diff --git a/sciencestuff.sty b/sciencestuff.sty index 25fc152..53d581d 100644 --- a/sciencestuff.sty +++ b/sciencestuff.sty @@ -266,58 +266,58 @@ %---- dot, bar, tilde, hat -\providecommand{\da}{\ensuremath{\dot{a}}\xspace} -\providecommand{\db}{\ensuremath{\dot{b}}\xspace} -\providecommand{\dc}{\ensuremath{\dot{c}}\xspace} -\providecommand{\dd}{\ensuremath{\dot{d}}\xspace} -\providecommand{\de}{\ensuremath{\dot{e}}\xspace} -\providecommand{\df}{\ensuremath{\dot{f}}\xspace} -\providecommand{\dg}{\ensuremath{\dot{g}}\xspace} -\providecommand{\dh}{\ensuremath{\dot{h}}\xspace} -\providecommand{\di}{\ensuremath{\dot{i}}\xspace} -\providecommand{\dj}{\ensuremath{\dot{j}}\xspace} -\providecommand{\dk}{\ensuremath{\dot{k}}\xspace} -\providecommand{\dl}{\ensuremath{\dot{l}}\xspace} -\providecommand{\dm}{\ensuremath{\dot{m}}\xspace} -\providecommand{\dn}{\ensuremath{\dot{n}}\xspace} -\providecommand{\do}{\ensuremath{\dot{o}}\xspace} -\providecommand{\dp}{\ensuremath{\dot{p}}\xspace} 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+\providecommand{\dotc}{\ensuremath{\dot{c}}\xspace} +\providecommand{\dotd}{\ensuremath{\dot{d}}\xspace} +\providecommand{\dote}{\ensuremath{\dot{e}}\xspace} +\providecommand{\dotf}{\ensuremath{\dot{f}}\xspace} +\providecommand{\dotg}{\ensuremath{\dot{g}}\xspace} +\providecommand{\doth}{\ensuremath{\dot{h}}\xspace} +\providecommand{\doti}{\ensuremath{\dot{i}}\xspace} +\providecommand{\dotj}{\ensuremath{\dot{j}}\xspace} +\providecommand{\dotk}{\ensuremath{\dot{k}}\xspace} +\providecommand{\dotl}{\ensuremath{\dot{l}}\xspace} +\providecommand{\dotm}{\ensuremath{\dot{m}}\xspace} +\providecommand{\dotn}{\ensuremath{\dot{n}}\xspace} +\providecommand{\doto}{\ensuremath{\dot{o}}\xspace} +\providecommand{\dotp}{\ensuremath{\dot{p}}\xspace} +\providecommand{\dotq}{\ensuremath{\dot{q}}\xspace} +\providecommand{\dotr}{\ensuremath{\dot{r}}\xspace} +\providecommand{\dots}{\ensuremath{\dot{s}}\xspace} +\providecommand{\dott}{\ensuremath{\dot{t}}\xspace} +\providecommand{\dotu}{\ensuremath{\dot{u}}\xspace} +\providecommand{\dotv}{\ensuremath{\dot{v}}\xspace} +\providecommand{\dotw}{\ensuremath{\dot{w}}\xspace} +\providecommand{\dotx}{\ensuremath{\dot{x}}\xspace} +\providecommand{\doty}{\ensuremath{\dot{y}}\xspace} +\providecommand{\dotz}{\ensuremath{\dot{z}}\xspace} +\providecommand{\dotA}{\ensuremath{\dot{A}}\xspace} +\providecommand{\dotB}{\ensuremath{\dot{B}}\xspace} +\providecommand{\dotC}{\ensuremath{\dot{C}}\xspace} +\providecommand{\dotD}{\ensuremath{\dot{D}}\xspace} +\providecommand{\dotE}{\ensuremath{\dot{E}}\xspace} +\providecommand{\dotF}{\ensuremath{\dot{F}}\xspace} +\providecommand{\dotG}{\ensuremath{\dot{G}}\xspace} +\providecommand{\dotH}{\ensuremath{\dot{H}}\xspace} +\providecommand{\dotI}{\ensuremath{\dot{I}}\xspace} +\providecommand{\dotJ}{\ensuremath{\dot{J}}\xspace} +\providecommand{\dotK}{\ensuremath{\dot{K}}\xspace} +\providecommand{\dotL}{\ensuremath{\dot{L}}\xspace} +\providecommand{\dotM}{\ensuremath{\dot{M}}\xspace} +\providecommand{\dotN}{\ensuremath{\dot{N}}\xspace} +\providecommand{\dotO}{\ensuremath{\dot{O}}\xspace} +\providecommand{\dotP}{\ensuremath{\dot{P}}\xspace} +\providecommand{\dotQ}{\ensuremath{\dot{Q}}\xspace} +\providecommand{\dotR}{\ensuremath{\dot{R}}\xspace} +\providecommand{\dotS}{\ensuremath{\dot{S}}\xspace} +\providecommand{\dotT}{\ensuremath{\dot{T}}\xspace} +\providecommand{\dotU}{\ensuremath{\dot{U}}\xspace} +\providecommand{\dotV}{\ensuremath{\dot{V}}\xspace} +\providecommand{\dotW}{\ensuremath{\dot{W}}\xspace} +\providecommand{\dotX}{\ensuremath{\dot{X}}\xspace} +\providecommand{\dotY}{\ensuremath{\dot{Y}}\xspace} +\providecommand{\dotZ}{\ensuremath{\dot{Z}}\xspace} \providecommand{\bara}{\ensuremath{\overline{a}}\xspace} \providecommand{\barb}{\ensuremath{\overline{b}}\xspace} diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index fa40fb8..18cce29 100644 --- a/sec/part1/introduction.tex +++ b/sec/part1/introduction.tex @@ -85,11 +85,11 @@ implies - \frac{1}{2 \pi \ap} \infinfint{\tau} \finiteint{\sigma}{0}{\sigma} - \sqrt{\dX \cdot \dX - \pX \cdot \pX} + \sqrt{\dotX \cdot \dotX - \pX \cdot \pX} = S_{NG}[X], \end{equation} -where $S_{NG}[X]$ is the Nambu--Goto action for the classical string, $\dX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$. +where $S_{NG}[X]$ is the Nambu--Goto action for the classical string, $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$. The symmetries of $S_P[\gamma, X]$ are the keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}. Specifically~\eqref{eq:conf:polyakov} displays symmetries under: diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index 219544b..2412f69 