From 23b62b0cd092fb00760b1f04187819ef9d6de5b4 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Tue, 10 Nov 2020 17:32:50 +0100 Subject: [PATCH] Typos Signed-off-by: Riccardo Finotello --- sec/part1/fermions.tex | 10 +++++----- sec/part2/divergences.tex | 2 +- thesis.pdf | Bin 3105791 -> 3105878 bytes thesis.tex | 10 +++++----- 4 files changed, 11 insertions(+), 11 deletions(-) diff --git a/sec/part1/fermions.tex b/sec/part1/fermions.tex index 708b30d..e134aa2 100644 --- a/sec/part1/fermions.tex +++ b/sec/part1/fermions.tex @@ -2039,7 +2039,7 @@ Using the algebra~\eqref{eq:ns-algebra} we compute the \ope of fermion fields as \qquad \abs{w} < \abs{z}, \end{equation} -where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}. +where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{\R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}. We then get the expression of the stress-energy tensor: \begin{equation} \begin{split} @@ -2189,7 +2189,7 @@ From the usual definition of the stress-energy tensor in terms of the Virasoro g \end{split} \end{equation} -We already hinted to the fact that the vacua state involved are not in general \SL{2}{R} invariant. +We already hinted to the fact that the vacua state involved are not in general \SL{2}{\R} invariant. In particular we can see that that the excited vacua \eexcvacket is a primary field \begin{equation} \begin{split} @@ -2394,7 +2394,7 @@ The last expression shows that the energy momentum tensor $\cT( z )$ is radial t First of all we notice that the vacuum $\Gexcvacket$ is actually $\GGexcvacket$, i.e.\ it depends only on $x_{(t)}$ and $\rE_{(t)}$. We can try to interpret the previous result in the light of the usual \cft approach. In particular we can refine the idea we discussed after~\eqref{eq:asymp_beha_Psi_on_exc_vac} that the singularity in the modes~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate} at the point $x_{(t)}$ is associated with a primary conformal operator which creates \eexcvacket with $\rE = \rE_{(t)}$. -By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{R} vacuum \regvacuum. +By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{\R} vacuum \regvacuum. Given the discussion in the previous section this is an excited spin field $\rS_{\rE_{(t)}}\qty( x_{(t)}) = e^{i \rE_{(t)} \phi( x_{(t)} )}$. The first order singularities in $x_{(u)} - x_{(t)}$ are then the result of the interaction between two of the previous excited spin fields. Using the \cft operator approach we postulate that the following identification holds @@ -2818,7 +2818,7 @@ Therefore we have \end{equation} which can be solved by \begin{equation} - \ln \left\langle + \left\langle \rR\qty[% \rS_{\rE_{(t)}}\qty(x_{(t)}) \prod\limits_{\substack{u = 1 \\ u \neq t}}^N @@ -2828,7 +2828,7 @@ which can be solved by = \cN_0 \qty( \qty{\rE_{(t)}} ) - \prod\limits_{\substack{t = 1}{t > u}}^N + \prod\limits_{\substack{t = 1 \\ t > u}}^N \qty( x_{(u)} - x_{(t)} )^{\rE_{(u)} \rE_{(t)}}. \end{equation} The constant $\cN_0\qty( \qty{\rE_{(t)}} )$ which depends on the $\rE_{(t)}$ only can be fixed by using the \ope diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index ff2b41a..3469fdd 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -12,7 +12,7 @@ In fact in~\cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplit \begin{equation} A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q ) \end{equation} -where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices). +where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{open}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices). Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen. The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory. In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution. diff --git a/thesis.pdf b/thesis.pdf index 0a44263cb2eb9b180b25193e405098a129fa8d9c..e1e1d021b705f7682e92c7825ffbad1826429588 100644 GIT binary patch delta 183244 zcmV(zK<2;y$Grj8$h`r8v;k(B0W_DPVkIem?Oa=P+r|}s*RNpTs7Z+IeF1sgnKl!r z$z(ch>`6Nl=K-c55iLo`AQVUW_5Jn&U>96cvM48U(ic%&EY6-i_jC3v$?SSYW?!ED z_0`#DFPNGYS)uUp>T1Rm&k90kS``_Wa`x(S_Whh@^n5`HA@hH<&CU6O<@2iT>gt?- z&Ufbv&c(dBGM{|@vTN(@wfq0os%qLh_kX#)bQ3>swpUHNDc64f-DXpDnDxV}ug|`C zb@uj*0x>e9GfH$ur9nhyQWdl1=Ir|)$m|jizn+nd7y0a+ow}I`sxnoW7uU0wXa77S z2X{9BKG?|xwJM&5B-_7eVhCiWcB}rKb-CNsOZNtuU;5#H)m`WD2$tjX zcXhYgUkp^kP8XDBMbE3HdAKUu>&iU+>;GuTXJh32`Pk?EWQ;Y<|5&DtiAFrvq1!6YL-dlJF~ z%lJ$)uvF+{@?jPogu@C>do5^Q;i!IO>nUyai)o$XK+@K(AC7ZOg7{ zcbVO;+h=?2H~j6FK%J5zK#D?tkVVdBJX2J;6_tP0>A+M0%&TIy2FT9NvRjp1 zg0E6!M9T?ZS#gXw#81itFghc75fbNft1q`Y^mhqUCnsU{by7$z_!c z0iEbLv49k8Q`_ZDCV2;#slb@lrib*P2~(V 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--- a/thesis.tex +++ b/thesis.tex @@ -54,11 +54,11 @@ \newcommand{\cint}[1]{\ensuremath{\oint\limits_{\ccC_{#1}}}} %---- states -\newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{R}}}\xspace} -\newcommand{\regvacuumin}{\ensuremath{\ket{0_{(\text{in})}}_{\SL{2}{R}}}\xspace} -\newcommand{\regvacuumout}{\ensuremath{\ket{0_{(\text{out})}}_{\SL{2}{R}}}\xspace} -\newcommand{\regvacuuminconj}{\ensuremath{\tensor[_{\SL{2}{R}}]{\bra{0_{(\text{in})}}}{}}\xspace} -\newcommand{\regvacuumoutconj}{\ensuremath{\tensor[_{\SL{2}{R}}]{\bra{0_{(\text{out})}}}{}}\xspace} +\newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{\R}}}\xspace} +\newcommand{\regvacuumin}{\ensuremath{\ket{0_{(\text{in})}}_{\SL{2}{\R}}}\xspace} +\newcommand{\regvacuumout}{\ensuremath{\ket{0_{(\text{out})}}_{\SL{2}{\R}}}\xspace} +\newcommand{\regvacuuminconj}{\ensuremath{\tensor[_{\SL{2}{\R}}]{\bra{0_{(\text{in})}}}{}}\xspace} +\newcommand{\regvacuumoutconj}{\ensuremath{\tensor[_{\SL{2}{\R}}]{\bra{0_{(\text{out})}}}{}}\xspace} \newcommand{\twsvacket}{\ensuremath{\ket{\mathrm{T}}}\xspace} \newcommand{\twsvacbra}{\ensuremath{\bra{\mathrm{T}}}\xspace} \newcommand{\excvacket}{\ensuremath{\ket{T_{\rE,\, \brE}}}\xspace}

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