-top: .5em;\n margin-right: 1em;\n margin-bottom: .5em;\n margin-left: .5em;\n font-size: 1em;\n padding: 1em;\n background: #aaa;}\n#sidebarOptions .sliderPanel a {\n display: inline;}\n#sidebarOptions .sliderPanel input {\n width: 1em;\n font-size: 1em;}\n#sidebar .inputFix input{width: 8em;}\n#sidebarTabs {\n width: 16em;\n margin: .5em .5em 1em .5em;\n font-size: 1em;}\n#sidebarTabs .tabset {padding: .3em 0em 0em 0em;}\n#sidebarTabs .tab {margin: .5em 0em 0em 0.5em;\n padding: 2px;\n border-top: 1px solid #ddd;\n border-left: 1px solid #ddd;\n border-right:1px solid #bbb;}\n#sidebarTabs .tabContents {\n width: 18em;\n margin: 0em 0em 0em 0em;\n padding: .25em 0em .25em 0em;\n border-top: none;\n border-left: none;\n border-bottom:1px solid #bbb;\n border-right:1px solid #bbb;\n overflow: hidden;}\n#sidebarTabs .tabContents ul{\n margin-left: .65em;\n overflow: hidden;}\n#messageArea {\n position:absolute;\n top:0;\n left:50em;\n right: 10em;\n margin:0.3em;\n padding: 0.4em;\n border: 1px solid #999;\n background: #ddd;\n color: #014;}\n#messageArea a{text-decoration: none; font-size: .75em; }\n*[id='messageArea'] {position:absolute !important; z-index:99;}\n.messageToolbar {padding: 0em;}\n#messageArea .button {\n border: 1px solid #555;\n padding: 0.1em 0.1em 0.1em 0.1em;\n color: #014;\n background: #aaa;}\n.popup {\n background: #fe8;\n border: 1px solid #db4;}\n.popup hr {\n color: #db4;\n background: #db4;\n border-bottom: 1px;}\n.popup li.disabled {\n color: #db4;}\n.popup li a, .popup li a:visited {\n color: #04b;\n border: none;}\n.popup li a:hover {\n background: #04b;\n color: #fe8;\n border: none;}\n#displayArea {\n margin-left: 18.5em;\n margin-right: 0em;\n margin-top: 0em;\n padding-top: 0em;\n padding-bottom: .25em;}\n.toolbar {\n float: right;\n font-size: 1em;}\n.note {\n border-bottom: 1px solid #bbb;\n border-right: 1px solid #bbb;\n margin: .5em .5em .5em .75em;\n padding-bottom: .5em;\n padding-top: .75em;\n background-color: white;}\n.shadow .title {\n color: #000; background: #fff;}\n.title {\n font-size: 1.5em;\n background: #fff;\n color: #000;}\nh1,h2,h3,h4,h5 {\n padding-left: 0em;\n padding-top: 0em;\n padding-bottom: 0em;\n margin-top: .5em;\n margin-bottom: .1em;\n color: #000;\n background: transparent;}\nh1 {font-size: 1.2em;}\nh2 {font-size: 1.15em;}\nh3 {font-size: 1.1em;}\nh4 {font-size: 1.05em;}\nh5 {font-size: 1em;}\n.subtitle {\n font-size: .8em;\n padding-right: 1em;\n padding-left: 3em;\n padding-top: 0em;\n margin-bottom: 0.25em;\n color: #888;}\n.tagged {margin: 0em 0em .25em 1.25em;}\n.selected .tagging, .selected .tagged {\n background-color: #eee;\n border: 1px solid #ccc;}\n.tagging .button, .tagged .button {color: #666;}\n.tagClear{margin-top: 0.1em;clear:both;}\n/* ie fix? */\n.viewer {\n line-height: 120%;\n font-size: 1.25em;}\n.viewer .listTitle {list-style-type:none; margin-left:-2em; background-color:white;}\n.viewer ul, .viewer ol {\n margin-left: 0.5em;\n margin-top: 0em;\n margin-bottom: 0em;\n padding-left: 1.5em;}\n.viewer pre {overflow: visible;}\n.viewer hr {\n border: 0;\n border-top: solid 1px #666;}\n.viewer table {\n text-align: center;\n margin-left: auto;\n margin-right: auto;}\n.viewer caption {\n text-align: center;\n margin-left: auto;\n margin-right: auto;}\n.editorFooter .button {padding-top: 0px; padding-bottom: 0px;}\n.references {\n font-size: 1em;\n text-align: center;\n color: #666;\n margin-top: .75em;\n padding: .25em;\n border: 1px solid #eee;\n background-color: #eee;}\n.selected .references {\n background-color: #eee;\n border: 1px solid #ccc;}\n.references .tiddlyLinkExisting {font-weight: normal;}\n/*}}}*/
@media print {\n#mainMenu, #sidebar, #messageArea, #displayControl {display: none ! important;}\n#displayArea {margin: 1em 1em 0em 1em;}\n}
!Header 1\n!!Header 2\n!!!Header 3\n!!!!Header 4\n!!!!!Header 5\n{{{\n!Header 1\n!!Header 2\n!!!Header 3\n!!!!Header 4\n!!!!!Header 5\n}}}\nA carriage return ends a paragraph, so a slightly larger (this should be made bigger) space appears between paragraphs. But the beginning of a new paragraph is not indented, even if tabs or spaces are inserted. So, for now, use extra carriage returns to separate paragraphs.\n\nAnd maybe implement tab indentation in the future.