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -49,7 +49,7 @@ We then go back to string theory and we verify that in the \nbo the open string We then introduce the Generalized Null Boost Orbifold (\gnbo) as a generalization of the \nbo which still has a light-like singularity and is generated by one Killing vector. However in this model there are two directions associated with $\cA$, one compact and one non compact. -We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation. +We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}. However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo. In the literature there are however also other attempts at regularizing the \nbo such as the Null Brane. This kind of orbifold was originally defined in \cite{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2004:TimedependentOrbifoldsString} and studied in perturbation theory in \cite{Liu:2002:StringsTimeDependentOrbifolds}. @@ -61,6 +61,1071 @@ The scalar eigenfunctions behave in time $t$ as $\abs{t}^{\pm i\, \frac{l}{\Delt In particular the scalar \qed on the \bo can be defined and the first term which gives a divergent contribution is of the form $\abs{\phi~\dphi}^2$, i.e.\ divergences are hidden into the derivative expansion of the effective field theory. Again three points open string amplitudes with one massive state diverge. +\subsection{Scalar QED on NBO and Divergences} +\label{sect:NOscalarQED} + +As discussed the four open string tachyons amplitude diverges in the \nbo. +The literature on the subject (see for instance~\cite{Cornalba:2004:TimedependentOrbifoldsString} and references therein) suggests that this can be cured by the eikonal resummation. +We therefore consider the scalar \qed on the \nbo as a first approach. +In this case all eigenmodes can be written using elementary functions thus making the issues even more evident. +Its action is given by +\begin{equation} + \rS_{\text{s}\qed} + = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g} + \qty( + - \qty(D^{\mu} \phi)^*\, D_{\mu} \phi + - M^2 \qty(\phi^*)\, \phi + - \frac{1}{4} f^{\mu\nu}\, f_{\mu\nu} + - \frac{g_4}{4} \abs{\phi}^4 + ), +\end{equation} +with +\begin{equation} + D_{\mu} \phi + = + \qty(\ipd{\mu} -i\, e\, a_{\mu}) \phi, + \qquad + f_{\mu\nu} + = + \ipd{\mu} a_{\nu} - \ipd{\nu} a_{\mu}. +\end{equation} +We reserve small letters for quantities defined on the orbifold and capital letters for those defined in flat space. +Moreover $\Omega$ denotes the orbifold. +We will construct directly both the scalar and the spin-1 eigenfunctions which we can use as a starting point for the perturbative computations. + + +\subsubsection{Geometric Preliminaries} +\label{sec:geometric_preliminaries_nbo} + +In Minkowski spacetime $\ccM^{1,D-1}$ with coordinates $\qty(x^{\mu}) = \qty(x^+,\, x^-,\, x^2,\, \vb{x})$ +and metric +\begin{equation} + \dss[2]{s} + = + - 2 \dd{x^+} \dd{x^-} + + \qty(\dd{x^2})^2 + + \eta_{ij} \dd{x}^i \dd{x}^j, +\end{equation} +we consider the following change of coordinates to $\qty(x^{\alpha}) = (u,\, v,\, z,\, \vb{x})$ +\begin{equation} + \begin{cases} + x^- & = u + \\ + x^2 & = \Delta u z + \\ + x^+ & = v + \frac{1}{2} \Delta^2 u z^2 + \end{cases} + \qquad + \Leftrightarrow + \qquad + \begin{cases} + u & = x^- + \\ + z & = \frac{x^2}{\Delta\, x^-} + \\ + v & = x^+ - \frac{1}{2} \frac{(x^2)^2}{x^-} + \end{cases}. + \label{eq:NBO_coordinates} +\end{equation} +Then the metric becomes: +\begin{equation} + \dss[2]{s} + = + - 2\, \dd{u}\, \dd{v} + + \qty(\Delta u )^2 (\dd{z})^2 + + \eta_{ij} \dd{x}^i \dd{x}^j, +\end{equation} +along with the non vanishing geometrical quantities +\begin{equation} + -\det g = \qty( \Delta u )^2, +\end{equation} +and +\begin{equation} + \tensor{\Gamma}{_z^v_z} = \Delta^2 u, + \qquad + \tensor{\Gamma}{_u^z_z} = u^{-1}. +\end{equation} +Riemann and Ricci tensor components however vanish since at this stage we only performed a change of coordinates from the original Minkowski spacetime. +Locally it is the same as the \nbo and they must have the same local differential geometry. + +The \nbo is introduced by identifying points along the orbits of the Killing vector: +\begin{equation} + \begin{split} + \kappa + & = + - i \qty(2 \pi \Delta) J_{+2} + \\ + & = + \qty(2 \pi \Delta)\, (x^2 \ipd{+} + x^- \ipd{2}) + \\ + & = + 2 \pi \ipd{z}, + \end{split} + \label{eq:nbo_killing_vector} +\end{equation} +in such a way that +\begin{equation} + x^{\mu} \equiv \cK^{n}\, x^{\mu}, + \qquad + n \in \Z, +\end{equation} +where $\cK^{n}= e^{n\kappa}$, leads to the identifications +\begin{equation} + x= + \mqty( x^- \\ x^2 \\ x^+ \\ \vb{x} ) + \equiv + \cK^{n} x + = + \mqty(% + x^- \\ + x^2 + n \qty(2 \pi \Delta) x^- \\ + x^+ + n \qty(2 \pi \Delta) x^2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 x^- \\ + \vb{x} + ) +\end{equation} +or to +\begin{equation} + \qty( u,\, v,\, z,\, \vb{x} ) + \equiv + \qty( u,\, v,\, z + 2 \pi n,\, \vb{x} ) +\end{equation} +in coordinates $\qty(x^{\alpha})$ where $\kappa = 2 \pi \ipd{z}$ is a global Killing vector. + +As a reference for the future, we notice that we could regularise the metric as +\begin{equation} + \dss[2]{s} + = + - 2\, \dd{u}\, \dd{v} + + \Delta^2 \qty(u^2 + \epsilon^2) (\dd{z})^2 + + \eta_{ij} \dd{x}^i \dd{x}^j. +\end{equation} +The non vanishing geometrical quantities are then: +\begin{equation} + -\det g = \Delta^2 \qty(u^2 + \epsilon^2), +\end{equation} +and +\begin{equation} + \tensor{\Gamma}{_z^v_z} = \Delta^2 u, + \qquad + \tensor{\Gamma}{_u^z_z} = \frac{u}{u^2 + \epsilon^2}, +\end{equation} +which lead to the following Riemann and Ricci tensor components: +\begin{equation} + \tensor{R}{^z_u_z_u} = - \frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2}, + \quad + \tensor{R}{^v_z_z_u} = - \frac{\Delta^2 \epsilon^2}{u^2 + \epsilon^2}, + \quad + \tensor{Ric}{_u_u} = -\frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2}. +\end{equation} +Since $\delta_{\text{reg}}(u) = \frac{1}{\pi} \frac{\epsilon}{u^2+ \epsilon^2}$ then $\tensor{R}{^z_u_z_u} = - \pi^2 \qty[ \delta_{\text{reg}}(u) ]^2$. + + +\subsubsection{Free Scalar Action} + +We study the eigenmodes of the Laplacian operator to diagonalize the scalar kinetic term given by:\footnotemark{} +\footnotetext{% + The factor $-g^{\alpha\beta}$ is due to the choice of the East coast convention for the metric, namely: + \begin{equation*} + - g^{\alpha\beta} + \ipd{\alpha} \phi^*\, \ipd{\beta} \phi + - + M^2 \phi^*\, \phi + \sim + \abs{\dot{\phi}}^2 - M^2 \abs{\phi}^2 + \sim + \rE^2 - M^2. + \end{equation*} +} +\begin{equation} + \begin{split} + \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \phi ] + & = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}~ + \qty(% + - g^{\alpha\beta} \ipd{\alpha} \phi^*\, \ipd{\beta} \phi + - M^2 \phi^*\, \phi + ) + \\ + & = + \int \dd[D-3]{\vb{x}}\, + \int \dd{u}\, + \int \dd{v}\, + \finiteint{z}{0}{2\pi} + \abs{\Delta u} + \\ + & \times + \qty(% + \ipd{u} \phi^*\, \ipd{v} \phi\, + + + \ipd{v} \phi^*\, \ipd{u} \phi\, + - + \frac{1}{\qty(\Delta u)^2} \ipd{z} \phi^*\, \ipd{z} \phi\, + - + \ipd{i} \phi^*\, \ipd{i} \phi + - + M^2 \phi^*\, \phi + ). + \end{split} +\end{equation} +The solution to the equation of motion is enough when we want to perform the canonical quantization. +Since we use Feynman diagrams we consider the path integral approach: we take off-shell modes and solve the eigenvalue problem $\square \phi_r = r \phi_r$. +Comparing with the flat case we see that $r$ is $2\, k_-\, k_+ - \norm{\vb{k}}^2$ when $k$ is the impulse in flat coordinates. +We therefore have +\begin{equation} + -2 \ipd{u} \ipd{v} \phi_r + - + \frac{1}{u} \ipd{v} \phi_r + + + \frac{1}{\qty(\Delta u)^2} \ipd{z}^2 \phi_r + + + \ipd{i}^2 \phi_r + = + r \phi_r. + \label{eq:nbo_eom} +\end{equation} +Using Fourier transforms it follows that the eigenmodes are +\begin{equation} + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}) + = + e^{i k_+ v + i l z + i \vb{k} \cdot \vb{x}}\, + \tphi_{\kmkr}(u), +\end{equation} +with +\begin{equation} + \tphi_{\kmkr}(u) + = + \frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}} + e^{ + - i \frac{l^2}{2 \Delta^2 k_+} \frac{1}{u} + + i \frac{\norm{\vb{k}}^2 + r}{2 k_+} u + }, +\end{equation} +and +\begin{equation} + \phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}) + = + \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vb{x}). +\end{equation} +We chose the numeric factor in order to get a canonical normalisation: +\begin{equation} + \begin{split} + & + \qty( \phi_{\kmkrN{1}},\, \phi_{\kmkrN{2}} ) + \\ + = & + \int \dd[D-1]{\vb{x}}\, + \int \dd{u}\, \int \dd{v}\, + \finiteint{z}{0}{2\pi} + \abs{\Delta u}\, + \phi_{\kmkrN{1}}\, \phi_{\kmkrN{2}} + \\ + = & + \delta^{D-3}( \vb{k}_{\qty(1)} + \vb{k}_{\qty(2)})\, + \delta( r_{(1)} - r_{(2)})\, + \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, + \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}. + \end{split} +\end{equation} +We can then perform the off-shell expansion +\begin{equation} + \phi\qty(u,\, v,\, z,\, \vb{x}) + = + \int \dd[D-3]{\vb{k}} + \int \dd{k_+} + \int \dd{r} + \infinfsum{l} + \cA_{\kmkr}\, + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}), +\end{equation} +such that the scalar kinetic term becomes +\begin{equation} + \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cA ] + = + \int \dd[D-3]{\vb{k}}\, + \int \dd{k_+} + \int \dd{r} + \infinfsum{l} + \qty(r - M^2)\, + \cA_{\kmkr}\, + \cA_{\kmkr}^*. +\end{equation} + + +\subsubsection{Free Photon Action} + +The action of the free photon can be written as +\begin{align} + \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ a ] + = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{-\det g}\, + \qty(% + - \frac{1}{2} g^{\alpha\beta} g^{\gamma\delta} + D_{\alpha} a_{\gamma} \qty( D_{\beta} a_{\delta} - D_{\delta} a_{\beta}) + ). +\end{align} +We choose to enforce the Lorenz gauge:\footnotemark{} +\footnotetext{% + Indeed it is exactly the usual Lorenz gauge since locally the space is Minkowski. +} +\begin{equation} + D^{\alpha} a_{\alpha} + = + - \frac{1}{u} a_{v} + - \ipd{u} a_{v} + - \ipd{v} a_{u} + + \frac{1}{\Delta^2 u^2} \ipd{z} a_z + + \eta^{ij} \ipd{i} a_j + = + 0. + \label{eq:Lorenz_gauge} +\end{equation} +As covariant derivatives commute since we are locally flat, the \eom read $\qty(\square a)_{\alpha} = 0$. +Explicitly we have: +\begin{equation} + \begin{split} + \qty( \square a )_u + & = + \frac{1}{u^2} a_{v} + - + \frac{2}{\Delta^2 u^3} \ipd{z} a_z + + + \qty[ + - + 2 \ipd{u} \ipd{v} + - + \frac{1}{u} \ipd{v} + + + \frac{1}{\Delta^2 u^2} \ipd{z}^2 + + + \eta^{ij} \ipd{i} \partial_j + ] + a_u, + \\ + \qty( \square a )_v + & = + \qty[ + - + 2 \ipd{u} \ipd{v} + - + \frac{1}{u} \ipd{v} + + + \frac{1}{\Delta^2 u^2} \ipd{z}^2 + + + \eta^{ij} \ipd{i} \partial_j + ] + a_v, + \\ + \qty( \square a )_z + & = + - + \frac{2}{u} \ipd{z} a_v + + + \qty[ + - + 2 \ipd{u} \ipd{v} + + + \frac{1}{u} \ipd{v} + + + \frac{1}{\Delta^2 u^2} \ipd{z}^2 + + + \eta^{ij} \ipd{i} \partial_j + ] + a_z, + \\ + \qty( \square a )_i + & = + \qty[ + - + 2 \ipd{u} \ipd{v} + - + \frac{1}{u} \ipd{v} + + + \frac{1}{\Delta^2 u^2} \ipd{z}^2 + + + \eta^{ij} \ipd{i} \partial_j + ] + a_i. + \end{split} +\end{equation} + +As in the previous scalar case we are actually interested in solving +the eigenmodes problem $\qty(\square a)_\alpha= r \,a_\alpha$. +We proceed hierarchically: first we solve for $a_v$ and $a_i$ whose equations are the same as in the scalar field, then we insert the solutions as a source in the equation for $a_z$ and eventually we solve for $a_u$.\footnotemark{} +\footnotetext{% + Notice that inside the square brackets of the differential equation for $a_z$ there is a different sign for the term $\frac{1}{u} \ipd{v}$ with respect to the equation for the scalar field. +} +We get the solutions: +\begin{equation} + \begin{split} + \norm{\tildea_{\kmkr\, \alpha}(u)} + \,= + \mqty(% + \tildea_u + \\ + \tildea_v + \\ + \tildea_z + \\ + \tildea_i + ) + & = + \sum\limits_{% + \underline{\alpha} + \in + \qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} } + } + \cE_{\kmkr\, \underline{\alpha}} + \norm{\tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)} + \\ + & = + \cE_{\kmkr\, \underline{u}} + \mqty( + 1 + \\ + 0 + \\ + 0 + \\ + 0 + )\, + \tphi_{\kmkr}(u) + \\ + & + + \cE_{\kmkr\, \underline{v}} + \mqty( + \frac{i}{2 k_+ u} + + + \frac{1}{2} \qty( \frac{l}{\Delta k_+} )^2 \frac{1}{u^2} + \\ + 1 + \\ + \frac{l}{k_+} + \\ + 0 + )\, + \tphi_{\kmkr}(u) + \\ + & + + \cE_{\kmkr\, \underline{z}} + \mqty( + \frac{l}{\Delta k_+ \abs{u}} + \\ + 0 + \\ + \Delta \abs{u} + \\ + 0 + )\, + \tphi_{\kmkr}(u) + \\ + & + + \cE_{\kmkr\, \underline{j}} + \mqty( + 0 + \\ + 0 + \\ + 0 + \\ + \delta_{\underline{ij}} + )\, + \tphi_{\kmkr}(u), + \label{eq:Orbifold_spin1_pol} + \end{split} +\end{equation} +then we can expand the off-shell fields as +\begin{equation} + a_{\alpha}\qty(u,\, v,\, z,\, \vb{x} ) + = + \int \ccD k + \sum\limits_{% + \underline{\alpha} + \in + \qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} } + } + \infinfsum{l} + \cE_{\kmkr\, \underline{\alpha}}\, + {a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vb{x} ), +\end{equation} +where ${a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vb{x}) = \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vb{k} \cdot \vb{x})}$ and $\int \ccD k = \int \dd[D-3]{\vb{k}} \int \dd{k_+} \int \dd{r}$. + +We can also compute the normalisation as +\begin{equation} + \begin{split} + \qty(a_{(1)},\, a_{(2)}) + & = + \int \dd[D-3]{\vb{x}} + \int \dd{u} + \int \dd{v} + \finiteint{z}{0}{2\pi} + \abs{\Delta u} + \\ + & \times + g^{\alpha\beta}\, + a_{\kmkrN{1}\, \alpha}\, a_{\kmkrN{2}\, \beta} + \\ + & = + \cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}} + \\ + & \times + \delta^{D-3}( \vb{k}_{\qty(1)} + \vb{k}_{\qty(2)})\, + \delta( r_{(1)} - r_{(2)})\, + \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, + \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}, + \end{split} +\end{equation} +where:\footnotemark{} +\footnotetext{% + We