The old symbol definitions of Garett....\nI've changed pretty much all of them, and to save space on the stingy physics webserver, I have note included the image based fonts. (They aren't that big, but many small files take up a lot of disk space!)\n\n|!Symbol|![[LaTeX]]|!Use|\n| ${\sbbold R} \s;\s; {\sbbold C} \s;\s; n \s;\s; \sud{a}$ | {{{ {\sbbold R} {\sbbold C} n \sud{a} }}} |[[real numbers|http://en.wikipedia.org/wiki/Real_numbers]], complex numbers, dimension, [[Grassmann number]] |\n| $M \s; \s; T_p M \s; \s; T_p^* M$ | {{{M T_p M T_p^* M }}} |[[manifold]], [[tangent space to M at point p|coordinate basis vectors]], [[cotangent space to M at point p|coordinate basis 1-forms]] |\n| $x^i \s; \s; \sve{\spa_i} \s; \s; \sve{v} \s; \s; \svv{l}$ | {{{x^i \sve{\spa_i} \sve{v} \svv{l} }}} |[[coordinates|manifold]] and [[coordinate basis vectors]] with coordinate [[indices]], [[tangent vector]], [[loop|vector-form algebra]] |\n| $t \s; \s; \sta$ | {{{ t \sta }}} |parameter time, [[proper time]] |\n| $\sf{dx^i} \s; \s; \sf{f} \s; \s; \sff{a} \s; \s; \sub{b}$ | {{{\sf{dx^i} \sf{f} \sff{a} \sub{b} }}} |[[coordinate basis 1-forms]], [[1-form]], [[2-form|differential form]], [[differential form]] of high or unspecified form grade |\n| $\spa_i \s;\s; \sf{\spa} \s;\s; \sf{d}$ | {{{\spa_i \sf{\spa} \sf{d} }}} |[[partial derivative]], partial derivative, [[exterior derivative]] |\n| $\sph \s; \s; \sph^* \s; \s; \sph_*$ | {{{\sph \sph^* \sph_* }}} |[[diffeomorphism]], [[pullback]], pushforward |\n| ${\scal L}_{\sve{v}} \s;\s; \slb\sve{v},\sve{u}\srb_L \s;\s; \sve{\sDe}$ | {{{{\scal L}_{\sve{v}} \slb\sve{v},\sve{u}\srb_L \sve{\sDe} }}} |[[Lie derivative]], [[Lie bracket|Lie derivative]] of two [[vector fields|tangent bundle]], [[distribution]] |\n| $\sub{\sve{A}} \s;\s; {\scal L}_{\sub{\sve{K}}} \s;\s; \slb\sub{\sve{K}},\sub{\sve{L}}\srb_L \s;\s; \sf{\sve{P}}$ | {{{ \sf{\sve{A}} {\scal L}_{ \sub{\sve{K}} } \slb\sub{\sve{K}},\sub{\sve{L}}\srb_L \sf{\sve{P}} }}} |[[vector valued form]], [[FuN derivative]], FuN bracket, [[vector projection]] |\n| $\sf{\sve{\scal A}} \s;\s; \sff{\sve{\scal F}} \s;\s; \sf{\scal D}$ | {{{ \sf{\sve{\scal A}} \sff{\sve{\scal F}} \sf{\scal D} }}} |[[Ehresmann connection]], [[FuN curvature]], [[Ehresmann covariant derivative]] |\n| $\sde_i^j \s;\s; \set_{\sal \sbe} \s; \s; \sep_{\sal \sdots \sbe} \s; \s; \sotimes$ | {{{ \sde_i^j \set_{\sal \sbe} \sep_{\sal \sdots \sbe} \sotimes }}} |[[Kronecker delta|http://en.wikipedia.org/wiki/Kronecker_delta]], [[Minkowski metric]], [[permutation symbol]], [[Kronecker product]] |\n| $G \s;\s; g^- \s;\s; T_A \s;\s; \slb{T_A,T_B}\srb$ | {{{G g^- T_A \slb{T_A,T_B}\srb }}} |[[Lie group]], [[inverse]] of a group element, [[Lie algebra]] generators, [[commutator]] bracket |\n| $\sf{\sna} \s;\s; \sf{A} \s;\s; \sff{F}$ | {{{\sf{\sna} \sf{A} \sff{F} }}} |[[covariant derivative]], [[connection]], [[curvature]] |\n| $\sf{\scal I} \s;\s; \sf{\sve{\scal I}} \s;\s; \sve{\sxi^L_A} \s;\s; \sve{\sxi^R_A}$ | {{{\sf{\scal I} \sf{\sve{\scal I}} \sve{\sxi^L_A} \sve{\sxi^R_A} }}} |[[Maurer-Cartan form]], Ehresmann-Maurer-Cartan form, [[left and right action vector fields|Lie group geometry]] |\n| $Cl \s; \s; Cl^*$ | {{{Cl Cl^* }}} |[[Clifford algebra]], [[Clifford group]] |\n| $\sga_\sal \s; \s; \sga_{\sal \sdots \sbe} \s; \s; \sga$ | {{{\sga_\sal \sga_{\sal \sdots \sbe} \sga }}} |[[Clifford basis vectors]], [[Clifford basis elements]], Clifford [[pseudoscalar]] |\n| $\shat{A} \s; \s; \stilde{A} \s; \s; \sbar{A} \s; \s; A^\sdagger \s; \s; \soverline{A}$ | {{{\shat{ \stilde{ \sbar{ \soverline{A}^\sdagger } } } }}} |[[Clifford involution, reverse, conjugate, Hermitian conjugate, Dirac conjugate|Clifford conjugate]] |\n| $\scdot \s; \s; \stimes$ | {{{\scdot \stimes }}} |symmetric and antisymmetric [[Clifford algebra]] product |\n| $\slb{A,\sdots,B}\srb_A \s;\s; a_{\slb{\sal\sdots\sbe}\srb}$ | {{{ \slb{A,\sdots,B}\srb_A a_{\slb{\sal\sdots\sbe}\srb} }}} |[[antisymmetric bracket]], [[index bracket]] |\n| $\sli{A}\sri_q \s; \s; \sli{A}\sri$ | {{{ \sli{A}\sri_q \sli{A}\sri }}} |[[Clifford grade]] $q$ part, [[scalar part|Clifford grade]] |\n| $\sf{A} \s; \s; \sff{b} \s; \s; \sve{e}$ | {{{ \sf{A} \sff{b} \sve{e} }}} |[[Lieform]]s or [[Clifform]]s |\n| $\slp{e_i}\srp^\sal \s;\s; \slp{e_\sal}\srp^i \s;\s; g_{ij} \s;\s; \slp\sve{u},\sve{v}\srp$ | {{{ \slp{e_i}\srp^\sal \slp{e_\sal}\srp^i g_{ij} \slp\sve{u},\sve{v}\srp }}} |co[[frame]] matrix, frame matrix, [[metric]], scalar product |\n| $\sf{e^\sal} \s;\s; \sve{e_\sal}$ | {{{ \sf{e^\sal} \sve{e_\sal} }}} |co[[frame]] 1-forms, orthonormal basis vectors |\n| $\sf{e} \s;\s; \sve{e}$ | {{{ \sf{e} \slp{e_i}\srp^\sal \sve{e} \slp{e_\sal}\srp^i g_{ij} }}} |co[[frame]], frame |\n| $\sub{e} \s;\s; \sll{e}\srl$ | {{{ \sub{e} \sll{e}\srl }}} |[[volume form]], frame [[determinant]] |\n| $\sf{e^s} \s;\s; \slp{e^s_i}\srp^\sal \s;\s; s$ | {{{ \sf{e^s} \s;\s; \slp{e^s_i}\srp^\sal \s;\s; s }}} |[[special frame]], special coframe matrix, conformal scalar |\n| $TM \s;\s; T^*M$ | {{{ TM T^*M }}} |[[tangent bundle]], [[cotangent bundle]] |\n| $\sGa^k{}_{ij} \s;\s; \sf{\sGa}^k{}_j \s;\s; \sff{R}^k{}_j$ | {{{ \sGa^k{}_{ij} \sf{\sGa}^k{}_j \sff{R}^k{}_j }}} |[[Christoffel symbols]], [[tangent bundle connection]], [[Riemann curvature]] |\n| $\sf{R}{}_j \s;\s; R$ | {{{ \sf{R}{}_j R }}} |[[Ricci curvature]], [[curvature scalar]] |\n| $L^\sbe{}_\sal \s;\s; \sf{w}^\sbe{}_\sal \s;\s; \sff{F}^\sbe{}_\sal$ | {{{ L^\sbe{}_\sal \sf{w}^\sbe{}_\sal \sff{F}^\sbe{}_\sal }}} |[[Lorentz rotation]], [[tangent bundle spin connection|tangent bundle connection]], [[Riemann curvature]] |\n| $ClM \s;\s; Cl^1M$ | {{{ ClM Cl^1M }}} |[[Clifford bundle]], [[Clifford vector bundle]] |\n| $\sf{A} \s;\s; \sf{\som} \s;\s; \sff{R}$ | {{{ \sf{A} \sf{\som} \sff{R} }}} |[[Clifford connection]], [[spin connection]], [[Clifford-Riemann curvature]] |\n| $\sf{R} \s;\s; R$ | {{{ \sf{R} R }}} |[[Clifford-Ricci curvature]], [[Clifford curvature scalar]] |\n| $\sff{T} \s;\s; \sf{\ska}$ | {{{ \sff{T} \sf{\ska} }}} |[[torsion]], contorsion |\n| $\sud{C} \s;\s; \sub{\sod{B}} \s;\s; \sudf{A} \s;\s; \sudf{\sudf{F}}$ | {{{ \sud{C} \sub{\sod{B}} \sudf{A} \sudf{\sudf{F}} }}} |[[BRST|BRST technique]] ghost, anti-ghost, extended connection, curvature |