use a shortened version of the polarizations $\cE$ for the sake of readability. + We write $\cE_{(n)\, \underline{\alpha}} = \cE_{\kmkrN{n}\, \underline{\alpha}}$ thus hiding the understood dependence of the components of $\cE_{(n)}$ on the momenta. +} +\begin{equation} + \begin{split} + \cE_{(1)} \circ \cE_{(2)} + = + - \cE_{(1)\, \underline{u}}\, \cE_{(2)\, \underline{v}} + - \cE_{(1)\, \underline{v}}\, \cE_{(2)\, \underline{u}} + + \cE_{(1)\, \underline{z}}\, \cE_{(2)\, \underline{z}} + + + \eta^{\underline{ij}}\, + \cE_{(1)\, \underline{i}}\, \cE_{(2)\, \underline{j}}. + \end{split} +\end{equation} +Finally the Lorenz gauge reads +\begin{equation} + \eta^{i \underline{j}}\, k_i\, \cE_{\kmkr\, \underline{j}} + - + k_+\, \cE_{\kmkr\, \underline{u}} + - + \frac{\vb{k}^2 + r}{2\, k_+} \cE_{\kmkr\, \underline{v}} + = + 0, + \label{eq:explicit_orbifold_Lorenz} +\end{equation} +which does not impose any constraint on the transverse polarization +$\cE_{\kmkr\, \underline{z}}$. +The photon kinetic term becomes +\begin{equation} + \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cE ] + = + \int \dd[D-3]{\vb{k}} + \int \dd{k_+} + \int \dd{r} + \infinfsum{l}\, + \frac{r}{2}\, + \cE_{\kmkr}\, + \circ + \cE_{\kmkr}^*. +\end{equation} + + +\subsubsection{Cubic Interaction} + +With the definition of the d'Alembertian eigenmodes we can now examine the cubic vertex which reads +\begin{equation} + \rS_{\text{s}\qed}^{(\text{cubic})}\qty[\phi,\, a] + = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}\, + \qty(% + -i\, e\, + g^{\alpha\beta} + a_{\alpha} + \qty(% + \phi^*\, \ipd{\beta} \phi + - + \ipd{\beta} \phi^*\, \phi + ) + ). +\end{equation} +Its computation involves integrals such as +\begin{equation} + \int \dd{u}\, + \abs{\Delta u}\, + \qty(\frac{l}{u})^2 + \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} + \sim + \int_{u \sim 0} \dd{u}\, + \qty(\frac{l^2}{\abs{u}^{\frac{5}{2}}}) + e^{% + -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +)}} + \frac{1}{u} + }, +\end{equation} +and +\begin{equation} + \int \dd{u}\, + \abs{\Delta u}\, + \qty(\frac{1}{u}) + \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} + \sim + \int_{u \sim 0} \dd{u}\, + \qty(\frac{1}{u\, \abs{u}^{\frac{1}{2}}}) + e^{% + -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +}} + \frac{1}{u} + }, + \label{eq:nbo_div_integral} +\end{equation} +which can be interpreted as hints that the theory is troublesome. +The first integral diverges if the exponential functions are all equal to unity. +Fortunately it happens when all factors $l_{\qty(i)}$ (where $i = 1,\, 2,\, 3$) vanish. +In this case however the integral vanishes if we set $l_{\qty(i)} = 0$ before its evaluation. +This however suggests that when all $l_{\qty(i)} = 0$, i.e.\ when the eigenfunctions are constant along the compact direction $z$, something suspicious is happening. +On the other side when at least one $l$ is different from zero we have an integral such as: +\begin{equation} + \int_{u \sim 0} \dd{u}\, + \abs{u}^{-\nu}\, e^{i \frac{\cA}{u}} + \sim + \int_{t \sim \infty} \dd{t}\, + t^{\nu-2}\, e^{i \cA t}. +\end{equation} +All $l_{\qty(i)}$ are discrete but $k_{\qty(i)\, +}$ are not thus $\cA$ has an isolated zero. +Otherwise it has continuous value and may be given a distributional meaning, similar to a derivative of the Dirac delta function. +The second integral has the same issues when all $l_{\qty(*)} = 0$ but, since it is not proportional to any $l$ as it stands, it is divergent unless we consider a principal part regularization. + +We can give in any case meaning to the cubic terms +and we get:\footnotemark{} +\footnotetext{% + The notation $(2) \rightarrow (3)$ meaning is that all previous terms inside the curly brackets appear again in exactly the same structure but with momenta of particle $(3)$ in place of those of particle $(2)$. +} +\begin{equation} + \begin{split} + \rS_{\text{s}\qed}^{(\text{cubic})}\qty[ \cA,\, \cE ] + & = + \finiteprod{i}{1}{3} + \qty[% + \int \dd[D-3]{\vb{k}_{\qty(i)}}\, + \dd{r_{\qty(i)}}\, + \dd{k_{\qty(i)\, +}} + \sum_{l_{\qty(i)}} + ]\, + \qty(2\pi)^{D-1} + \delta\qty(\finitesum{i}{1}{3} \vb{k}_{\qty(i)})\, + \delta\qty(\finitesum{i}{1}{3} k_{\qty(i)\, +})\, + \\ + & \times + e~ + \delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)},\, 0}\, + \qty(\cA_{\mkmkrN{2}})^*\, \cA_{\kmkrN{3}} + \\ + & \times + \left\lbrace + \cE_{\kmkrN{1}\, \underline{u}}\, + k_{\qty(2)\, +}\, + \cI_{\qty{3}}^{\qty[0]} + \right. + \\ + & + + \cE_{\kmkrN{1}\, \underline{z}}\, + \frac{% + k_{\qty(2)\, +} l_{\qty(1)} + - + l_{\qty(2)} k_{\qty(1)\, +} + }{\Delta