/***\nThe new SystemConfig feature allows arbitrary JavaScript code to be executed at startup from any note that is tagged with 'systemConfig', one of the new SpecialTags.\n***/\n/*{{{*/\nconfig.messages.messageClose.text = "\sxD7";\nconfig.messages.messageClose.tooltip = "Close this message";\nconfig.views.wikified.defaultText = "";\nconfig.views.wikified.tag.labelTags = "";\nconfig.views.wikified.tag.openTag = "";\nconfig.views.wikified.tag.tooltip = "Show other notes tagged with '%0'";\nconfig.views.editor.tagPrompt = "separated by spaces, using [[double square brackets]] if necessary";\n\nconfig.commands.closeNote.text = "\sxD7";\nconfig.commands.closeNote.tooltip = "Close this note";\nconfig.commands.closeOthers.text = "\sxA4";\nconfig.commands.closeOthers.tooltip = "Close all others";\nconfig.commands.jump.text = "\sxBB";\nconfig.commands.jump.tooltip = "Jump to another open note";\nconfig.commands.editNote.text = "+";\nconfig.commands.editNote.readOnlyText = " ";\nconfig.commands.editNote.tooltip = "Edit this note (double click)";\nconfig.commands.references.text = "\sxA5";\nconfig.commands.references.tooltip = "Notes that link to this one";\nconfig.commands.saveNote.text = "\sxA7";\nconfig.commands.saveNote.readOnlyText = ".";\nconfig.commands.saveNote.tooltip = "Save changes";\nconfig.commands.cancelNote.text = "\sxA2";\nconfig.commands.cancelNote.tooltip = "Cancel changes";\nconfig.commands.deleteNote.text = "\sxD8";\nconfig.commands.deleteNote.tooltip = "Delete this note";\n\nconfig.macros.timeline.dateFormat = "MMM DD, YYYY";\nconfig.macros.search.prompt = "Search this site";\nconfig.macros.search.successMsg = "%0 notes found matching %1";\nconfig.macros.search.failureMsg = "No notes found matching %0";\nconfig.macros.newNote.prompt = "Create a new note";\nconfig.macros.newNote.label = "+";\nconfig.macros.saveChanges.label = "Save";\nconfig.macros.saveChanges.prompt = "Save this whole wiki as an html file";\nconfig.macros.newJournal.prompt = "Create a new note from the current date and time";\nconfig.macros.permaview.label = "\sxA5";\nconfig.macros.permaview.prompt = "Make a URL showing all currently displayed notes";\nconfig.macros.closeAll.label = "\sxD7";\nconfig.macros.permaview.label = "\sxA5";\nconfig.macros.search.label = "\sxA7";\n/*}}}*/
*<<slider chkSliderphysicsF physicsF 'physics »' 'physics stuff'>>\n*<<slider chkSlidermathF mathF 'math »' 'math stuff'>>\n*<<slider chkSlidercomputerF computerF 'comp »' 'computer stuff'>>\n*<<slider chkSlidermetaF metaF 'meta »' 'describe the operation of this site'>>\n<<ListTagged none>>
<<list all>>
Existing [[Tags]] and note population:\n<<AllTagsExcept excludeLists folder plugin system systemConfig systemNotes template>>
<<timeline>>
sample table:\n|!th1111111111|!th2222222222|\n|>| colspan |\n| rowspan |left|\n|~| right|\n|bgcolor(#a0ffa0):colored| center |\n|caption|c\n{{{\nsample table:\n|!th1111111111|!th2222222222|\n|>| colspan |\n| rowspan |left|\n|~| right|\n|bgcolor(#a0ffa0):colored| center |\n|caption|c\n}}}
<<tabs txtFavourite\nUrgent "Priority 1" ToDo1\nImportant "Priority 2" ToDo2\n"Need To Do" "Priority 3" ToDo3\n>>\n{{{\n<<tabs txtFavourite\nUrgent "Priority 1" ToDo1\nImportant "Priority 2" ToDo2\n"Need To Do" "Priority 3" ToDo3\n>>\n}}}
*<<tag physics>>- physics stuff\n**<<tag qft>>- notes about and for quantum field theory\n***<<tag loops>>- notes about evaluating loop integrals\n***<<tag susy>>- supersymmetry\n*<<tag math>>- mathematics\n**<<tag group>>- group theory\n***<<tag su(n)>>- special unitary group\n*<<tag computer>>- notes about setting up and running various computer programs\n**<<tag latex>>- notes about LaTeX\n**<<tag mathematica>>- notes about Mathematica\n*<<tag meta>>- describes the operation of this site\n**<<tag editing>>- tips on editing and authoring notes, including all sorts of tools\n**<<tag 0>>- a note that's linked to but is empty or needs editing\n**<<tag system>>- control how the site operates and is layed out\n***<<tag systemConfig>>- this is a special tag, marking notes with code to be loaded at startup\n***<<tag xsystemConfig>>- deactivated code\n***<<tag systemNotes>>- system control notes loaded at startup, used to control content\n***<<tag plugin>>- code snippets enhancing functionality, and containing descriptions of what they do\n***<<tag template>>- custom css page template, used to describe layout\n***<<tag folder>>- a folder is a tag is a note\n***<<tag slider>>- all sliders are here\n**<<tag illus>>- notes containing illustrations\n**<<tag hidden>>- meta type notes that I don't want showing up in the meta folder!\n\nThe tags group collections of notes into sets -- they're adjectives, directories, or folders. By selecting a tag, visible in the upper right of each note, you can jump to any note labeled with that tag -- a navigational convenience. It's efficacious to build a flexible hierarchy of tagged content. Each note should be labeled by its most appropriate, "lowest" leaf tags -- more than one as appropriate. Click on the tag to see a popup menu of notes with that tag. Alternatively, these folders and their contents may be selectively displayed in the "Contents" tab to the left.\n\n