k_{\qty(1)\, +}}\, + \cJ_{\qty{3}}^{\qty[-1]} + \\ + & + + \cE_{\kmkrN{1}\, \underline{v}}\, + \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vb{k}_{\qty(2)}) + \\ + & - + \left. + \eta^{\underline{i}\, j}\, + \cE_{\kmkrN{1}\, \underline{i}}\, + k_{(2)_j}\, + \cI_{\qty{3}}^{\qty[0]}\, + - + \qty( (2) \rightarrow (3) ) + \right\rbrace, + \label{eq:sQED_cubic_final} + \end{split} +\end{equation} +where +\begin{equation} + \begin{split} + \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vb{k}_{\qty(2)}) + & = + \frac{\norm{\vb{k}_{\qty(2)}}^2 + r_{\qty(2)}}{2\, k_{\qty(2)\, +}} \cI_{\qty{3}}^{\qty[0]} + + + i + \frac{k_{\qty(2)\, +}}{2\, k_{\qty(1)\, +}} + \cI_{\qty{3}}^{\qty[-1]} + \\ + & + + \frac{1}{2} \frac{k_{\qty(2)\, +}}{\Delta^2} + \qty(% + \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} + - + \frac{l_{\qty(2)}}{k_{\qty(2)\, +}} + )^2 + \cI_{\qty{3}}^{\qty[-2]}. + \end{split} +\end{equation} +In the previous expressions we also defined for future use: +\begin{eqnarray} + \cI_{(1) \dots (N)}^{\qty[\nu]} + = + \cI_{\qty{N}}^{\qty[\nu]} + & = & + \infinfint{u}\, + \abs{\Delta u}\, u^{\nu}\, + \finiteprod{i}{1}{N} + \tphi_{\kmkrN{i}} + \\ + \cJ_{\qty{N}}^{\qty[\nu]} + & = & + \infinfint{u}\, + \abs{\Delta}\, \abs{u}^{1 + \nu} + \finiteprod{i}{1}{N} \tphi_{\kmkrN{i}}. +\end{eqnarray} +For the sake of brevity from now on we use +\begin{eqnarray} + \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}}, + \\ + \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}} +\end{eqnarray} +when not causing confusion. + + +\subsubsection{Quartic Interactions and Divergences} + +The issue with the divergent vertex is even more visible when considering the quartic terms: +\begin{equation} + \rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ] + = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}\, + \qty(% + e^2\, g^{\mu\nu}\, a_{\mu} a_{\nu}\, \abs{\phi}^2 + - + \frac{g_4}{4}\abs{\phi}^4 + ), +\end{equation} +which can be expressed using the modes as: +\begin{equation} + \begin{split} + \rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ] + & = + \finiteprod{i}{1}{4} + \qty[% + \int \dd[D-3]{\vb{k}_{\qty(i)}} + \dd{k_{\qty(i)\, +}} + \dd{r_{\qty(i)}} + \sum_{l_{\qty(i)}} + ]\, + \qty(2\pi)^{D-1} + \\ + & \times + \delta\qty( \finitesum{i}{1}{4} \vb{k}_{\qty(i)} )\, + \delta\qty( \finitesum{i}{1}{4} k_{\qty(i)\, +} )\, + \delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)} + l_{\qty(4)},\, 0} + \\ + & \times + \left\lbrace + e^2\, + \qty(\cA_{\mkmkrN{3}})^* \cA_{\kmkrN{4}} + \right\rbrace + \\ + & \times + \left[ + \qty(\cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}})\, + \cI_{\qty{4}}^{\qty[0]} + \right. + \\ + & - + \frac{i}{2} + \cE_{\kmkrN{1}\, \underline{v}}\, \cE_{\kmkrN{2}\, \underline{v}} + \qty(% + \frac{1 }{k_{\qty(2)\, +}} + + + \frac{1}{k_{\qty(1)\, +}} + )\, + \cI_{\qty{4}}^{\qty[-1]} + \\ + & + + \left. + \frac{1}{2}\, + \frac{\cE_{\kmkrN{1}\, \underline{v}} \cE_{\kmkrN{2}\, \underline{v}} }{\Delta^2} + \qty(% + \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} + - + \frac{l_{\qty(2)}}{k_{\qty(2)\, +}} + )^2\, + \cI_{\qty{4}}^{\qty[-2]} + \right] + \\ + & - + \left. + \frac{g_4}{4}\, + \ccA\qty(\qty{k_+,\, l,\, \vb{k},\, r})\, + \cI_{\qty{4}}^{\qty[0]} + \right\rbrace, + \end{split} +\end{equation} +where +\begin{equation} + \begin{split} + \ccA\qty(\qty{k_+,\, l,\, \vb{k},\, r}) + & = + \qty(\cA_{\mkmkrN{1}})^*\, + \qty(\cA_{\mkmkrN{2}})^*\, + \\ + & \times + \cA_{\kmkrN{3}}\, + \cA_{\kmkrN{4}}. + \end{split} +\end{equation} +When setting $l_{\qty(*)} = 0$ all the surviving terms are divergent. +The explicit behaviour is $\cI_{\qty{4}}^{\qty[0]} \sim \int \dd{u}\, \abs{u}^{1 -4 \times \frac{1}{2}}$ and $\cI_{\qty{4}}^{\qty[-1]} \sim \int \dd{u}\, u^{-1}\, \abs{u}^{1 - 4 \times \frac{1}{2}}$ since $\eval{\tphi}_{l = 0} \sim \abs{u}^{-\frac{1}{2}}$. +Higher order terms in the effective field theory have even worse behaviour. +This makes the theory ill defined and the string theory which should give this effective theory ill defined too. + + +\subsubsection{Failure of Obvious Divergence Regularizations} +\label{sec:saving} + +From the discussion in the previous section the origin of the divergences is the sector $l = 0$. +When $l = 0$ the highest order singularity of the Fourier transformed d'Alembertian equation vanishes. +Explicitly we have: +\begin{equation} + A\, \ipd{u} \tphi_{\kmkr} + + + B(u)\, \tphi_{\kmkr} + = + A\, e^{-\int^u \frac{B(u)}{A} du}\, + \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{\kmkr} ] + = + 0, +\end{equation} +with +\begin{equation} + A = \qty(-2\, i\, k_+), + \qquad + B(u) + = + -\qty(\vb{k}^2 + r) + - + i\, k_+\, \frac{1}{u} + - + \frac{l^2}{\Delta^2}\, \frac{1}{u^2}. +\end{equation} +This implies the absence of the oscillating factor $e^{i \frac{\cA}{u} }$ when $l$ vanishes. +It follows that any deformation which prevents the coefficient of the highest order singularity from vanishing will do the trick. + +The first and easiest possibility is to add a Wilson line along $z$, i.e.