Test some maths and [[jsMath|jsMathPlugin]] shortcuts\n$\sa \sb \sc \sd \se \sf \sg$\n\s[ \spint\sf \srme^{\srmi/\shbar \slp S[\sf]+J\scdot\sf \srp} \stag{tag} \s]\n\s[\shbar ~~ \sfrac\srmi\shbar\s]\n[[Character table for extra blackboard bold font: |http://www.math.union.edu/~dpvc/jsmath/download/extra-fonts/bbold10/bbold10.html]]\nNote, can not be installed on webserver due to lack of space!\nSo will use unicode for\n$\sdsR$ $\sdsC$ $\sdsN$ etc\nand a kludge for\n\s[ {\sLarge 1}x~~{\srm vs}~~ {\sLarge \sdsone}x\s]\n\namsMath??\n\s[\sbegin{split} \siint \s\s \siiint \send{split}\s]\n\n\n\sbegin{align}\n a&=b\s\s\n c&=d\n\send{align}
/***\n''TextAreaPlugin for TiddlyWiki version 2.0''\n^^author: Eric Shulman - ELS Design Studios\nsource: http://www.elsdesign.com/tiddlywiki/#TextAreaPlugin\nlicense: [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]^^\n\nThis plugin 'hijacks' the TW core function, ''Story.prototype.focusNote()'', so it can add special 'keyDown' handlers to adjust several behaviors associated with the textarea control used in the note editor. Specifically, it:\n* Adds text search INSIDE of edit fields.^^\nUse ~CTRL-F for "Find" (prompts for search text), and ~CTRL-G for "Find Next" (uses previous search text)^^\n* Enables TAB characters to be entered into field content^^\n(instead of moving to next field)^^\n* Option to set cursor at top of edit field instead of auto-selecting contents^^\n(see configuration section for checkbox)^^\n!!!!!Configuration\n<<<\n<<option chkDisableAutoSelect>> place cursor at start of textarea instead of pre-selecting content\n<<option chkTextAreaExtensions>> add control-f (find), control-g (find again) and allow TABs as input in textarea\n<<<\n!!!!!Installation\n<<<\nImport (or copy/paste) the following notes into your document:\n''TextAreaPlugin'' (tagged with <<tag systemConfig>>)\n<<<\n!!!!!Revision History\n<<<\n''2006.01.22 [1.0.1]''\nonly add extra key processing for TEXTAREA elements (not other edit fields).\nadded option to enable/disable textarea keydown extensions (default is "standard keys" only)\n''2006.01.22 [1.0.0]''\nMoved from temporary "System Tweaks" note into 'real' TextAreaPlugin note.\n<<<\n!!!!!Code\n***/\n//{{{\nversion.extensions.textAreaPlugin= {major: 1, minor: 0, revision: 1, date: new Date(2006,1,23)};\n//}}}\n\n//{{{\nif (!config.options.chkDisableAutoSelect) config.options.chkDisableAutoSelect=false; // default to standard action\nif (!config.options.chkTextAreaExtensions) config.options.chkTextAreaExtensions=false; // default to standard action\n\n// Focus a specified note. Attempts to focus the specified field, otherwise the first edit field it finds\nStory.prototype.focusNote = function(title,field)\n{\n var note = document.getElementById(this.idPrefix + title);\n if(note != null)\n {\n var children = note.getElementsByTagName("*")\n var e = null;\n for (var t=0; t<children.length; t++)\n {\n var c = children[t];\n if(c.tagName.toLowerCase() == "input" || c.tagName.toLowerCase() == "textarea")\n {\n if(!e)\n e = c;\n if(c.getAttribute("edit") == field)\n e = c;\n }\n }\n if(e)\n {\n e.focus();\n e.select(); // select entire contents\n\n // TWEAK: add TAB and "find" key handlers\n if (config.options.chkTextAreaExtensions) // add extra key handlers\n addKeyDownHandlers(e);\n\n // TWEAK: option to NOT autoselect contents\n if (config.options.chkDisableAutoSelect) // set cursor to start of field content\n if (e.setSelectionRange) e.setSelectionRange(0,0); // for FF\n else if (e.createTextRange) { var r=e.createTextRange(); r.collapse(true); r.select(); } // for IE\n\n }\n }\n}\n//}}}\n\n//{{{\nfunction addKeyDownHandlers(e)\n{\n // exit if not textarea or element doesn't allow selections\n if (e.tagName.toLowerCase()!="textarea" || !e.setSelectionRange) return;\n\n // utility function: exits keydown handler and prevents browser from processing the keystroke\n var processed=function(ev) { ev.cancelBubble=true; if (ev.stopPropagation) ev.stopPropagation(); return false; }\n\n // capture keypress in edit field\n e.onkeydown = function(ev) { if (!ev) var ev=window.event;\n\n // process TAB\n if (!ev.shiftKey && ev.keyCode==9) { \n // replace current selection with a TAB character\n var start=e.selectionStart; var end=e.selectionEnd;\n e.value=e.value.substr(0,start)+String.fromCharCode(9)+e.value.substr(end);\n // update insertion point, scroll it into view\n e.setSelectionRange(start+1,start+1);\n var linecount=e.value.split('\sn').length;\n var thisline=e.value.substr(0,e.selectionStart).split('\sn').length-1;\n e.scrollTop=Math.floor((thisline-e.rows/2)*e.scrollHeight/linecount);\n return processed(ev);\n }\n\n // process CTRL-F (find matching text) or CTRL-G (find next match)\n if (ev.ctrlKey && (ev.keyCode==70||ev.keyCode==71)) {\n // if ctrl-f or no previous search, prompt for search text (default to previous text or current selection)... if no search text, exit\n if (ev.keyCode==70||!e.find||!e.find.length)\n { var f=prompt("find:",e.find?e.find:e.value.substring(e.selectionStart,e.selectionEnd)); e.focus(); e.find=f?f:e.find; }\n if (!e.find||!e.find.length) return processed(ev);\n // do case-insensitive match with 'wraparound'... if not found, alert and exit \n var newstart=e.value.toLowerCase().indexOf(e.find.toLowerCase(),e.selectionStart+1);\n if (newstart==-1) newstart=e.value.toLowerCase().indexOf(e.find.toLowerCase());\n if (newstart==-1) { alert("'"+e.find+"' not found"); e.focus(); return processed(ev); }\n // set new selection, scroll it into view, and report line position in status bar\n e.setSelectionRange(newstart,newstart+e.find.length);\n var linecount=e.value.split('\sn').length;\n var thisline=e.value.substr(0,e.selectionStart).split('\sn').length;\n e.scrollTop=Math.floor((thisline-1-e.rows/2)*e.scrollHeight/linecount);\n window.status="line: "+thisline+"/"+linecount;\n return processed(ev);\n }\n }\n}\n//}}}
Wiki:\nblackboard bold??\narXiv links plugin?\n\nNotes:\nGaussian integrals\nFeynman integrals etc\nWess-Zumino Calculation\nand lots more!!
Some of the relevant papers to the calculation of the two-loop vacuum diagram (and non-vacuum diagrams). The first four present the main methods of calculating the diagram. The remaining links are a selection of useful/interesting papers.\n*[[Two Loop Large Higgs Mass Correction to the rho Parameter|http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B231,205]]\n**1984 paper by Veltman and van der Bij, \n***uses ''Homogeneity equation'' and Feynman parameters.\n***Is extended to non-vacuum diagrams in [[Massive two-loop diagrams: The Higgs propagator| http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/9405418]]\n***See the 1985 [[Hoogeveen paper|http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B259,19]] for a couple more details\n*[[The Standard model effective potential at two loops|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/0111190]]\n** 1992 paper by Jack, Jones and Ford\n*** Solves vacuum sunrise integral via ''method of characteristics''\n*** Ford and Jones published a [[paper|http://www.slac.stanford.edu/spires/find/hep/www?j=PHLTA,B274,409]] earlier that year that uses a similar ODE to the one by CCLR below (only in a simpler case)\n*[[The Master differential equations for the two loop sunrise selfmass amplitudes|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/9805118]]\n** 1998 Paper by Caffo, Czyz, Laporta, Remiddi\n*** Solves vacuum sunrise integral via an ''ODE'' (also treats non-vacuum diagrams)\n*** Is based on the "master equations" of [[ Differential equations for Feynman graph amplitudes|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/9711188]] by Remiddi.\n*[[Two loop selfenergy diagrams with different masses and the momentum expansion|http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B397,123]] \n**1993 paper by Davydychev and Tausk\n***Examines momentum expansion of two loop self-energy diagrams and solves the associated vacuum integrals using ''Mellin-Barnes'' representation of the propagators.\n***References diploma work by Scharf that I would like to see\n*[[A Magic connection between massive and massless diagrams|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/9504431]]\n**1995 paper by Davydychev and Tausk\n***Connection between one-loop massless off-shell triangle diagram and two-loop massive vacuum diagram.\n***Also see the [[continuation|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/9511261]]\n*Master integrals\n**[[Broadhurst(1990) The Master Two Loop Diagram With Masses|http://www.slac.stanford.edu/spires/find/hep/www?irn=1786652]]\n**[[Kriemer(1991) The Master two loop two point function: The General case|http://www.slac.stanford.edu/spires/find/hep/www?irn=2438844]]\n*[[Generalized recurrence relations for two loop propagator integrals with arbitrary masses|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/9703319]]\n**1997 paper by Tarasov\n*[[Hypergeometric representation of the two-loop equal mass sunrise diagram|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/0603227]]\n**2006 paper by Tarasov\n***Uses his dimensional recurrence relation for a 'simple' derivation of the result.\n*[[TARCER: A Mathematica program for the reduction of two loop propagator integrals|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/9801383]]\n**1998 implementation of .. by Mertig and Scharf\n*[[TSIL: A Program for the calculation of two-loop self-energy integrals|http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph/0501132]]\n**2005 Martin and Robertson\n----
<<slider chkSlider2LoopRefs 'Two Loop References' 'References for two-loop diagrams »' 'click here'>>\nIn this note we list basic properties about the two-loop vacuum sunset diagram. Sliders and links to other notes contain further information and derivations.\n\nWe will work in Euclidian, $d=4-2\seps$ space-time.\nWe have three independent masses, normally denoted as\n\s[x=m_1^2~,\squad y=m_2^2~,\squad z=m_3^2 ~.\s]\nOur normalisation conventions for momentum space can be summarised as\n\s[\sint\s!\srmd^d\sr\srme^{\srmi\sr\scdot k}=\sd^d(k)~,\n \sqquad \sint\s!\sfrac{\srmd^d k}{(2\sp)^d}\srme^{\srmi\sr\scdot k}=\sd^d(\sr)~.\n\s]\nThe Feynman integral that we want to find is\n\sbegin{align} \nI(x,y,z)&:=\sm^{4\seps}\sint\sfrac{\srmd^dk\srmd^dp}{(2\sp)^{2d}}\n \sfrac{1}{(k^2+x)(p^2+y)((p+k)^2+z)}\n\send{align}\nIn terms of proper-time or [[Schwinger Parameters]] the integral is, [[(derivation)|2LVacSPderiv]],\n\sbegin{align} I(x,y,z)&=\n \sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}\sint_0^\sinfty\s!\s!\n \sfrac{\srmd{s}~\srmd{t}~\srmd{u}}{(st+tu+us)^{2-\seps}}~\srme^{-sx-ty-uz}\s\s\n&=\sfrac{\sm^{4\seps}}{(4\sp)^{2(2-\seps)}}\sG(2\seps-1)\sint_0^\sinfty\s!\s!\n \sfrac{\srmd{s}\srmd{t}\srmd{u}}{(st+tu+us)^{2-\seps}}\n\sfrac{\sg g(s,t,u)^{1-1/\sg}\sd(1-g(s,t,u))}{(sx+ty+uz)^{2\seps-1}}\s\s\n&=\sfrac{\sm^{4\seps}\sG(2\seps-1)}{(4\sp)^{2(2-\seps)}}\sint_0^1\s!\s!\n\sfrac{\srmd\sa\srmd\sb}{(1-\sb)^{1-\seps}}\n\sfrac{(\sb x+\sa(1-\sb)y+(1-\sa)(1-\sb)z)^{1-2\seps}}{(\sb+\sa(1-\sa)(1-\sb))^{2-\seps}}~.\n\send{align}\nwhere we've followed the standard [[procedure|Schwinger Parameters]] of changing the parametrisation and $g$ is homogeneous of order $\sg$.\n\nNote that if we scale the masses we can compensate by rescaling the integrals\n(either in mtm space or proper-time), we find the homogeneity equation\n\s[I(\sl x,\sl y,\sl z)=\sl^{1-2\seps}I(x,y,z) \s]\nwhich is simply a scaling of the [[mass simplex|2LVac_Geom]]. The differential form of this scaling is\n\s[\slabel{homogDE} (1-2\seps-x\spd_x-y\spd_y-z\spd_z)I(x,y,z)=0~.\n\s]\nThe above DE is used as the starting point in the derivation of [[van der Bij and Veltman|http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B231,205]], which is repeated and elaborated on in [[this note|2LVacHomog]]. On top of the homogeneity equation, there is a whole suite of DEs that $I(x,y,z)$ has to satisfy. They are most easily found by using the <<slider chkSlider2LVacIBP '2LVacIBP' 'method of integration by parts. »' 'click here'>>\nThese DE's are then used to solve for $I(x,y,z)$ using the [[method of characteristics|2LVacMOC]] (integrating a one-parameter flow) or by solving a slightly less elegant (i.e. less symmetric) [[ODE|2LVacODE]].