\ $a = \theta \dd{z}$. +This shifts $l \rightarrow l - e\, \theta$ and regularises the scalar \qed. +Unfortunately this does not work in the string theory where Wilson lines on D25-branes are not felt by the neutral strings starting and ending on the same D-brane. +In fact not all interactions involve commutators of the Chan-Paton factors which vanish for neutral strings. +For instance the interaction of two tachyons with the first massive state involves an anti-commutator as we discuss later. +The anti-commutators are present also in amplitudes of supersymmetric strings with massive states and therefore the issue is not solved by supersymmetry. + +A second possibility is to include higher derivative couplings to curvature as natural in the string theory. +If we regularise the metric in a minimal way as shown at the end of~\Cref{sec:geometric_preliminaries_nbo}, only $\tensor{Ric}{_u_u}$ does not vanish. +We can introduce: +\begin{equation} + \begin{split} + & + S_{\mathrm{HE}}^{(\text{higher R})}\qty[ \phi,\, g ] + \\ + = & + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}\, + \qty(% + \finitesum{k}{1}{+\infty} + \qty(\ap)^{2 k-1}\, + \finiteprod{j}{1}{k}\, + g^{\mu_j \nu_j}\, g^{\rho_j\sigma_j}\, + \tensor{Ric}{_{\mu_j}_{\rho_j}}\, + \qty(% + \finitesum{s}{0}{2k} + c_{k s}\, \ipd{\nu_j}^{2k - s}\phi^*\, \ipd{\sigma_j}^s \phi + ) + ) + \\ + = & + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}\, + \qty(% + \ap\, + g^{\mu\nu}\, g^{\rho\sigma}\, + \tensor{Ric}{_{\mu}_{\rho}}\, + \qty(% + c_{12} + \phi^*\, \ipd{\nu\sigma}^2 \phi + + + c_{11} + \ipd{\nu} \phi^*\, \ipd{\sigma} \phi + + + c_{10} + \ipd{\nu\sigma}^2 \phi^*\, \phi + ) + ), + \end{split} +\end{equation} +where $\ap$ has been introduced after dimensional analysis and in order to have all adimensional $c$ factors. +Since only $\tensor{Ric}{_u_u}$ is non vanishing and it depends only on $u$, +the regularised d'Alembertian eigenmode problem now reads: +\begin{equation} + \begin{split} + - + 2 \ipd{u} \ipd{v} \phi_r + & - + \frac{u}{u^2 + \epsilon^2} \ipd{v} \phi_r + + + \frac{1}{\Delta^2 (u^2+ \epsilon^2)} \ipd{z}^2 \phi_r + \\ + & + + \finitesum{k}{1}{+\infty} \qty(\ap)^{2k-1}\, + C_k\, + \tensor{Ric}{_u_u}^k\, + \ipd{v}^{2k} \phi + + + \ipd{i}^2 \phi_r + - + r\, \phi_r + = + 0, + \end{split} +\end{equation} +with $C_k = \finitesum{s}{0}{2k} (-1)^s\, c_{k s}$. +We can perform the usual Fourier transform and the function $B(u)$ becomes +\begin{equation} + \begin{split} + B(u) + & = + - + (\vb{k}^2 + r) + - + i\, k_+\, \frac{u}{u^2 + \epsilon^2} + - + \frac{l^2}{\Delta^2}\, \frac{1}{u^2+\epsilon^2} + \\ + & + + \finitesum{k}{1}{+\infty} + \qty(\ap)^{2k-1}\, + C_k\, + \qty(\frac{\epsilon^2}{(u^2 + \epsilon^2)^2})^k + (-i k_+)^{2k}. + \end{split} +\end{equation} +When $u = 0$ we have: +\begin{equation} + B(0) + \sim + - \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2} + + + \finitesum{k}{1}{+\infty} + \qty(\ap)^{2k-1}\, + C_k\, + \frac{(-i k_+)^{2k}}{\epsilon^{2 k}}. +\end{equation} +Though the correction seems to lead to a cure for the divergence, ff we consider $\ap$ and $\epsilon^2$ uncorrelated we lose predictability. +However if $\ap \sim \epsilon^2$ as natural in string theory we do not solve the problem since +\begin{equation} + B(0) + \stackrel{\ap \sim \epsilon^2}{\sim} + - \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2} + + + \finitesum{k}{1}{+\infty} + C_k\, + (-i k_+)^{2k} + \epsilon^{2k - 2} +\end{equation} +and the curvature terms are no longer singular. + + +%%% TODO %%% +\subsection{A Hope from Twisted State Background} + +It is clear from the previous discussion that the true problem is +associated with the dipole string and its charge neutral states +since the charged ones can be cured rather trivially by a Wilson line. + +On the other side we know that the usual time-like orbifolds are well +defined because of a presence of a $B_{\mu\nu}$ background and this +field is sourced by strings. +So we can think of switching on such a background in the open string. +For open strings $B$ is equivalent to $F$ so we can consider what +happens to an open string in an electromagnetic background. + +The choice of such a background is