\n\nUsing these solutions we may perform the [[epsilon expansion]] to explore the divergent and finite parts of $I(x,y,z)$. (The details are contained in the notes for each method). The final result is\n\sbegin{align} \slabel{homog.eps.tot}\n (4\sp)^4I(x,y,z)&=-\sfrac{c}{2\seps^2}+\sfrac{\shat L_1}{\seps}\n-\shalf\sBigg(c\sleft(\sz(2)+\sfrac52\sright)+2\shat{L}_2+\stilde\sx(x,y,z)\sBigg)~,\n\send{align}\nwhere\n\s[ c=x+y+z~, \squad \n\shat{L}_n:= x\slog^n\sfrac{x}{\shat{\sm}^2}+y\slog^n\sfrac{y}{\shat{\sm}^2}+z\slog^n\sfrac{z}{\shat{\sm}^2}~,\n\s]\n$\shat\sm^2=4\sp\sm^2\srme^{3/2-\sg}$ and\n\sbegin{align}\n \stilde\sx(x,y,z)&=2\sleft[x f(y/x,z/x)+\scycl \sright] &\smbox{ homogeneity method} \s\s\n&=\sx(x,y,z)-\sleft[x\slog(y/x)\slog(z/x)+\scycl\sright] &\smbox{ODE and FlowDE}\n\send{align}\nTo get it into the form of JFJ just need choose renorm point $\sbar\sm^2=\shat\sm^2\srme^{-3/2}$ and note\n\sbegin{align} \shat L_2&=L_2- 3 L_1+\sfrac94c~,\squad {\srm with} \squad \n{L}_n:= x\slog^n\sfrac{x}{\sbar{\sm}^2}+\scycl\n\send{align}\nand then change one of the $L_2$'s using\n\sbegin{align}\n\sleft[(x-y-z)\slog\sfrac{y}{\sbar\sm^2}\slog\sfrac{z}{\sbar\sm^2}+\scycl\sright]\n &= L_2-\sleft[x\slog(\sfrac yx)\slog(\sfrac zx)+\scycl\sright]~.\n\send{align}\n
<div class='toolbar' macro='toolbar +editNote collapseNote collapseOthers closeOthers -closeNote'></div>\n<div class='title' macro='view title'></div>\n<div class='tagging' macro='tagging'></div>\n<div class='tagged' macro='tags'></div>\n<div class='viewer' macro='view text wikified'></div>\n<div class='references' macro="references"></div>\n<div class='tagClear'></div>
This is a personal wiki notebook in theoretical physics. \nMost of the content is either: \nreminders to myself, ie useful formulae / ideas\nor \nif I've LaTeXed up some summaries of my work I'll put them up here.\n\nThe notes are organized by [[Tags]] in an expandable list to the left of this window. Or you can type any phrase to search for in the field to the left. From any note you can follow links to others or click in the bottom list of notes that link to it. Or you can see which notes have been edited recently, under "[[Latest|TabTimeline]]." That's about it for basic orientation &mdash; you can read more [[About]] what you're looking at, or pick it up as you go. This site is best viewed with TeX fonts installed on your computer, rather than image fonts. (See that "jsMath" button down there? Click to check.)\n\nThe last two notes worked on are below.\n
$$\sint d^8z \sbar\sphi\sphi + \sBig[\sint d^6z\sbig(m\sphi^2+\sfrac\slambda{3!}\sphi^3\sbig) +c.c. \sBig] $$
yet another previewer!\na DVI viewer shipped with MiKTeX\n\nInverse search configured for [[CrimsonEditor]]\n<nowiki> "c:\sprogram files\sCrimson Editor\scedt.exe" /L:%l %f </nowiki>
Refs:\n*Reinhold A. Bertlmann\n**Anomalies in quantum field theory\n*Jeffrey A. Harvey\n**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]\n***anomalies from the particle physics point of view
*<<slider chkSliderlatexF latexF 'LaTeX »' 'latex stuff'>>\n*<<slider chkSlidermathematicaF mathematicaF 'Mathematica »' 'mathematica stuff'>>\n<<ListTagged computer>>
<<ListTagged editing>>
<<ListTagged folder>>
*<<slider chkSliderpbF pbF 'pb »' 'principal bundles'>>\n*<<slider chkSliderbrstF brstF 'brst »' 'BRST formalism'>>\n<<ListTagged gauge>>
<<ListTagged illus>>
//Use the first method in each example below, unless you have some reason not to.//\nMathematical symbols, such as \s(e^{x^2}\s), may be inserted inline.\n{{{\nMathematical symbols, such as $e^{x^2}$, may be inserted inline.\nMathematical symbols, such as \s(e^{x^2}\s), may be inserted inline.\n}}}\nOr as displayed math,$$e^{x^2}$$ on its own line.\n{{{\nOr as displayed math, \s[e^{x^2}\s] on its own line.\nOr as displayed math, $$e^{x^2}$$ on its own line.\nOr as displayed math, \sbegin{equation}e^{x^2}\send{equation} on its own line.\n}}}\nOr as an equation array,\n\sbegin{eqnarray}A &=& e^{x^2}\s\s&=&C\send{eqnarray}\n{{{\nOr as an equation array,\sbegin{eqnarray}A &=& e^{x^2}\s\s&=& C\send{eqnarray}\n}}}\n\nSome of the available TeX symbols can be found at [[jsMath|http://www.math.union.edu/~dpvc/jsMath/symbols/welcome.html]], the best method I could find for displaying TeX online. The small button in the lower right corner of this window opens its control planel. I'm not sure how many LaTeX and AMSTeX commands are supported -- play around.\n\nTeX substitution macros such as $\sf{A}$, ({{{$\sf{A}$}}}), may be inserted into the [[jsMathPlugin]] just before the jsMath.process call. See that plugin for abbreviated commands I've included.\n\nAlso see [[jsMath-talk|http://www.math.uwaterloo.ca/~pkates/LT3/jsMath-talk-June-05/jsMath-talk.html]] for some hints,\n[[Unicode blackboard bold|http://en.wikipedia.org/wiki/Blackboard_bold]] on wikipedia
/***\n|Name|Plugin: jsMath|\n|Created by|BobMcElrath (edited by Garrett and Simon)|\n|Email|my first name at my last name dot org|\n|Location|http://bob.mcelrath.org/tiddlyjsmath-2.0.3.html|\n|Version|1.3.g|\n|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] &ge; 2.1, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] &ge; 3.0|\n!Description\n[[LaTeX]] is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians. For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]]. This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser. Notice the small button in the lower right corner which opens its control panel.\n!Installation\nIn addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file. If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.\n!Examples\n|!Source|!Output|h\n|{{{The variable $x$ is real.}}}|The variable $x$ is real.|\n|{{{The variable \s(y\s) is complex.}}}|The variable \s(y\s) is complex.|\n|{{{This \s[\sint_a^b x = \sfrac{1}{2}(b^2-a^2)\s] is an easy integral.}}}|This \s[\sint_a^b x = \sfrac{1}{2}(b^2-a^2)\s] is an easy integral.|\n|{{{This $$\sint_a^b \ssin x = -(\scos b - \scos a)$$ is another easy integral.}}}|This $$\sint_a^b \ssin x = -(\scos b - \scos a)$$ is another easy integral.