limited first of all +by the request that it must be an exact string solution, i.e. that it +satisfies the e.o.m derived from the DBI. +If a closed string winds the compact direction $z$ +is coupled to $B_{z u}$, $B_{z v}$ and $B_{z i}$ but if we choose +\begin{equation} +\frac{1}{2\pi\ap} B = f(u) d u \wedge d z + \label{eq:F_bck} + . + \end{equation} +then + \begin{equation} + \det(g+ 2\pi \ap f ) = \det(g) + , + \end{equation} +therefore it is a solution of open string e.o.m. for any +$f(u,v,z,x^i)$. + Suppose that the action for a real neutral scalar $\phi$ is given by + (as the 2 tachyons -- 2 photons amplitude suggests) + \begin{align} + S_{\mbox{scalar kin}} + =& + \int_\Omega d^D x\, + \sqrt{- \det g} + \frac{1}{2} + \Bigl( + -g^{\alpha\beta} + \ipd{\alpha}\phi\, + \ipd{\beta} \phi + -M^2 \phi^2 + + c_1 + \qty(\ap)^2\, \ipd{\mu} \phi \ipd{\nu}\phi + f^{\mu\kappa} f^{\nu}_{~\kappa} + \Bigr) + \nonumber\\ + = + & + \int d^{D-3} \vb{x}\, + \int d u\, \int d v\, \int_0^{2\pi} d z\, + | \Delta u| + \frac{1}{2} + \Biggl( + 2\ipd{u} \phi\,\ipd{v} \phi\, + \nonumber\\ + & + - + \frac{1}{(\Delta u )^2} + (\ipd{z} \phi)^2 + - + (\vec \partial \phi)^2 + - + M^2 \phi^2 + +c_1 \qty(\ap)^2 + \frac{1}{(\Delta u)^2} (\ipd{v} \phi)^2 f^2(u) + \Biggr) + , +\end{align} +Performing the same steps as before we get +\begin{equation} + B(u) + = + (-\vb{k}^2-r) + + + (-i k_+) \frac{1}{u} + + + \frac{(-l^2 + c_1 \qty(\ap)^2 f(u)^2 k_+^2)}{\Delta^2\, u^2} + , +\end{equation} +so even for a constant $f(u)=f_0$ we get a solution which solves the issues. +Notice however that the ``trivial'' solution $f=f_0 d u \wedge d z$ is +not so trivial in Minkowski coordinates $f=\frac{f_0}{x^-} d x^- +\wedge d x^2$. +Though appealing, the study of the string in the presence of this non trivial background needs a deeper analysis which goes beyond the scope of this paper. \subsection{Summary and Conclusions} diff --git a/thesis.bib b/thesis.bib index 3e60c14..cb025ac 100644 --- a/thesis.bib +++ b/thesis.bib @@ -1003,6 +1003,23 @@ number = {1-2} } +@article{Estrada:2012:GeneralIntegral, + title = {A {{General Integral}}}, + author = {Estrada, Ricardo and Vindas, Jasson}, + date = {2012}, + journaltitle = {Dissertationes Mathematicae}, + shortjournal = {Dissertationes Math.}, + volume = {483}, + pages = {1--49}, + issn = {0012-3862, 1730-6310}, + doi = {10.4064/dm483-0-1}, + abstract = {We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if \$f\$ is locally distributionally integrable over the real line and \$\textbackslash psi\textbackslash in\textbackslash mathcal\{D\}(\textbackslash mathbb\{R\}\%) \$ is a test function, then \$f\textbackslash psi\$ is distributionally integrable, and the formula\% [{$<\backslash$}mathsf\{f\},\textbackslash psi{$>$} =(\textbackslash mathfrak\{dist\}) \textbackslash int\_\{-\textbackslash infty\}\^\{\textbackslash infty\}f(x) \textbackslash psi(x) \textbackslash,\textbackslash mathrm\{d\}\% x\textbackslash,,] defines a distribution \$\textbackslash mathsf\{f\}\textbackslash in\textbackslash mathcal\{D\}\^\{\textbackslash prime\}(\textbackslash mathbb\{R\}) \$ that has distributional point values almost everywhere and actually \$\textbackslash mathsf\{f\}(x) =f(x) \$ almost everywhere. The indefinite distributional integral \$F(x) =(\textbackslash mathfrak\{dist\}) \textbackslash int\_\{a\}\^\{x\}f(t) \textbackslash,\textbackslash mathrm\{d\}t\$ corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to \$f(x).\$ The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals --in the Ces\textbackslash `\{a\}ro sense--, mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.}, + archivePrefix = {arXiv}, + eprint = {1109.2958}, + eprinttype = {arxiv}, + file = {/home/riccardo/.local/share/zotero/files/estrada_vindas_2012_a_general_integral.pdf;/home/riccardo/.local/share/zotero/storage/34MFYX8V/1109.html} +} + @article{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes, title = {Generalised Supersymmetric Fluxbranes}, author = {Figueroa-O'Farrill, José and Simón, Joan}, diff --git a/thesis.tex b/thesis.tex index dbe5674..d15e228 100644 --- a/thesis.tex +++ b/thesis.tex @@ -70,6 +70,10 @@ %---- coordinates \newcommand{\pX}{\ensuremath{X'}\xspace} +\newcommand{\kmkr}{\ensuremath{\qty{k_+,\, l,\, \vb{k},\, r}}} +\newcommand{\kmkrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, l_{\qty(#1)},\, \vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}} +\newcommand{\mkmkr}{\ensuremath{\qty{-k_+,\, -l,\, -\vb{k},\, r}}} +\newcommand{\mkmkrN}[1]{\ensuremath{\qty{-k_{\qty(#1)\, +},\, -l_{\qty(#1)},\, -\vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}} %---- BEGIN DOCUMENT