|\n|{{{Block formatted equations may also use the 'equation' environment \sbegin{equation} \sint \stan x = -\sln \scos x \send{equation} }}}|Block formatted equations may also use the 'equation' environment \sbegin{equation} \sint \stan x = -\sln \scos x \send{equation}|\n|{{{Equation arrays are also supported \sbegin{eqnarray} a &=& b \s\s c &=& d \send{eqnarray} }}}|Equation arrays are also supported \sbegin{eqnarray} a &=& b \s\s c &=& d \send{eqnarray} |\n|{{{I spent \s$7.38 on lunch.}}}|I spent \s$7.38 on lunch.|\n|{{{I had to insert a backslash (\s\s) into my document}}}|I had to insert a backslash (\s\s) into my document|\n!Code\n***/\n//{{{\n\n// Define wikifers for latex\nconfig.formatterHelpers.mathFormatHelper = function(w) {\n var e = document.createElement(this.element);\n e.className = this.className;\n var endRegExp = new RegExp(this.terminator, "mg");\n endRegExp.lastIndex = w.matchStart+w.matchLength;\n var matched = endRegExp.exec(w.source);\n if(matched) {\n var txt = w.source.substr(w.matchStart+w.matchLength, \n matched.index-w.matchStart-w.matchLength);\n if(this.keepdelim) {\n txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);\n }\n e.appendChild(document.createTextNode(txt));\n w.output.appendChild(e);\n w.nextMatch = endRegExp.lastIndex;\n }\n}\n\nconfig.formatters.push({\n name: "displayMath1",\n match: "\s\s\s$\s\s\s$",\n terminator: "\s\s\s$\s\s\s$\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nconfig.formatters.push({\n name: "inlineMath1",\n match: "\s\s\s$", \n terminator: "\s\s\s$",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nvar backslashformatters = new Array(0);\n\nbackslashformatters.push({\n name: "inlineMath2",\n match: "\s\s\s\s\s\s\s(",\n terminator: "\s\s\s\s\s\s\s)",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath2",\n match: "\s\s\s\s\s\s\s[",\n terminator: "\s\s\s\s\s\s\s]\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath3",\n match: "\s\s\s\sbegin\s\s{equation\s\s}",\n terminator: "\s\s\s\send\s\s{equation\s\s}\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// These can be nested. e.g. \sbegin{equation} \sbegin{array}{ccc} \sbegin{array}{ccc} ...\nbackslashformatters.push({\n name: "displayMath4",\n match: "\s\s\s\sbegin\s\s{eqnarray\s\s}",\n terminator: "\s\s\s\send\s\s{eqnarray\s\s}\s\sn?",\n element: "div",\n className: "math",\n keepdelim: true,\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n/* S added this */\nbackslashformatters.push({\n name: "displayMath5",\n match: "\s\s\s\sbegin\s\s{align\s\s}",\n terminator: "\s\s\s\send\s\s{align\s\s}\s\sn?",\n element: "div",\n className: "math",\n keepdelim: true,\n handler: config.formatterHelpers.mathFormatHelper\n});\nbackslashformatters.push({\n name: "displayMath5",\n match: "\s\s\s\sbegin\s\s{align\s\s*\s\s}",\n terminator: "\s\s\s\send\s\s{align\s\s*\s\s}\s\sn?",\n element: "div",\n className: "math",\n keepdelim: true,\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// The escape must come between backslash formatters and regular ones.\n// So any latex-like \scommands must be added to the beginning of\n// backslashformatters here.\nbackslashformatters.push({\n name: "escape",\n match: "\s\s\s\s.",\n handler: function(w) {\n w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));\n w.nextMatch = w.matchStart+2;\n }\n});\n\nconfig.formatters=backslashformatters.concat(config.formatters);\n\n/* G updated this */\nwindow.wikify = function(source,output,highlightRegExp,note)\n{\n if(source && source != "") {\n var wikifier = new Wikifier(source,getParser(note),highlightRegExp,note);\n wikifier.subWikifyUnterm(output);\n jsMath.Process();\n }\n}\n\n/* insert jsMath LaTeX macros here */\n\n/* jsMath.Extension.Require('underset-overset'); */\n/* jsMath.Font.Load("bbold10"); */\njsMath.Extension.Require("AMSmath");\n\n/* Greek Letters */\njsMath.Macro('a','\s\salpha');\njsMath.Macro('b','\s\sbeta');\njsMath.Macro('g','\s\sgamma');\njsMath.Macro('d','\s\sdelta');\njsMath.Macro('e','\s\sepsilon');\njsMath.Macro('eps','\s\svarepsilon');\njsMath.Macro('z','\s\szeta');\njsMath.Macro('h','\s\seta');\njsMath.Macro('q','\s\stheta');\njsMath.Macro('i','\s\siota');\njsMath.Macro('k','\s\skappa');\njsMath.Macro('l','\s\slambda');\njsMath.Macro('m','\s\smu');\njsMath.Macro('n','\s\snu');\njsMath.Macro('r','\s\srho');\njsMath.Macro('s','\s\ssigma');\njsMath.Macro('t','\s\stau');\njsMath.Macro('u','\s\supsilon');\njsMath.Macro('f','\s\sphi');\njsMath.Macro('vf','\s\svarphi');\njsMath.Macro('c','\s\schi');\njsMath.Macro('j','\s\spsi');\njsMath.Macro('o','\s\somega');\njsMath.Macro('p','\s\spi');\njsMath.Macro('x','\s\sxi');\njsMath.Macro('G','\s\sGamma');\njsMath.Macro('D','\s\sDelta');\njsMath.Macro('Q','\s\sTheta');\njsMath.Macro('L','\s\sLambda');\njsMath.Macro('S','\s\sSigma');\njsMath.Macro('U','\s\sUpsilon');\njsMath.Macro('F','\s\sPhi');\njsMath.Macro('Ps','\s\sPsi');\njsMath.Macro('O','\s\sOmega');\njsMath.Macro('P','\s\sPi');\njsMath.Macro('pd','\s\spartial');\njsMath.Macro('na','\s\snabla');\n/* Caligraphic letters */\njsMath.Macro('cD','{\s\scal D}');\njsMath.Macro('cF','{\s\scal F}');\njsMath.Macro('cI','{\s\scal I}');\njsMath.Macro('cL','{\s\scal L}');\njsMath.Macro('cN','{\s\scal N}');\njsMath.Macro('cW','{\s\scal W}');\n/* Script letters */\njsMath.Macro('scD','{\s\smathscr D}');\njsMath.Macro('scF','{\s\smathscr F}');\njsMath.Macro('scG','{\s\smathscr G}');\njsMath.Macro('scH','{\s\smathscr H}');\njsMath.Macro('scJ','{\s\smathscr J}');\njsMath.Macro('scJb','{\s\smathscr{\s\sbar J}}');\n/* Romanised letters */\njsMath.Macro('rmd','{\s\srm d}');\njsMath.Macro('rme','{\s\srm e}');\njsMath.Macro('rmi','{\s\srm i}');\njsMath.Macro('rmv','{\s\srm v}');\n/* Dotted Letters */\njsMath.Macro('da','{\s\sdot a}');\njsMath.Macro('db','{\s\sdot b}');\njsMath.Macro('dg','{\s\sdot g}');\n/* Combination indices */\njsMath.Macro('ab','{ab}');\njsMath.Macro('adb','{a \s\sdot b}');\njsMath.Macro('daa','{\s\sdot a a}');\njsMath.Macro('ada','{a \s\sdot a}');\njsMath.Macro('dbb','{\s\sdot b b}');\njsMath.Macro('bdb','{b \s\sdot b}');\n/* Barred Letters */\njsMath.Macro('Bb','{\s\sbar B}');\njsMath.Macro('Db','{\s\sbar D}');\njsMath.Macro('Nb','{\s\sbar N}');\njsMath.Macro('Wb','{\s\sbar W}');\njsMath.Macro('Jb','{\s\sbar J}');\njsMath.Macro('cDb','{\s\sbar{\s\scal D}}');\njsMath.Macro('cNb','{\s\sbar{\s\scal N}}');\njsMath.Macro('cWb','{\s\sbar{\s\scal W}}');\njsMath.Macro('zetab','{\s\sbar \s\szeta}');\njsMath.Macro('abar','{\s\sbar a}');\njsMath.Macro('bbar','{\s\sbar b}');\njsMath.Macro('cbar','{\s\sbar c}');\n/* Blackboard Bold*/\njsMath.Macro('dsR','\s\sunicode{x211D}');\njsMath.Macro('dsC','\s\sunicode{x2102}');\njsMath.Macro('dsQ','\s\sunicode{x211A}');\njsMath.Macro('dsZ','\s\sunicode{x2124}');\njsMath.Macro('dsN','\s\sunicode{x2115}');\njsMath.Macro('dsI','\s\sunicode{1D540}');\njsMath.Macro('dsL','\s\sunicode{1D543}');\njsMath.Macro('dsK','\s\sunicode{1D542}');\njsMath.Macro('dsF','\s\sunicode{1D53D}');\njsMath.Macro('dsone','{1\s\s!\s\s!\s\s!1}');\n/* Circled letters */\njsMath.Macro('tcs','\s\smbox{\s{s\s}}');\njsMath.Macro('tct','\s\smbox{\s{t\s}}');\njsMath.Macro('tcu','\s\smbox{\s{u\s}}');\n/* Brackets */\njsMath.Macro('lb','\s\sleft[');\njsMath.Macro('rb','\s\sright]');\njsMath.Macro('lp','\s\sleft(');\njsMath.Macro('rp','\s\sright)');\njsMath.Macro('li','\s\sleft<');\njsMath.Macro('ri','\s\sright>');\njsMath.Macro('ll','\s\sleft|');\njsMath.Macro('rl','\s\sright|');\njsMath.Macro('lc','\s\sleft\s\s{');\njsMath.Macro('rc','\s\sright\s\s}');\njsMath.Macro('ld','\s\sleft.');\njsMath.Macro('rd','\s\sright.');\n/* Integrals */\njsMath.Macro('intx','\s\sint\s\s!\s\s!{\s\srm d}^4x\s\s,');\njsMath.Macro('intk','\s\sint\s\s!\s\s!\s\sfrac{{\s\srm d}^4k}{(2\s\spi)^4}\s\s,');\njsMath.Macro('intrho','\s\sint\s\s!\s\s!{\s\srm d}^4\s\srho\s\s,');\njsMath.Macro('intz','\s\sint\s\s!\s\s!{\s\srm d}^8z\s\s,');\njsMath.Macro('intc','\s\sint\s\s!\s\s!{\s\srm d}^6z\s\s,');\njsMath.Macro('intac','\s\sint\s\s!\s\s!{\s\srm d}^6{\s\sbar z}\s\s,');\njsMath.Macro('pint','\s\sint\s\s!\s\s!{\s\scal D}\s\s!#1\s\s,',1);\njsMath.Macro('intstu','\s\sint_0^\s\sinfty\s\s!\s\s!\s\s!\s\srmd{s}\s\srmd{t}\s\srmd{u}');\n/* Matrices */\njsMath.Macro('dpm','{\sbegin{pmatrix} \sdelta_+ & 0 \s\s 0 & \sdelta_- \send{pmatrix}}');\n/* Other symbols */\njsMath.Macro('Li','{\s\srm Li}');\njsMath.Macro('Tr','{\s\srm Tr}');\njsMath.Macro('tr','{\s\srm tr}');\njsMath.Macro('trF','{\s\srm tr_F}');\njsMath.Macro('trad','{\s\srm tr_{ad}}');\njsMath.Macro('Det','{\s\srm Det}');\njsMath.Macro('diag','{\s\srm diag}');\njsMath.Macro('ord','{\s\srm O}');\njsMath.Macro('const','{\s\srm const}');\njsMath.Macro('wrt','{\s\srm wrt}');\njsMath.Macro('ie','{\s\srm ie~}');\njsMath.Macro('cc','{\s\srm c.c.}');\njsMath.Macro('hc','{\s\srm h.c.}');\njsMath.Macro('ns','{\s\srm (no~sum)}');\njsMath.Macro('cycl','{\s\srm cycl.}');\njsMath.Macro('I','{\s\srm I}');\njsMath.Macro('II','{\s\srm II}');\njsMath.Macro('III','{\s\srm III}');\njsMath.Macro('IV','{\s\srm IV}');\njsMath.Macro('V','{\s\srm V}');\njsMath.Macro('half','{\s\ssmall \s\sfrac{1}{2}}');\njsMath.Macro('third','{\s\ssmall \s\sfrac{1}{3}}');\njsMath.Macro('quart','{\s\ssmall \s\sfrac{1}{4}}');\njsMath.Macro('eighth','{\s\ssmall \s\sfrac{1}{8}}');\njsMath.Macro('fr','{\s\ssmall \s\sfrac{#1}{#2}}',2);\n/* Structural */\njsMath.Macro('bem','\s\sbegin{pmatrix}');\njsMath.Macro('eem','\s\send{pmatrix}');\n/* Disabled latex */\njsMath.Macro('label','{}',1);\njsMath.Macro('non','{}');\n/* end of jsMath macros */\n//}}}
Mathematical typesetting
<<ListTagged latex>>
<<ListTagged loops>>
<<ListTagged math>>
Software by Wolfram Research
<<ListTagged mathematica>>
*<<slider chkSlidereditingF editingF 'editing »' 'tips on editing and authoring notes, including all sorts of tools'>>\n*<<slider chkSlider0F 0F '0 »' 'a note that is linked to but is empty or needs editing'>>\n*<<slider chkSlidersystemF systemF 'system »' 'control how the site operates and is layed out'>>\n*<<slider chkSlider sliderF 'sliders »' 'all sliders for all subjects'>>\n*<<slider chkSliderillusF illusF 'illus »' 'notes containing illustrations'>>\n<<ListTagged meta>>
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*<<slider chkSlidersusyF susyF 'susy »' 'supersymmetry'>>\n*<<slider chkSliderqftF qftF 'qft »' 'quantum field theory'>>\n*<<slider chkSliderotherF otherF 'other »' 'other physics'>>\n*<<slider chkSliderpaperF paperF 'paper »' 'notes about or containing links to a paper'>>\n<<ListTagged physics>>
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*<<slider chkSliderloopsF loopsF 'Loops »' 'Calculating Feynman integrals'>>\n<<ListTagged qft>>
<<ListTagged slider>>
<<ListTagged sugra>>
*<<slider chkSlidersusyalgF susyalgF 'sysyalg »' 'supersymmetry algebra>>\n*<<slider chkSlidersusyfieldrepsF susyfieldrepsF 'fieldreps »' 'field representations of supersymmetry>>\n*<<slider chkSlidersugraF sugraF 'sugra»' 'supergravity>>\n<<ListTagged susy>>
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<<ListTagged systemConfig>>
*<<slider chkSlidersystemConfigF systemConfigF 'systemConfig »' 'this is a special tag, marking notes with code to be loaded at startup'>>\n*<<slider chkSliderxsystemConfigF xsystemConfigF 'xsystemConfig »' 'deactivated code'>>\n*<<slider chkSlidersystemNotesF systemNotesF 'systemNotes »' 'system control notes loaded at startup, used to control system content'>>\n*<<slider chkSliderpluginF pluginF 'plugin »' 'code snippets enhancing functionality, and containing descriptions of what they do'>>\n*<<slider chkSlidertemplateF templateF 'template »' 'custom css page template, used to describe layout'>>\n*<<slider chkSliderfolderF folderF 'folder »' 'a folder is a tag is a note'>>\n<<ListTagged system>>
<<ListTagged systemNotes>>
<<ListTagged template>>
From Scott Pakin, posted on comp.text.tex\n{{{\snewcommand{\sxeq}[1]{\sstackrel{\s;#1}{\scleaders\shbox{=\s!}\shfill}\s:} }}}\n\nOR from http://www.biwako.shiga-u.ac.jp/sensei/kumazawa/tex\n{{{\n\smakeatletter\n\snewcommand{\sxequal}[2][]{\sext@arrow 0055{\sequalfill@}{#1}{#2}}\n\sdef\sequalfill@{\sarrowfill@\sRelbar\sRelbar\sRelbar}\n\smakeatother\n}}}\n\nor as part of the\n[[extarrows package|http://www.ctan.org/tex-archive/macros/latex/contrib/extarrows/]]
<<ListTagged xsystemConfig>>